Dissipative Vibrational Model for Chiral Recognition in Olfaction
aa r X i v : . [ phy s i c s . c h e m - ph ] A p r Dissipative Vibrational Model for Chiral Recognition in Olfaction
Arash Tirandaz, ∗ Farhad Taher Ghahramani, † and Afshin Shafiee
1, 2, ‡ Foundations of Physics Group, School of Physics,Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran Research Group On Foundations of Quantum Theory and Information,Department of Chemistry, Sharif University of Technology, P.O.Box 11365-9516, Tehran, Iran
We examine the olfactory discrimination of left- and right-handed enantiomers of chiral odorantsbased on the odorant-mediated electron transport from a donor to an acceptor of the olfactoryreceptors embodied in a biological environment. The chiral odorant is effectively described byan asymmetric double-well potential whose minima are associated to the left- and right-handedenantiomers. The introduced asymmetry is considered as an overall measure of chiral interactions.The biological environment is conveniently modeled as a bath of harmonic oscillators. The resultingSpin-Boson model is adapted by a polaron transformation to derive the corresponding Born-Markovmaster equation with which we obtain the elastic and inelastic electron tunneling rates. We showthat the inelastic tunneling through left- and right-handed enantiomers occurs with different rates.The discrimination mechanism depends on the ratio of tunneling frequency to localization frequency.
PACS numbers: 87.19.lt, 82.39.Jn, 03.65.Yz
I. INTRODUCTION
Olfaction, the sense of smell, is one of the most ancientyet intriguing senses of living organisms to being in con-tact with the environment. Smell is caused by the small,neutral, volatile molecules known as odorant. In Humanbeings, olfaction occurs when the odorant molecules bindto specific sites on olfactory receptors in nasal cavity. Inspite of considerable progress towards the understand-ing of structure of olfactory receptors that involved inthe early stages of olfactory process, the detailed mech-anisms of discrimination between vast number of odor-ants are not yet fully understood [1]. The most obviouscharacteristic of an odorant molecule is its shape. Ac-cordingly, Amoore first conjectured that the response toscent is initiated by a mutual structural fit between thereceptor and the odorant ( lock and key model) [2]. Amore flexible modification of this idea is that the wholesystem distorts to induce a more appropriate mutual fit( hand and glove model). A further modification of thisshape-based theory requires that a particular receptorresponds to only one structural feature, such as a func-tional group, as opposed to the main body of the odor-ant ( odotope model) [3]. In spite of the predictive powerof these shape-based models, there are some evidenceagainst them: some odorants smell the same althoughthey are structurally very different [4]. Furthermore,some odorants with almost identical shapes smell verydifferent [5, 6]. The only contender to the shape-basedmodels is the vibration-based model: an unique scentis attributed to the unique vibrational spectrum of theodorant [7, 8]. Motivated by this idea, Turin proposedthat mechanism of olfactory recognition is an odorant-mediated inelastic electron tunneling (ET) at the recep-tor: signal transduction is based on the success of an elec-tron tunneling from a donor (D) state of a receptor to an acceptor (A) state of the same or another receptor, fa-cilitated by a vibrational transition in the odorant corre-sponding to the energy difference between these states [9].Recently, Brookes and co-workers expanded this idea,and advanced a semi-classical model to show that such amechanism fits the observed features of smell [10]. Theyfound that the rate of odorant-mediated inelastic ET islarger than the rate of elastic one. Following this ap-proach, in a very recent work, Ch¸eci´nska and co-workersexamined dynamically dissipative role of environment invibration-based olfactory recognition and showed thatthe strong coupling to the environment can enhance theodorant frequency resolution in the ET rates [11].The vibration-based theories of olfaction are com-monly refused on the grounds that enantiomers of chi-ral odorants have the same vibrational spectra but dif-ferent smells [5]. The advocates of such theories