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what is the maximum weight a sabre can hold | Sabre (fencing) electrically separates the handle and the guard. The conventional handle of the sabre is shaped so that it may be held so that the hand may slide down to gain further extension of the weapon relative to the fencer. Other grips which form various shapes are incompatible and impractical with sabre as they limit the movement of the hand, and are likely to be ergonomically incompatible with the guard. The entire weapon is generally 105 cm (41 inches) long; the maximum weight is 500g, but most competition swords are closer to 400g. It is shorter than the foil or épée, | paq | 415 |
Method for manufacturing display device | According to one embodiment, it is disclosed a method of manufacturing a display device. The film material layer is formed on the supporting substrate. A first temperature to form a film layer, the second heating process for the second area surrounding the first area in the first heating step and a second temperature higher than the first temperature for the film material layer is made . The first region is provided in a central portion of the film layer. Forming a display layer on the first region and forms a peripheral circuit on at least a portion of the second area. A third temperature wherein the second temperature is higher than the film of the layer is at least a third heating step is performed for a portion of the film layer. In addition, the film layer is peeled from the supporting substrate. | s2orc_title_abstract | 97 |
Trader Joe's weaponizes idpol language to justify firing of employee for sending CEO letter on Covid safety precautions | Tweet gaining some traction, pretty clownish use of idpol language to paint safety concerns as harmful. And it seems to be a rallying cry for unionization at Trader Joe's to prevent it from happening again
| reddit_title_body | 604 |
Touches get stuck at bottom left of screen | I've had this problem for a while.
But quite often my phone thinks I'm touching and holding down on the very bottom left corner of the screen without me even going near it.
Since I have Show Touches enabaled, I can see that the phone is registering a touch at the bottom left
This problem can happen randomly or when the device starts heating up. The only way to stop it is it shut off my screen and hope it went away | reddit_title_body | 351 |
Who you all find harder between FC and TL? | Who do you all find harder between Failed Champion and Traitor Lord. For me personally I found them relatively the same difficulty which is probably why I beat Traitor Lord third try as I had beaten Failed Champion just before that | reddit_title_body | 632 |
DAE think guys calling their guy friends "the boys" is obnoxious | I dont know a better way to describe it but when my boyfriend says hes with the boys it makes me cringe. I think it might be part of the whole "Saturday is for the boyz" bro memes but it just grates at me for no real reason. On a related note I've always found the whole overly bro thing some guys do to be equally annoying. | reddit_title_body | 402 |
Probably the best consumer compound | I have to admit, this is probably the best compound one could purchase as a consumer. It's a little on the milder side but that's a good thing since you want to achieve results using the least aggressive method possible. My other favorite compound besides Meguiar's would probably be by 3M.
I did a complete pain restoration on my neglected paint by first claying it (you will be socked how much crud comes off your paint with clay). I then wet sanded some fine scratches with 2500 grit then followed with 3000 and used UC to buff out the sanded spots individually. UC was able to easily buff out 3000 grit, even by hand (if you feel like getting a workout) but I used the drill mount buffer which worked great. It will probably buff out 2500 grit as well with multiple applications but don't expect it to be very effective buffing out 2000 grit easily. After I fixed the paint defects, I hit the whole car with UC using a terry bonnet. I then followed with Swirl X (I like this because it's a very fine polish) then followed by wax. The paint looks and feels better the new.
You can use this to polish plastic headlights as well. Forget PlastX, save your money and just use this as it works just a well (I tested them both). My headlights were pretty bad so I sanded them with 2000, 2500 and finally 3000 then used a drill mounted buffer to buff them clear. They came out superb!
Keep in mind that this product has abrasives that *do not* break down as you work them into the clear coat. This means that the cutting is more or less consistent as you're buffing. However, I felt the cutting was fairly mild compared to professional products I've used in the past. I think this has partly to do with the oils in the product. Anyway, highly recommended!
Update:
I noticed that Ultimate Compound contains quite a bit of oil. This is good for lubrication and workability but it's a doubled edged sword as it hids scratches that were not removed. Once the oil dries up, he scratches will come back. To look at my work right after buffing, what I did was fill a spray bottle with water with some dishwashing liquid and sprayed and wiped the area. This removes the oils and reveals the true finish. You can then work more into the finish if you find it's not satisfactory.
Another note. I found out that SwirlX was too mild to polish after UC. I used some Meguiar's Fine Cut Cleaner which removed most of the heavier swirls then hit it with SwirlX to remove the very fine swirls. Going from UC to SwirlX is like going from 1000 grit to 4000 grit... It's not going to work that well. You need to step the cutting level to get a good final finish before wax. Again, keep a bottle of water and dish liquid handly to spray down your work to see what the true finish is. | amazon_reviews | 206 |
Abstract We have determined the primary structure of a 3500 base-pair part of the silkmoth chorion locus mapping in a region containing genes of late developmental specificity. This part of the locus was found to harbour a pseudogene related to one of the families of chorion genes encoding high cysteine proteins, HcB. The pseudogene exhibits an overall sequence identity of 84 % to the consensus coding region of HcB chorion genes. A 95 % identity was observed over a length of 190 base-pairs of its immediate 5′ upstream sequences and the corresponding part of the consensus 5′-intergenic sequences of Hc gene pairs, normally encompassing 270 base-pairs. Thus, the pseudogene has retained part of the promoter region that includes sequence elements whose presence is thought to be necessary for transcriptional competence of HcB genes. The pseudogene is also characterized by the elimination of part of its first exon containing most of the 5′ untranslated region, the ATG translation initiation codon and part of the signal peptide sequences. Its intron is longer than that of other HcB genes due to the insertion of a copy of a repetitive element that appears to be transcribed by RNA polymerase III. A previously characterized chorion cDNA clone, m2282, representing a rare mRNA sequence of late developmental specificity, was found to be identical to the pseudogene over its entirety spanning 65% of the pseudogene's second exon. Hybridizations of clones spanning a 260,000 base-pair domain of the chorion locus of Bombyx mori and of total genomic DNA to a subfragment of the cDNA clone containing relatively unique sequences, coupled to primer extension experiments, have demonstrated that m2282 mRNA originated from the pseudogene and that the pseudogene transcripts are initiated at the chorion cap site consensus sequence. We conclude that the 5′-flanking sequences retained by the pseudogene encompass elements necessary and adequate for correct transcriptional activation, but may not include those required for quantitative expression of the promoter. Possible reasons for the observed lack of random drift in the 5′-upstream sequences of the pseudogene and the maintenance of a functional promoter in a non-functional gene are discussed on the basis of the observation that elements resembling scaffold attachment sites are present in these sequences. | The part of the genetic locus of the domesticated silk moth, Bombyx mori, in which high cysteine (Hc) chorion genes of late developmental specificity reside contains regions encompassing gene-like sequences which exhibit properties distinct from those of functional Hc genes. One of these regions has been characterized and shown to contain a chorion pseudogene, psi HcB.15, which shares pronounced similarities with a transcribed chorion pseudogene, psi HcB.12/13, which was characterized previously. Both pseudogenes are homologous to HcB chorion genes but bear multiple single nucleotide substitutions and short segmental mutations (insertions and deletions) which introduce translational frame shifts and termination codons in the coding regions. Structural characteristics unique to the two pseudogenes suggest that psi HcB.15 was generated first from a functional HcB gene and gave rise subsequently to psi HcB 12/13 as a result of a sequence duplication event. The two pseudogenes can be distinguished from each other by the presence of distinct regions of similarity to the consensus sequence of functional HcB genes which appear to have arisen from gene-conversion-mediated correctional events. These findings lend support to the hypothesis that chorion pseudogene sequences represent reservoirs of genetic information that participates in the evolution of the chorion locus rather than relics of inactivated genes passively awaiting extinction. | s2orc_abstract_citation | 7 |
Double helix of the journal Cytology and Genetics: 50 years later | This survey paper contains a brief analysis of publications included in the current issue of the scientific journal Cytology and Genetics dedicated to its 50th anniversary. These papers reflect scientific achievements of their authors in the field of genetics and cell biology and underline the potential of these two biological disciplines, forming the double helix of the journal. | s2orc_title_abstract | 79 |
who sings from a chevy to a lincoln song | Roland called "Golden Tears" a "highly coindental release." Dave & Sugar frontman Dave Rowland, it seemed, had driven Chevrolets for most of his life, including the early period of Dave & Sugar's national success. However, Rowland had just purchased a new Lincoln when he heard the demo tape for the song. The song's first line in the refrain: "From a Chevy to a Lincoln ... ." The song is about a woman who marries a rich man for his money, but quickly realizes that money does not necessarily buy happiness. In one sense, it is also about life in the | paq | 329 |
who did the canadian forces fight in the war | Canadian War Museum the way in which armed conflict affected the evolution of the country and its peoples. It includes First Peoples warfare, the alliances and conflicts that marked the relationship between First Peoples and Europeans, and the imperial rivalries that marked most of North America's early history. Content includes the Seven Years' War, the American Revolution, the War of 1812, and conflicts in the Canadian West in 1870 and 1885. Canadian forces went abroad in 1899 and again in 1914 to fight in wars as part of the British Empire. This gallery covers the South African War and the First World War, | paq | 74 |
how much range will the hwasong-10 reduce | which will reduce its range by about half, after the experts acknowledged that the June 22 twin test range could be at 3,150 km if the missile was not launched in the lofted trajectory. North Korea sold a version of this missile to Iran under the designation BM-25. The number 25 represents the missile range (2500 km). The Iranian designation is Khorramshahr, and it was unveiled and test-fired in September 2017. Earlier test firing occurred in January 2017. According to IISS expert Dempsey, the missile looks very similar to Hwasong-10. It carries 1800 kg payload over 2000 km (Iran claims | paq | 127 |
The seller clearly didn't describe this book as well used ... | The seller clearly didn't describe this book as well used. As for the text and material covered it served it's purpose. | amazon_reviews | 247 |
Is queen beatrix of holland related to elizabeth 2nd? | Which is better queen elizabeth ii or queen beatrix? | wikianswers | 88 |
BlackBerry Q1 Expectations | What's everyone's thoughts on BB going into tomorrow Q1?
I've been holding for about a year now and have seen a number of these roller coaster quarterly results. I know Cylance is not yet profitable, but I'm wondering what the forecast will do for the stock price and if I should sell if/when the stock rallies tomorrow. | reddit_title_body | 410 |
Very entertaning and enlightening Slap at the Christ myth | The author uses very good evidence to prove that Chist was simply one of many "christs" from mythology. It would be a tough read for the faithful. | amazon_reviews | 228 |
At what a ge do maost boys voices change? | What age does a boy voice change? | wikianswers | 38 |
The emerging field of vibro-polaritonic chemistry studies the impact of light-matter hybrid states known as vibrational polaritons on chemical reactivity and molecular properties. Here, we discuss vibro-polaritonic chemistry from a quantum chemical perspective, go beyond the cavity Born-Oppenheimer (CBO) approximation and examine the role of electron-photon correlation under vibrational strong coupling (VSC). We first quantitatively review ab initio vibro-polaritonic chemistry based on the molecular Pauli-Fierz Hamiltonian in dipole approximation and a VSC Born-Huang expansion. We then derive cavity and nuclear non-adiabatic coupling elements from the generalized Hellmann-Feynman theorem, provide a detailed discussion of their properties and reevaluate the CBO approximation. In the second part, we introduce a crude VSC Born-Huang expansion based on adiabatic electronic states, which provides a foundation for widely employed effective Pauli-Fierz Hamiltonians in ground state vibro-polaritonic chemistry. The latter do not strictly respect the CBO approximation but an alternative scheme, which we name crude CBO approximation. We argue that the crude CBO ground state misses electron-photon entanglement relative to the CBO ground state due to neglected cavity-induced non-adiabatic transition dipole couplings to exited states. A perturbative connection between both ground state approximations is proposed, which identifies the crude CBO ground state as first-order approximation to its CBO counterpart. We provide an illustrative numerical analysis of the cavity Shin-Metiu model, which reveals the nature of non-adiabatic coupling under VSC, addresses the effect of electron-photon correlation on classical activation barriers and illustrates potential shortcomings of the electron-polariton Hamiltonian when employed in the VSC regime. |
I. INTRODUCTION
Vibro-polaritonic chemistry is a rapidly evolving field at the intersection of chemistry and quantum optics, which studies the impact of vibrational polaritons on chemical reactivity and molecular properties. [1][2][3] Vibrational polaritons are light-matter hybrid states, which form when molecular vibrations interact strongly with quantized light modes of optical Fabry-Pérot cavities operating in the infrared regime. [4] The field of vibropolaritonic chemistry was sparked by a series of seminal experiments, which reported on spectroscopic signatures of light-matter hybrid states [5][6][7][8][9][10] and, in particular, significantly altered reactive properties of chemical systems under vibrational strong coupling (VSC) [11][12][13].
A quantum mechanical ab initio description of molecular light-matter hybrid systems is based on the molecular Pauli-Fierz Hamiltonian underlying non-relativistic cavity quantum electrodynamics (cQED). [14][15][16][17][18][19] The Pauli-Fierz Hamiltonian accounts for interactions of electrons and nuclei forming molecules with quantized transverse * [email protected] field modes of an optical cavity and is usually formulated in dipole approximation and length-gauge representation. In contrast to the bare molecular many-body problem, the presence of quantized cavity modes renders the fully interacting light-matter hybrid problem significantly more complex. Motivated by experimental findings [12,13], especially an electronic ground state theory for the VSC regime has been of significant interest in recent years. This circumstance stimulated the formulation of a cavity Born-Oppenheimer (CBO) approximation [15,16], which extends concepts from quantum chemistry to the realm of molecular cQED. Within the CBO framework, both low-frequency cavity modes (e.g., infrared radiation) and nuclei are interpreted as "slow" degrees of freedom contrasted by "fast" electrons. [16] The CBO approximation manifests as truncation of an extended VSC Born-Huang expansion employing a basis of adiabatic electron-photon states, which is equivalently understood as neglecting both cavity and nuclear non-adiabatic coupling elements. [16] Small cavity non-adiabatic coupling has been motivated qualitatively by inquiring small momenta for cavity modes, which translated into small magnetic contributions to cavity mode energies. [16] A quantitative understanding and justification of the CBO approximation, however, requires knowledge about the explicit nature of cavity and nuclear non-adiabatic coupling effects under VSC, which have not been discussed in detail so far. Motivated by those aspects, non-adiabatic coupling under VSC constitutes the first main topic of the present work. We explicitly derive analytic expressions for cavity and nuclear nonadiabatic derivative coupling elements from the generalized Hellmann-Feynman theorem [65,66], which allows us to access the detailed character of cavity and nuclear non-adiabatic effects under vibrational strong coupling.
In spirit of the CBO approximation, two distinct routes to ground state vibro-polaritonic chemistry emerged. The first one aims at solving an extended electronic structure problem, which is capable of accounting for correlations between electrons and cavity modes [16],i.e., electron-photon correlation, by extending well established ab initio methods from quantum chemistry to the realm of molecular cQED [21][22][23][24][25][26][27]. An alternative and especially popular approach relies in contrast on effective model Hamiltonians, which are obtained by projecting the molecular Pauli-Fierz Hamiltonian on the adiabatic electronic ground state subsequently giving rise to even more approximate Tavis-Cummings [53] or Dickemodels [54] and variants thereof [20]. Thus, "only" the correlated electronic ground state problem is solved ab initio, which provides access to molecular potential energy and dipole moment surfaces by means of well established quantum chemistry methods. While the fully correlated approach to ground state vibro-polaritonic chemistry, here named VSC theory, has been consistently introduced in terms of an extended VSC Born-Huang expansion in combination with the CBO approximation [15,16], effective approaches commonly lack a derivation and rely on ad hoc formulations.
The second aim of this paper is to close the gap between fully correlated and effective approaches to ground state vibro-polaritonic chemistry. For this purpose, we introduce a consistent formulation of effective approaches, denoted as crude VSC theory, which employs a crude VSC Born-Huang expansion in terms of adiabatic electronic states. We identify the CBO approximation to be invalid with respect to effective ground state problems due to the emergence of non-adiabatic transition dipole couplings between adiabatic electronic states, which are not per se small for large energetic separation. In consequence, we formulate a crude CBO approximation, which leads to widely employed effective model Hamiltonians in ground state vibro-polaritonic chemistry. We argue that the crude CBO ground state misses electron-photon entanglement relative to the adiabatic electron-photon CBO ground state due to neglected non-adiabatic transition dipole couplings. In order to connect both ground state formulations, we propose a perturbative framework, which extends ideas from Herzberg-Teller theory to vibro-polaritonic chemistry. This new perspective allows us to perturbatively identify the crude CBO ground state as first-order approximation to the adiabatic CBO ground state. Moreover, we discuss systematic correc-tions in the crude VSC framework, which account for electron-photon correlation.
The paper is structured as follows: In Sec.II, we provide a consistent quantitative discussion of ab initio VSC theory beyond the cavity Born-Oppenheimer approximation and derive analytic expressions for nuclear and cavity non-adiabatic derivative couplings. In Sec.III, we introduce crude VSC theory as foundation for effective ground state models in vibro-polaritonic chemistry based on a crude CBO approximation. In Sec.IV, we identify differences in electron-photon entanglement accounted for in approximate CBO and crude CBO ground state theories and introduce a Herzberg-Teller inspired perturbative scheme for ground state VSC theory connecting both approaches. Sec.V numerically illustrates selected concepts via the cavity Shin-Metiu model. In Sec.VI, we summarize and conclude this paper.
II. VIBRATIONAL STRONG COUPLING THEORY FOR MOLECULES IN CAVITIES
We begin with a quantitative review of VSC theory based on the molecular Pauli-Fierz Hamiltonian in length-gauge representation and dipole approximation. [15,16] Starting from an extended VSC Born-Huang expansion, we explicitly derive coupled time-independent Schrödinger equations (TISE) for both adiabatic electron-photon states and nuclear-photon states along with non-adiabatic coupling elements for nuclei and cavity modes.
A. The Molecular Pauli-Fierz Hamiltonian
We consider a molecular system composed of N e electrons and N n nuclei coupled to 2N c quantized transverse field modes of an infrared Fabry-Pérot cavity. This lightmatter hybrid system is described by the molecular Pauli-Fierz Hamiltonian in minimal-coupling form [14,15,17]
H PF = Ne i p i + eÂ(r i ) 2 2m e + Nn a P a − Q a (R a ) 2 2M a + V (r, R) + 2Nc λ,k ω k â † λkâ λk + 1 2 .(1)
In the first line, we have kinetic energy contributions with electronic and nuclear momentum operators,p i = −i ∇ i , and,P a = −i ∇ a , electronic and nuclear masses, m e and M a , elementary charge, e, as well as nuclear charges, Q a = Z a e, with charge number, Z a . The first term in the second line is the molecular Coulomb-interaction potential
V (r, R) = Ne i>j e 2 4πǫ 0 |r i − r j | =Vee(r) + Nn a>b Q a Q b 4πǫ 0 |R a − R b | =Vnn(R) − Ne i Nn a Q a e 4πǫ 0 |r i − R a | =Ven(r,R) ,(2)
with electron-electron, V ee (r), nuclear-nuclear, V nn (R), and electron-nuclear, V en (r, R), Coulomb interaction terms. Electronic, r i , and nuclear coordinates, R a , are collected in vectors, r = (r 1 , . . . , r Ne ), and, R = (R 1 , . . . , R Nn ). The last term in Eq.(1) constitutes 2N c quantized transverse field modes characterized by mode index, k, polarization index, λ, and harmonic mode frequency, ω k , respectively.
[55] Cavity modes are represented by photon creation and annihilation operators, a † λk andâ λk , which satisfy, [â λk ,â † λ ′ k ′ ] = δ λλ ′ δ kk ′ . We restrict our discussion here to ideal cavities and neglect dissipative channels inducing spontaneous emission.
The quantized vector potential,Â(r), in Eq.(1) is purely transverse (Coulomb gauge), i.e., ∇ ·Â(r) = 0, and given by [14]
A(r) = 2Nc λ,k e λk ω k ω k 2ǫ 0 V c â † λk e −ik k ·r +â λk e ik k ·r , (3) ≈ 2Nc λ,k e λk ω k g k â † λk +â λk ,(4)
with vacuum permittivity, ǫ 0 , and cavity quantization volume, V c . Every cavity mode is doubly degenerate with respect to two orthogonal polarization directions related to polarization vectors, e λk , which satisfy the orthonormality condition, e λk · e λ ′ k ′ = δ λλ ′ δ kk ′ . In the second line, we employed the dipole approximation, e ±ik k ·r = 1 ± O(i k k · r) ≈ 1, which approximates,Â(r) ≈Â, as spatially uniform motivated by different length scales of infrared wavelengths and molecular diameters. [18] Further, we introduced a mode-specific light-matter interaction constant
g k = ω k 2ǫ 0 V c .(5)
From a quantum chemical perspective, the length-gauge representation of the molecular Pauli-Fierz Hamiltonian in dipole approximation as obtained via [14,22,60]
UŜ = exp i  · d en exp i π 2 2Nc λ,kn λk ,(6)
exhibiting electronic, d e , and nuclear, d n , components. Further,Ŝ in Eq.(6) relates to a unitary rotation in the cavity mode subspace, which is generated by the collective photon number operator, λ,kn λk = λ,kâ † λkâ λk , and induces a real light-matter interaction term as discussed below. [19] In length-gauge representation, the molecular Pauli-Fierz Hamiltonian in dipole approximation readŝ
H PF =Ŝ †Û †Ĥ PFÛŜ =T n +Ĥ e + 2Nc λ,k ω k â † λkâ λk + 1 2 =Ĥc + 2Nc λ,k g k e λk · d en â † λk +â λk =Ĥsc + 2Nc λ,k g 2 k ω k e λk · d en 2 =Ĥ dse .(8)
In the second line, we recover the nuclear KEO,T n , the electronic Hamiltonian,Ĥ e =T e + V (r, R), and the bare cavity Hamiltonian,Ĥ c . The third line constitutes the length-gauge representation of the light-matter interaction,Ĥ sc , determined by the polarization-projected molecular dipole moment, e λk ·d en . Finally, the last term in Eq.(8) resembles the dipole self-energy (DSE),Ĥ dse , which provides translational invariance, a bound ground state and gauge invariance ofĤ PF . [61,62] In what follows, we aim at describing infrared cavity modes on the same footing as nuclear DoF, which is achieved via a cavity coordinate representation [16] x λk = 2ω k â † λk +â λk ,
p λk = i ω k 2 â † λk −â λk .(9)
Here, x λk is a cavity displacement coordinate with conjugate momentum operator,p λk = −i ∂ ∂x λk , satisfying, [x λk ,p λk ] = i . In coordinate presentation, the cavity Hamiltonian readŝ
H c = 2Nc λ,k p 2 λk 2 + ω 2 k 2 x 2 λk =Tc+Vc(x) ,(11)
with KEO,T c , and harmonic cavity potential, V c (x), where, x, is a vector collecting all 2N c displacement coordinates. The cavity coordinate representation ofĤ PF is given then bŷ
H PF =T n +T c +Ĥ e + 2Nc λ,k ω 2 k 2 x λk + 2 ω 3 k g k e λk · d en 2 =Vc+Ĥsc+Ĥ dse ,(12)
which relates to a collection of displaced cavity modes shifted with respect to the polarization-projected molecular dipole moment, e λk · d en . One now introduces an electron-photon Hamiltonian [16]
H ec =Ĥ e + V c +Ĥ sc +Ĥ dse ,(13)
where the third term constitutes the length-gauge lightmatter interaction in coordinate representation,
H sc = 2Nc λ,k 2ω k g k e λk · d en x λk .(14)
The DSE term,Ĥ dse , remains as given in Eq.(8), since it is independent of the cavity mode representation.Ĥ PF in Eq. (12) can now be compactly rewritten aŝ
H PF =T n +T c +Ĥ ec ,(15)
such that the corresponding light-matter timeindependent Schrödinger equation (TISE) reads
T n +T c +Ĥ ec |Ψ i (R, x) = E i |Ψ i (R, x) ,(16)
with energies, E i , and corresponding orthonormal lightmatter many-body states, |Ψ i (R, x) . The ket -notation is employed with respect to the electronic subspace. In the remainder of this work, we are interested in finding approximate bound state solutions of Eq.(16).
B. The VSC Born-Huang Expansion
Ab initio VSC theory is based on an extended Born-Huang expansion for light-matter many-body states, |Ψ i (R, x) , in the following denoted as VSC Born-Huang expansion, which is given by [16]
|Ψ i (R, x) = µ χ (nc) iµ (R, x) |ψ (ec) µ (R, x) .(17)
Here, |ψ (ec) µ (R, x) , are adiabatic electron-photon states, which parametrically depend on both nuclear and cavity coordinates, (R, x), and provide a complete, orthonormal basis spanning the electronic subspace for each fixed configuration, (R, x). Moreover, χ (nc) iµ (R, x), are coefficient functions resembling nuclear-photon states. In order to indicate the hybrid character of states, we employ combined superscripts for electronic (e), nuclear (n) and cavity (c) contributions. We now insert Eq.(17) into the full light-matter TISE (16) and project on adiabatic electronphoton states, ψ (ec) ν |, which leads to
µ ψ (ec) ν |Ĥ PF χ (nc) iµ |ψ (ec) µ = E i χ (nc) iν .(18)
|ψ (ec) ν are taken to be eigenstates ofĤ ec in Eq.(13) and satisfy [16]
Ĥ ec |ψ (ec) ν = E (ec) ν (R, x) |ψ (ec) ν ,(19)
with cavity potential energy surface (cPES), E (ec) ν (R, x), for the ν th -state at fixed configuration, (R, x). Further, KEOs,T n andT c , inĤ PF act on both factors of the product, χ (nc)
iµ (R, x) |ψ (ec) µ (R, x)
, which leads to nuclear and cavity non-adiabatic coupling (NAC) elements between adiabatic electron-photon states [63]
C (n) νµ = − Nn a 2 2M a G (n) a,νµ + 2 F (n) a,νµ · ∇ a ,(20)C (c) νµ = − 2 2 2Nc λ,k G (c) λk,νµ + 2 F (c) λk,νµ ∂ ∂x λk .(21)
Nuclear contributions are determined by
G (n) a,νµ (R, x) = ψ (ec) ν (R, x)|∇ 2 a |ψ (ec) µ (R, x) r , F (n) a,νµ (R, x) = ψ (ec) ν (R, x)|∇ a |ψ (ec) µ (R, x) r ,(22)
and cavity contributions decompose into [63,64]
G (c) λk,νµ (R, x) = ψ (ec) ν (R, x)| ∂ 2 ∂x 2 λk |ψ (ec) µ (R, x) r , F (c) λk,νµ (R, x) = ψ (ec) ν (R, x)| ∂ ∂x λk |ψ (ec) µ (R, x) r ,(23)
where integration with respect to electronic coordinates is indicated by . . . r . Note, both cavity NAC elements are scalar in nature, in contrast to the first-order nuclear derivative term, F (n) a,νµ , which is a vector. The lengthgauge nuclear-photon TISE is eventually obtained as
T n +T c + E (ec) ν (R, x) χ (nc) iν + µ =νK νµ χ (nc) iµ = E i χ (nc) iν ,(24)
with generalized NAC elements given bŷ
K νµ =Ĉ (n) νµ +Ĉ (c) νµ .(25)
In the following, we give a detailed characterization of K νµ , which lies at the heart of the CBO approximation.
