Anharmonic Electron−Phonon Coupling in Condensed Media: 1

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Anharmonic Electron-Phonon Coupling in Condensed Media: 1. Formalism Mohamad Toutounji* College of Science, Department of Chemistry, United Arab Emirates University, Al-Ain, UAE ABSTRACT: Three different schemes for calculating anharmonic line shape functions are reported and discussed for the first time in this article using eigenstate representation. First, the linear dipole-moment time correlation function (DMTCF), homogeneous (single-site) absorption line shape function, and the respective Franck-Condon factors (FCF) are derived and explored as a molecule makes a transition from a harmonic to an anharmonic (Morse potential) electronic state. Second, the linear DMTCF, homogeneous absorption line shape function, and FCFs are also derived as a molecule makes a transition from one anharmonic to another linearly displaced anharmonic state; FCFs in this case are reported in an exact closed-form expressed in terms of Appell’s hypergeometric function. Third, same as the latter set of results are reported but with both linearly displaced and distorted shape of the upper Morse potential. FCFs of the zero-phonon line in all three cases are reported. The first case is rather mathematically complex as a result of taking the overlap integral of the Morse oscillator eigenfunctions, whose spatial decay is a simple exponential, with those of harmonic oscillator, whose decay is a Gaussian. This form of a functional disparity gives rise to some challenges. Model calculations are presented and discussed.

I. INTRODUCTION The electronic system may be coupled to a number of vibrational modes representing, e.g., intramolecular vibrations, local intermolecular modes, and collective solvent modes.1 Anharmonicity affects the molecular dipole moment magnitude and orientation in condensed media and thereby the associated dynamics change accordingly. This fact has become increasingly important in the advent of laser spectroscopy. Nuclear motions may be modeled as relatively small amplitude vibrations. They can couple to the electronic transition of a chromophore linearly, which may be modeled as harmonic modes whose equilibrium position is linearly displaced with respect to the initial electronic state, quadratically (Franck-Condon inactive modes),2-5 which may be viewed as independent distorted harmonic modes with respect to the initial state, or both linearly and quadratically.2-11 Normally, optical line shapes for condensed-phase systems are determined by the nature of electron-phonon coupling. As such, the type of dynamics projected out depends profoundly on the strength of that coupling. The multimode Brownian oscillator (MBO) model of the Mukamel group has been at the forefront in treating spectroscopy and dynamics of linearly coupled systems.1 In the harmonic approximation, phonons can couple linearly and quadratically (diagonally and off-diagonally) to the chromophore electronic degrees of freedom.2-11 More importantly, phonons also couple to the electronic transition anharmoincally, whereby anharmonic coupling effects become important as the bath temperature increases. However, unlike harmonic systems, anharmonic systems are not as feasible to handle analytically and simulation techniques such as path integral Monte Carlo methods would have to be utilized. The Mukamel group utilized the Liouville space generating function (LGF) approach to study dynamics of anharmonic molecules in the condensed phase, leading to very good results.1,14,15 r 2010 American Chemical Society

Understanding many important phenomena such as charge transfer,16 energy transfer,17,18 excitation dynamics,19 electron transfer dynamics,20 and spectral line shapes5-15,21-27 centrally depends on the time evolution of the electronic and nuclear dipole moment operators. It is needless to emphasize the central role time correlation functions (TCF) play in interpreting dynamics and spectroscopic measurements. (For a lucid account on the underlying physics of these time correlation functions, the reader is referred to refs 28-33.) There have been many attempts of analytically evaluating two- and four-point dipole moment time correlation functions (DMTCF) in anharmonic systems which have led to successful results,14,15,34-36 but there has not been an attempt reported on analytical evaluation of these time correlation functions with anharmonicity being exactly accounted for. Mukamel14,15 and Tanimura36,37 numerically, and successfully, calculated linear absorption spectra and pump-probe signals using the LGF approach having invoked different levels of approximations to solve the appropriate Heisenberg equations of motion. Although Tanimura reported sound nonlinear spectra of anharmonic molecules, the studied spectra involved only simple linear displacement (only bond length change upon optical excitation with no distortion of the final Morse potential, i.e., frequencies of both states were equal).36,37 Although the exact Morse oscillator eigenfunctions are available in the literature, integrating, manipulating, or dealing with them is, analytically, not straightforward, unlike their harmonic Special Issue: Shaul Mukamel Festschrift Received: May 24, 2010 Revised: July 5, 2010 Published: August 19, 2010 5121