suggestthat other molecular features such as structural flexibilityshould be included in the model to account this seeminglycounterintuitive behaviour [12]. To address this problem,we propose that the chiral recognition in olfaction is re-liant on detection of energy difference between two enan-tiomers of the chiral odorant. This energy difference isresulted from chiral interactions between the odorant andthe receptor. At sufficiently low temperatures, two enan-tiomers of a chiral molecule can be effectively describedas two localized states of an asymmetric double-well po-tential. This asymmetry portrays the energy differencebetween two enantiomers. The biological environment ofolfactory system can be represented as a collection of har-monic oscillators. When the oscillators couple linearly tothe double-well, the result is the Spin-Boson model stud-ied extensively in the literature, particularly by Leggettand co-workers [13]. Here, we examine the dynamics ofolfactory ET using the Spin-Boson model and therebyobtain the elastic and inelastic ET rates for each enan-tiomers of a typical chiral odorant. We show that theinelastic ET rates can be different for two enantiomers.The article is organized as follows. In section II, we de-scribe the olfactory Spin-Boson model for a chiral odor-ant. In section III, after an appropriate polaron trans-formation, we characterize the dynamics of the modelusing the Born-Markov master equation. In section IV,we conduct the resulting equations for the appropriatechoice of model parameters to obtain the corresponding(in)elastic tunneling rates. Finally, we summarize ourfindings in the last section. Throughout this article, wesuppose ~ = 1. II. CHIRAL OLFACTION MODEL
The free Hamiltonian of the total system consisting ofthe electron at the receptor, the chiral odorant and thesurrounding environment can be written asˆ H ◦ = ˆ H e + ˆ H o + ˆ H E (1)The electron state at donor and acceptor sites of receptorare represented by | D i with energy ε D and | A i with en-ergy ε A , respectively. We assume that ε D > ε A , so thatthe electron tunneling corresponds to a vibrational ab-sorption in the odorant. So, we can describe the electronat the receptor with Hamiltonianˆ H e = ε D | D ih D | + ε A | A ih A | (2)A chiral odorant can occur at least as two chiral enan-tiomers through the inversion at molecule’s center ofmass by a long-amplitude vibration known as contor-tional vibration [14, 15]. This vibration can be effec-tively described by an asymmetric double-well potential.The minima, associated to two chiral states | L i and | R i ,are separated by barrier V ◦ . In the limit V ◦ ≫ ω ◦ ≫ k B T ( ω ◦ is the vibration frequency in each well), state space ofthe molecule are effectively confined in two-dimensionalHilbert space spanned by two chiral states. For mostchiral molecules this limit holds up to room tempera-ture [14, 15]. The left- and right-handed enantiomers ofthe chiral odorant are associated to states | L i and | R i localized at left and right wells of the potential, respec-tively. If the barrier is high enough to prevent the tun-neling process, the molecule remains in its initial chiralstate. The corresponding Hamiltonian of the molecule inthe chiral basis is given by [16]ˆ H o = − δ σ x − ω z σ z (3)where δ is the frequency of tunneling between two chi-ral states and ω z , known as localization frequency, is theenergy difference between two chiral states. The localiza-tion frequency is an overall measure of chiral interactions. The odorant’s states of energy are described by a super-position of chiral states as | i = sin( θ | L i + cos( θ | R i| i = cos( θ | L i − sin( θ | R i (4)where we defined θ = arctan( δ/ω z ). The energies corre-sponding to these states are ∓ p δ + ω z .The biological environment is a kind of condensedbath, conveniently modelled as a collection of harmonicoscillators with Hamiltonianˆ H E = X i ω i ˆ b † i ˆ b i (5)where ˆ b † i and ˆ b i are the creation and annihilation opera-tors for modes of frequency ω i in the environment.The interaction Hamiltonian has three contributions:between donor and acceptor of the receptor with tunnel-ing strength ∆, between the donor (acceptor) and theodorant with coupling strength γ D ( γ A ), and betweenthe donor (acceptor) and i -th harmonic oscillator of theenvironment with coupling strength γ iD ( γ iA ). So, theinteraction Hamiltonian of total system is given byˆ H int = ∆( | A ih D | + | D ih A | )+ ( γ D | D ih D | + γ A | A ih A | )ˆ σ x + X i ( γ iD | D ih D | + γ iA | A ih A | )(ˆ b † i + ˆ b i ) (6) III. DYNAMICS