C. Non-Adiabatic Coupling under VSC and the CBO Approximation
We evaluate first-order derivative NAC elements,
F (c)
λk,νµ and F (n) a,νµ , analytically by means of the generalized (alternatively off-diagonal) Hellmann-Feynman theorem [65,66]
F (c) λk,νµ = ψ (ec) ν | ∂ ∂x λkĤ ec |ψ (ec) µ r E (ec) µ − E (ec) ν ,(26)F (n) a,νµ = ψ (ec) ν |∇ aĤ ec |ψ (ec) µ r E (ec) µ − E (ec) ν ,(27)
where,Ĥ ec , is the electron-photon Hamiltonian in Eq. (13). For the cavity contribution, we obtain (cf. Appendix A for details)
F (c) λk,νµ = 2ω k g k e λk · D νµ E (ec) µ − E (ec) ν ,(28)
with transition dipole elements between adiabatic electron-photon states
D νµ (R, x) = ψ (ec) ν (R, x)|d en |ψ (ec) µ (R, x) r .(29)
Eq.(28) can be traced back to contributions from the light-matter interaction term,Ĥ sc . Further, since F λk,νµ , becomes particularly large for close lying adiabatic states.
For nuclear derivative NAC elements in Eq.(27), we find (cf. Appendix A)
F (n) a,νµ = ψ (ec) ν |∇ aV en |ψ (ec) µ r E (ec) µ − E (ec) ν + 2Nc λ,k 2 Q a g 2 k ω k e λk · D νµ e λk E (ec) µ − E (ec) ν ,(30)
where the first term resembles the vibronic non-adiabatic coupling in the subspace of adiabatic electron-photon states. The second term constitutes a cavity-induced matter contribution, which emerges from the dipole selfenergy as apparent from the quadratic dependence on the light-matter interaction constant, g 2 k . This term therefore depends also on transition dipole moments between adiabatic electron-photon states, D νµ , in analogy to the cavity NAC element and relates to the transverse polarization operator (cf. Appendix A). Thus, bright and dark adiabatic electron-photon states do not only behave differently with respect to cavity-induced non-adiabatic coupling but also with respect to nuclear contributions under VSC.
When the adiabatic electron-photon ground state is energetically well separated from the excited state manifold, i.e., |E (ec) µ − E (ec) 0 | ≫ 0, both cavity and nuclear NAC elements are small and the VSC Born-Huang expansion can be truncated to the first term [16] |Ψ (ec)
cbo (R, x) = χ (nc) 0 (R, x) |ψ (ec) 0 (R, x) .(31)
This approach resembles the cavity Born-Oppenheimer (CBO) approximation to the light-matter many-body ground state, |Ψ 0 (R, x) ≈ |Ψ
H ec |ψ (ec) 0 (R, x) = E (ec) 0 (R, x) |ψ (ec) 0 (R, x)
.
Two approximate schemes concerning NAC elements are now possible: The first one, here denoted as cavity adiabatic approximation in analogy to vibronic notation [69], neglects off-diagonal NAC elements and accounts for diagonal CBO ground state corrections. The corresponding approximate adiabatic ground state cPES reads λk,00 are defined in Eqs. (22) and (23). The second scheme is realized by additionally setting diagonal non-adiabatic contributions to zero, i.e.,K νµ = 0, for all ν, µ, such that, E (ec) 0 (R, x) ≈ E (ec) 0 (R, x), in Eq.(32) is the bare cPES. Note, our definition slightly differs from the original formulation in Ref. [16], where the CBO approximation took into account diagonal NAC contributions, G
E (ec) 0 (R, x) = E (ec) 0 (R, x) + G (n) 00 + G (c) 00 ,(33)
III. CRUDE VSC THEORY FOR MOLECULES IN CAVITIES
A. The Crude VSC Born-Huang Expansion VSC theory, as discussed in the last section, crucially relies on the knowledge of adiabatic electron-photon states obtained from TISE (19). We now introduce an alternative approach, denoted as crude VSC theory, which circumvents this issue by employing a basis of adiabatic electronic states. In crude VSC theory, we consider a crude VSC Born-Huang expansion
|Ψ i (R, x) = µχ (nc) iµ (R, x) |ψ (ec) µ (R, x 0 ) ,(35)
where adiabatic electron-photon states, |ψ (ec) µ (R, x 0 ) , are considered at a fixed cavity displacement coordinate reference value, x 0 . In order to make the difference to VSC theory explicit, we write nuclear-cavity states,
χ (nc)
iµ (R, x), augmented by a tilde sign. We now choose reference cavity displacement coordinates as
x 0,λk = − 2 ω 3 k g k e λk · d en ,(36)
such that the electron-photon Hamiltonian reduces to the bare electronic Hamiltonian,Ĥ ec (x 0 ) =Ĥ e , for the cavity reference configuration, x 0 . Hence, the electronphoton TISE (19) evaluated at x 0 reduces to the bare electronic TISÊ
H e |ψ (e) ν = E (e) ν (R) |ψ (e) ν ,(37)
which is solved by adiabatic electronic states, i.e.,
|ψ (ec) ν (R, x 0 ) = |ψ (e)
ν (R) , with molecular potential energy surfaces, E (e) ν (R). Accordingly, the crude VSC expansion in Eq. (35) can equivalently be formulated as
|Ψ i (R, x) = µχ (nc) iµ (R, x) |ψ (e) µ (R) ,(38)
where the adiabatic basis is spanned by conventional adiabatic electronic states, which only show a parametric dependence on nuclear coordinates.
B. Coupled TISE in Crude VSC Theory
We now derive coupled TISE in crude VSC theory by inserting Eq.(38) into the light-matter TISE (16), which leads after projection on ψ
(e) ν | to µ ψ (e) ν |Ĥ PF χ (nc) iµ |ψ (e) µ = E iχ (nc) iν .(39)
Evaluation of matrix elements, as shown in Appendix B, eventually provides the bare electronic TISE (37) and the crude nuclear-photon TISE
T n +T c + V ν (R, x) χ (nc) iν + µ =νK νµχ (nc) iµ = E iχ (nc) iν ,(40)
with crude cPES
V ν (R, x) = E (e) ν (R) + V c (x) + 2Nc λ,k 2ω k g k e λk · d νν x λk + 2Nc λ,k g 2 k ω k e λk · d 2 νν + α =ν e λk · d 2 να ,(41)
and dipole matrix elements
d νµ = ψ (e) ν |d en |ψ (e) µ r .(42)
The crude cPES is determined by bare molecular and cavity potentials in the first line, the light-matter interaction term in the second line and the DSE contribution in the third line. The latter contains a "virtual" interaction involving adiabatic states, |ψ
(e) ν
, and, |ψ
(e)
α , which is commonly neglected in ground state problems with ν = 0. Eventually, generalized NAC elements read
K νµ =Ĉ (n) νµ + 2Nc λ,k 2ω k g k e λk · d νµ x λk + 2Nc λ,k g 2 k ω k α e λk · d να e λk · d αµ ,(43)
where the bare nuclear derivative coupling element,
C (n)
νµ , is augmented by two non-adiabatic coupling terms related to polarization-projected transition dipole moments, e λk · d νµ . Those terms emerge as off-diagonal contributions of light-matter interaction and DSE terms in the adiabatic electronic basis.
C. Crude Non-Adiabatic Coupling and the Crude CBO Approximation
We shall now examine the detailed character of crude non-adiabatic coupling elements,K νµ , introduced in Eq. (43) and the notion of a CBO approximation in crude VSC theory. We first recall, that nuclear derivative NAC elements follow from the generalized Hellmann-Feynman theorem as [66]
F (n) a,νµ = ψ (e) ν |∇ aĤe |ψ (e) µ r E (e) µ − E (e) ν ,(44)
which become small for energetically well separated states. However, for cavity-induced NAC contributions in Eq.(43), we found expressions crucially different from Eq.(44): Those terms are proportional to transition dipole moments, d νµ , which are in general not necessarily small for energetically well separated states. Hence, energetic arguments valid for invoking the CBO approximation in VSC theory do not strictly hold in crude VSC theory.
In order to make progress, we introduce a truncation of the crude VSC expansion, Eq. (38), in analogy to Eq.(31) as
|Ψ (e) ccbo (R, x) =χ (nc) 0 (R, x) |ψ (e) 0 (R) ,(45)
which we denote as crude CBO (CCBO) approximation. The latter equivalently resembles, |Ψ 0 (R, x) ≈ |Ψ (e) ccbo (R, x) , where we in particular assume that,K νµ = 0, i.e., all transition dipole NAC elements are neglected. The relevance of the crude CBO approximation manifests in an appealing sum-of-products form of the ground state crude cPES, V 0 (R, x), in Eq.(41) with respect to molecular and cavity degrees of freedom. This representation lies at the heart of effective vibro-polaritonic model Hamiltonians. Thus, according to the arguments presented here, most studies on ground state vibropolaritonic chemistry do actually not rely on the CBO approximation as formulated in Ref. [15,16], but its crude version as introduced in this work.
IV. ELECTRON-PHOTON ENTANGLEMENT AND CRUDE VSC PERTURBATION THEORY
In previous sections, we have seen that approximate crude CBO and CBO ground state theories differ in certain aspects, which we will discuss now in closer detail. First, based on a reduced density matrix (RDM) analysis, we argue that the crude CBO ground state accounts only for a fraction of electron-photon entanglement relative to the CBO ground state. Second, we introduce a Herzberg-Teller inspired perturbative scheme, denoted as crude VSC perturbation theory (cVSC-PT), which allows us to identify the crude CBO ground state as firstorder approximation to the CBO ground state. Higherorder perturbative corrections are subsequently shown to account for non-adiabatic electron-photon correlation effects, which manifest as cavity-induced transition dipole type couplings between adiabatic electronic states.
A. Electron-Photon Entanglement in Approximate CBO Ground States
In absence of electron-photon entanglement, an approximate light-matter ground state, Ψ 0 , is separable in the electron-cavity subspace such that the corresponding electronic RDM,ρ (e) , does not contain any vibropolaritonic contributions. [67,68] Based on this notion, we examine RDM,ρ (i) , for electrons, nuclei and cavity modes with i = e, n, c, as shown in Tab.I for three approximate ground states: The CBO ground state, Ψ We first recall, that both factors in Ψ (ec) cbo , Eq.(31), depend on nuclear and cavity coordinates. We then realize that the adiabatic electron-photon ground state, ψ (ec) 0 , can be expanded in the adiabatic electronic basis, Eq.(C.2), which replaces cavity coordinate dependence of the adiabatic contribution to Ψ I. Reduced density matrices for electrons,ρ (e) (r, r ′ ), nuclei,ρ (n) (R, R ′ ), and cavity modes,ρ (c) (x, x ′ ), for CBO, Ψ
µν , are defined in Appendix C. ρ (e) (r, r ′ )ρ (n) (R, R ′ )ρ (c) (x, x ′ ) Ψ (ec) cbo (r, R, x) µνρ (e) µν µνS (n) µν O (n) µν µS (c) µµ Ψ (e) ccbo (r, R, x)ρ (e) 00S (n) 00 O (n) 00S (c) 00 Ψ ref (r, R, x) ρ (e) 00s (n) 00s (c) 00 traced back to overlap integrals,S (n) µν andS (c)
µµ , between nuclear-photon states,χ
(nc) µ (cf. Appendix C).
The crude CBO ground state, Ψ (ec) ccbo , and related crude CBO-RDM (Tab.I, second line) emerge now naturally by truncating adiabatic basis expansions after the first term. This procedure resembles the crude CBO approximation in the wave function picture, which is equivalent to neglecting cavity-induced non-adiabatic transition dipole type couplings as discussed in Subsec.III C. We thus argue that electron-photon entanglement in Ψ (ec) ccbo due to cavity-induced non-adiabatic contributions from excited electronic states is missing relative to Ψ ccbo , but "indirectly" entangled via nuclei.
Illustratively, the reference state's electronic RDM,
ρ (e)
00 , is independent of vibro-polaritonic contributions (Tab.I, third line), which reflects the fully separable product structure of Ψ ref , Eq. (C.5), in the electron-cavity subspace: Here, electrons are disentangled with respect to both nuclei and cavity modes (although the latter are entangled).
B. Ground State VSC Theory from a Perturbative Perspective
We propose to connect ground state CBO and ground state crude CBO theories via a perturbative scheme, which extends ideas from Herzberg-Teller theory [69] to molecular cQED in the VSC regime. This approach is intended to systematically capture non-adiabatic electronphoton correlation effects accounting for entanglement in the respectively corrected ground state and will be abbreviated as cVSC-PT(n) at n th -order of perturbation theory.
According to Sec.II, the adiabatic electron-photon ground state, |ψ
(ec) 0 (R, x) , satisfies the TISE (32) with cPES, E (ec) 0 (R, x) = E (ec) 0 (R, x)
, in the CBO approximation. We now rewrite the electron-photon Hamiltonian asĤ
ec =T e + V (r, R, x 0 ) + ∆V (r, R, x) ,(46)
with difference potential
∆V (r, R, x) = V (r, R, x) − V (r, R, x 0 ) ,(47)
and recall that the cavity reference configuration, x 0 , in Eq. (36) was chosen such that,Ĥ ec (x 0 ) =Ĥ e . Hence,
V (r, R, x 0 ) = V (r, R)
, is just the bare molecular potential in Eq. (2) and ∆V (r, R, x) contains all cavity contributions toĤ ec . Subsequently, we apply Rayleigh-Schrödinger perturbation theory to approximately solve Eq.(32) with zeroth-order electronic Hamiltonian
H 0 =T e + V (r, R, x 0 ) =Ĥ e ,(48)
and perturbation, ∆V (r, R, x). Perturbative expansions of the CBO ground state and cPES are given by well known expressions
|Ψ (ec) cbo (R, x) = |Ψ (0) 0 (R, x) + λ |Ψ (1) 0 (R, x) + O(λ 2 ) ,(49)
and
E (ec) 0 (R, x) = E (0) 0 (R, x) + λ E (1) 0 (R, x) + λ 2 E (2) 0 (R, x) + O(λ 3 ) . (50) At zeroth-order, we find H e |Ψ (0) 0 (R, x) = E (0) 0 (R, x) |Ψ (0) 0 (R, x) ,(51)
which identifies the bare adiabatic electronic ground state, |Ψ
(0) 0 (R, x) = |ψ (e) 0 (R) , as zeroth-order state with corresponding molecular PES, E (0) 0 (R, x) = E (e) 0 (R).
The first-order corrected cPES is then given by
V (1) 0 (R, x) = E (e) 0 (R) + ψ (e) 0 |∆V |ψ (e) 0 r =E (1) 0 (R,x) ,(52)
where the difference potential, ∆V , reads
∆V = V c +Ĥ sc +Ĥ dse ,(53)
such that we recover for, V
0 (R, x), the correspond- ing ground state crude cPES, Eq.(41). Since E (1) 0 (R, x)(1)
is fully determined by the zeroth-order wave function, i.e., the adiabatic electronic ground state, in line with the crude CBO ground state in Eq.(45), we can accordingly interpret ground state crude VSC theory as firstorder approximation to ground state VSC theory, denoted by cVSC-PT(1). Further, we recall from Subsec.IV A, that electrons and cavity modes in the crude CBO ground state are only indirectly entangled via nuclei since electron-photon correlation induced by nonadiabatic coupling terms involving electronic transition dipole moments, d 0µ , is absent in cVSC-PT(1).
Non-adiabatic corrections accounting for electronphoton correlation enter in cVSC-PT(2) as
V (2) 0 (R, x) = V (1) 0 (R, x) + µ =0 |∆V 0µ (R, x)| 2 ∆E (e) 0µ (R) =E (2) 0 (R,x) ,(54)with, ∆V 0µ = ψ (e) 0 |∆V |ψ (e) µ r , and, ∆E (e) 0µ = E (e) 0 −E (e) µ ,
as determined by the first-order corrected state
|Ψ (1) 0 (R, x) = µ =0 ∆V 0µ (R, x) ∆E (e) 0µ (R) |ψ (e) µ (R) .(55)
In order to illustrate how electron-photon correlation corrections manifest perturbatively, we discuss E (53), we recognize the second-order correction to be independent of the bare cavity potential, since matrix elements, ∆V 0µ , are strictly off-diagonal. Hence, we have explicitly
(2) 0 (R, x) in more detail. From Eq.E (2) 0 (R, x) = µ =0 |H sc 0µ | 2 + 2 H sc 0µ H dse 0µ + |H dse 0µ | 2 ∆E (e) 0µ (R) , (56) with matrix element, H sc 0µ = ψ (e) 0 |Ĥ sc |Ψ (e)
µ r (equivalently for, H dse 0µ ). Further, we recall scaling with respect to the light-matter interaction constant as,Ĥ sc ∝ g, and, H dse ∝ g 2 (we dropped mode indices for brevity), such that, |H sc 0µ | 2 ∝ g 2 , in Eq.(56) provides the leading-order contribution to E (2) 0 (R, x), which reads explicitly
|H sc 0µ | 2 = 2 λ,λ ′ Nc k,k ′ 2ω k 2ω k ′ g k g k ′ × e λk · d 0µ e λ ′ k ′ · d 0µ x λk x λ ′ k ′ . (57)
This term accounts for correlations between electrons and cavity modes via a cavity-induced transition-dipole type interaction between adiabatic electronic states, which in turn couples cavity modes with potentially distinct mode and polarization indices, k, k ′ , and, λ, λ ′ . Interestingly, a similar observation has been recently reported from a non-perturbative perspective. [27] V. NUMERICAL ANALYSIS
We now numerically analyze several aspects of VSC and crude VSC theory discussed above for the cavity Shin-Metiu (CSM) model [15]. The CSM model provides a numerically exactly solvable model, which has been proven versatile to capture the rich physics of nonadiabatic phenomena involving moving electrons and nuclei coupled to quantized cavity modes.
A. The Cavity Shin-Metiu Model
The cavity Shin-Metiu (CSM) model describes a "molecular" model system composed of a moving nucleus and a moving electron [71], which couple via their charges to a single quantized cavity mode [15] as schematically shown in Fig.(1). Both particles move along a molecular axis connecting two fixed nuclei at a given distance, L.
T n +T c = − 1 2M a ∂ 2 ∂R 2 − 1 2 ∂ 2 ∂x 2 ĉ H ec =Ĥ e + ω 2 c 2 x c + 2 ω 3 c g c d en (r, R) 2 H e = − 1 2 ∂ 2 ∂r 2 + V (r, R) ,(58)
with single-mode cavity displacement coordinate, x c , molecular dipole moment
d en (r, R) = −r + Q a R ,(59)
and Shin-Metiu potential [71] V (r, R) = V n (R) + V en (r, R) .
The first term relates to a purely repulsive Coulomb interaction between the moving and fixed nuclei positioned at ±L/2
V n (R) = Q a Q |L/2 − R| + Q a Q |L/2 + R| ,(61)
with nuclear coordinate, R, and nuclear charges, Q a = Q = 1, respectively. The attractive "softened" Coulomb interaction between the electron and all nuclei reads
V en (r, R) = − Q erf |L/2 − r| R f |L/2 − r| − Q erf |L/2 + r| R f |L/2 + r| − Q a erf |R − r| R c |R − r| ,(62)
with error function, erf(. . . ), besides screening lengths for fixed, R f , and moving nuclei, R c . The latter determines the nuclear non-adiabatic coupling strength. [71] Following earlier work [16], we consider a model with L = 10 Å and R f = 1.5 Å, where we set, R c = 1.5 Å, for weak and, R c = 2.0 Å, for strong non-adiabatic coupling. CSM model parameters, which determine the VSC scenario, are obtained by first solving the electronic TISE withĤ e given in Eq.(58)
H e ψ (e) 0 (r; R) = E (e) 0 (R) ψ (e) 0 (r; R) ,(63)
for the ground state molecular PES, E (e) 0 (R), and subsequently the corresponding nuclear TISE
T n + E (e) 0 (R) ϕ (n) v (R) = ε v ϕ (n) v (R) ,(64)
for vibrational eigenstates, ϕ
B. Adiabatic Electron-Photon States
We begin our discussion by examining the impact of vibrational strong coupling on adiabatic electron-photon states. We restrict ourselves to the ground and first excited adiabatic electron-photon states of the CSM model with quantum numbers, ν = 0, 1, and consider reduced densities in the molecular coordinate subspace
ρ (ec) ν (r, R) = dx c |ψ (ec) ν (r; R, x c )| 2 ,(65)ρ (e) ν (r, R) = |ψ (e) ν (r; R)| 2 ,(66)
where, ψ ν ), we introduce a state-dependent difference density [16]
∆ρ (ec) ν (r, R) = ρ (ec) ν (r, R) − ρ (e) ν (r, R) ,(67)
which vanishes in the non-interacting limit, i.e., ν (r, R) are shown for VSC with η = 0.04, and quantum numbers ν = 0, 1 for both the weak and strong non-adiabatic coupling regimes. We observe cavity-induced density redistribution in both the weakly and the strongly non-adiabatic CSM model for the ground and first excited electronphoton state: Red coloured regions in ∆ρ (ec) ν (r, R) indicate a density increase and blue coloured regions a density decrease. New maxima emerge along the electron coordinate, r, in the weakly non-adiabatic regime, while density is dominantly shifted along the nuclear coordinate, R, under strong non-adiabatic coupling.
Density variations with η indicate non-trivial electronphoton correlation effects in energetically low lying adi-abatic electron-photon states of the CSM model, despite the fact that cavity photon energies lie significantly below characteristic electronic excitation energies. Based on those observations, an approximation of the adiabatic electron-photon ground state by the bare electronic ground state, i.e., ρ (ec) 0 (r, R) ≈ ρ (e) 0 (r, R) (crude CBO approximation), does not necessarily seem to be an always well justified simplification. We will provide further arguments supporting this statement in Subsec.V D.
C. Nuclear and Cavity Non-Adiabatic Coupling
We now turn to non-adiabatic coupling under VSC at η = 0.04 and discuss both nuclear and cavity derivative NAC elements for weak and strong NAC regimes of the CSM model.
As observed before in the weak non-adiabatic coupling regime [16], both the ground and first excited state cPES, V 0 (R, x c ) and V 1 (R, x c ), are distorted in the (R, x c )plane under VSC (cf. Figs.3a and 3d). Both states are subject to nuclear (mass-weighted) and cavity derivative NAC elements, F (n) 10 and F (c) 10 , which are found to behave qualitatively similar. According to Figs.3b and 3e, two dominant minima are observed, where the distorted cPES are energetically close. Minima are located close to the nuclear coordinate's origin and take values of x c ≈ ±100 √ m e a 0 for the cavity displacement coordinate corresponding to roughly, n c = 13, photons as estimated from classical turning points of the harmonic cavity potential. The cavity NAC element for the CSM model
F (c) 10 (R, x c ) = 2 ω 3 c d 30 η D 10 (R, x c ) ∆E (ec) 10 (R, x c ) ,(68)
is observed to be two orders of magnitude larger compared to the nuclear derivative NAC, F
F (n) en (R, x c ) = ψ (ec) 1 (R, x c )| ∂ V en ∂R |ψ (ec) 0 (R, x c ) r ∆E (ec) 10 (R, x c ) ,(69)F (n) dse (R, x c ) = 2 Q a η 2 d 30 D (ec) 10 (R, x c ) ∆E (ec) 10 (R, x c ) .(70)
From Figs.3c and 3f, we observe the DSE-induced term,
F (n)
dse , to be roughly one order of magnitude larger compared to the vibronic contribution, F 10 (R, x c ). The difference in magnitude is explained through different scaling with respect to η. Consequently, nuclear derivative NAC elements turn out to be significantly influenced by light-matter interaction, when transitions between two adiabatic states have a strong bright component (cf. Subsec.II C). This finding contrasts the bare molecular picture, where optical properties of adiabatic states are not relevant for non-adiabatic coupling. Moreover, since cavity NAC contributions are here found to be larger in magnitude than their molecular counterparts, the notion of a weak NAC regime does interestingly not simply transfer from the molecular to the light-matter hybrid scenario, at least in the herein presented model.
For the strong NAC regime under VSC, cPES and derivative NAC elements are presented in Fig.4. The ground state cPES shows here a pronounced double minimum, whereas the energetically close lying first excited state exhibits a single minimum close to the ground state's transition state. [16] NAC elements exhibit a sharp maximum with evenly spaced, large amplitude peaks along a seam in the (R, x c )-plane, which is parallel to the separating surface between the two minima of the ground state cPES. A similar observation has been reported very recently. [64] The maximal amplitude is naturally larger compared to the weak NAC regime and the cavity contribution is again roughly two orders of magnitude larger than the vibronic coupling. This observation points at another difference in the notion of weak and strong NAC regimes compared between light-matter hybrid systems and the bare molecular picture: For the system studied here, cavity weak non-adiabatic coupling is characterized by relatively localized peaks in NAC elements, i.e., where cPESs come close, which only becomes relevant when the system is excited in the cavity subspace. In contrast, cavity strong non-adiabatic coupling exhibits substantially extended non-adiabatic interactions between ground and first excited state cPESs as soon as the hybrid system enters the transition state region, irrespective of the energy stored in the cavity mode.
D. Electron-Photon Correlation and Cavity PES
Cavity potential energy surfaces are conceptually appealing, since they directly generalize well established ideas from quantum chemistry to the realm of vibropolaritonic chemistry. Hence, common quantum chemical concepts of (classical) activation barriers [35,38,40,41] and minimum energy paths (MEP) [38,72,73] are naturally transferred to mechanistic arguments in thermal vibro-polaritonic chemistry. However, correlated CBO and crude CBO approaches are commonly not distinguished, which motivates us to discuss here how the different treatment of electron-photon correlation impacts those properties.
In Fig.5a, we show the energy difference between the ground state cPES and its crude counterpart introduced in Eq. (41) under VSC at η = 0.04
∆E 0 (R, x c ) = E (ec) 0 (R, x c ) − V 0 (R, x c )(71)
for the weakly non-adiabatic CSM model. We find pronounced maxima in ∆E 0 close to the transition state along the nuclear coordinate and negative regions along the cavity displacement coordinate. The transition state is determined by classical activation barriers, which are defined as the energy difference between the global min-imum and the transition state energy
∆E a cl = E (ec) 0 (R ‡ , x ‡ c ) − min E (ec) 0 (R, x c ) ∆E a cl = V 0 (R ‡ , x ‡ c ) − min V 0 (R, x c ) ,(72)
for both CBO and crude CBO ground state cPES. In Tab.II, classical activation barriers are shown for the ground state cPES of the CSM model for different lightmatter interaction strengths, η, as obtained under the crude CBO approximation, ∆E a cl , and the CBO approximation, ∆E a cl . We observe classical activation barriers in crude VSC theory, ∆E a cl , to be nearly independent of η for the values studied here, which is in line with results reported in literature, fully accounting for the DSE term. [33,35] In contrast, classical barriers obtained from fully correlated calculations, ∆E a cl , increase with η, here about 4 %. We note, this can be already qualitatively observed in Ref. [16]. Hence, electron-photon correlation can lead to cavityinduced changes in classical activation barriers on CBO cPES, which in turn will potentially influence the reactive system's kinetics. This observation is naturally assumed to be system dependent. Further, since crude cPES do not exhibit a light-matter interaction dependent classical activation barrier, a comparison to CBO cPES should be taken with care. However, crude cPES studied here are indeed able to capture some light-matter interaction effects, e.g., barrier broadening and transition state valley narrowing. [35,38] Thus, further research will be required to map out the strength and weakness of the two perspectives.
Eventually, we consider differences in the cavity minimum energy path (MEP), another concept related to reaction kinetics beyond transition state arguments only: Along the (cavity) MEP, the energy is minimized with respect to variations along all orthogonal valley coordinates. [72] In Fig.5b, we compare cavity MEPs obtained for both ground state crude and CBO cPES at η = 0.04 and observe a rather close match, i.e., the path's curvature is qualitatively similar on CBO and crude cPES: Both MEPs show a linear behavior close to the transition state, whereas the crude version is subject to a slightly stronger curvature, i.e., the valley coordinates couple slightly stronger to motion along the crude reaction coordinate. Accordingly, electron-photon correlation seems to be less important for the cavity MEP, although the corresponding reaction potential still suffers from drawbacks in the barrier's description.