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The Journal of Physical Chemistry B counterparts. With the right transformation such as Wigner,38-40 Liouville, or canonical transformation,41 one can map Morse potential into a harmonic one, and the converse is true. The Morse oscillator has proven to be a reasonably good model for understanding anharmonic dynamics,23,43-46 particularly local modes in the high vibrations of C-H and O-H in polyatomic molecules.47,48 Nuclear dynamics of Morse clusters was utilized to address the stability of multicharged finite systems driven by Coulomb forces.49 Very recently, an exhaustive account of Morse analytic dynamics has rendered useful information.50 Although Morse potential seems to serve as an accurate model of a true electronic energy curve, it does not feature the simplifying factors of the harmonic oscillator, such as symmetry of the potential and definite parity of the eigenfunctions. As a result, Morse oscillator eigenfunctions do not easily lend themselves to use analytically or numerically; i.e., Laguerre polynomials weighted by a simple decaying exponential are not as friendly as Hermite polynomials weighted by Gaussian. Moreover, while the finite bound states of Morse anharmonic oscillator are neither complete nor evenly spaced, those of the harmonic oscillator are complete and uniformly spaced. While the Morse oscillator, with positive anharmonicities, models diatomic molecules very accurately, the Poche-Teller potential, with negative anharmonicities, models polyatomic vibrational modes.23 Very recently, Toutounji managed to obtain the closed-form expression of the canonical vibrational partition function for each of the aforementioned potentials.23 The intimate connection between the momentum, position, and DMTCFs and optical line shapes was irrevocably demonstrated in ref 13, especially the linear line shape was explicitly expressed in terms of the position correlation function. The dipole moment changes as a function of the molecular motion, which is projected out well in the corresponding momentum and position autocorrelation functions. There has been tremendous research activity to calculate different dynamical DMTCFs of different complex systems where anharmonicity is of significance; vide supra. The challenging part in these calculations has been evaluating the time evolution operator. The most popular approach to deal with the time evolution operator is path integral techniques. However, other groups often resort to algebraic treatment whereby matrices, Campbell-Baker Hausdorff theory, in which canonical transformation is often needed, differential equations, or perturbative treatment in some cases to deal with time evolution operator.11,41,50-55 There have been many reports in the past regarding Morse potential; however, all dealt with having Morse potential in both the ground and excited states. This is comparatively a simple problem. Nevertheless, there has not been a report of a closedform expression for FCF at any level of approximation. Looking back at a sizable number of different works since 1964, one could infer (as elucidated in section V) that this is a first time attempt with a report of successful analytical results and in-depth analysis to the problem whereupon the ground and excited electronic states are harmonic and anharmonic (Morse- or Kratzer-like), respectively. Although FCF will fall out of our derivation, the motivation of this work is not calculating FCF so much as deriving electronic DMTCFs, thereby extracting dynamical and structural information from the system of interest. The theme of this article is about how one can probe anharmonic nuclear dynamics through linear and nonlinear DMTCF in condensed media using different schemes of implementing anharmonicity. Although eigenstate representation is a

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fairly old technique, it has not yet been applied, or even attempted, to anharmonic molecular systems; this might be ascribed to the heavy involvement of the respective eigenfunctions. For this reason it is analytically and numerically worth attempting herein as many important properties can be observed, inferred, and confirmed (in case worked out by other groups). Three main results will be reported and discussed for the first time in this article. First, the linear DMTCF, absorption line shape function, and the corresponding FCF are derived as a molecule makes a transition from a harmonic to an anharmonic state. Second, the linear DMTCF, absorption line shape function, and FCF are also derived as a molecule makes a transition from an anharmonic to another linearly displaced anharmonic state. FCF in this case is reported exactly in a closed-form. Third, same results as the second set are reported but with both linearly displaced and distorted shape of the upper Morse potential. This paper is organized as follows. After the background on the eigenfunctions of harmonic and Morse oscillators in the coordinate representation is provided in section II, the electronic dipole moment time correlation function is derived for a harmonic ground state and a Morse excited state. Sections III and IV treat electronic dipole moment correlation function of systems whose ground and excited states are Morse potentials under different conditions. Discussion and model calculations are provided in section V. In section VI we conclude.