In order to avoid the undesired transitions from donorto acceptor when the odorant is absent, the couplingstrength ∆ should be small compared to other energyscales in the model. In this regime, we conveniently moveinto a polaron transformed reference frame. We definethe unitary polaron transformation as U = exp h i λ ′ | D ih D | − λ | A ih A | )ˆ σ y i × exp h X i ( γ iD ω i | D ih D | + γ iA ω i | A ih A | )(ˆ b † i − ˆ b i ) i (7)where we defined λ = tan − (cid:0) δ − γ D ω z (cid:1) λ ′ = tan − (cid:0) δ + γ A ω z (cid:1) (8)Under the polaron transformation, the free Hamiltonianand interaction Hamiltonian take the form e H ◦ = ( ε D + µ ′ ˆ σ z ) | D ih D | + ( ε A + µ ˆ σ z ) | A ih A | + X i ω i ˆ b † i ˆ b i (9)with µ = − ω z λ h (cid:16) δ − γ D ω z (cid:17) i µ ′ = − ω z λ ′ h (cid:16) δ + γ A ω z (cid:17) i (10)and e H int =∆ (cid:16) | A ih D | e i ( λ + λ ′ )ˆ σ y ˆ ̥ + + | D ih A | e − i ( λ + λ ′ )ˆ σ y ˆ ̥ − (cid:17) (11)with ˆ ̥ ± = e ± P i ( γiD − γiAωi )(ˆ b † i − ˆ b i ) (12)The dynamics of the reduced density matrix of the recep-tor and the odorant (hereafter receptor+odorant) can bedescribed by the so-called Redfield equation [17] ∂ t ˆ ρ ( t ) = − Z t dt ′ T r E (cid:2) e H int ( t ) , [ e H int ( t ′ ) , ˆ ρ tot ( t ′ )] (cid:3) (13)If we assume that the interaction between the recep-tor+odorant and the environment is sufficiently weak,the total density matrix remains at all times in an ap-proximate product form (ˆ ρ tot ( t ) ≈ ˆ ρ ( t ) ⊗ ˆ ρ E ( t )). Also,because the environment is large in comparison with thesize of the receptor+odorant, the temporal change of theenvironment density matrix can be neglected (Born ap-proximation) (ˆ ρ E ( t ) ≈ ˆ ρ E (0)). At sufficiently high tem-peratures, moreover, it is assumed that the environmentquickly forget any internal self-correlations established inthe course of the interaction with the receptor+odorant(Markov approximation). After some mathematics, thenthe Born-Markov master equation is obtained as ∂ t ˆ ρ ( t ) = − ı (cid:2) ˆ H RO ◦ , ˆ ρ ( t ) (cid:3) − £ ρ ( t ) (14)with £ ρ ( t ) = Z ∞ dτ n C ( τ ) (cid:2) ˆ H ROint , ˆ H ROint ( − τ )ˆ ρ ( t ) (cid:3) + C ( − τ ) (cid:2) ˆ ρ ( t ) ˆ H ROint ( − τ ) , ˆ H ROint (cid:3)o (15)where ˆ H ROint ( − τ ) = e itε | A ih D | ˆΞ( τ ) + h.c (16)in which ε = ε D − ε A and we definedˆΞ( τ ) = (cid:18) Ξ ( τ ) Ξ ( τ )Ξ ( τ ) Ξ ( τ ) (cid:19) (17)where Ξ ( τ ) = cos( λ + λ ′ e − iτ ( µ − µ ′ ) Ξ ( τ ) = sin( λ + λ ′ e − iτ ( µ + µ ′ ) Ξ ( τ ) = − sin( λ + λ ′ e iτ ( µ + µ ′ ) Ξ ( τ ) = cos( λ + λ ′ e iτ ( µ − µ ′ ) (18) and the self-correlation functions of the environment C ( τ ) are obtained by C ( τ ) = C ∗ ( − τ ) = tr E n ˆ ̥ ± ( − τ ) ˆ ̥ ∓ (0) o = e − (cid:8) ν ( τ )+ iη ( τ ) (cid:9) (19)where ν ( τ ) and η ( τ ) are the noise and dissipation kernels ν ( τ ) = Z ∞ dω J ( ω ) ω h sin ( ωτ ω k B T ) i η ( τ ) = Z ∞ dω J ( ω ) ω sin( ωτ ) (20)Here, J ( ω ) is the spectral density of a continuous spec-trum of environmental frequencies, ω . It encapsulatesthe physical properties of the environment. For moreconvenience, we employ an ohmic spectral density withan exponential cut-off as J ( ω ) = X i ( γ iD − γ iA ) δ ( ω − ω i ) ≡ J ◦ ωe − ω/ Λ (21)in which J ◦ is a measure of system-environment couplingstrength and Λ is a high-frequency cut-off. If we re-place the explicit form of spectral density into (20), andfor the high-temperature environment, approximate theterm coth( ω/ k B T ) by 2 k B T /ω , we obtain C ( τ ) = e − kBTJ ◦ Λ (cid:2) Λ τ tan − (Λ τ ) − ln(1+Λ2 τ (cid:3) + i tan − (Λ τ ) (22)If we expand the the exponential function to second orderin τ we arrive at C ( τ ) ≈ e − k B T J ◦ Λ τ − i Λ τ (23)The elastic ET coincides with a situation in which thechiral odorant is present in the receptor but does notundergo any transition. The inelastic ET through left-and right-handed enantiomers are accompanied by a vi-brational transition from left- and right-handed states tothe first excited energy state, respectively (FIG. 1). FIG. 1: Relevant transitions in chiral odorant.