E. Perturbatively Corrected Ground State Crude cPES
In order to improve the crude description, we now examine perturbative corrections to the ground state crude cPES based on the cVSC-PT formalism introduced in Subsec.IV B. We discuss second-and third-order corrected cPESs of the weakly non-adiabatic CSM model in the CBO approximation, as given by
V (2) 0 (R, x) = V (1) 0 (R, x) + E (2) 0 (R, x c ) ,(73)
and
V (3) 0 (R, x) = V (2) 0 (R, x) + E (3) 0 (R, x c ) ,(74)
where, E
0 , is explicitly given in Appendix E. According to Eq.(56), the second-order correction decomposes into three contributions, E
(2) 0 = 3 i=1 E (2) 0,i , which read for the CSM model E (2) 0,1 = µ =0 |H sc 0µ | 2 ∆E (e) 0µ = µ =0 2ω c g 2 d 2 0µ x 2 c ∆E (e) 0µ ,(75)E (2) 0,2 = µ =0 2 H sc 0µ H dse µ0 ∆E (e) 0µ , = µ =0 α 8 ω c g 3 d 0µ (d µα d α0 ) x c ∆E (e) 0µ ,(76)E (2) 0,3 = µ =0 |H dse 0µ | 2 ∆E (e) 0µ , = µ =0 α,β g 4 ω 2 c (d 0α d αµ ) (d µβ d β0 ) ∆E (e) 0µ ,(77)
where, g = ωc d30 η. We suppress nuclear and cavity coordinate dependence for brevity and indicate contributions from DSE expectation values by brackets, e.g., (d 0α d αµ ). Numerically converged results are obtained via inclusion of all neutral states of the weakly non-adiabatic CSM model at fixed η.
We illustratively discuss the sensitivity of classical activation barriers to electron-photon correlation effects due to their potential relevance in thermal vibro-polaritonic chemistry. In Tab.II, second-order, ∆E (2) cl , and thirdorder, ∆E (3) cl , corrected classical activation barriers of the CSM model are provided for selected values of η. We immediately recognize sizable differences with respect to the crude barrier energy, ∆E a cl , which points at the relevance of beyond-crude CBO electron-photon correlation in low-frequency cavity mode settings. While the secondorder correction leads to a reduction of the barrier, we recover the correct trend and a significant fraction of barrier energy increase relative to the fully correlated result at third-order.
Finally, we note a significantly corrected classical activation barrier, ∆E (2,1) cl , when accounting only for the leading order term of cVSC-PT(2) in Eq. (75). Such a result is very appealing since the numerical effort towards a good approximation is rather low. However, this correction has to be treated carefully, since it always leads to a barrier increase. This statement is rationalized by not-ing that E (2) 0,1 vanishes at the transition state located at x c = 0.0, since only there the cavity coordinate gradient tends to zero as required by the minimum energy path. Further, since the numerator is strictly positive, the denominator leads to a negative E (2) 0,1 , which indicates an energy decrease at the reactant configuration and therefore an effective barrier increase. In this context, it is an interesting open question, which we here leave for future studies, whether classical activation barriers on cPES can actually be lowered due to light-matter interaction and under which circumstance this scenario appears to happen.
F. The Electron-Polariton Hamiltonian in Vibro-Polaritonic Chemistry
In this last section, we discuss potential issues occurring when the electron-polariton Hamiltonian,Ĥ p = T c +Ĥ ec , is employed in the VSC regime (cf. Appendix F for details). InĤ p , the electron-photon Hamiltonian, H ec , is augmented by the cavity KEO,T c , which motivates its applicability in the electronic strong coupling regime [25], where both electrons and cavity modes are treated as "fast" degrees of freedom due to similar energy scales. Here, we shall analyze a VSC scenario for the CSM model, where we calculate cPES based on,Ĥ p , motivated by a recent study [26].
In Fig.6a, we show polaritonic cPES, E (p) µ (R), obtained fromĤ p for forty states of the weakly nonadiabatic CSM model under VSC with η = 0.04 (cf. Appendix F for numerical details). We find a dense manifold of surfaces (colored orange), which show an energetic separation of approximately ω c . At elevated energies several avoided crossings appear (colored red), which can be traced back to the first excited adiabatic electronic state becoming energetically dominant.
In Fig.6b, the two energetically lowest lying cPES (colored blue) and the corresponding mass-weighted, nuclear derivative NAC element, − 1 Ma F (p) 10 (R), are shown as functions of the nuclear coordinate. The NAC element is subject to two symmetric maxima at around
R ≈ ±1.5 a 0 , where − 1 Ma F (p)
en (R) = 0.0012 a.u. and cPES are separated by ∆E 10 = 0.0024 a.u. = 527 cm −1 . From the latter, we have, ∆E 10 < ω c , which indicates strong light-matter coupling effects, since the first excited state's cPES is not just shifted by the cavity frequency relative to the ground state cPES. Further, the nuclear derivative NAC element shows a significant magnitude compared to the energy gap. We find, F Based on the non-vanishing non-adiabatic coupling and the small energetic spacing of polaritonic cPES obained fromĤ p , the CBO approximation for the ground state cPES seems not to be well justified from this perspective. Furthermore, due to small energetic separation of polaritonic cPES, corresponding "vibrational" eigenstates are assumed to hybridize, such that the bare ground state polaritonic cPES does not contain all relevant information. From a dynamics perspective, a ground state truncation additionally implies that the light-matter hybrid system will dynamically evolve only on a single surface, which effectively relates to suppression of excitations in the cavity subspace. Consequently, the electron-polariton Hamiltonian,Ĥ p , seems not straightforwardly applicable to ground state problems under the CBO approximation as compared to VSC theory discussed in Sec.II.
We like to close by emphasizing that our arguments apply to cavities, which operate at low energies compared to electronic excitations, i.e., infrared or terahertz cavities with, ω c ≈ ∆ vib ≪ ∆ e . In particular, our arguments do not transfer to scenarios, where cavity mode frequencies acquire a significant fraction of electronic excitation energies, as recently studied computationally in context of cavity-altered ground state reactions [74][75][76].
VI. SUMMARY AND CONCLUSION
In this paper, we extended ab initio vibro-polaritonic chemistry beyond the cavity Born-Oppenheimer approximation [15,16] and discussed the role of electronphoton correlation in the vibrational strong coupling regime. In the first part, we quantitatively reviewed VSC theory by derivation of coupled TISE for adiabatic electron-photon states and nuclear-photon states. Electrons are here described by an extended electronic structure problem, which accounts for correlations between electrons and cavity modes. [16] We furthermore derived explicit expressions for nuclear and cavity non-adiabatic derivative coupling elements via the generalized Hellmann-Feynman theorem. In combination with a numerical analysis of non-adiabatic coupling in the cavity Shin-Metiu model under VSC, our findings can be formulated as:
1. Cavity-induced non-adiabatic coupling is characterized by transition dipole moments between adiabatic electron-photon states and therefore relies crucially on their optical character. Nuclear nonadiabatic coupling is altered in presence of cavity modes by acquiring a transition dipole mediated correction to the vibronic coupling of adiabatic electron-photon states. The additional term emerges from the dipole self-energy and can be understood in terms of the transverse polarization operator.
2. The CBO approximation is well justified, when both adiabatic electron-photon states are energetically well separated and light-matter non-adiabatic coupling is small. The latter relates in particular to vanishing transition dipole moments between electron-photon states. Numerically, we find cavity-induced non-adiabatic coupling relevant if the system is highly excited in the cavity mode, which can become relevant even at low energies, when the non-interacting molecular system is already subject to strong vibronic coupling.
3. Numerical investigation concerning the applicability of the electron-polariton Hamiltonian in the VSC regime revealed the emergence of many close lying polaritonic cPES, which can be locally subject to cavity-induced non-adiabatic coupling. Thus, the CBO approximation seems not simply applicable in this setting. We argue that a separation of electrons as "fast" and cavity modes in combination with nuclei as "slow" DoFs finds in VSC theory a more suitable theoretical approach to vibropolaritonic chemistry. In contrast,Ĥ p , applies well in scenarios, where both electrons and cavity modes resemble "fast" high-energy degrees of freedom and nuclei are "slowly" moving (electronic strong coupling).
In the second part of this paper, we introduced a consistent approach to effective ground state models in vibro-polaritonic chemistry, named crude VSC theory, which can be understood as an approximation to the CBO model. Our formulation relies on a crude VSC Born-Huang expansion that employs a basis of adiabatic electronic states and does not satisfy the CBO but a crude CBO approximation. Accordingly, in crude VSC theory, only the bare electronic structure problem is solved ab initio, which motivates our argument that crude CBO and CBO ground state theories differ in the way they account for electron-photon correlation. We identified correlations partially missing in the crude framework to stem from neglected non-adiabatic coupling terms related to transition dipole moments in the adiabatic subspace, which in turn leads to different electronphoton entanglement character in Ψ (ec) cbo and Ψ (ec) ccbo . We furthermore proposed a perturbative formulation of CBO ground state theory inspired by molecular Herzberg-Teller theory, denoted as crude VSC perturbation theory (cVSC-PT). In the cVSC-PT framework, we characterized the crude CBO ground state as first-order approximation to the CBO ground state, which accounts only for electron-photon entanglement arising indirectly from interactions of electrons and cavity modes with nuclei. Correlation corrections manifest then as cavityinduced, transition dipole-mediated non-adiabatic couplings between adiabatic electronic ground and excited states, which are shown to initially enter at second-order of cVSC-PT. Further characteristics of electron-photon correlation effects based on a numerical analysis of the CSM model summarize as follows:
4. Electron-photon correlation in low-lying adiabatic electron-photon states manifests in terms of density variations in the molecular subspace relative to light-matter uncorrelated adiabatic electronic states (non-interacting limit of light-matter hybrid system).
5. Classical activation barriers on CBO cPES are subject to electron-photon correlation effects, which are absent on crude cPES. In the CSM model, electron-photon correlation induces a barrier increase of 4 % relative to the crude cPES under VSC at η = 0.04. Minimum energy paths on cPES are in contrast well reproduced by effective ground state models concerning path curvature, despite shortcomings in related reaction potentials. From this perspective, a comparison between fully correlated and effective results should be handled carefully, since the relevance of electron-photon correlation effects under VSC is a priori not clear. 6. Perturbative electron-photon correlation corrections at second-and third-order of cVSC-PT were explicitly evaluated for the weakly non-adiabatic CSM model under VSC and illustrated the relevance of electron-photon correlation beyond the crude CBO approximation. Further, cVSC-PT corrections allowed us to obtain significantly improved classical activation barriers relative to the crude cPES of the CSM model.
We conclude by noting, that it remains an open question, when an effective ground state description in spirit of the crude CBO approximation can qualitatively capture trends of the fully correlated CBO approach. Since ab initio approaches to the extended electronic structure problem underlying fully correlated VSC theory are still in their infancy, it would be desirable to identify suitable scenarios, where one can benefit from the appealing nature of crude model approaches to vibro-polaritonic chemistry. Further, non-adiabatic effects under vibrational strong coupling will become relevant for reactive systems subject to energetically low-lying excited states like for example the thermal isomerization process of azobenzene [77,78]. Such "beyond CBO" scenarios would be furthermore conceptually interesting, since they point at a deeper understanding of strong interactions between electrons, nuclei and cavity modes. Eventually, collective effects in vibro-polaritonic chemistry and their inclusion in an ab initio description of light-matter hybrid systems under VSC pose a contemporary theoretical issue relevant for connecting theoretical predictions with experimental findings. In particular, potentially non-local, cavity-induced interactions [79,80], which are able to shape the local "making-and-breaking" of chemical bonds in the VSC regime as well as the relevance of electron-photon correlation effects in this context requires further research. The applicability of crude VSC theory relative to fully correlated approaches can also here constitute a beneficial first step towards collective strong coupling scenarios due to its inherently simpler character.
ACKNOWLEDGMENTS
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
CONFLICT OF INTEREST
The authors have no conflicts to disclose.
APPENDIX
A. Details on Derivative NAC Elements
We analytically evaluate derivatives of electron-photon Hamiltonians with respect to cavity displacement and nuclear coordinates in VSC theory for derivative NAC elements as discussed in Sec. II C.
Cavity Derivative NAC Elements
For the cavity displacement coordinate derivative, we obtain
∂ ∂x λkĤ ec = ∂ ∂x λk Ĥ e + V c +Ĥ sc +Ĥ dse , (A.1) = ∂ ∂x λk 2Nc λ ′ ,k ′ ω 2 k ′ 2 x 2 λ ′ k ′ (A.2) + 2ω k ′ g k ′ e λ ′ k ′ · d en x λ ′ k ′ + g 2 k ′ ω k ′ e λk · d en 2 , = ω 2 k x λk + 2ω k g k e λk · d en , (A.3)
such that the corresponding matrix element evaluates to
ψ (ec) ν | ∂ ∂x λkĤ ec |ψ (ec) µ r = ω 2 k x λk δ νµ + 2ω k g k e λk · D νµ . (A.4)
Here, the diagonal term in the first line vanishes since, ν = µ.
Nuclear Derivative NAC Elements
For the nuclear coordinate derivative, we find
∇ aĤ ec = ∇ aĤ e + ∇ a V c + ∇ aĤ sc + ∇ aĤ dse , (A.5)
where we immediately recognize, ∇ a V c = 0. The interaction term's derivative gives
∇ aĤ sc = ∇ a 2Nc λ,k 2ω k g k e λk · d en x λk , (A.6) = 2Nc λ,k 2ω k g k ∇ a e λk · d n x λk , (A.7) = 2Nc λ,k 2ω k g k ∇ a b Q b e λk · R b x λk , (A.8)
where in the second line, ∇ a e λk · d e = 0. With,
d n = b Q b R b ,
in the third line and evaluation of nuclear gradients as
∇ a e λk · R b = e λk ∇ a · R b = e λk δ ab , (A. 9) it follows that
∇ aĤsc = 2Nc λ,k 2ω k g k Q a e λk x λk . (A.10)
Turning to the dipole self-energy term, the nuclear derivative follows as where the second term corresponding to the light-matter interaction contribution, ∇ aĤsc , is independent of electronic coordinates and vanishes therefore due to orthogonality of adiabatic electronic states.
∇ aĤ dse = ∇ a
B. Details on Crude VSC Theory
Starting from Eq.(39), the matrix element with respect toĤ ec is evaluated as
ψ (e) ν |Ĥ ecχ (nc) iµ |ψ (e) µ = E (e) ν (R) + V c (x) χ (nc) iν (B.1) + ψ (e) ν |Ĥ sc |ψ (e) µ rχ (nc) iµ + ψ (e) ν |Ĥ dse |ψ (e) µ rχ (nc) iµ ,
where the first term constitutes the diagonal contribution determined by the molecular PES, E (e) ν (R), and the cavity potential, V c (x). The second term resembles the matrix element for the light-matter interaction,Ĥ sc , which evaluates to
ψ (e) ν |Ĥ sc |ψ (e) µ rχ (nc) iµ = 2Nc λ,k 2ω k g k e λk · d νµ x λkχ (nc) iµ , (B.2)
and has in general both diagonal and off-diagonal contributions with dipole matrix elements as given in Eq. (42). The third term in Eq.(B.1) involves the dipole self-energy term,Ĥ dse , with matrix element
ψ (e) ν |Ĥ dseχ (nc) iµ |ψ (e) µ = 2Nc λ,k g 2 k ω k α e λk · d να e λk · d αµ χ (nc) iµ , (B.3)
where we inserted the resolution-of-the-identity in the adiabatic subspace, α |ψ which holds, when e λk projects out only a single element of the molecular transition dipole moment, d να . This can be always achieved for linearly polarized cavity modes as considered here: Vectors, {e λk , e λ ′ k , k}, span a cavity coordinate frame, which can be chosen parallel to the molecule fixed frame with molecular center of mass located at the origin. Then, only single elements of d να are addressed with respect to polarization vectors, e λk , and, e λ ′ k , parallel to the molecular frame's axis.
C. Details on Reduced Density Matrices
A beneficial connection between CBO and crude CBO ground states in Eqs. (31) and (45) can be introduced by realizing that the adiabatic electron-photon ground state, |ψ (ec) 0 (R, x) , at a fixed nuclear configuration, R, can be expanded in an orthonormal basis of adiabatic electronic states, |ψ (e) µ (R) . [68] Accordingly, the CBO ground state can be written as
|Ψ (ec) cbo (R, x) = χ (nc) 0 (R, x) c µ (x) |ψ (e) µ (R) (C.1) = µχ (nc) µ (R, x) |ψ (e) µ (R) , (C.2)
where we absorb expansion coefficients, c µ (x), which account for the cavity displacement coordinate dependence, in the second line into,χ where expansion coefficients, u i (R), now accounting for nuclear coordinate dependence, are absorbed into, x). By truncating the expansion in Eq.(C.4) after the first term, we obtain a zeroth-order reference (ref) state
(nc) µ (R, x) = c µ (x) χϕ (nc) i (R, x) = u i (R)χ (nc) 0 (R,|Ψ ref (R, x) =φ (nc) 0 (R, x) |ψ (e) 0 , (C.5)
which is a product state with respect to electronic and nuclear-cavity DoF, i.e., electrons are by definition disentangled from nuclei and cavity modes. In the following, electronic, nuclear and cavity reduced density matrices are derived for the CBO ground state, the crude CBO ground state and the partially disentangled reference state in Eqs.(C.2)-(C.5).
Electronic Reduced Density Matrices
Starting with electronic RDM, we obtain for the CBO ground statê
ρ (e) cbo (r, r ′ ) = dR dx Ψ (ec) cbo (r, R, x) Ψ (ec)⋆ cbo (r ′ , R, x) , = µν dR ψ µ (r; R) ψ ⋆ ν (r ′ ; R) × dxχ µ (R, x)χ ⋆ ν (R, x) , = µν dR ρ µν (r, r ′ ; R)S µν (R, R) , = µνρ µν (r, r ′ ) =ρ (e) µν , (C.6)
with vibro-polaritonic overlap integral
S µν (R, R) = dxχ µ (R, x)χ ⋆ ν (R, x) =S (n) µν , (C.7)
and
ρ µν (r, r ′ ) = dR ρ µν (r, r ′ ; R)S µν (R, R) =ρ (e) µν , (C.8) where ρ µν (r, r ′ ; R) = ψ µ (r; R) ψ ⋆ ν (r ′ ; R) . (C.9)
The crude CBO electronic RDM follows immediately by truncating the sum over adiabatic electronic states in Eq.(C.6) asρ For the reference state, one findŝ
ρ (e) ref (r, r ′ ) = dR dx Ψ ref (r, R, x) Ψ ⋆ ref (r ′ , R, x) , = ψ 0 (r) ψ ⋆ 0 (r ′ ) dR dx |φ 0 (R, x)| 2 =1 , = ψ 0 (r) ψ ⋆ 0 (r ′ ) =ρ (e) 00 , (C.11)
where normalization ofφ 0 (R, x) was employed in the second line. Note, the third line resembles a bare (crude adiabatic) electronic contribution.
Nuclear Reduced Density Matrices
The nuclear RDM for the CBO state follows aŝ .
ρ (n) cbo (R, R ′ ) = dr dx Ψ (ec) cbo (r, R, x) Ψ (ec)⋆ cbo (r, R ′ , x) , = µν dxχ µ (R, x)χ ⋆ ν (R ′ , x) × dr ψ µ (r; R) ψ ⋆ ν (r; R ′ ) , = µνS µν (R, R ′ ) O µν (R, R ′ ) =S (n) µν O (n) µν
(C.14)
Eventually, for the reference state, we havê
ρ (n) ref (R, R ′ ) = dr dx Ψ ref (r, R, x) Ψ ⋆ ref (r, R ′ , x) , = dxφ 0 (R, x)φ ⋆ 0 (R ′ , x) dr ψ 0 (r) ψ ⋆ 0 (r) =1 , =s 00 (R, R ′ ) =s (n) 00 , (C.15)
where normalization of ψ 0 (r) was employed in the second line. Note, in contrast to CBO and crude CBO nuclear RDM,ρ
ref is independent of adiabatic electronic contributions, which directly follows from the product nature of the reference state.
Cavity Reduced Density Matrices
Finally, the cavity RDM for the CBO ground state is obtained aŝ 16) and the crude CBO cavity RDM is straightforwardly found to beρ
ρ (c) cbo (x, x ′ ) = dr dR Ψ (ec) cbo (r, R, x) Ψ (ec)⋆ cbo (r, R, x ′ ) , = µν dRχ µ (R, x)χ ⋆ ν (R, x ′ ) × dr ψ µ (r; R) ψ ⋆ ν (r; R) =δνµ , = µ dRχ µ (R, x)χ ⋆ µ (R, x ′ ) , = µS µµ (x, x ′ ) =S (c) µµ , (C.(c) ccbo (x, x ′ ) =S (c) 00 . (C.17)
For the reference state, we find an expression similar to the nuclear RDM witĥ
ρ (n) ref (x, x ′ ) = dr dR Ψ ref (r, R, x) Ψ ⋆ ref (r, R, x ′ ) , = dRφ 0 (R, x)φ ⋆ 0 (R, x ′ ) dr ψ 0 (r) ψ ⋆ 0 (r) =1 , =s 00 (x, x ′ ) =s (c) 00 . (C.18)
As in the nuclear case, the latter is independent of adiabatic electronic contributions.
D. Numerical Details on the VSC-CSM Model
We follow earlier work and consider the two fixed nuclei in the CSM Hamiltonian separated by L = 10 Å = 18.9 a 0 with screening length, R f = 1.5 Å and nuclear charges, Q a = Q = 1 e. Further, we discuss weak (strong) nonadiabatic coupling regimes characterized by mobile nucleus screening lengths, R c = 1.5 Å (R c = 2.0 Å).
For the numerical solution of the length-gauge CSM model Hamiltonian, we employ a Colbert-Miller discrete variable representation (DVR) for all DoF, i.e., the electron, the moving nucleus and the cavity mode, with KEO matrix elements
T ij = 2 2∆s 2 (−1) i−j π 2 3 , i = j 2 (i − j) 2 , i = j , (D.1)
for coordinates, s = r, R, x c . Up to vibrational strong coupling with η = 0.04, we obtain converged results on grids r ∈ [−2L, 2L], R ∈ [− L 2 , + L 2 ], and, x c ∈ [−300, 300] √ m e a 0 . We employ grid points, M e = M n = M c = 201, for electrons, nucleus and cavity mode in both the weak and strong non-adiabatic coupling regimes.
E. Details on cVSC-PT(3)
We provide an explicit expression for the thirdorder energy correction in cVSC-PT(3) when applied to the CSM model. Following conventional Rayleigh-Schrödinger perturbation theory, the third-order energy correction is given by
E (3) 0 = µ,ν =0 ∆V 0µ ∆V µν ∆V ν0 ∆E (e) 0µ ∆E (e) 0ν − ∆V 00 µ =0 |∆V 0µ | 2 ∆E (e) 0µ 2 , (E.1)
where we suppress coordinate dependence for brevity. We first show, that contributions involving the cavity potential, V c , cancel. Taking into account the orthonormal character of adiabatic electronic states, the first term can be written as
0 is independent of V c and can be decomposed into eight contributions, E
0 = 8 i=1 E(3)
0,i , by grouping equivalent products of expectation values. The leading order term scales as g 3 and reads .
(E.7)
The quintic contribution scaling as g 5 is determined by three terms E F. Details on the Electron-Polariton Hamiltonian
Extended Born-Huang Expansion and TISEs
The electron-polariton Hamiltonian is given aŝ
H p =T c +Ĥ ec , (F.1)
and augments the length-gauge electron-photon Hamiltonian by the cavity KEO. A generalized Born-Huang expansion can be introduced as
Ψ i (r, R, x) = µ φ (n) iµ (R) ψ (p) µ (r, x; R) , (F.2)
with orthonormal, adiabatic electron-polariton states, ψ (p) µ (r, x; R), and nuclear states, φ (n) iµ (R), providing expansion coefficients. Adiabatic states treat electrons and cavity modes here on equal footing and only the nuclear coordinate dependence is understood to be parametric in nature. The molecular Pauli-Fierz Hamiltonian takes the form,Ĥ PF =T n +Ĥ p .
After projecting on adiabatic electron-polariton states, ψ (p) ν (R)|, we obtain a system of two coupled TISEs, where adiabatic states solvê
H p |ψ (p) ν (R) = E (p) ν (R) |ψ (p) ν (R) , (F.3)
with polaritonic cPES, E (p) ν (R), which only depends on nuclear coordinates, R, and provides a potential for a nuclear TISE Although F (p) a,νµ (R) is formally similar to F (n) a,νµ (R) in Eq.(30) (note different superscripts (p) and (n)), it differs in two relevant aspects: First, for infrared Fabry-Pérot cavities
T n + E (p) ν (R) φ (n) iν (R) + µ =νĈ (p) νµ φ (n) iµ (R) = E i φ|E (p) ν − E (p) ν+1 | ≈ ω c ≪ ∆ e , (F.7)
with characteristic fundamental electronic transition energy, ∆ e , and second, D (p) νν+1 , is not necessarily small for energetically close-lying cPES. We discuss those aspects in some detail in Subsec.V F.
Numerical Details
We solve the electron-polariton TISE (F.3) illustratively for the CSM model in the weak NAC regime with R c = 1.5 Å and η = 0.04. Convergence of the 40 energetically lowest lying polaritonic cPES is obtained with grids, r ∈ [−2L, 2L], x c ∈ [−300, 300] √ m e a 0 and R ∈ [− L 2 , + L 2 ], discretized by grid points, M e = 91, M c = 91 and M n = 101. Note, we restrict the electronic grid to the region where the electron-nuclear potential dominates and do not account for regions far from nuclei as determined by a harmonically-confining electronic DSE contribution. The latter may be relevant if the electron is driven far from the three nuclei due to an external driving field for example, which is beyond the scope of the present work.
proportional to polarization-projected transition dipole moments, e λk · D νµ , the optical character of adiabatic electron-photon states plays a central role: Cavity non-adiabatic coupling elements vanish for dark transitions characterized by, D νµ = 0, and are non-zero for bright transitions with D νµ = 0, where, F (c)
R, x) , where the corresponding electron-photon ground state satisfieŝ
and a partially disentangled reference state, Ψ ref (cf. Appendix C for details).
contributions from all adiabatic electronic states (cf. Appendix C). Thus, CBO-RDM (Tab.I, first line) are determined by all adiabatic electronic states and furthermore contain vibro-polaritonic contributions (indicated by tilde signs)
only apparently separable in the electron-cavity subspace, since both factors depend on nuclear coordinates, which in turn results in vibro-polaritonic contributions to the electronic crude CBO-RDM,ρ
cf. Appendix C). From this perspective, electrons and cavity modes are actually not disentangled in Ψ (ec)
FIG. 1 .
1Schematic representation of the cavity Shin Metiu model with a moving positively charged nucleus (blue) with coordinate, R, and a moving negatively charged electron (yellow) with coordinate, r, both interacting with a single cavity mode. Fixed positively charged nuclei are colored in red and located at a distance L from each other, while mobile particles move along the related molecular axis connecting fixed nuclei.The length-gauge Pauli-Fierz Hamiltonian, Eq.(15), reads for the CSM model (in atomic units)
v
(R), with energies, ε v . For the CSM model under VSC, we tune the cavity resonant to the fundamental vibrational transition, ω c = ε 3 − ε 0 . For the weak (strong) NAC regime, we have a cavity frequency, ω c = 585.16 cm −1 (516.74 cm −1 ), with vibrational transition dipole element, d 30 = 0.089 ea 0 (0.26 ea 0 ). The light-matter interaction constant, g = ωc d30 η, is furthermore characterized by a dimensionless parameter, η.[20] Further model parameters and numerical details on the CSM model are provided in Appendix D.
ν
(r; R, x c ), are eigenstates ofĤ ec in Eq.(58). In order to access cavity-induced changes in reduced light-matter hybrid densities, ρ(ec) ν , relative to the bare FIG. 2. Contour plots of reduced adiabatic electron-photon densities, ρ (ec) ν (r, R), (a1-d1), reduced adiabatic electron densities, ρ (e) ν (r, R), (a2-d2) and adiabatic difference densities, ∆ρ
, R), (a3-d3) in electron-nuclear coordinate space for adiabatic ground and first excited states (ν = 0, 1) in weak and strong non-adiabatic coupling (NAC) regime of the CSM model under vibrational strong coupling with η = 0.04.