II. BRIEF DIPOLE MOMENT CORRELATION FUNCTION BACKGROUND Normally one would calculate optical response functions, from which linear and nonlinear spectral line shapes or the corresponding DMTCFs may easily be obtained, starting with the Hamiltonian of the system, the bath and their coupling (in case of the MBO model, spin-boson Hamiltonian was used to describe the coupling); this will in turn lead to the spectral density from which relaxation functions can be found.20 Consider an electronic system with two states, a ground state |gæ and an excited state |eæ, interacting with nuclear degrees of freedom denoted by xj. The adiabatic electronic Hamiltonian is described by ð1Þ ^ ¼ Hg jgæÆgj þ He jeæÆej H where Hg and He are the respective nuclear Hamiltonians. (Note the quantities with “̂” are electronic operators.) The primary coordinates x are coupled to the solvent via some system-bath coupling Hamiltonian, whereby decoherence, dephasing, and other dynamical processes can be accounted for. The collective bath coordinate ð2Þ U ¼ He - Hg represents the coupling of the two-level system to its environment (bath) and is responsible for spectral shifts and broadening. The quantum mechanical linear response function S(t) defined through the first-order polarization P(r,t) expressed to first order in the radiation field E(r,t) Z ¥ dτ SðtÞ Eðr;t - τÞ ð3Þ Pðr;tÞ ¼ 0

Using cumulant expansion to second order, one obtains 2 SðtÞ ¼ - Im JðtÞ p 5122

ð4Þ

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III. BRIEF BACKGROUND ON HARMONIC AND MORSE OSCILLATOR EIGENSTATES The vibrational harmonic oscillator Hamiltonian for the ground state reads P 2 ω g 2 x2 ð5Þ þ Hg ¼ 2μr 2 of which the eigenfunctions are Ψn ðxÞ ¼ Nn Hn ðRxÞe-R x =2 2 2

ð6Þ

where Hn(x) are the Hermite polynomials, defined explicitly as Hn ðxÞ ¼

½n=2 X l¼0

ð-1Þn

n! ð2xÞn-2l l!ðn - 2lÞ!

1 R2 Nn ¼ pffiffiffiffiffiffiffiffi ν 2 n! π

ð6aÞ

!1=4 ð6bÞ

and R = (μrωg/p)1/2 with μr and ωg being the reduced mass and the ground-state angular frequency. The vibrational Morse oscillator Hamiltonian for the excited state reads P2 þ De ðe-2aðx-xo Þ - 2e-aðx-xo Þ Þ ð7Þ He ¼ 2μ defined over -¥ < x < ¥, with P and x being the momentum and position. The eigenfunctions of He are expressed in terms of the generalized Laguerre polynomials (GLP) Φm ðxÞ ¼ Am ð2νe-aðx-xo Þ Þν-ðmþ1=2Þ e-νe  ð2νe-aðx - xo Þ Þ where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aΓðm þ 1Þ2s Am ¼ Γð2ν - mÞ

-aðx-xo Þ

Lm2ν-2m-1 ð8Þ

ð9Þ

and ν  (2μrDe)1/2/pa. (It should be noted the constraint condition for this system is 2s = 2ν - 2m - 1, as dictated by Landau and Lifshitz.56) The normalization condition is Z þ¥ jΦm ðxÞj2 dx ¼ 1 ð10Þ -¥

To simplify notation and the forthcoming mathematical operations, we set y  2ν e-a(x-xo) (often called the Morse coordinate) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aΓðm þ 1Þ2s s -y=2 2s ðyÞ e Lm ðyÞ ð11Þ Φm ðyÞ ¼ Γð2ν - mÞ of which the normalization condition with respect to Morse coordinate becomes Z þ¥ jΦm ðyÞj2 dy ¼ 1 ð12Þ ay 0 Note the set {Φm}m=0 are only the bound states, as the continuum states of Morse oscillator are unnormalizable.57 While this set is incomplete, it still forms the basis in its respective discrete subspace in terms of which the vibrational wave function at hand may be expanded. We shall assume here that the molecule in question is not dissociating, as this will allow us to employ the discrete subspace spanned by the basis in evaluating anharmonic correlation functions.39,58