Now we can derive the explicit expressions of the cor-responding elastic and inelastic ET rates. We maydecompose the receptor+odorant density matrix asˆ ρ ( t ) = P i,j ; X,X ρ ijXX ′ | i, X ih j, X ′ | where i, j ∈ L, R and
X, X ′ ∈ D, A . In accordance with previous works, weassume that the donor state of the receptor only couplesto the lower state of the odorant at all times. In the de-coherence regime where the unitary dynamics is ignored,the population dynamics of acceptor state | A i and odor-ant state | j i is given by h A, j | £ ρ ( t ) | A, j i . We may thenobtain the elastic and inelastic ET rate for transition | D, i i → |
A, f i byΓ DiAj = 2∆ ℜ Z ∞ dτ n C ( τ ) e iετ h f | ˆΞ( τ ) | i ih i | e − i ( λ + λ ′ )ˆ σ y | j i o (24)Using the explicit form of the environmental correlationfunction C ( τ ) and the transition matrix Ξ( τ ), we finallyobtain the inelastic ET rates asΓ DLA ≈ cos (cid:0) λ + λ ′ − θ (cid:1) sin (cid:0) λ + λ ′ (cid:1) sin (cid:0) θ (cid:1) √ π ∆ √ k B T J ◦ Λ exp n − [ ε + ( µ + µ ′ ) − Λ]4 k B T J ◦ Λ o Γ DRA ≈ sin (cid:0) λ + λ ′ − θ (cid:1) sin (cid:0) λ + λ ′ (cid:1) cos (cid:0) θ (cid:1) √ π ∆ √ k B T J ◦ Λ exp n − [ ε − ( µ + µ ′ ) − Λ]4 k B T J ◦ Λ o (25)The elastic ET rate Γ DiAi ( i = L, R ), which is equal fortwo enantiomers, is given byΓ
DiAi = cos (cid:0) λ + λ ′ (cid:1) √ π ∆ √ k B T J ◦ Λ exp n − ( ε − Λ)4 k B T J ◦ Λ o (26)In the limit δ →
0, our expressions for the (in)elasticET rates reduce to the corresponding expressions for anodorant with one configuration obtained by Ch¸eci´nskaand co-workers [11].