ν (r, R) = 0 , for, η = 0
00
FIG. 3 .
3Weakly non-adiabatic CSM model under VSC with η = 0.04: a) Ground, V0(R, xc), and d) first excited state cPES, V1(R, xc), as function of nuclear, R, and cavity displacement, xc, coordinates with energies in atomic units (a.u.). b) Mass-weighted nuclear derivative NAC element, − 1 Ma F (n) 10 (R, xc), and e) cavity derivative NAC element, − F (c) 10 (R, xc), in a.u. as function of coordinates under VSC. c) Molecular, − 1 Ma F (n) en (R, xc), and f) DSE-induced nuclear NAC contributions, − 1 Ma F (n) dse (R, xc), to mass-weighted nuclear NAC element in b) with same parameters.
(n) 10 .
10Note, d 30 , is the vibrational transition dipole moment, whereas ∆E (ec) 10 relates to the energy difference between adiabatic electron-photon states. The nuclear term, F (n) 10 , has according to Eq.(30) a vibronic, F (n) en , and a cavity-induced contribution, F (n) dse , which explicitly read here
which can be traced back to the bright transition dipole moment, D
only quadratic corrections in η (cf. Subsec.V E). All energies are given in wave numbers for selected values of η.
FIG. 4 .
4Strongly non-adiabatic CSM model under VSC with η = 0.04: a) Ground, V0(R, xc), and c) first excited state cPES, V1(R, xc), as function of nuclear, R, and cavity displacement, xc, coordinates with energies in atomic units (a.u.). b) Massweighted nuclear derivative NAC element, − 1 Ma F (n) 10 (R, xc), and d) cavity derivative NAC element, − F (c) 10 (R, xc), in a.u. as function of coordinates under VSC as a) and d).
FIG. 5 .
5Differences between ground state cPES and ground state crude cPES for the weakly non-adiabatic CSM model under VSC with η = 0.04. (a) Energy difference, ∆E0(R, xc), in wave numbers (cm −1 ) and (b) numerical approximations to cavity minimum energy paths for E (ec) 0 (R, xc) in red and for V0(R, xc) in blue (contour plot shown in cm −1 ).
to be dominantly determined by the cavity-induced contribution to the nuclear derivative NAC, − 1 Ma F (p) dse (R) (cf. Appendix F), as shown in dashed-blue, i.e., non-adiabatic coupling is an effect of strong interaction between electrons and cavity modes. The molecular component, − 1 Ma F (p) en (R), is very small and not observable from Fig.6b. FIG. 6. a) 40 energetically lowest lying adiabatic polariton cPES, E (p) µ (R), as function of nuclear coordinate, R, obtained from, Hp, for the CSM model in the weak NAC regime under vibrational strong coupling with η = 0.04. Lowest two surfaces are colored in blue and high energy cPES manifold subject to avoided crossings is colored in red. b) Two energetically lowest lying polaritonic cPES, E (R), as function of nuclear coordinate, R.
e λk · d en ∇ a e λk · d en , e λk · d en Q a e λk , (A.13) where we employed the chain rule in the second line and took advantage of the result found for the interaction term in the third line. The nuclear derivative of the DSE term is related to the transverse polarization operatore λk · d en e λk , e λk · d en e λk , (A.16) where the second line follows with, α = (λ, k), for the summation index and, λ λk = λ k e λk , whereas in the thirdline, λ k = 2 ω k g k .For the corresponding matrix element, we find ψ (ec) ν |∇ aĤ ec |ψ (ec)
α
| = 1 (e) , between projected dipole moments. For diagonal matrix elements with ν = µ, it is instructive to rewrite the sum over adiabatic states as α e λk ·d να e λk ·d αν = α e λk ·d 2 να , (B.4)
(nc) 0 (
0R, x). Eq.(C.2) is formally appealing, since the crude CBO ground state, |Ψ
R, x) , introduced in Eq.(45) follows directly by truncating Eq.(C.2) after the first term.Further, one can similarly expand the crude CBO ground state in an orthonormal basis of crude adiabatic electronic states, |ψ
R 0 ) , with nuclear reference configuration, R 0 , leading to |Ψ (e) ccbo (R, x) =χ
(r, r ′ ) =ρ (e) 00 .(C.10)
O
µν (R, R ′ ) = dr ψ µ (r; R) ψ ⋆ ν (r; R ′ ) = O (n) µν . (C.13) Note, the adiabatic matrix element, O µν (R, R ′ ), is nonzero for R = R ′ . The crude CBO equivalent follows again by truncating the sum over adiabatic electronic states in Eq.(C.12) to the ground state contribution ρ
F
, are formally similar to Eq.(20), however, act now in the subspace of adiabatic electron-polariton states. Nuclear derivative NAC elements take the form (cf. Sec.II C) (the second line relates to a contribution of the transverse polarization operator (cf. Appendix A) in the basis of adiabatic electron-polariton states with transition dipole moments D (p) νµ (R) = ψ (p) ν |d en |ψ (p)
TABLE
TABLE II .
IIClassical activation barriers on the ground state cPES of the weakly non-adiabatic CSM model as obtainedfrom crude VSC theory, ∆E a
cl , VSC theory, ∆E a
cl , cVSC-
PT(2), ∆E
We acknowledge fruitful discussions with Tillmann Klamroth, Christoph Witzorky and Foudhil Bouakline (all Potsdam). This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC 2008/1-390540038. E.W. Fischer also acknowledges support by the International Max Planck Research School for Elementary Processes in Physical Chemistry.
∆V 0µ ∆V µν ∆V ν0where the first contribution is diagonal in adiabatic state indices µ, ν, such that it cancels with the first term in ∆V 00 |∆V 0µ | 2 due to different signs in Eq.(E.1). Thus, E∆E
(e)
0µ ∆E
(e)
0ν
=
V c H sc
0µ + H dse
0µ
H sc
ν0 + H dse
ν0
δ µν
∆E
(e)
0µ ∆E
(e)
0ν
+
H sc
0µ + H dse
0µ
H sc
µν + H dse
µν
H sc
ν0 + H dse
ν0
∆E
(e)
0µ ∆E
(e)
0ν
, (E.2)
∆E
(e)
0µ
2
=
V c H sc
0µ + H dse
0µ
H sc
µ0 + H dse
µ0
∆E
(e)
0µ
2
+
H sc
00 + H dse
00
H sc
0µ + H dse
0µ
H sc
µ0 + H dse
µ0
∆E
(e)
0µ
2
, (E.3)
µ , ν
,=0Terms with two interaction and a single dipole self-energy expectation value scale quartic in the light-matter interaction constant, g 4 , and read explicitly∆H sc
0µ ∆H sc
µν ∆H sc
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H sc
00
µ =0
|∆H sc
0µ | 2
∆E
(e)
0µ
2 =
µ,ν =0
(2ωc)
3
2 g 3 x 3
c d0µ dµν dν0
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0
(2ωc)
3
2 g 3 x 3
c d00 d 2
0µ
∆E
(e)
0µ
2
, (E.4)
E
(3)
0,2 =
µ,ν =0
∆H sc
0µ ∆H sc
µν ∆H dse
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H sc
00
µ =0
∆H sc
0µ ∆H dse
µ0
∆E
(e)
0µ
2
=
µ,ν =0 α
2 g 4 x 2
c d0µ dµν (dναdα0)
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α
2 g 4 x 2
c d00 d0µ (dµαdα0)
∆E
(e)
0µ
2
,
(E.5)
E
(3)
0,3 =
µ,ν =0
∆H sc
0µ ∆H dse
µν ∆H sc
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H sc
00
µ =0
∆H dse
0µ ∆H sc
µ0
∆E
(e)
0µ
2
=
µ,ν =0 α
2 g 4 x 2
c d0µ (dµαdαν) dν0
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α
2 g 4 x 2
c d00 (d0αdαµ) dµ0
∆E
(e)
0µ
2
,
(E.6)
E
(3)
0,4 =
µ,ν =0
∆H dse
0µ ∆H sc
µν ∆H sc
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H dse
00
µ =0
|∆H sc
0µ | 2
∆E
(e)
0µ
2 =
µ,ν =0 α
2 g 4 x 2
c (d0αdαµ) dµν dν0
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α
2 g 4 x 2
c (d0αdα0) d 2
0µ
∆E
(e)
0µ
2
and contain a single interaction and two DSE expecta-tion values. Eventually, the remaining term scales as g 6 and contains three DSE expectation values0,5 =
µ,ν =0
∆H sc
0µ ∆H dse
µν ∆H dse
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H sc
00
µ =0
∆H dse
0µ ∆H dse
µ0
∆E
(e)
0µ
2
,
=
µ,ν =0 α,β
2
ω 3
c
g 5 xc d0µ (dµαdαν) (d νβ d β0 )
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α,β
2
ω 3
c
g 5 xc d00 (d0αdαµ) (d µβ d β0 )
∆E
(e)
0µ
2
,
(E.8)
E
(3)
0,6 =
µ,ν =0
∆H dse
0µ ∆H sc
µν ∆H dse
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H dse
00
µ =0
∆H sc
0µ ∆H dse
µ0
∆E
(e)
0µ
2
,
=
µ,ν =0 α,β
2
ω 3
c
g 5 xc (d0αdαµ) dµν (d νβ d β0 )
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α,β
2
ω 3
c
g 5 xc (d0αdα0) d0µ (d µβ d β0 )
∆E
(e)
0µ
2
,
(E.9)
E
(3)
0,7 =
µ,ν =0
∆H dse
0µ ∆H dse
µν ∆H sc
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H dse
00
µ =0
∆H dse
0µ ∆H sc
µ0
∆E
(e)
0µ
2
=
µ,ν =0 α,β
2
ω 3
c
g 5 xc (d0αdαµ) (d µβ d βν ) dν0
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α,β
2
ω 3
c
g 5 xc (d0αdα0) (d 0β d βµ ) dµ0
∆E
(e)
0µ
2
,
(E.10)
(3)
0,8 =
µ,ν =0
∆H dse
0µ ∆H dse
µν ∆H dse
ν0
∆E
(e)
0µ ∆E
(e)
0ν
− ∆H dse
00
µ =0
∆H dse
0µ ∆H dse
µ0
∆E
(e)
0µ
2
,
=
µ,ν =0 α,β,γ
g 6
ω 3
c
(d0αdαµ) (d µβ d βν ) (dνγ dγ0)
∆E
(e)
0µ ∆E
(e)
0ν
−
µ =0 α,β,γ
g 6
ω 3
c
(d0αdα0) (d 0β d βν ) (dµγ dγ0)
∆E
(e)
0µ
2
.
(E.11)
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Well built product, but one needs to know that ... | Well built product, but one needs to know that because this is item has a single anchor point the is some jiggling on rougher roads. Used in my 2014 F350 crew can for grandkids who watch movies from my iPad. Must have worked well, because heard no complaints from them concerning movies they were watching. | amazon_reviews | 323 |
where does the water from evaporation ponds come from | Evaporation pond Evaporation ponds are artificial ponds with very large surface areas that are designed to efficiently evaporate water by sunlight and exposure to the ambient temperatures. Evaporation ponds have several uses. Salt evaporation ponds produce salt from seawater. They are also used to dispose of brine from desalination plants. Mines use ponds to separate ore from water. Evaporation ponds at contaminated sites remove the water from hazardous waste, which greatly reduces its weight and volume and allows the waste to be more easily transported, treated and stored. Evaporation ponds can also be used to evaporate the precipitation that falls | paq | 227 |
Does Investor-Ownership of Nursing Homes Compromise the Quality of Care? | Quality problems have long plagued the nursing home industry. While two-thirds of U.S. nursing homes are investor-owned, few studies have examined the impact of investor-ownership on the quality of care. The authors analyzed 1998 data from inspections of 13,693 nursing facilities representing virtually all U.S. nursing homes. They grouped deficiency citations issued by inspectors into three categories (“quality of care,” “quality of life,” and “other”) and compared deficiency rates in investor-owned, nonprofit, and public nursing homes. A multivariate model was used to control for case mix, percentage of residents covered by Medicaid, whether the facility was hospital-based, whether it was a skilled nursing facility for Medicare only, chain ownership, and location by state. The study also assessed nurse staffing. The authors found that investor-owned nursing homes provide worse care and less nursing care than nonprofit or public homes. Investor-owned facilities averaged 5.89 deficiencies per home, 46.5 pe... | s2orc_title_abstract | 71 |
Environmental Sensitivity Indexes in oil hydrocarbons: Its application in SIROCO | In this study, the Environmental Sensitivity Indexes (ESI) are analyzed through SIROCO (Island Response Systems and Operations against ocean pollutants), which allow us to determine the behavior of the substrates that constitute the coastal littoral in front of the polluting action originated by hypothetical accidental spills of hydrocarbons that may affect the islands of Lanzarote and Fuerteventura. | s2orc_title_abstract | 43 |
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A comparison of fire-resistant hydraulic fluids for hazardous industrial environments. Part II. Stability, fluid life, and environment | Fire-resistant hydraulic fluids have been developed since the Second World War. In addition to phosphate esters, there are various other fluids (e.g. oil-in-water emulsions, water glycols) available. This paper provides a technical comparison of the different fluid types, in terms of fire resistance, lubricating properties, and viscosity. The second part of the study examines the corrosion properties, thermal and oxidative stability, shear stability, materials compatibility, and other physical properties. | s2orc_title_abstract | 224 |
Bungie is working on a fix to solar based dot effects that were unintentional | The nerf to solar grenades and incendiary grenades seems to be unintentional. Bungie is working on it.
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who called ivan ivanovich a goose in 1835 | Ivans agreeing to fight a duel. Ivan Ivanovich, as the challenged party, has the choice of weapons, so he chooses the Turkish rifle, but the duel degenerates into a struggle for the rifle. It goes off in the struggle, having been overloaded with gunpowder, and the two Ivans are killed. They both go to Heaven, but upon seeing Ivan Ivanovich's outspread wings Ivan Nikiforovich again calls him "a goose", which sets off the squabble all over again. The Tale of How Ivan Ivanovich Quarreled with Ivan Nikiforovich "The Tale of How Ivan Ivanovich Quarreled with Ivan Nikiforovich" (, 1835), also | paq | 408 |
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Plasmid Profile and Virulence Analysis of Salmonella Gallinarum Indian Isolates | Epidemiological Characterization of Avian Salmonella enterica Serovar Infections in India | s2orc_citation_titles | 52 |
My left nipple has a slight bulge. | Hi I am a 13 yo male, 5’ 1” and about 90lbs (small kid). Sometime in the past few weeks a little bulge appeared behind my left nipple, but not my right. Putting to much pressure against it produces pain (i would say it’s a dull pain, coming in at a four depending on where I push on it). It almost feels loose, like I can get my fingers under it or push it around slightly, but it’s stuck behind my nipple. The bulge itself is not very big, probably unnoticeable to the untrained eye, but I can see it and while I’m not looking at it right now, it may be about half a centimeter high and 2-3 cm across. I don’t really want to ask my mom but I want to know what it is. Thanks in advance! | reddit_title_body | 416 |
when did sweetbay open in florida | personal information, such as names or addresses, was accessed. The intrusion affected Hannaford stores, Sweetbay stores in Florida and certain independently owned retail locations in the Northeast that carry Hannaford products. They say they are aware of about 1,800 cases of fraud related to the data intrusion and about 4.2 million unique account numbers were exposed. Sweetbay Supermarket Sweetbay Supermarket was a chain of American supermarkets located entirely in Florida. The first Sweetbay Supermarket to open was in Seminole, Florida, in November 2004. The company was headquartered near Tampa, in unincorporated Hillsborough County, Florida, in the Tampa Bay Area, and | paq | 466 |
where did ben stapleton serve as mayor of | Benjamin F. Stapleton Locke. Rumors of Stapleton's Klan membership circulated during the mayoral campaign. Stapleton responded by denying that he was a Klan member and condemning the Klan, "to appease his Jewish and Catholic supporters." Stapleton declared, "True Americanism needs no mask or disguise. Any attempt to stir up racial prejudices or religious intolerance is contrary to our constitution and is therefore un-American." The voters believed Stapleton's denial and he was elected, defeating an unpopular incumbent, Dewey Bailey. Stapleton then appointed fellow Klansmen to multiple positions in Denver government, though he initially resisted Klan pressure to appoint a Klansman as chief of police. | paq | 52 |
How to Calculate the Value/Price of Monero? | I am wondering how the value/price of one Monero is determined. It probably is connected to a lot of factor and not just made up by a random person. I just can imagine how I could calculate the value myself. I would like to know a way to determine the price by myself if that is even possible. | reddit_title_body | 218 |
What explains why vote buying occurs in some elections but not others? The phenomenon of vote buying is under-studied in authoritarian, single-party-dominant regimes, especially in non-partisan elections in which competition is candidatecentered rather than party centered. Village elections in China provide a valuable window on the dynamics of vote buying in these conditions. Employing both an in-depth case study and an original, panel survey to provide new, systematic measures of rents and vote buying, we develop and test the following hypothesis: the availability of non-competitive rents accessible by winning candidates explains the variation in the incidence of vote buying in local elections. Our causal identification strategy exploits the timing of land takings and the exogenous nature of formal land takings authorized in state land-use plans at higher administrative levels to test the vote-buying-as-rent-seeking hypothesis. We find that the lure of rents, mainly from government takings of village land, is a key driver of vote buying by non-partisan candidates for the office of village leader. The evidence suggests that vote buying provides information to the authoritarian state about which local elites it should recruit into the rent-sharing coalition.Keywords Vote buying · Rent seeking · Village elections · China · Land What explains why vote buying occurs in some elections but not others? The existing literature most commonly analyzes vote buying as a strategy of political parties to mobilize electoral support(Bratton 2008;Corstange 2018;Hicken 2011;Mares and Young 2016;Stokes et al. 2013). Parties pursue vote buying to increase their tallies at the polls. Vote buying is less commonly studied in electoral contexts in which parties and political machines are weakly institutionalized or absent (e.g., |
1 3 Keefer and Khemani 2005;Muñoz 2014) and in which elections are implemented by authoritarian regimes (e.g., Blaydes 2011;Frye et al. 2019). Village elections in China provide a valuable window on the dynamics of vote buying in these contexts. 1 Employing both an in-depth case study and an original, panel survey to provide new, systematic measures of vote buying and rents, we develop and test the following hypothesis: the availability of non-competitive rents accessible by elected officials explains the variation in the incidence of vote buying in local elections. We find that the lure of rents, mainly from government takings of village land, is a key driver of vote buying by non-partisan candidates for the office of village leader. Our causal identification strategy exploits the timing of land takings and the exogenous nature of formal land takings authorized in state land-use plans at higher administrative levels to test the vote-buying-as-rent-seeking hypothesis.
Our findings build on and extend the existing literature on comparative vote buying. Scholars note that more personalized competition is less likely to be programmatic and more likely to rely on the exchange of individual benefits (Kitschelt and Wilkinson 2007). In non-partisan competition, individual candidates' promises to voters about future benefits lack the institutional commitment mechanisms that political parties provide, making pre-election vote buying an important electoral strategy (Hanusch and Keefer 2013). We examine rent-generating state interventions at the local level and make a theoretical contribution by showing that candidates invest personal resources to buy votes as an individual investment strategy repaid by the capture of rents once in office. Our study also makes an empirical contribution by providing novel and systematic evidence on how land takings-a common phenomenon in the developing world-generate rents and drive vote buying in grassroots elections, where parties do not structure or finance competition and where government intervention in the local economy varies across locales. In testing hypotheses involving rent seeking and vote buying, one challenge is measurement (Lehoucq 2003;Kramon 2016). Existing work examining the link between rent seeking and vote buying relies on anecdotal accounts or formal models. 2 Combining an in-depth case study with an original multi-wave survey that covers 1,240 household in 62 villages allows us to identify a scalable measure of rents and develop a unique measure of vote buying based on household reports.
While most studies of vote buying focus on multi-party elections, the prominence of vote buying in non-party elections has important implications for our understanding of authoritarianism. Existing studies demonstrate the informational value of vote buying to voters in democratic contexts (Kramon 2016;Muñoz 2014). This study highlights how successful vote-buying candidates reveal information to the state about their social and financial capacity to command support from the local community. Authoritarian regimes can use this information about local notables to decide whom to allow into the rent-sharing coalition and how to coopt these local elites for regime purposes (Blaydes 2011). Fieldwork evidence shows that the Chinese Communist Party (CCP) has tapped successful, non-partisan vote-buying candidates after their election to take on party roles within the village.
The paper proceeds as follows. The next section grounds the analysis in the comparative literature on vote buying. The following sections detail our argument that vote buying is best explained as a means of capturing rents and provide background on the institutional context in which rents are generated. We then present an in-depth case study, which allows us to identify a scalable measure of rent-seeking opportunities and to generate specific, testable hypotheses about vote buying. The remaining sections present the data and findings, discuss implications of the study, and conclude with an agenda for future research.
Vote Buying in the Comparative Literature
Vote buying is a particular form of clientelism, in which voters receive a non-programmatic benefit-like cash or gifts-in exchange for a future vote for a particular candidate (Hicken 2011,292). Vote buying is prevalent in elections across globe. Roughly half of respondents in the sixth wave (2010)(2011)(2012)(2013)(2014) of the World Values Survey report that "voters were bribed very or fairly often in their country's elections"; in light of its prevalence, the question of "why politicians undertake vote buying in the first place" has begun to receive more scholarly attention (Keefer and Vlaicu 2017,775). While vote buying is typically analyzed as a political strategy to mobilize voters in multi-party competition (e.g., Diaz-Cayeros et al. 2016;Greene 2010;Lust-Okar 2006;Magaloni 2008;Stokes et al. 2013), it is also a common phenomenon in contexts where electoral competitions do not develop along partisan lines, such as local elections in one-party regimes (e.g., Zhao 2018). Theoretical analyses suggest that vote buying is more likely to take place when competition is candidatecentered rather than party-centered (Hicken 2007) and when the role of parties is weak or absent (Keefer and Vlaicu 2017;Muñoz 2014). Voters are less likely to trust candidates' policy promises that yield benefits only after the election. When party organizations that could discipline candidates are not involved in the elections, individual candidates' policy promises are less credible. Candidates instead choose to deliver pre-election benefits (i.e., vote buying) to mitigate the commitment problem (Hanusch and Keefer 2013). In an empirical analysis, Heath and Tillin (2018) find that when governance institutions are weak and inefficient, voters prefer benefits from direct, clientelistic exchanges (e.g., vote buying) over promises of uncertain programmatic rewards.
Elections and vote buying, where it occurs, can distort citizens' wishes but provide useful information to autocrats. Focusing on China, Martinez-Bravo et al. (forthcoming) show that local, non-partisan elections in autocratic regimes delegate selection and monitoring of village officials to citizens. They argue that rulers introduce elections when they lack information about local officials. However, they do not address the existence of vote buying or how it distorts the information that 1 3 regimes receive from elections. In describing parliamentary elections in Mubarak's Egypt, Blaydes instead suggests that vote buying in elections "works as a kind of market mechanism for the selection of individuals who will be allowed to extract state rents (2011,49)." Local elites who enter and win in the electoral marketplace reveal information to the regime about whom to coopt. Thus, vote buying serves as a mechanism to select local elites to be drawn into the rent-sharing coalition. As important and prevalent as vote buying is, the empirical evidence on the relationship between rent seeking and vote buying in non-partisan, authoritarian elections remains scant.
Vote Buying as Rent Seeking
We hypothesize that the incidence of vote buying in non-partisan elections is patterned on the availability of rents. We build our argument on the comparative literature that highlights the prominence of rents in shaping vote buying (Diaz-Cayeros et al. 2016;Keefer and Vlaicu 2017;Kitschelt 2000). Chandra (2007), in her conceptual overview of vote buying, begins by establishing the existence of rents, which stem from the state's "monopolistic control" over valued resources. Officials can sell resources for artificially high prices, generating non-competitive rents (Bates 1987;Buchanan et al. 1980). Thus, where the state controls significant resources, political control of the state is a valuable prize.
Identifying rents presents both a theoretical and empirical challenge. In his study of clientelism, Kitschelt (2000,862) is critical of studies that conceptualize rentseeking opportunities in terms of macro-level trade policies or the overall size of the public sector, since they fail to specify micro foundations or allow for sub-national variation. Hellman associates the existence of rents and rent seeking with partial reform in post-Communist transitions. "Partial reforms were expected to generate rent-seeking opportunities arising from price differentials between the liberalized sectors of the economy and those still coordinated by nonmarket mechanisms … these arbitrage opportunities have generated rents to those in a position to take advantage of these market distortions (Hellman 1998,219)." In the next section on China's institutional context, we establish the existence of non-competitive rents associated with village leadership positions in China.
Having established the existence of rents, scholars analyze how political actors seek to access them. 3 While bribery of regulators is a paradigmatic example of rent seeking (Aidt 2016), this paper highlights buying votes as an important pathway to controlling lucrative rents. Posner describes how shifting monopoly profit to officials "draws real resources into the activity of becoming the official (1975,812)." Politicians' access to rents rewards election winners with personal benefits. According to Aidt, "Controlling these rents is a contestable prize (2016,150)," and individuals will seek positions in government in order to gain access to them. Elections in authoritarian regimes are widely understood as a mechanism for sharing rents or spoils with elites who have the potential to threaten regime survival (Gandhi 2008).
Recent literature suggestively links vote buying and rent seeking. Blaydes argues that Egypt's non-partisan "rent-seeking elite" buy votes with cash or goods in a quest to win seats in parliament in order to access rents-not policy influence or party representation (2011,8). As she notes, "parliamentary hopefuls spend millions to reap billions (10)." High spending on vote buying is a rational choice, since elections in Egypt are a "sorting mechanism" for "access to state largesse," and "those who spend the most on campaigning are most likely to prevail (54)." While this account reflects comments from Egyptian informants, the study offers no test of the intuition. Zhao (2018), in a case study of village elections in China, similarly relates the availability of rents to vote buying on the part of candidates for local office. Keefer and Vlaicu (2017) develop a formal voting model in which parties use various strategies to win votes. They suggest that the larger the available rents, the greater the incentive to buy votes in order to control those rents. These studies derive their conclusions from anecdotes (Blaydes 2011;Ahram 2012), case studies (Zhao 2018), and theoretical models (Keefer and Vlaicu 2017).
A direct challenge in systematically testing the vote-buying-as-rent-seeking hypothesis is the measurement of rent-seeking opportunities and vote-buying activities; both remain largely hidden in most electoral contexts. We use an in-depth case study to reveal the nature of rent seeking and vote buying. Then, employing original survey data, we provide a quantitative test of the relationship between rent seeking and vote buying in China's village elections. Our measures of rents address Kitschelt's (2000) call for specifying micro foundations and accounting for subnational variations in rents in the context of elections.
Institutional Context
This section introduces the relevant institutions in which rents are generated and elections are contested. In contemporary rural China, rent seeking occurs overwhelmingly in the context of formal land takings, in which the state exercises a statutory monopoly over the sale of rural land into the urban market, making available non-competitive rents to state authorities, including village leaders (Zhao 2018). 4 The county land management bureau requisitions land from village collectives at below-market value, calculated in terms of the agricultural use-value of the land, and sells it to commercial developers at higher, market prices, making the rents quite sizeable (Chen and Kung 2016). Indeed, land takings have become a significant source of revenue for county governments (Su and Tao 2017). The difference between the price received from developers and the compensation paid to farmers, minus any public investment in basic infrastructure, represents the rent the state captures. Village leaders, who manage compensation funds for the village, share in these land rents by under-compensating rural households and by controlling compensation paid for land collectively owned and operated by the village.