While the eigenvalues of the harmonic oscillator are En = pωg(n þ 1/2), with n being the number of the harmonic phonons, that of Morse potential are      1 1 1 2 pωe m þ ð13Þ Em ¼ pωe m þ 2 4De 2 where m is the number of anharmonic phonons with m = 0, 1, ..., (ν - 1); i.e., m cannot exceed ν. Additionally, a  (ke/2De)1/2 with ke being the force constant at the minimum of the Morse potential; and while De signifies the depth of Morse well, or the dissociation energy of a diatomic molecule, the parameter a (often called Morse parameter) denotes its breadth and shape. Spectroscopically, eq 12 can be recast as "   # 1 1 2 -χ mþ ð14Þ Em ¼ pωe m þ 2 2 where the anharmonicity constant χωe = pωe2/4De. The force constant ke for the Morse well is different from that for the harmonic system since ke is given by the second derivative of the potential. The number of states is then given by M = [ν þ 1/2] = [(1 - χ)/2χ], where [...] denotes maximum integer. Hence the number of bound states increases as anharmonicity decreases; when χ ∼ 0, M is infinite, the system is now harmonic, which possesses an infinite number of bound states and no continuum. Interestingly, one may observe that the anharmonicity χ is totally governed by the number of bound states: χ = 1/2ν. Pollak reported dynamical calculations with two- and five-state Morse oscillators to probe vary large anharmonic effects.58 While Laguerre polynomials are given by the Rodrigues representation   xm dm m -x x e Lm ðxÞ ¼ ð15Þ m! dxm the corresponding representation for GLP reads 0 1 μ m X μ þ λ Ax ð-1Þm @ Lλμ ðxÞ ¼ μ - m m! m¼0 ð16Þ μ X Γðμ þ λ þ 1Þ xm ð-1Þm ¼ ðμ mÞ!Γðλ þ m þ 1Þm! m¼0 where the binomial coefficient ! a a! ¼ b ða - bÞ!b!

ð16aÞ

GLP may also0be expressed in terms of confluent hypergeometric  1  a functions 1 F1 @ b xA  0  1  -μ  A Γðλ þ μ þ 1Þ λ @ x Lμ ðxÞ ¼ ð17Þ 1 F1 λþ1 μ!Γðλ þ 1Þ where Γ(.) is the Gamma function. If the electronic Hamiltonian of the system reads ^ ¼ Hg jgæÆgj þ He jeæÆej H

ð18Þ

J(t) is the linear DMTCF is given by _

JðtÞ ¼ 5123

_

^ ^ TrðeiHt=p d e-iHt=p d e-βH^ Þ _2

Trðd e-βH^ Þ

ð19Þ

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where β is the inverse temperature, Tr(...) denotes the trace over the entire system (electronic and nuclear degrees of freedom), _ and d is the electronic transition dipole moment operator given by _

d ¼ deg ðxÞjeæÆgj þ dge ðxÞjgæÆej

ð20Þ

Again, in the spirit of the Born-Oppenheimer approximation deg(x) is a matrix element in the electronic subspace and a nuclear operator in the nuclear subspace. Upon expanding eq 14 in the electronic basis set, assuming electronic adiabatic gap much greater than kT and invoking the Condon approximation, one obtains TrN ðeiHg t=p e-iHe t=p e-βHg Þ ð21Þ JðtÞ ¼ Z where the partition function is Z ¼ TrN ðe-βHg Þ

ð21aÞ

where TrN(...) denotes the trace over the nuclear degrees of freedom and is usually evaluated by employing some complete nuclear basis set. On the basis of the above, one can assume that the equilibrium density operator of the entire system is -βHg jgæÆgj F^eq ¼ e Z

step to understanding the dynamics would be calculating DMTCF. Upon expanding the nuclear trace in terms of number states, and using trace invariance, {|Ψgnæ}n=0, eq 21 reads JðtÞ ¼ Z-1

In the following sections we shall assume a single-mode excitation while the rest of the modes remaining in the ground state, and similarity of normal coordinates in the ground and excited electronic states. The disparity in these coordinates in space (symmetry change) upon excitation gives rise to Duschinsky mixing effect (vide infra), which will not be accounted for in this study.

IV. HARMONIC GROUND STATE AND ANHARMONIC (MORSE POTENTIAL) EXCITED STATE Treating the initial state harmonically and the final state anharmonically upon optical transition can in some cases be helpful and useful in extracting some dynamical information. This is one of the cases that was used by Heller’s group to generalize Tully-Preston surface hopping method to include larger quantum jumps in case the potential surfaces do not cross.59 This and other schemes of the potential surfaces were used to study such jumps in the C-H local stretching modes in benzene. Mukamel14,15 and Pollak58 have also used similar schemes to test the accuracy and applicability of this approach to semiclassical wavepackets dynamics. The interesting feature about studying dynamics while making the transition from a harmonic environment to that of anharmonic is one can see how anharmonicity affects the harmonic dynamics over time. The first

g

g

e-βEn eiEn t=p ÆΨgn je-iHe t=p jΨgn æ

ð24Þ

n¼0

To proceed, one would need to surmount the dilemma of {|Ψgnæ}n=0 not being eigenstates of e-iHet/p. The easiest way to do this would be expanding the ket |Ψgnæ in terms of the eigenbasis set of the excited-state nuclear Hamiltonian He (vide supra), which will involve some formidable integrals. ÆxjΨgn æ ¼