IV. RESULTS
To calculate the numerical values of corresponding(in)elastic ET rates, we first estimate the parametersrelevant to our analysis. The magnitude of odorant’stunneling frequency, δ , extracted from the spectroscopicdata, ranges from the inverse of the lifetime of the uni-verse to millions of hertz [18]. We consider this parameteras the signature of the chiral odorant. The localizationfrequency of the odorant, ω z , represents an overall mea-sure of all chiral interactions involved. A chiral interac-tion in general transforms as a pseudo-scalar [19]. Forour system, these chiral interactions include the disper-sion intermolecular interactions between the odorant andthe receptor. Since the structural characteristics of odor-ants and especially receptors are complex by nature, toour knowledge, the molecular interactions between them are not characterized yet. Since both odorant and recep-tor are chiral, nevertheless, we assume that the contri-bution of intermolecular chiral interactions is significant.Note that if chiral interactions are strong enough to over-come tunneling process, say ω z ≫ δ , they confine themolecule in the chiral state corresponding to the deeperwell (the Hamiltonian (3) coincide only with the right-handed enantiomer). This situation may be realized inchiral biomolecules with only one stable enantiomer, e.g.,the building blocks of life, i.e., L-proteins and D-sugars.For chiral odorants with two stable enantiomers, we ex-clude this situation. On the other hand, as the discrim-ination factor of our model is the localization frequency, ω z , in the limit δ ≫ ω z , we obtain the same inelasticET rates for the left- and right-handed enantiomers. Ac-cordingly, we consider the case in which tunneling andlocalization compete with each other. We assume thatthe DA energy gap, ε , is relatively close to resonancewith corresponding transitions in the odorant. The en-ergy difference between left-handed (right-handed) enan-tiomer and the first excited state is p δ + ω z − ω z / p δ + ω z + ω z / ε = p δ + ω z . Also, we takethis value as the characteristic frequency of the odor-ant. While the odorant’s structure is known, the detailedstructure of donor and acceptor sites of the receptor in-teracting with it is relatively unknown. Since the elas-tic ET increases with the DA tunneling frequency, ∆,which is undesired as far as the odorant-assisted mech-anism is concerned, we keep ∆ small in comparison tothe characteristic frequency of the odorant. So, we es-timate ∆ ≃ . p δ + ω z . The estimation of couplingfrequency between the DA pair and the odorant moderequires more consideration. The coupling strength be-tween donor (acceptor) γ D ( γ A ) is proportional to changeof the odorant’s intrinsic dipole moment due to the inter-action with the electric field of the transferred electronlocalized at the donor (acceptor) site of the receptor. Ifwe suppose that the force on the odorant molecule as aresult of electron’s electric field at the donor site is op-posite of the of the corresponding force at the acceptorsite, we have γ D = − γ A (see [10]). On the other hand,the difference between DA-odorant couplings γ D − γ A ,calculated from the associated Huang-Rhys factor, tran-scribed to our system, is given as 0 . p δ + ω z [10]. So,we estimate γ D = − γ A = 0 . p δ + ω z . The spectraldensity is obtained from the microscopic details of themodel under consideration. The simplest model ariseswhen the odorant is treated as a point dipole inside auniform, spherical protein surrounded by a uniform po-lar solvent. For a Debye solvent and a protein with astatic dielectric constant, the parameters of the spectraldensity (20) is obtained as [20] J ◦ = (∆ ν ) πǫ ◦ b ǫ p ( ǫ s − ǫ ∞ )(2 ǫ s + ǫ p )(2 ǫ ∞ + ǫ p )Λ (27)where Λ = 2 ǫ s + ǫ p ǫ ∞ + ǫ p τ − D (28)in which ∆ ν is the difference between the dipole momentof the odorant in the ground and excited states, b is theradius of the protein containing the odorant, ǫ p is thedielectric constant of the protein environment, ǫ s and ǫ ∞ are the static and high-frequency dielectric constants ofthe solvent, respectively and τ D is the Debye relaxationtime of the solvent. For an odorant in water, we have J ◦ ≈ ≈ Hz . Accordingly, we summarize therelevant parameters in TABLE I. Parameter Value Parameter Value ε √ δ + ω z J ◦
1∆ 0 . √ δ + ω z Λ 10 Hzγ D (= − γ A ) 0 . √ δ + ω z T If we choose the characteristic frequency of the chiralodorant close to the typical vibrational frequency of anodorant, our results would be numerically consistent withthe results of previous works [10, 11]: for δ = 10 Hz and ω z = 10 Hz , we obtain the inelastic ET time for left-and right-handed enantiomers as 0 . ns and 8 . ns , andthe elastic tunneling time as 80 ns . The ratio of inelasticrate to elastic rate against the tunneling frequency forthe left- and right-handed odorant with a constant local-ization frequency is plotted in FIG. 2. It obviously showsthat the inelastic to elastic ratio increases with the ratioof tunneling frequency to localization frequency, but withdifferent rates for two enantiomers. Since we are primar-ily interested in the odorant-assisted inelastic tunneling,the desired regime would be δ > ω z . ´ ´ ´ ´ ´ ∆ G DRA2 G DRAR