Land takings are part of a top-down planning process. Since the 1990s, the Ministry of Land and Resources has assigned "annual land requisition quotas to each locality in the country (Su et al. 2019,761)." As concerns about loss of arable land have mounted, the ministry has established even tighter guidelines on land takings (Brandt et al. 2017(Brandt et al. ,1041. Regulations protecting arable land place strict limits on the area of farmland that may be converted into construction land in any given year (World Bank and Development Research Center 2014,113). Land-use plans received by the county are highly detailed, and "alteration [is] very rare (Su et al. 2019,762)." Ma and Mu (2020, 4) and Su et al. (2019,762) conclude that village leaders have no influence on formal state decisions to requisition land. Only after a decision is made, officials from the county, township, and village hold joint meetings to announce the decision and to present information to villagers regarding the purpose and size of the taking as well as the basis on which households are to be compensated (Brandt et al. 2017). Compensation funds, to be paid by the developer (the state in the case of takings for rail and highway projects, e.g.), reach the village via the local government late in the course of the project, since funds are generated only after land is taken, cleared, and sold through auction or negotiation by the state to the developer.
A related source of rents for village officials comes from illegal, informal sales of farmland by the village. Informal sales occur outside the purview of local government agencies that regulate the taking of village land for lucrative non-agricultural uses. Such informal sales are one response to state limits on rural-to-urban land conversion intended to protect arable land. In a survey designed to enumerate all changes to rural land holdings in a two-province sample between 1996 and 2011, Brandt et al. (2017Brandt et al. ( ,1036 find that only 12 percent of land takings recorded at the village level were initiated informally, i.e., by the township or village.
Village-run enterprises have historically been a source of rents, stemming from privileged access to capital from local state-run banks and credit cooperatives and soft tax obligations (Hu 2005;Oi and Rozelle 2000;Takeuchi 2013). 5 However, Whiting (2001,289-295) shows that, following the banking and fiscal reforms of the 1990s, township-and village-run enterprises (TVEs) struggled to compete and survive. "TVEs, instead of being a tremendous asset, turned into a liability (Su and Tao 2017,240)."
Leaders tasked with governing villages in China are selected through popular election for three-year terms. Village elections take place under the auspices of the 1998 Organic Law of Villagers' Committees and select the village head and members of the village leadership committee. 6 The entire village constitutes the electoral district, and any adult whose household registration is in the village may stand for election and vote. Elections are directly managed by election steering committees, which are themselves selected by popular bodies, including village assemblies, village groups, or village representative assemblies. According to O'Brien and Han, election steering committees are quite "independent [from the Chinese Communist Party] and responsive [to villagers], even though committees in most locations continue to be presided over by village party secretaries (2009,." Moreover, under rules in force in most provinces of China, "[CCP] control over the nominating process has been gradually loosened," and every voter is entitled to nominate candidates (364). Electoral rules mandate that there must be at least two candidates for village leader. Elections are candidate-centered and self-funded; party affiliation does not appear on the ballot, and the party does not finance campaigns of candidates who are party members. Indeed, CCP members may compete against each other. Most candidates campaign by making personal appeals in speeches before village assemblies and during visits to voters' homes.
In an intensive study of one county in Yunnan, Landry et al. (2010) show that the ruling party does not structure competition in village elections. They "confidently conclude that in practice both party and nonparty members were allowed [to] run in the …village elections and that nonparty candidates emerged from strata of villagers that were unlikely to be incorporated in the formal, non-electoral political institutions that dominated village life before the introduction of elections (2010,(13)(14)." Kennedy further argues that vote buying is unlikely to occur in elections where competition is absent or manipulated by the local party-state. "[Vote buying] is a problem that exists only because of reduced township interference and a fairer election process (2009,394)," i.e., absence of direct interference by the CCP in the selection of candidates and electoral choices of voters.
Scholars highlight the ambiguity of the 1998 Organic Law with respect to the legality of vote buying (Kennedy,(620)(621). The law does not mention vote buying (贿选); it states that the occurrence of bribery (贿赂) may invalidate an election, without defining what bribery entails. A 2009 central party-state notice acknowledges this problem and calls for "further clarification of the bounds of vote buying ( 进一步明确贿选的界限)" without final resolution. 7
Case-Study Evidence on Village Elections
Working within this institutional context, we combine an in-depth case study with an original survey in order to identify scalable measures of key concepts-rents and vote buying-and to generate specific, testable hypotheses relating rent seeking and vote buying. 8 The experience of villages in one satellite county outside a metropolis in an eastern coastal province of China is illustrative. In 2013, one of the authors conducted more than seventy in-depth interviews, including elected village leaders, unsuccessful candidates, villagers (voters), and township officials as well as informal discussions with county officials, focusing on village governance in more than a dozen villages in the county. 9
Land Rents
Land development in Baijia Village (a pseudonym) was spurred in 2005 by the county's decision to site the new campus of a highly regarded secondary school on land near the village. The county government's economic development plan, designed to attract outside investment to the county, included mixed residential and commercial development of roughly 500 mu 10 of village land in the vicinity of the new school, which would be the anchor for a new, planned community. In the case of Baijia Village, urban-planning regulations necessitated dividing the entire 500-mu project into multiple, smaller sub-developments to be included in consecutive annual land-use plans in order to comply with the letter-if not the spirit-of provincial regulations limiting the pace of urbanization.
Three levels of local authorities-the county, township, and village 11 -shared in the lucrative rents generated by the series of land takings implemented over the following years. The county government controlled the lion's share of rent, while the township and village received smaller shares. Although the amount the developer actually paid to the county for the land was undisclosed, compensation to be paid to Baijia villagers was publicly announced and set at 18,000 yuan per mu. This amount reflects the average value of agricultural output on the land over the preceding three years (1,000 yuan), calculated over the number of years remaining on farm households' 30-year land-use contracts with the village (18 years, in this case). Interviews reveal that the village leadership received at least 21,000 yuan per mu, representing a direct rent to the village leadership of 3,000 yuan per mu. Thus, the village leader would have access to approximately 1.5 million yuan in rents at the village level over the course of the 500-mu project as a result of the lower compensation to villagers.
A related source of rent came from compensation paid directly to the village for collective land under village management, including land previously used for village offices, plazas, and roads as well as forest land managed directly by the village. Compensation for this land, roughly 50 mu at 21,000 yuan per mu, totaled approximately one million yuan and was paid to the village and managed by the village leader but not distributed among village households.
The village leader's role in negotiations between the government and real estate developers, on the one hand, and members of the village collective, on the other, presented further opportunities for generating rents in the process of the land takings. Villagers who lost housing in addition to arable land were slated to receive Studies in Comparative International Development (2022) 57:337-360 new housing provided by the developer based on a per-meter exchange negotiated between the developer and the village leader, acting on behalf of village households. For example, an exchange rate of 1:1.5 meant that 100 square meters of villagers' old housing could be exchanged in the future for 150 square meters of new housing in the new residential community. However, in return for side payments from the developer, the village leader reportedly lowered the negotiated rate. In 2009, he agreed with the developer to an exchange rate of 1:1.25, while other affected villages had settled on a rate of at least 1:1.3. Villagers believed that the leader was willing to accept a lower exchange rate because he had received more than two million yuan in side payments from the developer.
In sum, interviews suggest that the leader of Baijia Village could control nearly five million yuan from the development project from these three sources alone. Land takings represent the greatest opportunity for a village official to access rents.
Vote Buying as Rent Seeking
Prior to the advent of land takings, elections in Baijia Village had been a sleepy affair. However, the 2008 election for village leader saw unprecedented competition, with six candidates contesting the election, along with the introduction of vote buying to the village. Two leading candidates emerged. One was the incumbent village leader, who reportedly offered targeted voters 600 yuan (about 86 US dollars) each for their votes. The leader's wife's successful retail business in the township provided funds to buy votes. His main challenger was the local owner of a construction company who had never before stood for election. The owner operated his construction company out of the county seat, but because his household registration remained in the village, he was eligible to compete for the position of village leader. Through paid vote brokers, well-connected and respected members of the village, he offered 800 yuan (about 116 US dollars) to targeted voters in order to counter the appeal of the incumbent. This sum is equivalent to approximately ten percent of an individual's average annual income (8,000 yuan) at that time. 12 The remaining candidates, including a middle school teacher who reportedly borrowed money to buy votes, were quickly outbid and abandoned any attempt at vote buying. As one villager commented, "Nobody in our village would spend money on running for office in the past… But since we've had land takings, people have become crazy spending money on the election. They know that no matter how much money they invest, they can easily recoup it from the land takings if they win the election." 13 Another villager made a similar assessment: "What could make an owner of a construction company give up his comfortable life in the county and move back to our poor village? It can only be because he expected to earn more money from his position." 14
Vote-Buying Process
Vote-buying candidates took a number of steps to try to ensure electoral success. They targeted swing voters, recognizing that relatives and allies of competing candidates were unlikely to be swayed by payments of cash from other candidates. Swing voters targeted in the vote buying schemes reportedly cast their votes for the highest bidder. 15 Vote-buying candidates and their brokers used a variety of mechanisms to monitor vote choice. For example, brokers instructed recipients of cash for votes to include their own names as write-in candidates for the less important position of village committee member. These voters' names would be read in the public vote count but would not be elected with a single vote. In other cases, brokers instructed recipients of cash for votes to take pictures of their completed ballots with their cellphones and to send them to the brokers. 16 The construction company owner took other steps to try to ensure electoral success. He offered bribes of gift cards and expensive liquor to township officials overseeing the election to look the other way as vote buying occurred and to send cooperative officials to monitor the election itself. Interviews suggest that these actions inhibited township officials from reporting vote buying to the county civil affairs bureau, the government agency responsible for overseeing village elections. Rumors repeated by villagers in interviews suggested that the construction company owner also sought to curry favor with the deputy county leader overseeing elections by providing the finish work on his house for free. At the same time, these reports also indicate that township and county officials were aware of vote buying in the village election.
Aftermath of Vote Buying
In the aftermath of vote buying, competition in the 2008 village election dwindled to two effective candidates, and victory went to the highest bidder-the construction company owner. After his victory in the village-wide election, the new village head was recruited by the local party apparatus to serve concurrently as village party secretary. 17 Finally, once land was formally requisitioned for the mixed residential and commercial development, the new village leader controlled the rents from the taking of land. As chairman of the village committee, his signature was required to implement the land taking and to access compensation funds deposited in the village account. As detailed above, the village leader controlled up to five million yuan (three quarters of a million US dollars) in rents through the land taking. The new village head's actions were not without controversy. Rumors circulated among villagers about under-compensation for their lost land. A small group of villagers-initially 5-6 people, ultimately growing to 15-mobilized to protest the low compensation by petitioning township and county authorities. They sought both higher compensation for land lost to urban development and dismissal of the elected village leader. However, neither township nor county officials took any action against the village leader. In the conclusion, we return to the issue of accountability in the context of vote buying in the agenda for future research.
Hypothesis Generation
The case of Baijia Village provides the basis for the following testable hypothesis. Specifically, we hypothesize that availability of rents is associated with the occurrence of vote buying in non-partisan, village elections. Candidates buy votes in order to gain access to rents controlled by the state. The case study also highlights the political context: in the absence of parties to structure and finance political competition, vote buying by candidates is a calculated, individual investment to capture rents.
Data and Identification Strategy
Survey
We test this hypothesis with an original dataset. The dataset includes two waves of a multi-wave panel survey; the relevant waves were conducted in 2008 and 2014, in which two of the authors directly participated. The 2008 wave is based on a sample of 2,000 households in 100 villages selected from five provinces: Jilin, Hebei, Shaanxi, Sichuan, and Jiangsu. 18 Within each of the five provinces is a stratified random sample; five sample counties were randomly selected-one from each income quintile. Within each of the 25 counties, two townships were selected-one each from the top and bottom halves of the income distribution. Within each of the 50 townships, two villages were randomly selected for a total of 100 villages. In each village, 20 randomly selected households were interviewed, along with village and small group leaders. 19 The 2014 wave of the panel survey is based on a 60-village sub-sample of the original panel. To select the sub-sample, the counties from the high-and low-income quintiles were retained, and a third county was randomly selected from among the three remaining sample counties in each province. Two additional sample villages were included, bringing the total number of villages to 62. 20 The present study employs the 62-village sub-sample, with data from 1,240 households. Village leaders and household representatives were surveyed in faceto-face interviews. The 2008 wave included detailed batteries on village elections and other aspects of the household and village experience. The 2014 wave included extensive batteries on changes in household and village assets, including land. The unit of analysis for this study is the village election. Appendix I includes individuallevel analysis as a robustness check (Table A4); results are consistent with those reported, below. 21
Dependent Variable: Vote Buying
The dependent variable is operationalized through questions asked at the household level about the overall situation in the village's most recent election, specifically: Did anyone approach you [literally, go to your house] to buy your vote? (有人到 你家拉选票吗?) In the village-level analysis, the dependent variable is the share of households in each village responding affirmatively to the question about vote buying. The use of a common euphemism for vote buying (literally, "pulling votes" (O'Brien and Han 2009,365)) is intended to reduce its sensitivity for respondents. While official statements often use the more formal term "huixuan (贿选)" to refer to vote buying, "lapiao (拉票)" is the more colloquial term used by villagers in faceto-face interviews to refer to the payment of cash to influence vote choice. Lu (2011) reports that lapiao is a common usage by villagers in referring to material exchange for votes. A special feature on elections in the official People's Daily newspaper uses the combined term "lapiao huixuan" in analytical context but the more informal "lapiao" in providing specific accounts of vote buying in village elections (Yang and Li 2016). To check for sensitivity, we include an indirect measure of vote buying ("Did you hear about vote buying in the village?) (Appendix I, Tables A2, A3).
Independent Variables: Availability of Rents
The main independent variable of interest is availability of rents, measured in terms of whether there was a land taking within one term of a village election. 22 We use village responses to the following question: Since the second round of land contracting [in the mid-1990s], have there been any land takings in this village? Enumerators recorded all instances of land takings.
The identification strategy is two-fold. We are able to leverage multiple sources of variation, including formal and informal land takings and those preceding and following elections. Formal land takings involve decisions made at higher levels of government and are exogenous to the village. We contrast formal takings with all takings, including informal, illegal ones initiated by village leaders. We also exploit the timing of the land taking process, comparing takings initiated before and 21 Note that land takings and elections vary systematically at the village level. 22 Both village and household representatives were surveyed, and the data on land takings are mutually consistent. See Appendix I, Table A1. after the election. In order to influence the election, vote-buying candidates, who, we hypothesize, are motivated to control land rents that pass through the hands of village leadership, must know about land takings before the election. Moreover, delivery of compensation funds to the village comes late in the process. Thus, we test the relationship between land takings (available rents) and vote buying in three time periods: when land takings occur within one term before or after the election, three years before, and three years after the election. We also include other possible sources of rent, namely the number of enterprises in the village.
Control Variables
Scholars of clientelism identify a range of other factors, including income level, population size, and electoral rules that are conducive to vote buying. Comparative studies find that vote buying is more common among poorer communities (Stokes et al. 2013;Brusco et al. 2004), as these voters "discount the future, rely on short causal chains, and prize instant advantages (Kitschelt 2000,857)." By contrast, Kennedy notes with respect to the Chinese case "that vote buying is most prevalent in wealthier villages close to urban centers, such as large metropolitan areas (Beijing, Shanghai, or Shenzhen), provincial capitals, district cities, and even smaller county seats (Kennedy 2010,625)." However, the apparent association between wealthier, peri-urban communities and vote buying is also likely to be a reflection of the locus of land takings rather than the level of income. We address this debate by including an income measure in tests of our hypothesis.
Consistent with variants of clientelist theory that emphasize face-to-face interactions, smaller communities are also more likely to experience vote buying (Cox 1987). Candidates incur lower costs in smaller villages, because the number of votes to be bought is smaller in these villages (Takeuchi 2013). In what follows, we control for village population.
There is disagreement in the literature regarding the impact of the secret ballot on vote buying. Since the informational requirements of clientelist vote buying are high, secret ballot reforms are frequently associated with declines in vote buying (Schaffer 2007;Kam 2017). However, "many voters share the belief that their electoral choices can't be kept secret," even though they do not necessarily know how their votes are monitored (Stokes et al. 2013,101). Blaydes (2011,106) describes the ways that brokers in Egyptian parliamentary elections observe voters' choices on ostensibly secret ballots. To address this issue, we control for the use of secret ballots in the analysis. Other procedural factors arise in the case of village elections in China; some villages permit proxy voting, in which another person-often a husband or wife-may cast a ballot on behalf of another voter. Brandt and Turner indicate that "a village where looser proxy voting is permitted can engage in considerably more rent-seeking behavior than in a village where proxy voting is more restricted (Brandt and Turner 2007, 477)." Therefore, we control for proxy voting as well as the use of a secret ballot.
The intensity of competition is also commonly associated with vote buying (Golden and Chang 2001;Cox and Thies 1998). If candidates can easily win 1 3 elections or are not challenged by other candidates, they have less incentive to pursue vote buying. In studies of village elections in China, scholars use the number of candidates as an indicator to show that, in general, village elections have become more competitive over time (He 2007;O'Brien and Han 2009). We control for competitiveness using this measure.
Scholars have also noted the impact of higher-level interference on village elections (e.g., Fang and Hong 2020). To account for this possibility, we control for whether candidates are approved by local government officials and the numbers of official monitors on the day of election.
We also examine the effect on reports of vote buying when the winning candidate for village head is the CCP village party secretary. To the extent that the party influences electoral competition through vote buying, we would expect the election of the party secretary to be positively associated with vote buying.
Descriptive Statistics
The descriptive data (Table 1) allow us to paint a picture of both available rents and vote buying in the context of village elections. In 69 percent of the villages, one or more villagers reported vote buying. In the average village, 11 percent of respondents experienced vote buying. The highest percentage of respondents in any single village experiencing vote buying was 50 percent. 23 When asked less directly, 20 percent of villagers reported having heard of vote buying taking place. According to the case-study research, vote buying is limited by the fact that candidates believe their relatives and friends will vote for them without vote buying, while the relatives and friends of opposing candidates will behave the same way. 24 Over the period 1996-2013, 85 percent of the sample villages have reported at least one instance of land taking. These land takings affect more than 22 percent of all households in our sample. The average village experienced two takings during that period, and the average area of land lost was 133 mu. With compensation for arable land ranging in the tens of thousands of yuan per mu in sample villages, these takings presented clear opportunities to access rents.
The descriptive data also show that, in the period covered by the study, elections are meaningful contests for the selection of village leader. 25 For example, villages held between one and four rounds of nominations for candidates, with eight candidates on average coming from open nominations and at least 23 These percentages are not extreme by international norms. Keefer and Vlaicu (2017) note that, "Approximately 52 percent of respondents to Wave Six of the World Values Survey (2010)(2011)(2012)(2013)(2014) reported that voters were bribed very or fairly often in their country's elections." They also note significant cross-national variation: "The fraction of respondents who said that vote buying occurred often in their country ranged from 12 percent in the OECD to 56 percent in South Asia; and from 4.3 percent of Dutch respondents to 75.8 percent of Brazilian (773).". 24 Author's interviews. See also Nichter (2008) and Stokes (2005). Mattingly (2016) argues that lineage leaders, particularly those who become village leaders, exercise effective influence over their kinship group. 25 See full descriptive data in Appendix I Table A1. two candidates contesting the final election. Typically, not all candidates were members of the Chinese Communist Party. In 58 percent of sampled villages, candidates made campaign speeches. All elections were supervised by at least one representative of the township government. Seventy-five percent of villages employed secret ballots. Eight-five percent of villages allowed proxy voting; for voters using proxies, their proxy was a relative in the vast majority (85 percent) of cases. On average, turnout was over 80 percent of eligible voters. In all sample villages, the winner was decided by simple majority. The elected village head was concurrently party secretary in less than a third of villages.
Analysis and Findings: Vote Buying as a Form of Rent Seeking
In this section, we report tests of the hypothesis that the presence of one or more land taking within one term of a village election will be associated with a higher incidence of vote buying.
We estimate the following specification using ordinary least square to test our hypothesis:
where VB i is the proportion of voters reporting vote buying in a given village, land_taking i is the number of land takings within three years (one term) before and after the election in that village, X includes controls, and K denotes county dummies. Column 1 of Table 2 provides a baseline by examining the correlation between all land takings-formal and informal-within one term-both before and after an election-and vote buying. In the bivariate model, the number of rent-seeking opportunities in the three-year period both before and after the election is positively and significantly associated with the prevalence of vote buying. Each additional rent-seeking opportunity is associated with a 2.6 percent increase in the share of villagers reporting vote buying. 26 Columns 2-5 show that the number of firms in the village, hypothesized to be another possible source of rents to office holders, is negatively related to vote buying, substantively small, and not significant. This result is consistent with Su and Tao's (2017) assessment that village-run enterprises have become a liability. Moreover, in our dataset, the majority of firms registered in sample villages are privately owned and operated, meaning that village leaders cannot extract rents from the firms as easily. In case-study research, one of the authors finds that the presence of more private firms, especially ones that make significant contributions to the local tax base, is associated with closer monitoring of the village leader by the county and township governments. 27 This context makes it more difficult for village leaders to capture rents in regulating firms.
VB i = c + land_taking i + XB + K + i
Column 2 also introduces controls for village characteristics. As hypothesized in both the comparative and China-specific literature, the larger the population of the electoral district, the less likely the occurrence of vote buying. However, this factor is not significant when county fixed effects are included, while land takings remain significant. The results also show no significant relationship between income levels in the community and vote buying. While the finding that poorer communities are more likely to experience vote buying is well established in the literature (Stokes et al. 2013), it does not find support here. One possible reason is that income measures in surveys in rural China may be unreliable due to social desirability bias. Kennedy's (2010) suggestion that vote buying is prevalent in wealthier communities is not supported by the analysis; rather, his hypothesis likely reflects the role of land takings, which are more likely in peri-urban areas. In this dataset, land takings are significantly negatively correlated with distance from the village to the county seat. 28 The analysis also controls for multiple aspects of electoral rules and design (columns 3-5). None of these variables is consistently significant, while land takings are positive and significant and robust to all specifications. Although measures of electoral competitiveness-number of candidates and number of rounds of nominations-are positively associated with vote buying as expected, they do not attain statistical significance. Similarly, winning candidate vote share, a measure of (non) competitiveness, is negatively and significantly related to vote buying, as expected, but loses significance after controlling for county fixed effects. Two other variables, higher-level government approval of candidates and the number of government officials monitoring village elections, reflect local government intervention or oversight of the election, but neither is significant. Based on one author's interviews with township officials, monitors arrive in villages only on polling day to supervise elections, while candidates or their brokers usually make their deals with villagers at earlier times. 29 As a result, it is difficult for officials to know about the existence of these deals. The officials also point out that, even when they do know about vote buying, they may still prefer to not report it so as to avoid unnecessary trouble. In some situations, if the government receives complaints from villagers about vote buying prior to the election, it might send a higher number of officials to monitor elections to assure that the elections will not be disrupted by conflicts between villagers and candidates. 30 Among electoral rules, the positive (but insignificant) relationship between secret ballots and vote buying is most striking. This finding stands in contrast to much of the existing literature, which documents the importance of secret balloting in undermining the ability of candidates and their brokers to monitor vote choice (Lehoucq 2007). However, qualitative evidence produced by one author's fieldwork reveals strategies that candidates use to monitor votes even in the context of secret balloting. Blaydes (2011) describes similar techniques in Egyptian elections. Thus, candidates, motivated by the desire to capture large rents while in office, buy votes even when monitoring is difficult and devise new strategies to monitor votes. As Kam concludes, the secret ballot is "a powerful institutional force in improving… elections but not a decisive one (2017,626)." In columns 3 and 4, proxy voting is negatively and significantly related to vote buying. It remains negative but loses significance when controls for county fixed effects are introduced in column 5. Proxies reduce the number of targets for vote buying, since relatives cast ballots on behalf of their family members.
Finally, column 4 introduces a variable to assess whether the election of the party secretary as village head is associated with vote buying. When the party secretary runs in the election for village head, he must compete with other candidates (including, potentially, other party members) and may do so by buying votes. The relationship is positive, although after controlling for county fixed effects, the coefficient is not significant. This result is consistent with vote buying being an individual rather than institutional practice. Using the less sensitive measure of vote buying (Appendix I, Tables A2, A3), this variable remains positive and regains significance. This suggests that voters may hesitate to report having directly experienced vote buying in the election for village head when the party secretary participates in vote buying but are more willing to report having heard about vote buying in this situation.
Formal Land Takings as an Exogenous Source of Variation
The hypothesis-testing strategy employed here relies on the grounded assumption that land takings are exogenous to village electoral politics. If village leaders themselves can initiate informal land takings outside of official land-use plans, then the causality may be reversed: successful vote-buying candidates may drive land takings.
To further explore the hypothesized causal relationship between land rents and vote buying, Table 3 limits the analysis to only formal land takings, as measured in the survey by the presence of a permit accompanying the land taking. These takings are unambiguously initiated outside the village; formal land takings are approved and included in land-use plans at the county level and above and are implemented with the oversight of the county land bureau. Columns 1 and 2 of Table 3 present both simple bivariate results and control variables for all elections that occur within one electoral term of a land taking. In the first bivariate model, the number of formal land takings within one electoral term is positively and significantly associated with the prevalence of vote buying. Each additional formal land taking is associated with a 2.9 percent increase in the share of villagers reporting vote buying. In the fully specified model, each additional formal land taking is associated with a 3.1 percent increase in the share of villagers reporting vote buying. This effect appears to be driven by formal land takings that were initiated outside the village and not other factors.
Land-Taking Information Prior to an Election
The information environment in which electoral candidates operate is important. Our hypothesis-testing strategy also relies upon the logic that candidates for village office have information about the land rents available to office holders before they engage in vote buying. Columns 3-6 of Table 3 disaggregate rent-seeking opportunities into the electoral term immediately preceding and following an election. Column 3 reports the positive bivariate relationship between rent-seeking opportunities during the term before the election and the proportion of villagers reporting vote buying. Column 4 includes variables controlling for other possible causes of vote buying identified above and including county fixed effects. In this specification, only available rents through land takings, among all other relevant factors, is significant. Each additional formal land taking is associated with a 2.9 percent increase in the share of villagers reporting vote buying. This result reinforces the initial findings that the presence of land rents motivates vote buying on the part of individual candidates. The number of formal land takings in the three years after the election is not significantly correlated with vote buying. 31 Reliable information about land takings to be initiated in the future is not likely to be known or available to election candidates and therefore is unlikely to influence their electoral ambitions. This result is consistent with the information environment and the process of formal land takings described in the case study and supports the hypothesis that the lure of land rents drives vote buying.
Conclusion
This study makes an empirical contribution to the vote buying literature by introducing a new measure of available rents that captures sub-national variation and reflects the micro-foundations of vote buying. In theoretical terms, rent-seeking opportunities created by state intervention in the local economy shape vote-buying behavior by candidates. We find that the availability of rents-as reflected in the occurrence of lucrative state land takings-explains variation in vote buying in village elections in China. Furthermore, the Chinese case reveals the dynamics of elections in which competition is non-partisan and candidate-centered, rather than structured and financed by any political party. The evidence suggests how hegemonic political parties use authoritarian elections. The party as an institution does not use vote buying to elect its preferred candidate. Rather, vote buying provides information to the authoritarian state about local elites' economic capacity to buy votes and their socio-political standing to win elections. These distorted electoral contests provide information to the authoritarian state about whom it should recruit into the rent-sharing coalition. As reflected in the qualitative data, local economic elites who successfully buy their way into the office of village head may be recruited to the post of village party secretary. Similarly, village party secretaries may position themselves as individuals to try to win elections for village head in order to control the rents flowing through the village.