M X

cm ÆxjΦm æ

ð25Þ

m¼0

where the expansion coefficient cmn is given by Z ¥ cmn ¼ Φm ðxÞ Ψgn ðxÞ dx -¥

ð26Þ

(Note that while the integration along the x-axis extends from -¥ to þ¥, it only goes from 0 to þ¥ over the y-axis (cf. eq 12); vide infra.) Equation 24 thus reads ¥ M X X g g e e-βEn eiEn t=p e-iEm t=p jcmn j2 ð27Þ JðtÞ ¼ Z-1

ð22Þ

That is, the entire system is in thermal equilibrium with the electronic ground state. Once the DMTCF is determined, the linear electronic absorption line shape function may be calculated via Z 1 ¥ IðωÞ ¼ dt JðtÞeiωt ð23Þ 2π -¥

¥ X

n¼0

m¼0

M is the number of the bound anharmonic vibrational states in the Morse potential, which can be as little as one vibrational state, depending on the anhrmonic frequency and De of the molecule in question. Using the ground and excited-state wave functions in eq 26 yields Z ¥ 2 2 ðHn ðRxÞe-R x =2 ð2νe-aðx-xo Þ Þs cmn ¼ Nn Am -¥ ð28Þ -νe-aðx-xo Þ 2s -aðx-xo Þ e Lm ð2νe ÞÞ dx Using eq 16 in eq 28, completing the square, and carrying out the integral yield ! pffiffiffiffiffiffi Ξ ¥ X m m þ 2s 2π 3inπ=2 X lþλ ðνΔÞ e cmn ¼ Nn Am ð-1Þ m-λ R λ!l! l¼0 λ¼0  expððaΞ=RÞ2 =2ÞHn ½iaΞ=R

ð29Þ

where Δ  e , Nn, and Am are the normalization constants of the respective eigenfunctions of both the ground and excitedstate nuclear Hamiltonians, and Ξ  λ þ l þ s. | cmn|2 yield the Franck-Condon factors (FCF) of the system of interest, cmn = Æm|næ. An interesting case would be obtaining the FCF for the 0-0 transition, of which the jargon in molecular crystals is zerophonon line (ZPL), axo

jÆ0j0æj2 ¼  2 pffiffiffiffiffiffi ¥ lþs   2π X   l ðΔνÞ 2 expððaðl þ sÞ=RÞ =2Þ ð-1Þ N0 A 0   R l¼0 l!

ð30Þ

The infinite series has a convergence problem caused by completing the square while evaluating the integrand. This divergence shows up more clearly in eq 30. The root cause of this convergence problem is interchanging the power series of -a(x-x ) in eq 28 and the integral; i.e., uniform expanding e-νe convergence is not satisfied. This is an interesting observation o

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-a(x-xo)

as e-νe is an analytic function and yet only converges absolutely but not uniformly. This nonuniform convergence bars the sum-integral-interchange property, and hence eqs 29 and 30 are very ill-behaved or even wrong. Therefore, a strategy would be needed so as to warrant uniform convergence, thereby mathematically legitimatizing the sum-integral-interchange property. Before embarking on this strategy, it is now the proper juncture to point out well-documented convergence/divergence problems that arise upon modeling vibrations with Morse oscillator. Appendix B, for further ratification of the above solution, shows that one can equally obtain equivalent result of the above overlap integral with respect to y, by employing the commonly used transformation in solving the Morse potential problem, namely y = 2Δνe-ax. The Morse oscillator, without loss of generality, canonical partition function has the potential to diverge if caution is not exercised.23,42,60-62 (For a lucid rigorous discussion of the possible divergence associated with the Morse oscillator, the reader is referred to ref 23.) While this divergence problem crops up upon going from harmonic motion to that of anharmonic, it does not while making the transition from harmonic to harmonic or anharmonic to anharmonic geometry—configuration is preserved. The reason behind this is that the eigenfunctions of each oscillator have a different spatial argument in the decaying exponential (e.g., harmonic eigenfunctions have a spatial Gaussian decay whereas the Morse oscillator has a simple exponential decay, which will mandate completing the square, giving rise to a positive quadratic exponential) or that a more profound justification is yet to be found. As an aside, the same evaluation (not shown) was carried out using harmonic potential for the ground state and using Kratzer potential (Morse-like) for the excited state, a similar divergence problem arose. The above said convergence problem does not arise during a transition in a full harmonic or anharmonic environment, namely a transition from a Morse to another Morse oscillator or from a harmonic to another harmonic oscillator; vide infra. The challenge now is to surmount the above divergence. This can be done by considering the classical turning points (ctps) of either state; however, in this case the harmonic oscillator ctps should be accurate to take since the overlap integral contains the product of the two states eigenfunctions whose end points on the nuclear axes are governed by the harmonic oscillator ctps (since the Morse eigenfunctions tend to die out faster at each end). In light of the above, the integration limits in eq 28 will rather change from -¥ to þ¥ to a finite interval where the integrand is finite and zero outside this interval and uniform convergence is certain; in this case this interval would be the ctps. As such, eq 28 may equivalently be written as ¥ m X ðΔνÞsþl X ð-1Þl Bλ cmn ¼ Nn Am l! l¼0 λ¼0 Z xR 2 2  ðHn ðRxÞe-R x =2 e-axΞ Þ dx ð31Þ xL