FIG. 2: The ratio of inelastic rate to elastic rate versus thetunneling frequency δ for left-handed enantiomer (blue) andright-handed enantiomer (red) at ω z = 10 Hz . The ratio of the inelastic rate for the left-handed enan-tiomer to the inelastic rate for the right-handed enan-tiomer with the same localization frequency, as plottedin FIG. 3, shows a minimum at δ ≈ ω z . In the limit δ > ω z where inelastic ET is favored over elastic one, theratio increases with the ratio of tunneling frequency tolocalization frequency. We suggest that this difference isthe basis of chiral discrimination in olfactory system. ´ ´ ´ ´ ´ ∆ G DLA2 G DRA2
FIG. 3: The ratio of inelastic rates for the left-handed enan-tiomer Γ
DLA to the right-handed enantiomerΓ DRA versusthe tunneling frequency δ at ω z = 10 Hz . V. CONCLUSION
A common reason to reject vibration-based theories ofolfaction is the existence of enantiomers of chiral odorantswith the same vibrational spectra, but different smells.Despite the same vibrational spectra, there is an energydifference between the ground states of two enantiomersresulted from t the chiral intermolecular interactions be-tween the odorant and the receptor. These chiral inter-actions might be the origin of chiral discrimination inolfactory system. Motivated by this idea, we have devel-oped a dynamical model to examine the chiral odorant-mediated electron transport in the context of a proposedolfactory process. We modeled the chiral odorant effec-tively as an asymmetric double-well potential. The intro-duced asymmetry, introduced as localization frequency,is considered as an overall measure of all chiral interac-tions. The resulting (in)elastic ET rates are presented in(25) and (26). Our results show that if we choose the pa-rameters of our model close to the corresponding parame-ters of previous works on vibrational olfaction, we obtainconsistent values for the inelastic rates, yet different fortwo enantiomers. More specifically, the ratio of tunnelingfrequency to localization frequency can be considered asthe discrimination factor of our model. We showed thatan increase in this factor intensifies the inelastic tunnel-ing for both enantiomers (FIG. 2). At the region whereinelastic tunneling is dominant, chiral discrimination in-creases with the discrimination factor (FIG. 3). It is anexperimental fact that the majority of enantiomer pairshave very similar smells. In the model we proposed thisis realized by the fact that majority of chiral odorantshave low ratio of tunneling frequency to localization fre-quency, and therefore weak chiral discrimination by theolfactory system. ∗ [email protected] † [email protected] ‡ shafi[email protected][1] M. Zarzo, The sense of smell: molecular basis of odorantrecognition, Biol. Rev. , 455 (2007).[2] J. Amoore, The stereochemical theory of olfaction, Na-ture , 912 (1963).[3] K. Mori and G. Shepard, Emerging principles of molec-ular signal processing by mitral/tufted cells in the olfac-tory bulb, Semin. Cell. Biol. . 65 (1994).[4] L. Turin and F. Yoshii, Struture-odor relations: a modernperspective, in Handbook of Olfaction and Gustaion , R.Doty, Marcel Dekker, New York (2003).[5] R. Bentley, The nose as a stereochemist: Enantiomersand odor, Chem. Rev. , 4099 (2006).[6] J. C. Brookes, A. P. Horsfield, and A.M. Stoneham,Odour character differences for enantiomers correlatewith molecular flexibility, J. R. Soc. Interface , 75(2009).[7] G. Dyson, The scientific basis of odour, Chem. Ind. ,647 (1938).[8] R. Wright, Odor and molecular vibrations: neural coding of olfactory information, J. Theor. Biol. , 473 (1977).[9] L. Turin, A spectroscopic mechanism for primary olfac-tory reception, Chem. Senses , 773 (1996).[10] J. C. Brookes et al. , Could humans recognize odor byphonon-assisted tunneling? Phys. Rev. Lett. , 038101(2007).[11] A. Ch¸eci´nska et al. , Dissipation enhanced vibrationalsensing in an olfactory molecular switch, J. Chem. Phys. , 025102 (2015).[12] J. C. Brookes, A. P. Horsfield and A. M. Stoneham,Odour character differences for enantiomers correlatewith molecular flexibility, J. R. Soc. Interface , 75(2009).[13] A. J. Leggett et al. , Dynamics of the dissipative two-statesystem, Rev. Mod. Phys. , 1 (1987).[14] C. H. Townes and A. L. Schawlow, Microwave Spec-troscopy , McGraw-Hill, New York (1955)[15] G. Herzberg,
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