Recent policy changes since 2018 calling for village party secretaries to serve concurrently as the-presumably elected-heads of village committees raise new questions about the future of village elections. 32 One possible scenario is suggested by the findings of the present study. The lure of lucrative land rents may continue to drive vote buying as an individual strategy among competing village notables inside and outside the party. Non-party elites may seek to buy victory in competitive elections for the position of village head in the hopes of being tapped to join the party, undertaking the concurrent role of village party secretary, and thereby entering the rent-sharing coalition. Alternatively, vote buying may become a prominent phenomenon in intra-party elections for village party secretary.
The emergence of vote buying also has important implications for governance; comparative studies highlight the effect of vote buying on political accountability. Leight et al. (2019) find that voters who receive payment for their votes are more likely to tolerate takings by politicians and to tolerate expropriation in larger amounts. Moreover, politicians who pay for votes are likely to expropriate more resources than they would in the absence of vote buying.While multiple studies of village elections in China highlight their positive impact on accountability of leaders and on the quality of governance (e.g., Birney 2007; Kennedy et al. 2004;Manion 2006), the ways in which vote buying erodes local accountability remain an unexplored topic.
Further research is needed to assess the accountability of elected leaders who engage in both vote buying and rent appropriation. Two types of data will be useful in this regard. First, in future work, data on subsequent electoral contests involving incumbent leaders who previously bought votes, presided over the distribution of land takings compensation, and captured sizeable rents will shed light on the willingness and ability of voters to hold leaders accountable. Second, data on the presence or absence of villager protests following land takings will allow us to 32 See note 1.
3
test the results of laboratory experiments suggesting that voters who are paid for their votes are more likely to tolerate rent appropriation by the leaders they elect (Leight et al. 2019). Given the scale of land grabbing and rent appropriation under electoral authoritarianism as well as in electoral democracies in Asia, Africa, and Latin America (White et al. 2012), it is important to explain not only the causes of vote buying but also the relationship among rents, vote buying, and political accountability.
Supplementary Information
The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s12116-022-09355-y.
Table 1
1Descriptive statistics of variables in baseline analysis (unit: village)N
Mean
SD
Min
Max
Proportion of villagers experienced vote buying
62
0.11
0.12
0
0.5
Frequency of land takings from 1996 to 2013
62
2.29
2.22
0
12
Frequency of land takings within 3 years of election
62
1.13
1.32
0
7
Formal Land takings within 3 years of election
62
0.63
1.16
0
7
No. of enterprises
61
4.64
12.23
0
86
Total population (unit: persons)
62
1535
972
161
4800
Average annual income of villagers (unit: yuan)
62
3531
1865
350
8000
Secret ballot (0 = no, 1 = yes)
61
0.75
0.43
0
1
Proxy voting (0 = no, 1 = yes)
61
0.85
0.36
0
1
No. of higher level Monitoring Officials
61
4.41
3.2
1
16
Rounds of nomination
59
2
0.7
1
4
Election winner is concurrent party secretary
62
0.32
0.47
0
1
Winning candidate vote share
60
79.62
11.19
55
98
Candidates approved by government (yes = 1, no = 0)
60
0.9
0.3
0
1
Number of final candidates
59
2.2
0.91
1
6
Table 2
2Rent-seeking opportunities and vote buying*p < 0.1; **p < 0.05; ***p < 0.01
Table 3
3Formal rent-seeking opportunities (exogenous to village) and vote buying *p < 0.1; **p < 0.05; ***p < 0.01Independent variableDependent variable: proportion of households reporting vote buyingWithin one electoral
term
Term before election
Term after election
Availability of rents
Land taking(s) in village
0.029**
0.031**
0.039**
0.029*
0.007
0.035
(0.014)
(0.013)
(0.017)
(0.016)
(0.024)
(0.023)
Firm(s) in village
− 0.002
− 0.002
− 0.001
(0.001)
(0.001)
(0.001)
Village characteristics
Log (population)
0.034
0.041
0.019
(0.027)
(0.029)
(0.029)
Log (income)
− 0.042
− 0.020
− 0.038
(0.039)
(0.039)
(0.042)
Election characteristics
Number of candidates
0.008
0.004
0.009
(0.016)
(0.017)
(0.017)
Rounds of nominations
0.026
0.017
0.028
(0.026)
(0.027)
(0.028)
Candidates approved by gov
0.036
0.034
0.064
(0.051)
(0.054)
(0.053)
Number of official monitors
− 0.002
− 0.002
− 0.002
(0.006)
(0.006)
(0.006)
Secret ballot
0.067
0.060
0.064
(0.050)
(0.052)
(0.053)
Proxy voting
− 0.029
− 0.048
− 0.037
(0.049)
(0.050)
(0.052)
Party influence
Party secretary as candidate
0.007
0.014
0.007
(0.042)
(0.044)
(0.045)
Constant
0.089***
0.195
0.095***
− 0.036
0.104***
0.187
(0.018)
(0.349)
(0.016)
(0.346)
(0.018)
(0.388)
County fixed effects
No
Yes
No
Yes
No
Yes
Observations
62
57
62
57
62
57
R 2
0.067
0.791
0.083
0.773
0.002
0.766
Adjusted R 2
0.051
0.609
0.067
0.575
− 0.015
0.564
Prior to 2018, the ruling Chinese Communist Party (CCP) played little or no institutional role in structuring electoral competition among candidates for village leadership (Landry, Davis, and Wang 2010). For the new role for CCP secretaries in village elections beginning in 2018, see CCP Central Committee and State Council (2018; 2019) in Appendix II, Documents. 2 Ahram (2012, 158) notes the anecdotal nature of Blaydes' analysis of vote buying as rent seeking; Keefer and Vlaicu (2017) employ a formal model.
We define rent seeking as the expenditure of effort/resources to capture returns by seeking political advantage in an institutional setting characterized by political allocation. SeeBuchanan et al. (1980,4).
Land Management Law (see Appendix II, Documents).
Ang (2016) shows that material payoffs constitute an important incentive for grassroots officials in China. 6 See Appendix II, Documents. In 2018, the electoral term was extended to five years.
CCP Central Committee Office, State Council Office 2009 (see Appendix II, Documents). 8 This analysis echoesMuñoz's (2014, 95) call for multi-method studies of electoral clientelism.
The case study was conducted separately in a province and county not included in the survey sample. 10 One mu, the standard unit of land area in China, is equal to 0.067 hectares or 0.16 acres. 11 There was a clear division of work among the village leadership in Baijia Village. While the party secretary was responsible for party affairs, the village head was responsible for public affairs involving ordinary villagers. Land requisition fell within the bailiwick of the village head. Author's interviews.
The price of votes varies. Kennedy reports that the "amount can vary from 5 yuan (US $0.75) to 1500 yuan (US$220) for a single ballot (2010,621)."Zhao (2018,283) finds that the price that candidates paid to villagers for each ballot ranged from 400 to 1800 yuan, with an average of 880 yuan.13 Author's interview 2013. 14 Author's interview 2013.
Author's interview 2013.16 Author's interview 2013. 17 A single person serving as both village committee chairman and village party secretary is referred to as "carrying on a single set of shoulders (一肩挑).".
The provinces are broadly representative of all of China's major regions with the exception of the south-southeast.19 On average, sample villages have about four hundred households; the random sample includes approximately 5 percent of households in each village. 20 The two-village over-sample was included in case a village withdrew from the sample.
Using the less direct measure, each additional land taking leads to a 3.2 percent increase in proportion of villagers who heard about vote buying (Appendix I,Table A2).27 Author's interview 2013.28 When land takings and distance from the county are both controlled in the specification, only land takings register a statistically significant result.
Author's interview 2013. 30 Author's interview 2013.
Findings are robust to the less direct measure of vote buying (Appendix I,Table A3).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.Authors and Affiliations
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| s2orc_abstract_body | 12 |
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what was the original b-side for this is love | This Is Love (George Harrison song) "This Is Love" is a song by English rock musician George Harrison that was released on his 1987 album "Cloud Nine". Harrison co-wrote the song with Jeff Lynne, who also co-produced the track. In June 1988, it was issued as the third single from "Cloud Nine", peaking at number 55 on the UK Singles Chart. The original B-side for the single was going to be "Handle with Care", a collaboration between Harrison, Lynne, Roy Orbison and Tom Petty recorded at Bob Dylan's studio in Santa Monica, California. When executives at Harrison's distributor Warner Bros. | paq | 344 |
A recent trend of research on direction-of-arrival (DOA) estimation is to localize more uncorrelated sources than sensors by using a proper sparse linear array (SLA) and the Toeplitz covariance structure, at a cost of robustness to source correlations. In this paper, we make an attempt to achieve the two goals simultaneously by using a single algorithm. In order to statistically efficiently localize a maximal number of uncorrelated sources, we propose an effective algorithm for the stochastic maximum likelihood (SML) method based on elegant problem reformulations and the alternating direction method of multipliers (ADMM). We prove that the SML is robust to source correlations though it is derived under the assumption of uncorrelated sources. The proposed algorithm is usable for arbitrary SLAs (e.g., minimum redundancy arrays, nested arrays and coprime arrays) and is named as maximumlikelihood estimation via sequential ADMM (MESA). Extensive numerical results are provided that collaborate our analysis and demonstrate the statistical efficiency and robustness of MESA among state-of-the-art algorithms. |
I. INTRODUCTION
Direction-of-arrival (DOA) estimation is a fundamental problem in statistical and array signal processing. It refers to the problem of estimating the directions of a number of sources impinging on a sensor array given a series of snapshots of the output of the sensor array [3]. In this paper, we consider DOA estimation for far-field narrowband sources using a uniform or sparse linear array (ULA or SLA), resulting in a DOA estimation problem equivalent to multiple-snapshot spectral analysis, a topic at the core of wireless channel estimation [4], radar signal processing [5], structural health monitoring [6] and fluorescence microscopy [7]. The SLA corresponds to the missing data case in the language of spectral analysis [8] or compressive data in the language of compressed sensing [9], [10] and brings new challenges to theoretical analysis and algorithm design.
The use of SLAs for DOA estimation dates back to [11] and has been extensively studied in the past two decades with an emphasis of localizing O(M 2 ) sources using M sensors only.
The key to achieving such a goal is that under the assumption of uncorrelated sources the data covariance matrix regarding a Part of this paper was presented in the 2021 CIE IEEE International Conference [1] and part will be presented in the 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) [2].
The authors are with the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China (e-mail: [email protected]).
ULA becomes Toeplitz and thus can be determined by a few of its entries. With this in mind, different array geometries for SLAs, e.g. minimum redundancy arrays (MRAs) [11], nested arrays [12]- [14] and coprime arrays [15], [16], have been proposed that determine which entries (indexed by the coarray) are sampled to reconstruct the whole or a shrunk version of the Toeplitz covariance matrix. The assumption of uncorrelated sources is crucial to guarantee the Toeplitz covariance structure, however, it is not always satisfied. In fact, correlated and coherent (fully correlated) sources usually occur in practice due to multipath propagations and other effects, and dealing with them has always been a central topic in DOA estimation (see, e.g., [17]- [22]). Consequently, the goal of localizing more uncorrelated sources than sensors by using the Toeplitz covariance structure seemingly contradicts with the one of robust localization of highly correlated and coherent sources, making the practical use of previous methods questionable in correlated environments. In the present work, we make the first attempt to achieve the two goals simultaneously and resolve the above concern.
It is well-known that the maximum likelihood (ML) method, if solvable, provides benchmark performance for DOA estimation. Under the assumption of uncorrelated sources, the stochastic ML (SML) method can be used to localize the maximal number of sources with statistical efficiency. Its asymptotic performance, in terms of the Cramér-Rao bound (CRB), has been well understood [23], [24]. However, few algorithms have been proposed for the SML method since it resorts to a highly nonconvex optimization problem. The challenges arise due to the nonlinearity with respect to the DOAs, the nonconvex log-det term in the SML criterion function and the source number constraint, and it becomes even worse in the SLA case. In this work, in order to achieve the aforementioned two goals simultaneously, we present an effective algorithm for the SML and prove that the SML for uncorrelated sources is robust to source correlations. Our main contributions are summarized below. 1) We start with the specialized ULA case and formulate the SML optimization problem as a rank-constrained Toeplitz covariance estimation problem, which is further transformed as sequential rank-constrained semidefinite programs (SDPs) by applying a majorizationminimization (MM) technique [25]. An elegant reformulation of the rank-constrained SDP is derived to fit and solved using the alternating direction method of multipliers (ADMM) encouraged by its successes in solving nonconvex problems [26], [27]. The resulting algorithm is named as maximum-likelihood estimation via sequential ADMM (MESA) (see Section III). 2) In the general SLA case, we repeat the above derivations and show that the SML problem can be similarly solved, extending MESA to this case (see Section IV). 3) While the SML method concerned in the present paper is derived under the assumption of uncorrelated sources, we prove that it produces consistent estimates of the DOAs and the source powers (regardless of source correlations) as the noise vanishes, implying its robustness to correlated and coherent sources (see Section V). 4) Numerical results are provided confirming statistical efficiency of MESA for uncorrelated source localization, in cases when the source number is less than or greater than the sensor number, and its robustness to correlated and coherent sources (see Section VI).
A. Relations to Prior Art
The SML method is known also as unconditional ML and has a long history of research. Its asymptotic performance in the ULA case, in terms of the CRB, is well documented in literature [28], [29]. To solve the SML optimization problem, expectation maximization (EM) and Newton-type algorithms have been proposed in earlier works [30], [31] but their performance heavily depends on the initialization step. Instead of solving for an exact ML estimator, great efforts have been made to develop algorithms that have the same asymptotic performance as the SML. Such examples include multiple signal classification (MUSIC) [32], method of direction estimation (MODE) [33] and weighted subspace fitting (WSF) [34]. Good reviews can be found in [35], [36]. But it is worth noting that these algorithms usually consider the ULA and assume deterministic (as opposed to uncorrelated) sources and thus cannot be used to localize more sources than sensors with an SLA.
Sparse optimization and compressed sensing methods [37]- [39], which have become popular since early of this century, do not use explicitly the array geometry and fit into the SLA case. The recent atomic norm and gridless compressed sensing methods [40]- [44] are remedies of earlier compressed sensing methods by working with continuous (as opposed to on-grid) DOAs and providing theoretical guarantees. These methods do not make statistical assumptions on the sources and are robust to source correlations. But correspondingly, they cannot localize more sources than sensors. Readers are referred to [45] for a review.
To localize more uncorrelated sources than sensors, a coarray-based averaging/selection (CBA/S) step is usually adopted to explicitly use the Toeplitz covariance structure and transform the sample covariance matrix regarding the SLA as an output regarding an enlarged virtual ULA, followed by a DOA estimation method for ULAs; see such two-step estimation approaches in [46]- [51], to name just a few. It is shown in [23] that CBA combined with spatial-smoothing (SS) MUSIC results in strictly non-efficient solutions. A state-ofthe-art method is proposed in [52] that uses a weighted least square (WLS) criterion for the vectorized sample covariance and is shown to yield an asymptotically efficient estimator. An iterative algorithm is also proposed to solve the resulting nonconvex optimization problem. While these methods are tailored for uncorrelated sources, it is confirmed by numerical results in this paper that they are indeed sensitive to highly correlated sources. In contrast to this, MESA achieves statistical efficiency and robustness to source correlations simultaneously.
Several algorithms have been proposed to deal with a mixture of uncorrelated and coherent sources given the source coherence structure [20]- [22], [53]. Differently from these algorithms, MESA allows the sources to be correlated but noncoherent and needs only the total number of sources (as opposed to the detailed coherence structure).
The SML method in the ULA case is closely related to structured (to be specific, Toeplitz) covariance estimation (see e.g., [54]- [57]) because the data covariance matrix is the sum of a low-rank Toeplitz covariance and the noise covariance, where the rank is specified by the source number and the DOAs are uniquely determined by the Toeplitz covariance matrix. While the low-rank constraint is a major challenge for Toeplitz covariance estimation, it is explicitly considered in [58], [59]. In [58], the Toeplitz structure is relaxed initially and then used to obtain a Toeplitz approximation of an intermediate solution, which does not result in an exact SML estimator. In [59], the noise variance is assumed known and the Carathéodory-Fejér theorem [45,Theorem 11.5] is invoked to approximate the Toeplitz covariance by a Vandermonde decomposition in which the frequency nodes of the Vandermonde matrix are restricted on a fixed grid so that the original problem is transformed as one of nonnegative sparse vector recovery. In contrast to these methods, we make no approximations or relaxations and MESA solves the exact SML. Moreover, MESA is usable in the SLA case.
Since the difficulty in solving the SML problem partly comes from the source number constraint, which is known as signal sparsity in compressed sensing, it is relaxed in sparse Bayesian learning (SBL) methods [60]- [62]. Similar relaxation techniques are also used in covariance fitting methods [63]- [67], which are approximate versions of the SML method by using convex surrogates for its criterion function. Interestingly, it has been empirically observed in [60], [64], [65] that the resulting algorithms are robust to source correlations though they are derived by assuming uncorrelated sources. In the recent work [68], the case of two correlated sources is considered and it is shown that if the DOAs and the noise variance are known a priori, then the source powers can be stably estimated from the SML method. In contrast to this, our result on robustness is applicable to any source number and shows that the DOAs can be accurately estimated jointly with the source powers, at least in the high SNR regime. It also partially explains the observations in [60], [64], [65].
B. Notation
The sets of real and complex numbers are denoted by R and C respectively. For vector x, diag (x) denotes a diagonal matrix with x on the diagonal. The jth entry of vector x is x j . For matrix A, A T , A H , |A|, A −1 , rank (A), tr (A) and A F denote the matrix transpose, conjugate transpose, determinant, inverse, rank, trace and Frobenius norm of A, respectively. The complex conjugate of scale x is denoted by x. The notation A ≥ 0 means that A is Hermitian positive semidefinite. For index set Ω and matrix A, A Ω represents a submatrix of A obtained by keeping only the rows indexed by Ω unless otherwise stated. The inner product is represented by ·, · . For matrices Y and C ≥ 0, we define
tr Y H C −1 Y = min X tr (X) , subject to X Y H Y C ≥ 0
(1) whenever C is positive definite or not. The expectation of a random variable is denoted by E[·]. Assume that K far-field narrowband sources impinge on the ULA in which adjacent sensors are placed by half a wavelength apart. The output of the sensor array at each snapshot composes an N × 1 complex vector y that can be modeled as [3], [36]:
y(l) = K k=1 a (f k ) x k (l) + e(l), l = 1, . . . , L,(2)
where L is the number of snapshots, x k (l) is the kth (complex) source signal at the lth snapshot, f k ∈ [− 1 2 , 1 2 ) has a one-toone connection to the kth DOA θ
k ∈ [−90 • , 90 • ) by f k = 1 2 sin θ k , a (f k ) denotes an N × 1 steering vector given by a (f ) = [1, e i2πf , . . . , e i2(N −1)πf ] T ,(3)
and e(l) is the vector of complex noise. It is seen that all snapshots share the same parameters {θ k } and {f k }. By stacking {x k (l)} , {f k } into vectors x(l), f and defining the steering matrix (2) is written compactly as:
A (f ) = [a (f 1 ) , . . . , a (f K )] that is N × K Vandermonde, the data model iny(l) = A(f )x(l) + e(l), l = 1, . . . , L.(4)
In the general SLA case, the array output at one snapshot is a subvector of y(l), denoted by y Ω (l). The data model in (4) thus becomes
y Ω (l) = A Ω (f )x(l) + e Ω (l), l = 1, . . . , L,(5)
which encompasses (4) as a special case. Our objective is to estimate the DOAs {θ k } K k=1 , or equivalently {f k } K k=1 , given the multiple-snapshot data {y Ω (l)}
K l=1
under certain statistical assumptions on the source signals {x(l)} and noise {e(l)}. Since each f k is the frequency of a sinusoid, the DOA estimation problem that we concern is equivalent to multiple-snapshot spectral estimation with missing data. We focus on the estimation of {f k } K k=1 throughout this paper.
B. The SML Method for DOA Estimation
We make the following assumptions to derive the SML method for DOA estimation. A1: The sources {x(l)} are spatially and temporally independent and follow a complex Gaussian distribution with zero mean and covariance P = diag (p 1 , . . . , p K ), where p k > 0 denotes the kth sources power; A2: The noises {e(l)} are spatially and temporally independent and each entry follows a complex Gaussian distribution with zero mean and variance σ > 0; A3: The sources and noises are independent. It follows immediately that {y Ω (l)} are i.i.d. Gaussian with zero mean and covariance
R Ω = A Ω (f )P A H Ω (f ) + σI.(6)
By maximizing the likelihood criterion, or equivalently minimizing the negative log-likelihood function, we obtain the SML optimization problem as:
min f ,p,σ ln |R Ω | + tr R −1 Ω R Ω ,(7)
where
R Ω = 1 L L l=1 y Ω (l)y H Ω (l)(8)
is the sample covariance matrix.
The SML method has good statistical properties. But the SML problem in (7) is nonconvex and complicated to solve due to the log-det term and the nonlinearity of R Ω with respect to {f k } K k=1 . Moreover, the SML is derived under the assumption of uncorrelated sources and its performance is unclear in presence of source correlations.
C. The ADMM Algorithm
The ADMM algorithm solves the following optimization problem:
min x∈D1,q∈D2 g(x) + h(q), subject to Ax + Bq = c, (9)
where D 1 , D 2 defines the feasible domain of x, q respectively. Write the augmented Lagrangian function as:
L µ (x, q, λ) = g(x) + h(q) + Ax + Bq − c, λ + µ 2 Ax + Bq − c 2 2 = g(x) + h(q) + µ 2 Ax + Bq − c + µ −1 λ 2 2 + C,(10)
where λ is a Lagrangian multiplier, µ > 0 is a penalty coefficient and C is a constant independent of x, q. ADMM consists of the iterations:
x ← arg min
x∈D1 L µ (x, q, λ) ,(11)q ← arg min q∈D2 L µ (x, q, λ) ,(12)λ ← λ + µ (Ax + Bq − c) ,(13)
where the latest values of the other variables are always used. The ADMM algorithm has been extensively studied and practically used due to its global optimality in solving convex problems, simplicity in dealing with nonsmooth functions, and good scalability for solving high-dimensional problems [26]. Good performance has also been frequently achieved for nonconvex problems; see [27] and references therein. See also [69]- [71] for theoretical progresses on this topic. It is worth noting that the key to using ADMM to solve a specific problem is to provide an elegant problem formulation within the ADMM framework so that the two subproblems in (11) and (12) can be simply and efficiently solved.
III. MESA IN THE ULA CASE
In this section, we derive the MESA algorithm for the SML optimization problem in (14) in the specialized ULA case. In this case, we write the data and sample covariance matrices R Ω , R Ω into R, R for simplicity and the problem to solve becomes: min
f ,p,σ ln |R| + tr R −1 R ,(14)
where
R = A(f )P A H (f ) + σI.(15)
The MESA algorithm consists of re-parameterization, majorization-minimization, problem reformulation and ADMM steps which are detailed below.
A. Re-parameterization
The data covariance matrix R is a highly nonlinear function of {f k }. To overcome such nonlinearity, a common scheme is to utilize the fact that the first term A(f )P A H (f ) in (15) is rank-K positive-semidefinite Hermitian Toeplitz and do the re-parameterization:
R = T t + σI, T t ≥ 0, rank (T t) = K,(16)where T t = (t i−j ) N ×N with t = [t 1−N , . . . , t N −1 ]
T and t −j =t j , j = 0, . . . , N − 1. It follows from the Carathéodory-Fejér theorem [45,Theorem 11.5] that the T t above has a one-to-one connection to {f , p} given K < N . Consequently, the original SML problem (15) is transformed into a rankconstrained Toeplitz covariance estimation problem in which R is a linear function of the variables {t, σ}. Once t is solved for, the variables {f , p} can be computed from T t by a subspace method such as root-MUSIC [72].
B. Majorization Minimization
The objective function in (14) is nonconvex with respect to R since the log-det function ln |R| is concave on the positive semidefinite cone. A commonly used locally convergent method is the majorization-minimization (MM) algorithm (see, e.g., [73]) that drives the objective function downhill by minimizing a simple surrogate function. At the jth iteration of MM, the SML objective function is linearized (and thus majorized) at the previous iterate R j−1 = T t j−1 + σ j−1 I, yielding the problem (by omitting constant terms):
min tr R −1 j−1 R + tr R −1 R .(17)
Substituting (16) into (17), we obtain the problem to solve at the jth iteration as:
min t,σ≥0 tr(R −1 j−1 (T t + σI)) + tr (T t + σI) −1 R , subject to T t ∈ S K + ,(18)
where S K + is the set of positive semidefinite matrices of rank no greater than K.
C. Problem Reformulation
Let W = R −1 j−1 and Y be any matrix satisfying that R = Y Y H ,(19)
where Y has at most min(L, N ) columns. The objective function in (18) then becomes:
tr(W (T t + σI)) + tr Y H (T t + σI) −1 Y .(20)
Making use of the following identity [74,Lemma 5]:
tr X H (R 1 + R 2 ) −1 X = min Z tr Z H R −1 1 Z + tr (X − Z) H R −1 2 (X − Z) ,(21)where R 1 , R 2 ≥ 0, the function in (20) becomes a function of t, σ, Z: tr(W (T t+σI))+tr Z H [T t] −1 Z +σ −1 Y −Z 2 F . (22)
Since in (22) the optimizer to σ is given in close-form by:
σ * = 1 tr(W ) Y − Z F ,(23)
the objective in (22) can be concentrated with respect to t, Z, yielding the following problem to solve:
min t,Z tr(W T t) + tr Z H [T t] −1 Z + 2 tr(W ) Y − Z F , subject to T t ∈ S K + ,(24)
or equivalently,
min t,X,Z tr(W T t) + tr(X) + 2 tr(W ) Y − Z F , subject to X Z H Z T t ≥ 0, rank (T t) ≤ K.(25)
We next show that the problem in (25) is equivalent to the following:
min t,X,Z tr(W T t) + tr(X) + 2 tr(W ) Y − Z F , subject to X Z H Z T t ≥ 0, rank X Z H Z T t ≤ K.(26)
In particular, it is easy to see that the constraint rank(T t) ≤ K in (25) is implied by rank (26). To
X Z H Z T t ≤ K in
show the equivalence between (25) and (26), it suffices to show that the latter constraint is feasible for any optimizer to (25). Denote by (t * , X * , Z * ) an optimizer to (25) and suppose
T t * = T T H where T is an L ×K matrix with full column rankK ≤ K. Since X * Z * H Z * T t * ≥ 0, we have Z * = T V
for some V due to the column/row inclusion property and
X * = Z * H (T t * ) † Z * = V H V . Therefore, X * Z * H Z * T t * = V H V V H T H T V T T H = V H T V H T H(27)
whose rank isK ≤ K, completing the proof.
D. Using ADMM
We introduce an auxiliary matrix variable Q and rewrite (26) as
min t,X,Z,Q∈S K + tr(W T t) + tr(X) + 2 tr(W ) Y − Z F , subject to Q = X Z H Z T t ,(28)
which is exactly in the form of (9) by identifying that
x = {t, X, Z}, q = Q, g(x) = tr(W T t) + tr(X) + 2 tr(W ) Y − Z F , h(q) = 0 and D 2 = S K + .
Following the procedures of ADMM, we introduce the Hermitian Lagrangian multiplier Λ and write the augmented Lagrangian function as:
L µ (t, X, Z, Q) = tr(W T t) + tr(X) + 2 tr(W ) Y − Z F + tr((Q − Ax)Λ) + µ 2 Q − Ax 2 F = tr(W T t) + tr(X) + 2 tr(W ) Y − Z F + µ 2 Q − Ax + µ −1 Λ 2 F − 1 2µ Λ 2 F .(29)
The remaining task is to solve the two subproblems in (11) and (12).