where Bλ ¼ ð-1Þλ

Γðm þ 2s þ 1Þð2ΔνÞλ ðm - λÞ!Γð2s þ λ þ 1Þλ!

ð31aÞ

The finite interval xL e x e xR is typically defined as xL , -(2En/μω2)1/2 and xR . (2En/μω2)1/2, i.e., xL,R = -[(p/ μω)(2n þ 1)]1/2 - x0, where x0 is the upper potential linear

displacement (i.e., the bond length in the final state). Replacing the infinite limits of the integral gives the integrand as an analytic function over the interval defined by these limits and therefore uniform convergence is warranted, allowing us to take the sum outside the integral. The above integral is formidable due to the spatial dependence disparity of the initial and final wave functions. The evaluation mechanics of the integral is sketched in Appendix A. Upon evaluating the integral in eq 31, the overlap integral cmn reads cmn

¼ Nn Am

½n=2 ¥ m X X ðΔνÞsþl X ð-1Þl Bλ ð-1Þn l! λ¼0 p¼0 l¼0

n-2p n - 2p n!ð2RÞn-2p X  ð-ε=R2 Þn-2p v p!ðn - 2pÞ! v¼0

!

 ! 2 2 eε =2R k þ 1 R2 ε 2 ;  kþ1 pffiffiffi Γ xL þ 2 2 2 R R 8   !! k þ 1 R2 ε 2 ; -Γ xR þ 2 2 2 R

ð32Þ

where Γ(.,.) is the upper incomplete Gamma function. Although eq 32 might appear formidable with quadruple sum, practically it only has one infinite sum, of which terms terminate rapidly depending on the temperature, and the other inner sums have very few terms, especially for the cold transitions (on the high energy side) related terms. For example, the overlap integral for 0-0 transition (ZPL), reads  ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  R2 1=4 ðν - 1Þ X ¥ ðΔνÞsþl eðaΞÞ =2R  2 pffiffiffi ð-1Þl jÆ0j0æj ¼   π ΓðνÞ l¼0 l! R 8 !     !!2 1 R2 aΞ 2 1 R2 aΞ 2   Γ ; xL þ 2 xR þ 2 -Γ ;   2 2 R 2 2 R ð33Þ Equation 33 can be used to treat electronic dephasing as laid out in ref 5. Alternatively, eq 33 may be written in terms of error function.64 Despite the infinite sum in the ZPL FCF in eq 33, the series converges rapidly. (The profile of the ZPL obtained using the MBO model of Mukamel1 has a similar mathematical structure, infinite sum.)13,65,66 The FCF for anharmonic progression members, which are responsible for vibrational relaxation and eventually form the phonon side band of the absorption profile, are 2 2 ¥ m X ðΔνÞsþl X eε =2R ð-1Þl Aλ pffiffiffi cm0 ¼ N0 Am l! λ¼0 R 8 l¼0 !     !! ð34Þ 1 R2 ε 2 1 R2 ε 2  Γ ; xL þ 2 xR þ 2 -Γ ; 2 2 R 2 2 R The cold transitions in eq 34 may be used as the only contributing absorption/emission profiles (multiphononic profiles)5,6 in a line shape at low temperatures as hot bands would not arise, thereby tremendously simplifying the expression for linear line shape and the calculations of the corresponding nonlinear 5125