To solve (11), by partitioning Q =
Q 1 Q H 2 Q 2 Q 3 and Λ = Λ 1 Λ H 2 Λ 2 Λ 3 as X Z H Z T t ,
we note that the objective function L µ is separable in {t, X, Z} and thus they can be solved for separately. To solve for t, we equate the derivative of L µ with respect to t to zero and obtain the update:
t ← (T * T ) −1 T * Q 3 + µ −1 Λ 3 − µ −1 W ,(30)
where T * is the Hermitian adjoint of T . Similarly, we have that
X ← Q 1 + µ −1 Λ 1 − µ −1 I.(31)
For Z, the optimization problem to solve is given by:
min Z tr(W ) Y − Z F + µ 2 Q 2 + µ −1 Λ 2 − Z 2 F ,(32)
yielding the update:
Z ← Y − 1 − tr(W ) µ L F + L,(33)
where (x) + max(x, 0) and L Y − Q 2 − µ −1 Λ 2 . The detailed derivations of (33) are deferred to Appendix A.
Solving (12) results in the update:
Q ← P S K + X Z H Z T t − µ −1 Λ ,(34)
where P S K + denotes the orthogonal projection onto S K + that can be computed by the truncated eigen-decomposition by keeping only the largest K positive eigenvalues and associated eigenvectors.
Finally, Λ is updated according to (13) as:
Λ ← Λ + µ Q − X Z H Z T t .(35)
The ADMM algorithm runs (30), (31), (33), (34) and (35) iteratively.
The overall MESA algorithm consists of the outer MM loop and the inner ADMM loop. Its computations are dominated by the truncated eigen-decomposition and the matrix inverse to compute W . Consequently, MESA has a computational complexity of O N 3 per inner iteration that is affordable in DOA estimation where the array aperture N is usually small.
IV. MESA IN THE SLA CASE
In this section, we consider the general SLA case and derive the MESA algorithm by repeating the same steps as in the previous section.
A. Re-parameterization
For a general SLA designated by Ω, let Γ ∈ {0, 1} M×N be the row-selection matrix satisfying that
y Ω = Γy(36)
for any N × 1 vector y and its subvector y Ω . Then, we have
R Ω = Ey Ω y H Ω = ΓRΓ T ,(37)
which is an M × M principal submatrix of R, defined in (15), and is re-parameterized as a linear function of (t, σ) following from (16).
B. Majorization Minimization
In this case, the objective function at the jth iteration becomes:
tr R −1 Ω,j−1 R Ω + tr R −1 Ω R Ω = tr Γ T R −1 Ω,j−1 ΓR + tr R −1 Ω Y Ω Y H Ω = tr (W R) + tr Y H Ω R −1 Ω Y Ω ,(38)
where R Ω,j−1 denotes the (j − 1)st iterate of R Ω , W = Γ T R −1 Ω,j−1 Γ, and Y Ω is any matrix satisfying that R Ω = Y Ω Y H Ω and has at most min(L, M ) columns.
C. Problem Reformulation
Making use of [74, Lemma 6], we obtain
tr Y H Ω R −1 Ω Y Ω = min Y Ω tr Y H R −1 Y ,(39)
where the set Ω denotes the complement of Ω. By substituting (39) and (16) into (38), the objective function becomes
tr(W (T t + σI)) + tr Y H (T t + σI) −1 Y(40)
with respect to t, σ, Y Ω , which is exactly in the form of (20). Consequently, the same derivations as in the ULA case can be applied, yielding the following optimization problem:
min t,X, Y Ω ,Z tr(W T t) + tr(X) + 2 tr(W ) Y − Z F , subject to X Z H Z T t ∈ S K + ,(41)
or equivalently,
min t,X,Z tr(W T t) + tr(X) + 2 tr(W ) Y Ω − Z Ω F , subject to X Z H Z T t ∈ S K +(42)
by noting that the solution to Y Ω is exactly Z Ω . In this process, as in (23), we have
σ * = 1 tr(W ) Y Ω − Z Ω F .(43)
D. Using ADMM
The only difference between (42) and (26) is the inclusion of the index set Ω in the term Y − Z F . Consequently, the only difference in the ADMM algorithm occurs in the update of Z. In particular, the objective function to minimize regarding Z changes from that in (32) to the following:
tr(W ) Y Ω − Z Ω F + µ 2 Q 2 + µ −1 Λ 2 − Z 2 F = tr(W ) Y Ω − Z Ω F + µ 2 Q 2 + µ −1 Λ 2 Ω − Z Ω 2 F + µ 2 Q 2 + µ −1 Λ 2 Ω − Z Ω 2 F .(44)
Therefore, it follows from (33) that
Z Ω ← Y Ω − 1 − tr(W ) µ L Ω F + L Ω ,(45)Z Ω ← Q 2 + µ −1 Λ 2 Ω ,(46)
where L is as defined below (33). The ADMM algorithm in this case runs (30)
V. ROBUSTNESS TO SOURCE CORRELATIONS
The assumption of uncorrelated sources is crucial to derive the SML method concerned in the present paper, for which the MESA algorithm is proposed. In this section, we show that the SML method is robust to source correlations, which implies robustness of MESA.
We consider the SLA case that consists of the ULA case when M = N . In order to show the robustness to source correlations, we will not use the statistical assumptions A1-A3 in Subsection II-B. Instead, we make the following (deterministic) assumptions, where f o , S o , σ o denote the true values of the parameters.
Y o Ω = A Ω (f o ) S o ; A5: Y Ω = A Ω (f o ) S o + E, where E is random noise satisfying that E 2 F = N Lσ o > 0 and inf f ,S Y Ω − A Ω (f ) S 2 F is strictly positive; A6:
A Ω (f ) has full column rank in a neighborhood of f o .
Assumption A4 seems necessary if no statistical assumptions are made on the source signals in S o . Note that A4 implies K < M . Given A4, A5 is trivial given random noise E since otherwise, E, translated by a constant vector A (f o ) S o , must be in a K-dimensional subspace. A6 is a technical assumption ensuring that the estimates of source powers are consistent. The following proposition is a result of combining [
K < Spark (Ω) + rank (S o ) − 1 2 ,(47)
where Spark (Ω) is defined as the smallest number of atoms in {a Ω (f )} that are linearly dependent. Our main result is stated in the following theorem. Theorem 1: Under assumptions A4-A6 and letting (f * , p * , σ * ) be the solution to the nominal SML optimization problem given by:
min {f k },{p k ≥0},{σ≥0} ln A Ω P A H Ω + σI + 1 L tr Y H Ω A Ω P A H Ω + σI −1 Y Ω ,(48)
we have that σ * > 0 and
lim σ o →0 σ * = 0,(49)
lim
σ o →0 f * = f o ,(50)lim σ o →0 p * k = 1 L S o k 2 ,(51)
where S o k denotes the kth row of S o .
Proof: See Appendix B. It is shown in Theorem 1 that the SML method produces consistent estimates of the DOAs and source powers as the noise vanishes regardless of (spatial and temporal) source correlations, implying its robustness to (spatially) correlated or coherent sources, at least in the high SNR regime.
Remark 1: Theorem 1 is related to [43,Theorem 2] which is concerned with the problem
min t ln |T t + ǫI| + tr Z H [T t] −1 Z , subject to T t ≥ 0,(52)
where Z = A (f o ) S o is noiseless and ǫ > 0 is a fixed small constant. In contrast to this, Theorem 1 is on the noisy case in which the noise variance σ is a variable to optimize. Another difference is that the source number is explicitly given in Theorem 1, while there is no a corresponding rank constraint on T t in (52). All these differences arise due to the fact that the problem in (52) was introduced in [43] as a surrogate function for the spectral sparsity of Z, rather than a consequence of the SML as in the present paper.
Remark 2: While the focus of this paper is on DOA estimation using SLAs, we note that Theorem 1 is applicable to arbitrary linear arrays for which the sensors are not necessarily located on a regular grid. Moreover, it can easily be extended to general joint sparse recovery problems [60], [75].
VI. NUMERICAL RESULTS
A. Experimental Setup
In this section, we present numerical results to illustrate the performance of the proposed MESA algorithm for DOA estimation using SLAs. In our implementation of MESA, R Ω is initialized with R Ω in general (a small scalar matrix is added if R Ω tends to be singular), while it is initialized as an identity matrix if the ratio of the Kth greatest eigenvalue and the smallest eigenvalue of R Ω is smaller than a threshold (set to 5) for better performance in presence of highly correlated sources. As for the first ADMM loop, Y Ω = R 1 2 Ω and R Ω are used to initialize Z Ω and the corresponding principal submatrix of Q 3 , respectively. Other variables are initialized with zero. The outer MM loop is terminated if the relative change of the negative log-likelihood function at two consecutive iterations is lower than 10 −5 or the number of iterations reaches 20. The ADMM iteration is terminated if the relative and absolute errors are below 10 −5 and 10 −4 , respectively (see [26,Section 3.3.1] for details), or a maximum number 1000 of iterations is reached. To better understand the performance of MESA, we also present its performance with a single MM iteration, termed as MESA-1. Note that the criterion of MESA-1 (when initializing R Ω with R Ω ) has been used in [64], [65] where the source number or the rank constraint is relaxed.
The methods that we use for comparison include SS-MUSIC [46], [47], WLS [52], multiple-snapshot Newtonized orthogonal matching pursuit (MNOMP) [44], [76], reweighted atomicnorm minimization (RAM) [43] and maximum-likelihood estimation of low-rank Toeplitz (MELT) [59]. SS-MUSIC is a popular CBA method and is implemented with forwardbackward SS and root-MUSIC. WLS is the only asymptotically efficient algorithm prior to this work when more uncorrelated sources than sensors are present. It is initialized with SS-ESPRIT following from [52]. MNOMP is a greedy algorithm for the deterministic ML and does not require a complex initialization. RAM tries to minimize the number of sources subject to data fidelity. MNOMP and RAM do not make statistical assumptions on the sources and cannot localize more sources than sensors. RAM requires the noise power rather than the source number. MELT solves the same SML problem as MESA but is usable only for ULAs. We also compare with the CRB that is computed following from [24] for uncorrelated sources, or based on a standard routine in presence of correlated sources.
All sources and noise are generated by using complex Gaussian distributions. All sources have unit powers. The signal-to-noise ratio (SNR) is defined as the ratio of the source power to noise power. The root mean squared error (RMSE) of the frequency estimates is computed as
1 K f − f o
B. Convergence and Optimality
In this subsection, we test the numerical performance of MESA in convergence and optimality by computing the negative log-likelihood function value at each outer MM iteration. To compare with MELT [59], we consider the ULA in (53). As in [59], the frequency domain [− 1 2 , 1 2 ) is approximated by a set of 2N − 1 uniform gridding points and the true frequencies are selected from the gridding points to achieve the best performance for MELT, which though is not required in MESA. We also compare with the function value computed with the true values of parameters and denoted by "ground truth". While the globally optimal function value is hard to obtain, it is expected that good accuracy is achieved if the obtained function value is smaller than the "ground truth". The true noise power σ o is also fed into MELT as in [59].
In Experiment 1, K = 3 sources are generated with DOAs such that the frequencies are taken as the 5th, 7th and 18th gridding points of MELT. We set SNR = 10dB and the number of snapshots L = 100. The curve of the function value with respect to the index of the MM iteration of MESA is plotted in Fig. 1. It is seen that the function value decreases monotonically and converges in 9 iterations. MESA produces a function value smaller than the ground truth and MELT, indicated by the two horizontal dashed lines. MESA takes 534 inner iterations in total.
We tried a total number of 500 Monte Carlo runs and MESA always converges and produces a function value smaller than the ground truth. It performs better than MELT in 498 out of 500 runs.
C. Statistical Efficiency for Uncorrelated Sources
In this subsection, we use the MRA in (54) and test the statistical efficiency of MESA for uncorrelated sources. RAM and MNOMP are considered only in the case of K < M .
In Experiment 2, We consider K = 7 sources with DOAs satisfying that the frequencies are taken in {−0.43, −0.28, −0.21, −0.05, 0.1, 0.26, 0.42}. We fix the number of snapshots L = 200 and vary the SNR from −10 to 20 dB. Our simulation results are presented in Fig. 2. It is seen that WLS and and MESA attain the CRB as SNR ≥ −5dB, while SS-MUSIC always produces an error greater than the CRB. It is interesting to note that the results of MESA-1 and MESA are almost indistinguishable. In this case, the sample covariance is a good estimate of the data covariance and a single outer loop of MESA suffices to produce an accurate estimate. In Experiment 3, we fix SNR = 10dB, L = 100 and vary K from 2 to N − 1 = 13. The K sources are generated with f k = −0.44 + 0.98(k − 1)/K, k = 1, . . . , K. Our simulation results are presented in Fig. 3. Again, WLS and MESA (and MESA-1) attain the CRB or even better whenever the number of sources is smaller or greater than the number of sensors, while SS-MUSIC cannot. NMOMP and RAM can accurately localize only a small number of sources, as expected. In Experiment 4, we consider K = 3 sources with frequencies given by {−0.2, 0.1, 0.1 + δ} and vary δ ∈ {0.001, 0.003, . . . , 0.029}. We fix the number of snapshots L = 100 and SNR = 10dB. It is seen in Fig. 4 that MESA attains the CRB for very closely located sources and thus has a higher resolution than the other methods. MNOMP fails to resolve the closely located sources. A gap is shown between MESA and MESA-1, implying that the MM iterations of MESA are useful to improve the resolution.
D. Robustness to Source Correlations
In Experiment 5, we consider K = 3 sources with frequencies in {−0.2, −0.1, 0.2}, where the first two sources are coherent with the correlation coefficient e iπ/3 (that can be changed to any other value on the unit circle). We use the nested array in (55) for DOA estimation to make sure that the assumptions of Theorem 1 are satisfied, which can be verified according to Proposition 1 since the source matrix S has rank 2 and Spark (Ω) ≥ 6 by noting that the nested array contains a 5-element ULA. We fix L = 100 and vary the SNR from −10 to 20dB. Our numerical results are presented in Fig. 5. It is seen that MESA has stable performance when the SNR is above 0dB, which is consistent with Theorem 1. Remarkably, MESA attains the CRB as in the case of uncorrelated sources. In contrast to this, SS-MUSIC and WLS do not have the same robustness as MESA. Satisfactory performance is also obtained by MNOMP and RAM. In this case, MESA performs better than MESA-1 since the solution to R Ω is significantly different from the sample covariance and an accurate initialization is unavailable. We present in Fig. 6 results of one Monte Carlo run of the previous experiment (at SNR = 20dB). It is seen that MESA can accurately estimate the frequencies/DOAs and the source power, validating Theorem 1. Interestingly, SS-MUSIC and WLS can accurately localize the two coherent sources but mislocate the third source that is uncorrelated with the other two. The reason underlies this behavior needs further investigation. In Experiment 6, we repeat Experiment 5 by changing the nested array to the MRA in (54). In this case, it is difficult to verify the assumptions of Theorem 1. We present our results in Fig. 7. It is seen that all algorithms are affected to a larger extent by the source correlation as compared to the nested array case presented in Fig. 5. Differently from SS-MUSIC and WLS, MESA remains to be robust to source correlations. MNOMP has a poor performance in this case with a small array size. In Experiment 7, we repeat Experiment 2 by fixing SNR = 10dB and letting the first and the fourth sources to be correlated with a correlation coefficient ρe iπ/4 , where the modulus ρ changes from 0 (uncorrelated) to 1 (coherent). MNOMP and RAM are not usable since more sources than sensors are present. It is seen in Fig. 8 that the performance of all methods becomes worse as the correlation increases. In contrast to a steady performance loss of MESA, a sharp loss is shown for SS-MUSIC and WLS in the regime of highly correlated sources. As ρ = 1, in fact, SS-MUSIC and WLS mislocate at least one of the sources in over 10% of the Monte Carlo runs, while MESA always accurately localize the sources. To sum up, we have shown by numerical results that MESA can statistically efficiently localize more uncorrelated sources than sensors and has robust performance in the presence of correlated and coherent sources. This makes it unique among existing coarray-based methods tailored for uncorrelated sources and sparse methods usable for a small number of sources. MESA-1 is a good accelerated approximation of MESA in general, while the latter has improved resolution and robustness. More simulation results can be found in our conference papers [1], [2].
VII. CONCLUSION In this paper, we showed that more sources than sensors can be localized using proper SLAs without sacrificing robustness to source correlations. This is realized by studying the robustness property of the ML method derived under the assumption of uncorrelated sources and proposing the MESA algorithm for the ML method based on elegant problem reformulations. Extensive numerical results are provided that validate our theoretical findings and demonstrate superior performance of MESA in terms of statistical efficiency, resolution and robustness to highly correlated sources as compared to stateof-the-art algorithms.
It is shown in this paper that MESA can localize more sources than sensors even in presence of highly correlated or coherent sources. A theoretical understanding of this behavior is a future work. Moreover, both algorithm-dependent andindependent analyses are of interest to investigate how the number of localizable sources leverages with source correlations. For a particular algorithm, it is shown that using the assumption of uncorrelated sources does not necessarily contradict with its robustness to correlated sources. Therefore, it is of great interest to investigate their robustness for both existing and future algorithms proposed with the uncorrelated setup. It is also shown that the array geometry is another factor affecting the robustness to source correlations. It is interesting to take the robustness into consideration for future array geometry design. APPENDIX A. Proof of (33) To show (33), it suffices to show that
1 − β L F + L = arg miň Z β Ž F + 1 2 Ž − L 2 F(56)
by identifying thatŽ = Y − Z and β = µ −1 . To this end, observe that
g(Ž) β Ž F + 1 2 Ž − L 2 F ≥ β Ž F + 1 2 Ž F − L F 2 = 1 2 Ž F − L F + β 2 + β L F − 1 2 β 2 ,(57)
where the equality is achieved ifŽ has the sign of L. Since the last expression is minimized if Ž F = ( L F − β) + , the overall function g(Ž) is therefore minimized ať
Z = ( L F − β) + sgn (L) = 1 − β L F + L,(58)
completing the proof.
B. Proof of Theorem 1
We first show the following lemma. Lemma 1: Assume A = [a 1 , . . . , a K ] and that P is positive-definite diagonal. Then, it holds for any σ > 0 and k = 1, . . . , K that
a H k AP A H + σI −1 a k < p −1 k .(59)
If, further, A has full column rank, then
lim σ→0 a H k AP A H + σI −1 a k = p −1 k .(60)
Proof: Note that the matrix
p −1 k a H k a k AP A H + σI is posi-
tive definite since so is p −1 k and its Schur complement
AP A H + σI − p k a k a H k ≥ σI.(61)
Consequently, the Schur complement regarding AP A H + σI is positive, yielding (59). Without loss of generality, we next show (60) for k = 1. For any ǫ > 0, diag (−ǫ, p 2 , . . . , p K ) is indefinite and so is Adiag (−ǫ, p 2 , . . . , p K ) A H since by assumption A has full column rank. It follows that for any 0 < σ < −λ min Adiag (−ǫ, p 2 , . . . , p K ) A H , (63) is the Schur complement regarding (p 1 + ǫ) −1 > 0. Since AP A H + σI is positive definite, consequently, its Schur complement must be negative, i.e.,
(p 1 + ǫ) −1 − a H 1 AP A H + σI −1 a 1 < 0,(64)
which combined with (59) yields that
(p 1 + ǫ) −1 < a H 1 AP A H + σI −1 a 1 < p −1 1 ,(65)
of which a direct consequence is (60).
We are ready to prove Theorem 1. For notational simplicity, we omit the subscript Ω in Y Ω , A Ω hereafter and write them as Y , A without ambiguity. It follows from Lemma 5 and Lemma 4 in [74] that
tr Y H AP A H + σI −1 Y = min Z tr Z H AP A H −1 Z + σ −1 Y − Z 2 F = min S tr S H P −1 S + σ −1 Y − AS 2 F ,(66)
which can also be shown directly. Define L (f , p, σ, S) = ln AP A H + σI + 1 L tr S H P −1 S
+ 1 Lσ Y − AS 2 F .(67)
It follows that (f * , p * , σ * , S * ) = arg min f ,p,σ,S L (f , p, σ, S) ,
and we let L * denote the optimal value. We first show σ * > 0. It suffices to show that lim σ→0 L (f , p, σ, S) = +∞ (69) for any (f , p, S). To do so, note by (67) that
L (f , p, σ, S) ≥ ln |σI| + 1 Lσ Y − AS 2 F = M ln σ + 1 Lσ Y − AS 2 F ,(70)
and the only stationary point of the lower bound above regarding σ, which is the global minimizer, is given by
σ = Y −AS 2 F N L
that is bounded from below by a positive number for any (f , S) by Assumption A5. Consequently, the lower bound above always approaches infinity as σ → 0, resulting in (69).
Inserting the ground truth (f o , p o , σ o , S o ) into L and conditioning on σ o ≤ 1, we have that
L * ≤ L (f o , S o , p o , σ o ) = ln A o P o A oH + σ o I + 1 L tr S oH P o−1 S o + 1 Lσ o Y − A o S o 2 F ≤ (M − K) ln σ o + K k=1 ln λ k A o P o A oH + 1 + K + M,(71)
where λ k denotes the kth greatest eigenvalue. Combining (71) and the inequality L * ≥ ln A * P * A * H + σ * I ≥ M ln σ *
yields that
σ * ≤ Cσ o M −K M ,(73)
where C = e 1+K/M K k=1 λ k A o P o A oH + 1 1/M is a constant. A direct consequence of (73) is (49). Inserting the stationary point σ =
Y −AS 2 F N L
into the lower bound in (70) yields that
L (f , p, σ, S) ≥ M ln σ + 1 Lσ Y − AS 2 F ≥ M ln Y − AS 2 F N L + M.(74)
Consequently,
L * = L (f * , p * , σ * , S * ) ≥ M ln Y − A * S * 2 F N L + M.(75)
Combining (75) and (71), we obtain that
Y − A * S * 2 F N L ≤ Ce −1 σ o M −K M .(76)
Therefore, lim
σ o →0 A * S * = A o S o ,(77)
implying consistence of (f * , S * ) (up to perturbations of entries) by Assumption A4. Finally, note by (67) that p * = arg min p ln A * P A * H + σ * I + 1 L tr S * H P −1 S * ,
(78) where all rows of S * are nonzero and A * has full column rank if σ o is small enough due to their consistency as σ o → 0 and Assumption A6. It follows that the derivative of the above objective function with respect to p k vanishes at p k = p * k , yielding that
a H (f * k ) A * P * A * H + σ * I −1 a (f * k ) − S *
is bounded from below by a universal positive number if σ o is small enough. We further apply the second part of Lemma 1 to obtain by (79) and (49) that
0 = lim σ o →0 a H (f * k ) A * P * A * H + σ * I −1 a (f * k ) − S * k 2 Lp * 2 k = lim σ o →0 1 p * k − S o k 2 Lp * 2 k ,(81)
which results in (51) and completes the proof.
II. PRELIMINARIES A. DOA Estimation Using SLAsAn M -element SLA of aperture N −1 composes a subset of an N -element virtual ULA. Let the index set Ω ⊂ {1, . . . , N }, of cardinality M ≤ N , denote the SLA. We first consider the specialized ULA case when Ω = {1, . . . , N } and M = N .
,(31),(45),(46),(34) and(35) iteratively. Again, the overall MESA algorithm consists of the outer MM loop and the inner ADMM loop and it is illustrated in Algorithm 1. It has a computational complexity of O(N 3 ) per inner iteration and degenerates into MESA in the previous section in the specialized ULA case.
SLA Ω, sample covariance R Ω , source number K. Output: Estimates of frequencies f , source powers p and noise power σ. Initialize R Ω , Q, Λ, t, X, Z and calculate W = Γ T R in(16)and W = Γ T R −1 Ω Γ; 9: end while 10: Calculate the estimates of f and p by computing the decomposition T t = A (f ) P A H (f ) using root-MUSIC. A4: f o , S o are uniquely identifiable from their product
averaged over 200 Monte Carlo runs, where f is the vector of estimated frequencies. We consider three different types of SLAs consisting of a 10-element ULA, a 6-element MRA and an 8-element nested array given respectively by Ω ULA = {1, 2, . . . , 10} , (53) Ω MRA = {1, 2, 7, 10, 12, 14} , (54) Ω Nested = {1, 2, 3, 4, 5, 10, 15, 20}.
Fig. 1 .
1Negative log-likelihood function value versus the number of MM iterations of MESA.
Fig. 2 .
2Frequency estimation error for K = 7 uncorrelated sources using the MRA in (54), with L = 200.
Fig. 3 .
3Frequency estimation error for K ∈ {2, . . . , 13} sources using the MRA in(54), with L = 100 and SNR= 10dB.
Fig. 4 .
4Frequency estimation error for K = 3 uncorrelated sources, two of which are separated by δ, with L = 100 and SNR = 10dB.
Fig. 5 .
5Frequency estimation error for K = 3 sources using the nested array in(55), where the first two sources are coherent.
Fig. 6 .
6Results of one Monte Carlo run inFig. 5at SNR = 20dB, where the first two sources are coherent.
Fig. 7 .
7Frequency estimation error for K = 3 sources using the MRA in(54), where the first two sources are coherent.
Fig. 8 .
8Frequency estimation error for K = 7 sources using the 6-element MRA in(54), where the first and the fourth sources are correlated.
41, Theorem 1] and [75, Lemma 1]. Proposition 1: Assumptions A4 and A6 hold true if
the matrix
matrixAP A H + σI − (p 1 + ǫ) a 1 a H 1 = Adiag (−ǫ, p 2 , . . . , p K ) A H + σI(63)is indefinite. Hence, the matrix (p 1 + ǫ) AP A H + σI is indefinite by observing that the matrix in−1
a H
1
a 1
ACKNOWLEDGMENTThe authors would like to thank Prof. Jiang Zhu of Zhejiang University for providing the code of MNOMP.
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Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices. M Fazel, H Hindi, S P Boyd, American Control Conference. 3M. Fazel, H. Hindi, and S. P. Boyd, "Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices," in American Control Conference, vol. 3, 2003, pp. 2156-2162.
On gridless sparse methods for line spectral estimation from complete and incomplete data. Z Yang, L Xie, IEEE Transactions on Signal Processing. 6312Z. Yang and L. Xie, "On gridless sparse methods for line spectral estimation from complete and incomplete data," IEEE Transactions on Signal Processing, vol. 63, no. 12, pp. 3139-3153, 2015.
Rank awareness in joint sparse recovery. M E Davies, Y C Eldar, IEEE Transactions on Information Theory. 582M. E. Davies and Y. C. Eldar, "Rank awareness in joint sparse recovery," IEEE Transactions on Information Theory, vol. 58, no. 2, pp. 1135- 1146, 2012.