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signals. (“m” in eq 34 signifies the number of anharmonic phonons created or annihilated during a transition.) For example, the oneand two-phonon profiles inclusion, in addition to the ZPL, should give a very accurate description of the linear/nonlinear absorption at low temperatures in molecular crystals and glasses.2,3 Furthermore, eq 33 will prove very useful should the need arise for adopting a particular model to account for electronic dephasing, as was rigorously done in refs 5, 6, 13, 65, and 66. As such, the ZPL FCF in eq 33 may be eliminated to be substituted by that of the MBO model13,65-67 or the models that appeared in refs 5 and 6. Despite the apparent complexity of eq 34 it is feasibly calculable and can be used in further computing linear and nonlinear spectral signals. Moreover, one can readily do spectroscopic and dynamical calculations using one- and two-phonon profiles at low temperatures, as most probably only these profiles will be appreciable in condensed systems; vide supra. This should tremendously simplify calculating nonlinear spectral signals. Equations 28-34 are herein first time reported. Substituting eq 31 in eq 27 and using eq 21 yields DMTCF for the system of interest with a harmonic ground state JðtÞ ¼ 2 sinh ðβpωg =2Þ

¥ X M X

g

g

jcmn j2 e-βEnþiEn t=p-iEm t=p e

n¼0 m¼0

Expanding eq 37 in Morse oscillator eigenstates (cf. eqs 8 and 11) yields ¼

m¼0

¼

IðωÞ

¼ ¼

1 2π

Z

¥

-¥

JðtÞeiωt-γjtj=2 dt

γ sinhðβpωg =2Þ

¥ X M X n¼0

g

jcmn j2 e-βEn =π e 2 g 2 m¼0 ðω þ En - Em Þ þ ðγ=2Þ

V. ANHARMONIC GROUND STATE AND DISPLACED MORSE EXCITED STATE (NO SHAPE DISTORTION) In this section we shall assume that both the ground and excited states are Morse potentials in which the latter is linearly displaced with respect to the former. This is a common scheme for studying anharmonic dynamics. This case is relatively easy, vide infra, to deal with mathematically, and while an analytical expression will be obtained for the DMTCF, a closed-form one will result for the corresponding FCF, which is a first ever reported. The ground and excited nuclear Hamiltonians are, respectively, taken to be p2 þ De ðe-2ax - 2e-ax Þ 2μr p2 0 0 þ D0 e ðe-2a ðx-xo Þ - 2e-a ðx-xo Þ Þ He ¼ 2μr Hg ¼

ð37Þ

TrN ðeiHg t=p e-iHe t=p e-βHg Þ Z

g

Mg X Me X

g

g

e

-βEm þiEm t=p-iEme 0 t=p

Z

¥ -¥

ð39Þ Φem0 ðxÞ

Φgm ðxÞ

dx

yg ¼ 2νg e-ax 0 ye ¼ 2Δ0 νe e-a x 0 Δ0  ea xo

ð40Þ

Following the above procedure in section IV, one will need to evaluate this integral, which serves as the expansion coefficient, Z ¥ e Ψg c m0 m ¼ m ðxÞ Ψm0 ðxÞ dx -¥ Z ¥ 0 -ax 0 g e ¼ Nm Nm0 ð2νg e-ax Þs e-νg e ð2Δ0 νe e-a x Þs -¥ 0

0 -a x -ax  e-Δ νe e L2s Þ m ð2νg e

0

0

0 -a x L2s Þ dx m0 ð2Δ νe e

ð41Þ

where 2s = 2νg - 2m - 1 and 2s0 = 2νe - 2m0 - 1, and m and m0 signify the number of anharmonic phonons of the ground and excited states, respectively. Also, pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2μDe 2μD0 e and νe  νg  ð41aÞ pa0 pa When the variable change, u = yg, is introduced, the integral reads Z

c

m0 m

¼

Nmg Nme 0

¥

0

0

0

0

0 2s us-1 e-u=2 ðΔ0 ruÞs e-Δ ru=2 L2s m ðuÞ Lm0 ðΔ ruÞ du

ð42Þ with r  νe/νg. It has been assumed while the each potential well would have the same width, ag = ae = a, the anharmonic phonons are different (r 6¼ 1). Evaluating the above integral yields a closedform expression in terms of Appell’s function,68,69 ! ! m0 þ 2s0 Γðs þ s0 Þ Nmg Nme 0 0 s0 m þ 2s ðΔ rÞ c m0 m ¼ 0 m0 m a δsþs ! 0 1 Δ r  F2 s þ s0 ;-m;-m0 ;2s þ 1;2s0 þ 1; ; ð43Þ δ δ where δ = -Δ0 (1 þ r). F2 is called Appell’s hypergeometric function and is given by70 F2 ðσ;R1 ;R2 ;β1 ;β2 ;x;yÞ ¼