Newtonized orthogonal matching pursuit: Frequency estimation over the continuum. B Mamandipoor, D Ramasamy, U Madhow, IEEE Transactions on Signal Processing. 6419B. Mamandipoor, D. Ramasamy, and U. Madhow, "Newtonized orthog- onal matching pursuit: Frequency estimation over the continuum," IEEE Transactions on Signal Processing, vol. 64, no. 19, pp. 5066-5081, 2016.
| s2orc_abstract_body | 57 |
Stony Brook Business College | Is business management major competitive at Stony Brook? What are the grade requirements to get into the business college? I applied on the 28th of December but there is no decision yet. | reddit_title_body | 655 |
Magnetic sensors with polysilicon TFTs | Characteristics Research of Hall Magnetic Sensor Based on Nano-Polysilicon Thin Film Transistors | s2orc_citation_titles | 56 |
We consider dynamical systems given by interval maps with a finite number of turning points (including critical points, discontinuities) possibly of different critical orders from two sides. If such a map $f$ is continuous and piecewise $C^2$, satisfying negative Schwarzian derivative and some summability conditions on the growth of derivatives and recurrence along the turning orbits, then $f$ has finitely many attractors whose union of basins of attraction has total probability, and each attractor supports an absolutely continuous invariant probability measure $\mu$. Over each attractor there exists a renormalization $(f^m,\mu)$ that is exact, and the rates of mixing (decay of correlations) are strongly related to the rates of growth of the derivatives and recurrence along the turning orbits in the attractors. We also give a sufficient condition for $(f^m,\mu)$ to satisfy the Central Limit Theorem. In some sense, we give a fairly complete global picture of the dynamics of such maps. Similarly, we can get similar statistical properties for interval maps with critical points and discontinuities under some more assumptions. | For piecewise C 1 interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recur- rence along critical orbits, we prove (1) the nonexistence of wandering intervals, (2) the existence of absolutely continuous invariant measures, and (3) the bounded backward contraction property. The proofs are based on the method of proving the existence of abso- lutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien. 1. Introduction and main results The concept of wandering intervals plays an important role in studying dynamical behavior of non-uniformly hyperbolic dynamical system. In the area of interval dynamics, most important results are related to the absence of wandering intervals. Our main aim in this paper is to obtain a condition on the orbits of the critical values to ensure the absence of wandering inter- vals for piecewise C 1 interval maps with critical points and discontinuities, and to give a sufficient condition to the nonexistence of wandering intervals for multimodal maps whose orders of critical points are different to the left and to the right. The motivation to show nonexistence of wandering intervals are well known. Firstly, it is relevant to the isomorphism problem of dynamical system. In the 1880s, Poincare proved that each orientation preserving homeomorphism of the circle without periodic points is semi-conjugate to an irrational rotation. Denjoy showed that for the C 2 diffeomorphism of the circle without periodic points such wandering interval cannot exist, and this semiconjugacy is indeed a conjugacy. Analogue of Denjoy's theory also holds for a C 3 unimodal map f with a non-flat critical point (whose orders are equal to the left and to the right) and negative Schwarzian derivative, | s2orc_abstract_citation | 11 |
Food52 Mighty Salads by the Editors of Food52 et. ... | Food52 Mighty Salads by the Editors of Food52 et. al. is a free NetGalley ebook that I read in early April.
From the website/book/blog/podcast/merchantile conglomerate that is Food52 comes a contributor-compiled salad cookbook that categorizes them by leafy, non-leafy, grain/legume, pasta, fish & seafood, and meat-based salads. With the mixture of flavors, origins, non-cooked & cooked textures, and choices of homemade dressing, there's changes in the styles of photographs (a recipe in progress at home or ready for placement on a bistro table). | amazon_reviews | 336 |
I need a dress for a Wedding size 11 anyone want to swap or be nice enough to let me borrow a dress? | Hey Girls I need a dress for a wedding, I hate buying dresses because I only wear them once and then they just sit in my closet. I am willing to trade or someone could be nice enough to let me borrow a dress I am an honest and trustworthy.
Thanks,
P.S. I do reddit gift so you can see I don't flake. | reddit_title_body | 4 |
Moroccan Dreams: Oriental Myth, Colonial Legacy | Morocco has long been a mythic land, firmly rooted in the European colonial imagination. For more than a century it has been appropriated by travellers, explorers, writers and artists. It is just these images and imaginings that are now being reconstructed for nostalgic consumption. In Moroccan Dreams, Claudio Minca examines this aestheticised re-enactment of the colonial, exploring the ways in which Moroccans themselves have become complicit in the re-writing of their homes and lives. Richly illustrated, the book provides a fascinating journey that will engage and delight all those enamoured of Morocco and its extraordinary geographies. | s2orc_title_abstract | 13 |
Our dog didn't like it at first | Wow, this takes a ton of hair off my dog, which thankfully won't end up in my carpets! We weren't expecting our dog to shed so much, so this cuts down on vacuuming a lot. Our dog didn't like it at first, but is getting used to it after a week or so. It's very sturdy and well made. Should last a long time. | amazon_reviews | 43 |
Occlusion of the left main coronary artery and collateral circulation via the conus branch. | We report the case of a 71-year-old-man, a smoker, admitted for unstable angina. Subsequent investigation revealed complete proximal occlusion of the left main coronary with an unusual collateral circulation. The left coronary artery was filled by a large conus branch originating from the right sinus of Valsalva. This case shows the importance of looking for atypical collateral circulation in patients with chronic occlusion of the left main coronary artery and normal left ventricular function. | s2orc_title_abstract | 264 |
which country has won the most global management challenges | Global Management Challenge Italy, Kazakhstan, Latvia, Macau SAR (China), Mexico, Morocco, People's Republic of China, Poland, Portugal, Romania, Russia, Singapore, Slovakia, Spain, Turkey, Ukraine, United Kingdom and Venezuela. For the 2010-2011 session, two new African countries are entering the contest: Ivory Coast, Benin, in partnership with Educarriere.net, a leading website in Ivory Coast which have been famous because it used to publish the result of many exams. Each country organizes its own National competition, the model producing all material in the language of that country. The winners of the National competition meet to contest the International Semi-Finals and, if successful, go on to | paq | 469 |
I hope these hold up better than the last pair | I had a pair that I bought from a store years ago. They lasted about 4 or 5 years. I finally decided that it was too me to buy some new ones (exact same style and brand) and they fell apart in 3 months. I just bought yet another pair... I hope these hold up better than the last pair. The glue comes apart. | amazon_reviews | 58 |
Purification and cloning of aggrecanase-1 | Purification and cloning of aggrecanase-1: a member of the ADAMTS family of proteins. | s2orc_citation_titles | 27 |
Pike Service - LSC Needle O-Ring | Was servicing my Pike and the last step was bleeding the damper. While doing so I removed the o-ring on the lsc needle, went to install new o-ring from kit and no luck as the o-ring is larger in diameter and is loose in the groove in the needle.
Everything points to me having the correct kit. Just that maybe this o-ring isnt in the kit or wasnt included. Anybody know for certain on this?
Picture for reference:
| reddit_title_body | 245 |
what are thermionic valves in the heath robinson | Heath Robinson (codebreaking machine) by a battery of ten photocells, an eleventh for the sprocket holes and two additional ones for the "stop" and "start" signals that were hand-punched between the third and fourth and fourth and fifth channels. This was designed by Tommy Flowers of the Post Office Research Station at Dollis Hill in North London. It used thermionic valves (vacuum tubes) to implement the logic. This involved the Boolean "exclusive or" (XOR) function in combining the various bit-streams. In the following "truth table", 1 represents "true" and 0 represents "false". (At Bletchley Park these were known as x and • respectively.) Other | paq | 195 |
Taxonomic comments on some protobranch bivalves from the northeastern Atlantic | Abstract Most of the small northeastern Atlantic protobranchs classically referred to Leda, Portlandia, Yoldiella, Yoldia etc., are discussed and figured to facilitate identification. Yoldiella solidula sp.n. was previously included in Y. fraterna Verrjll & Bush, but is here considered a distinct, arctic species. The following new synonymies are made (the valid name mentioned first): Yoldiella lucida (Loven, 1846) = Y. subangulata Verrill & Bush, 1898 = Y. iris Verrill & Bush, 1898. - Yoldiella nana (M. Sars, 1865) = Yoldiella fraterna Verrill & Bush, 1898 = Yoldiella inconspicua Verrill & Bush, 1898. - Yoldiella propinqua (Leche, 1878) = Yoldia pygmaea var. symmetrica Friele, 1878 = Portlandia persei Messjatsev, 1931. - Yoldiella messanensis (Seguenza, MS, Jeffreys, 1879) = Leda acuminata Jeffreys, 1873. - Yoldiella philippiana (Nyst, 1845) (neotype designated) = Yoldiella frigida mediterranea Nordsieck, 1974. - Megayoldia Verrill & Bush, 1987 = Microyoldia Verrill & Bush, 1987. - Megayoldia thraciaeform... | s2orc_title_abstract | 104 |
Optimization of epoxy passivation process to fabricate silicon radiation detectors | Passivation of surface using twocomponent epoxy at room temperature is followed to fabricate silicon surface-barrier type of detectors. Proper curing of the epoxy resin using the right amount of amine-based hardener is crucial to minimize moisture intake and hence obtain stable detector performance. A simple experiment was conducted to test hardening and water intake of the cured epoxy by varying the resin to hardener ratio. The data obtained helped us to fix the proper resin to hardener ratio and fabricate charged-particle detectors with lower leakages and better stability. | s2orc_title_abstract | 181 |
Rhabdomyolysis after bariatric surgery: a potentially fatal complication | Rhabdomyolysis in Bariatric Surgery: a Systematic Review | s2orc_citation_titles | 51 |
countries that require citizens to renounce their citizenship | about to relinquish their citizenship do not become a stateless person and many countries require evidence of another citizenship or an official promise to grant citizenship before they release that person from citizenship. Some countries may not allow or do not recognize renunciation of citizenship or establish administrative procedures that are essentially impossible to complete. Mexico requires renunciation of all other citizenships as a condition of naturalization. Israel allows dual citizenship upon naturalization through the Law of Return, but Knesset (Israeli parliament) members are required to renounce all foreign citizenships. Renunciation of citizenship is most straightforward in those countries which | paq | 326 |
Robert Permane | Robert Constantin "Bobby" Permane (January 21, 1924 – October 24, 2017) was a Thoroughbred horse racing jockey whose successful career included riding future Hall of Fame inductee Stymie to thirteen wins. Fittingly, in 1951 Permane won the Stymie Purse at Bowie Race Track in Maryland.
Entertainment career
Robert Permane was born in Camden, New Jersey to parents who were vaudeville performers. At age eight he was competing in equestrian events for ponies. Prior to embarking on his career as a jockey in Thoroughbred racing he had success as a singer, performing on radio and touring the United States and Australia.
Jockey career
In August 1943 Permane made his professional riding debut at Garden State Park Racetrack in Cherry Hill, New Jersey where on August 31 he earned his first win. The following year he was the leading jockey at Tropical Park Race Track in Miami, Florida. In April, during the track's 1944 spring meet, Permane rode a record five winners three days in a row. Permane was also second in total wins in New York State in 1944 and third in 1948. He was severely injured in March 1949 in a racing accident at Gulfstream Park which would require eight operations that put him out of racing for fifteen months, not returning until June 3, 1950.
Among his other major successes, Robert Permane won the 1944 Jockey Club Gold Cup aboard Bolingbroke, and the 1946 Santa Anita Derby for Maine Chance Farm aboard Knockdown. Then, in the U.S. Triple Crown races, he rode Knockdown to a fifth place finish in the 1946 Kentucky Derby and to fourth in the Preakness Stakes. In 1947 Permane won the Hollywood Gold Cup and Sunset Handicap aboard Cover Up and in 1950 rode the filly Busanda to victory in the Alabama Stakes.
When his racing career was over, Robert Permane and his wife made their home in Florida where they built Carlyn Estates Trailer Park in Palmetto. He died on October 24, 2017 at age 93.
References
1924 births
2017 deaths
American male singers
Vaudeville performers
American businesspeople
American jockeys
People from Camden, New Jersey | wikipedia | 35 |
A FULLY AUTOMATED TEST BENCH FOR THE MEASUREMENT OF THE FIELD DISTRIBUTION IN RFQ AND OTHER RESONANT CAVITY | As part of the RF studies of the resonant cavities of the “IPHI” accelerator project (“Injector of Proton High Intensity”), a fully computer controlled bench for the field distribution measurement has been developed. Based on the perturbation method, the acquisition of the s21 transmission coefficient phase shift is synchronized with the displacement of a bead. A systematic study of the noise and uncertainties has led to a clearly enhanced signal over noise. The measured raw data are then converted to a quantity proportional to the electromagnetic field magnitude. Specifically for the RFQ tuning study, different transverse section positions where to guide the bead have been tested on our RFQ cold model. | s2orc_title_abstract | 356 |
Is discipline really superior to motivation ? | I wanna start by saying that I perfectly understand why discipline is an important quality to have. Discipline is the thing that makes you go to the gym even if you feel lethargic and depressed so that you go there, lift heavy things and end up doing a fantastic training.
Discipline is the thing that allows you to wake up in the morning to do the job that will pay for your bills
But what about the severely depressed people ? The ones who have neither motivation nor discipline?
The low T, depressed betas. Can they really strive with discipline alone, when their discipline is so weak ? I think not
I think people like that need to believe at 200% that they can change. They need to believe that they can be happy, they need to become incredibly optimist, so that they can now understand why discipline has a purpose and how things will get better if they stick to it and never give up.
So how does one get motivated ? Swallowing the red pill can be very bleak for blue pilled addicts who got their views totally shattered.
Have we underestimated the roles of motivation, testosterone and realistic optimism in this road of se elf-improvement? | reddit_title_body | 366 |
Keynote Speaker 2 Intelligent Pattern Recognition and Applications - An Interactive E-Learning System, Modelling and Simulation | This talk deals with fundamental aspects of Intelligent Pattern Recognition (IPR) and applications. It basically includes the following: Overview of 3D Biometric Technology and Applications, Importance of Security: A Scenario of Terrorists Attack, What are Biometric Technologies? Biometrics: Analysis vs Synthesis, Analysis: Interactive Pattern Recognition Concept, Importance of Measurement, How it works: Fingerprint Extraction and Matching, Iris, and Facial Analysis, Authentication Applications, Thermal Imaging: Emotion Recognition. Synthesis in Biometrics, Modelling and Simulation, and more Examples and Applications of 3D Biomedical Imaging and Vision in Interactive Web/Video Networking Fuzzy e-Learning Environment. Finally, some future research directions are discussed. | s2orc_title_abstract | 48 |
White tree ? ( Garden of Tranquillity ) | Did they keep the white tree with the remake of the white wolf mountain ?
I need it for the Garden of Tranquillity quest and can't seem to find it. | reddit_title_body | 205 |
It is unfortunate that the the inspiration tree, designed for "creative rule bending," has kind of just turned into the "free gold with some perks" tree. | The most meta choices from this tree include:
Klepto- most of the power comes from free gold, though i wont pretend the items arent nice.
Slightly magical boots- free 300 gold boots, 50 gold discount to upgrade. Movespeed is nice.
Stopwatch- i would say the power is actually mostly in the active but quite a bit is the free 300 gold (or a bit less if you sell it)
The above three are the most meta choices from inspiration. I guarantee whatever source you use, if you look at champs that are running inspiration primary, klepto is nearly always their keystone with the two other runes (with a few exceptions) i mention, probably with the 5% cdr as well.
Champs that go inspiration secondary usually take free boots and free stopwatch.
I know that some supports will take hexflash for the mid pressure. I fully admit this.
The other exception is that lots of supports like futures market, which kind of also boils down to FREE GOLD!
I just feel like this tree had a ton of promise, but just turned into the free gold tree and thats kind of sad. | reddit_title_body | 310 |
what is the life of the nissan fuga hybrid | Nissan Fuga and the transmission, produces from 1.3kWh lithium-ion batteries that are expected to have a service life of 10 years. The batteries are installed upright behind the rear seats. The Fuga Hybrid introduced Nissan's first in-house developed electric hybrid technology, and Nissan claims it will double the fuel economy of its gasoline-powered version. A driving mode selector knob has been installed as standard equipment on all models, situated below the transmission gear lever and between the heated and ventilated front seat controls, providing four selections labeled "Standard", "Sport", "Eco", and "Snow", allowing the 7-speed transmission, engine and various systems to optimize | paq | 464 |
to refill over and over would get expensive after awhile and may as well just buy a better bag with better filling | for the price it was decent, but the filling went flat quick. to refill over and over would get expensive after awhile and may as well just buy a better bag with better filling. | amazon_reviews | 285 |
This textbook was interesting and easy to read | I used this book for my Intro to Cinema class. This textbook was interesting and easy to read. | amazon_reviews | 186 |
Price of scooty pep plus? | Price of tvs scooty pep? | wikianswers | 5 |
Virtual reality as a support tool in the shoe life cycle | The life cycle of an industrial product goes through several stages, from the initial idea to the finished product, and on to the purchase and recycling. Virtual reality environments (VRE) allow the user to interact with a digital representation of a product, and so can be used to perform aesthetical, ergonomic, functional tests as well to support customer decisions in the selling process; mainly for customized products. In this contribution, ITIA-CNR presents two important applications in the shoes context: VRShoe and MagicMirror. VRShoe is a VRE for designing shoe aesthetics. MagicMirror, currently under development, is an augmented reality (AR) system for supporting the decision-making processes of customer customized shoes. | s2orc_title_abstract | 232 |
Purchased as a gift for a friend whom had one ... | Purchased as a gift for a friend whom had one and it tore so I found her another one. She really likes using it wanted one where her belongings were covered and couldn't fall out. | amazon_reviews | 177 |
Traffic design method of expressway ramp with adjacent intersection | Problems of vehicle interleaving between a ramp and a ground,the loss of green time,and the shortage of the capacity of adjacent intersections were introduced.Spatial-temporal design methods were applied to avoid interleaving and conflict,a lane functionality replacement was used for the import lanes of intersection in the space,and for the time according to the principle of saturation equilibrium in traffic flow.A design idea about signal combination phase was put forward,which can decrease the loss of effective green time.One example was introduced to prove the methods in the pretext.The results indicate that grade separation capacity was improved,the average vehicle delay was greatly reduced.Moreover,the vehicle interleaving between a ramp and a ground was almost eliminated.The benefits were improved obviously. | s2orc_title_abstract | 207 |
33 [M4F] Tampa area - Looking for sugar baby on the side... | Hello, I'm a 33 white male business owner in the tampa area. I work late nights at my office. I do have a GF but shes always back at home and I would like some extra fun on the side. Message me and we can work out the details. Also my kik is the same as my username here if you have kik. Message me anytime, i'm up all night. | reddit_title_body | 578 |
Sex pheromone in the shore crab Carcinus maenas, and the site of its release from females | Primer and Short-Range Releaser Pheromone Properties of Premolt Female Urine from the Shore Crab Carcinus maenas | s2orc_citation_titles | 47 |
Great for support of paper plates | Outstanding. Great for support of paper plates. Do not advise use in microwave, but will last a while. | amazon_reviews | 69 |
Tailoring of nickel silicide contacts on silicon carbide | Co-deposition technique by means of simultaneous ion beam sputtering of nickel and silicon onto SiC was performed for tailoring of Nisilicide/SiC contacts. The prepared samples were analysed by means of XRD and XPS in order to obtain information about the surface and interface chemistry. Depth profiling was used in order to analyse in-depth information and chemical distribution of the specimens. XRD results showed that the main phase formed is Ni2Si. The XPS analysis confirmed the formation of the silicide on the surface and showed details about the chemical composition of the layer and layer/substrate interface. Moreover, the XPS depth profiles with detailed analysis of XPS peaks suggested that tailoring of C distribution could be monitored by the co-deposition technique employed. (C) 2007 Elsevier B.V. All rights reserved. | s2orc_title_abstract | 111 |
where does years of living dangerously air | Environmental Content. Years of Living Dangerously Years of Living Dangerously is an American documentary television series focusing on global warming. The first season was broadcast in the US in 2014 on "Showtime". It won an Emmy Award as Outstanding Documentary or Nonfiction Series. The second season aired on the National Geographic Channel in 2016. Executive producers included James Cameron, Arnold Schwarzenegger, and series creators Joel Bach and David Gelber (formerly of "60 Minutes"). Joseph Romm and Heidi Cullen were the chief science advisors. The weekly episodes featured celebrity hosts with a history of environmental activism and well-known journalists with a | paq | 517 |
Kim will always be Kris's true favorite. | I know it's an open joke in the family that Kris's favorite is whichever kid is most famous at the moment -- aka Kim for many years, Khloe for a bit when Lamar was doing good in the Lakers, Kendall when her modeling career was looking really promising, and now Kylie because of how well she's doing financially.
But honestly throughout it all, I really believe no matter what that Kim will always be her favorite. Kim is exactly what Kris wanted - a gorgeous daughter who is as business minded as she is but also deferential to Kris.
Kim always wants to please Kris (Khloe & Kourtney aren't this way). Kim is incredibly ambitious and a go-getter (Kendall isn't this way). Kim spent enough time around her mom to develop a really close relationship (Kylie didn't get this).
Kim is the daughter that Kris can discuss business with, make plans with, but also have a mother-daughter like bond because Kim loves her so much too. | reddit_title_body | 525 |
how many pounds of respiratory toxicant emissions did dominion have in 2002 | Hope Bay. In 2002, Dominion was responsible for 1,110,703 pounds of gastrointestinal or liver toxicant emissions, 1,440,000 pounds of musculoskeletal toxicant emissions, and 1,489,763 pounds of suspected respiratory toxicant emissions, and 1,478,383 pounds of suspected skin or sense organ toxicant emissions among other emissions that are suspected to be hazardous. Dominion Energy Dominion Energy, Inc., commonly referred to as Dominion, is an American power and energy company headquartered in Richmond, Virginia that supplies electricity in parts of Virginia and North Carolina and supplies natural gas to parts of West Virginia, Ohio, Pennsylvania, and eastern North Carolina. Dominion also has generation | paq | 423 |
how many female servants are there in hogarth's servants | Hogarth's Servants the work. The three female figures have similarly youthful appearances, not children but also not old. The male figures progress in age from a boy at the top centre, through a mature man at the bottom centre, to a more elderly man at the top right. The older man is possibly Ben Ives; the other servants may be "Samuel"; Mary Lewis, Hogarth's wife's cousin, who later inherited the painting; and Mrs Chappel, who was known to have worked for the Hogarths in Chiswick. Ronald Paulson believes the servants featured could be a coachman, valet, page, housekeeper and two housemaids. The | paq | 267 |
High temperature superconducting components for microwave systems | Abstract Starting with an overview of the various requirements placed on substrates for superconducting microwave applications, the current state of the art of high temperature superconducting thin films on different substrate types is discussed. Recent results for several superconducting microwave components that could form important building blocks for advanced microwave systems are presented, including performance data on experimental devices such as narrow band-pass and band-stop filters, superconducting circulators and miniature, low-frequency spiral resonators. Some examples of system types that can most likely benefit from high temperature superconductivity are identified. | s2orc_title_abstract | 300 |
When was rome sacked by goths? | What year was rome is sacked by the vandals? | wikianswers | 93 |
Subsea oil loading system for tankers | The offshore oil loading systems known up to the present have many disadvantages with respect to safety and costs. Feasibility studies have therefore been carried out to examine the concept of a subsea oil loading system. This work was followed up by design of the individual components. The investigations are being carried out by DEMINEX, TNSW and AEG. This has resulted in new solutions which enable economical and reliable operations with a high utilization factor even in marginal fields and in water depths down to 200 meters. The fact that the fixed components of the loading system are located on the seabed means that application of the system is also feasible in ice hazard areas. | s2orc_title_abstract | 268 |
Very useful and handy tool plus the quality of Leatherman is amazing | This is my first multi tool from Leatherman. It's great for everyday fixes and handy enough to handle serious jobs. The quality is excellent and the feel of the metal is what you expect from Leatherman. | amazon_reviews | 235 |
how many parts are there in octavio paz's labyrinth of solitude | the subject that you are studying so that the argument remains critical yet rational and objective. As the intellectual gets more involved with the political environment, his arguments can often become influenced by other factors such as political motivation and pressure to conform. The Labyrinth of Solitude The Labyrinth of Solitude () is a book-length essay by Octavio Paz, first published in 1950. One of his most famous works, it consists of nine parts: "The Pachuco and other extremes", "Mexican Masks", "The Day of the Dead", "The Sons of La Malinche", "The Conquest and Colonialism", "From Independence to the Revolution", | paq | 285 |
still fits though but I naturally have a very slim waste, I just have a little jiggly tummy from ... | Really tight! I bought a large and it is VERY tight...I am 5'7 and 160...still fits though but I naturally have a very slim waste, I just have a little jiggly tummy from have two babies so thats really what I wanted flattened but it doesn't do a great job at that...also the straps are really awkward, like practically in my armpits....its still wearable with certain dresses, but not amazing....I'd buy a different shaper if I could reorder :/ | amazon_reviews | 340 |
Is anyone else getting Dark Cloud 2 vibes? | Been playing Xenoblade since last night, and between the music and some aspects.. I can't help but keep attaching my mind to Dark Cloud 2 xD.
Curious if anyone else has felt the same! | reddit_title_body | 256 |
AP Chemistry | Advanced Placement Chemistry (also known as AP Chemistry or AP Chem) is a course and examination offered by the College Board as a part of the Advanced Placement Program to give American and Canadian high school students the opportunity to demonstrate their abilities and earn college-level credit. AP Chemistry has the lowest test participation rate, with around half of AP Chemistry students taking the exam.
Course
AP Chemistry is a course geared toward students with interests in chemical biologies, as well as any of the biological sciences. The course aims to prepare students to take the AP Chemistry exam toward the end of the academic year.
AP Chemistry covers most introductory general chemistry topics (excluding organic chemistry), including:
Reactions
Chemical equilibrium
Chemical kinetics
Stoichiometry
Thermodynamics
Electrochemistry
Reaction types
States of matter
Gases, Ideal gases and Kinetic theory
Liquids
Solids
Solutions
Structure of matter
Atomic theory, including evidence for atomic theory
Chemical bonding, including intermolecular forces (IMF)
Nuclear chemistry (removed for May 2014 test)
Molecular geometry
Molecular models
Mass spectrometry
Laboratory and chemical calculations
Thermochemistry
Chemical kinetics
Chemical equilibrium
Gas laws calculations
Exam
The annual AP Chemistry examination, which is typically administered in May, is divided into two major sections (multiple-choice questions and free response essays).
Old test (2013 and earlier)
The old test was composed of two sections: a multiple-choice section consisting of 75 questions with five answer choices each, and a free-response section consisting of six essay prompts that required the authoring of chemical equations, solution of problems, and development of thoughtful essays in response to hypothetical scenarios.
Section I, the multiple-choice portion, did not allow the use of a calculator, nor did it provide any additional reference material, other than a periodic table. Each question contained five answer choices. 90 minutes were allotted for the completion of Section I. Section I covered the breadth of the curriculum.
Section II, the free response section, was divided into two sections: Part A, requiring the completion of three problems, and Part B, also containing three problems. Part A, lasting 55 minutes, allowed the use of calculators, while Part B, lasting 40 minutes, did not. The first problem in Part A concerned equilibrium related to solubility, acids and bases, or pressure/concentration. The first question of Part B was a chemical equation question in which 3 scenarios were presented and the student was required to work all 3 scenarios, authoring a balanced net ionic chemical equation for each scenario and answering questions about the equations and scenarios. If time permitted, students may have edited their responses from Part A during the time allotted for responding to Part B, though without the use of a calculator. The student needed to have completed all six questions.
While the use of calculators was prohibited during Section I and Section II Part B, a periodic table, a list of selected standard reduction potentials, and two pages of equations and conventions are available for use during the entirety of Section II.
New test (2014 and later)
The 2014 AP Chemistry exam was the first administration of a redesigned test as a result of a redesigning of the AP Chemistry course. The exam format is now different from the previous years, with 60 multiple choice questions (now with only four answer choices per question), 3 long free response questions, and 4 short free response questions. The new exam has a focus on longer, more in depth, lab-based questions. The penalty for incorrect answers on the multiple choice section was also removed. More detailed information can be found at the related link.
Topics
Structure and Matter, 20%
States of Matter, 20%
Reactions, 35–40%
Descriptive Chemistry, 10–15%
Laboratory, 5–10%
Grade distribution
The score distributions since 2007 were:
Prerequisites
The College Board recommends successful completion of High School Chemistry and precalculus; however, requirement of this may differ from school to school. AP Chemistry usually requires knowledge of precalculus; however, some schools allow students to take Pre-CalcI concurrently with this class. The requirement of regular or honors level High School Chemistry may also be waived, but usually requires completion of a special assignment or exam.
See also
Chemistry
Glossary of chemistry terms
References
External links
Test format change in 2007
AP Chemistry Exam Overview
Chemistry education
Advanced Placement | wikipedia | 26 |
Where is the best place to level up in ff7? | Final Fantasy 7 where is the best place to level up for a level 65? | wikianswers | 50 |
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Wildey (surname) | Wildey is a surname. Notable people with the surname include:
Doug Wildey (1922–1994), American cartoonist and comic book artist
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How do i stop leaking into air intake?
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How do I stop my gurgling and any other advice for a relatively new vape owner
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