We need to evaluate JðtÞ ¼

g

e-βEm eiEm t=p ÆΦgm je-iHe t=p jΦem0 æ

While Me is the number of bound states of the upper Morse potential given by [1 - χe/2χe] with χe being the anharmonicity constant of the upper state potential, Mg is the same for the ground state. Introducing the following variables is helpful,

ð36Þ where γ signifies the homogeneous line width.

m0¼0

m¼0 m0¼0

ð35Þ A single-site (homogeneous) absorption line shape can impart useful dynamical and structural information. This line shape, precluding the zero-point energies in both energy surfaces, may be obtained as

Mg X Me X

ð38Þ 5126

Z 1 R1 -1 Γðβ1 Þ u ð1 - uÞβ1-R1-1 ΓðR1 Þ Γðβ1 - R1 Þ 0 ð1 - xuÞσ 0 1  σ;R2  y A  2 F1 @  β2 1 - xu

ð44Þ

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with 2F1(...) being the Gauss hypergeometric function.64 An extensive, lucid account of Appell’s function may be found in ref 71. DMTCF in this case reads Mg Me g g 1 XX e jcm0 m j2 e-βEm þiEm t=p-iEm0 t=p ð45Þ JðtÞ ¼ Q m¼0 m0 ¼0 where the canonical partition function Q = Tr[e-βHg]. Closedform expressions for the canonical partition function of Morse oscillator have been derived and reported in closed forms.23,60-63 FCFs for the system in eq 42 are given by the closed-form formula jcm0 m j2 ¼ jÆm0 jmæj2  N g N e 0 0  ¼  m m ðΔ0 rÞs  a  F2

and z2 ¼ 1 þ 2s - F - ~rl

0

! m þ 2s m0 þ 2s0 Γðs þ s0 Þ 0 m0 m δsþs !2  0 1 Δ r  s þ s0 ;-m;-m0 ;2s þ 1;2s0 þ 1; ;  ð46Þ δ δ 

¼ λ1

¼-

Nmg Nme 0 ag

Z

¥ 0

ð2νg uÞs -νg u 2s e Lm ð2νg uÞ u ~

0 r ~ 0 0 0 ~r  ð2Δ0 rur Þs e-Δ ru L2s m0 ð2Δ ru Þ

du

ð47Þ

where ~r = ae/ag and ag 6¼ ae. The above integral is formidably difficult to evaluate; however, after some algebraic manipulation and extensive utilization of orthogonal polynomials properties,64,72 the above integral evaluates to pffiffiffiffiffiffiffi ¥ m X ð-1Þk ð Δ0 r Þ2k 2k ð2n - 1Þ!! X ð-2Δ0 rÞl Γðz1 ÞΓðF þ ~rlÞ cm0 m ¼ ~k Fþ~rl ð2kÞ! l! m!Γðz2 Þνg k¼0 l¼0

ð48Þ

¥ νk=2 X g

0

Hk ð0Þ

Γðλ2 þ k=2~rÞΓðm0 þ 2s0 þ 1Þ ~

m0 !Γð2s0 þ 1ÞðΔ0 νe Þðλ1þk=2rÞ 0  1 -m0 ; λ2 þ k=2~r  ð49Þ  2F 1@ 2 A β2 

k¼0

Equation 46 is a first time report of a closed-form formula of FCF that is derived for ground and excited states modeled by Morse potentials. Equation 46 is much simpler than it looks as it encompasses simple finite values and an extensively tabulated function, Appell’s hypergeometric function.71

-¥

0

0  ð2Δ0 νe uÞs e-Δ νe u L2s m0 ð2Δ νe uÞ du

!

VI. ANHARMONIC GROUND STATE AND DISPLACED MORSE EXCITED STATE WITH SHAPE DISTORTION Now we move to evaluating a harder case in which the upper Morse potential is both displaced and distorted with respect to the Morse ground state, leading to shape and curvature change. Proceeding along the same lines of evaluating J(t) in the preceding two subsections, one will first need to evaluate the expansion coefficient cm0 m (inner product of the upper and lower Morse wave functions, each of which will have different displacements and frequencies). As such, cm0 m in this case reads Z ¥ Ψgm ðxÞ Ψem0 ðxÞ dx c m0 m ¼

k!

0 ð2νg Þs ð2Δ0 νe Þs -ae s λ2 ¼ ~ þ s0 r

λ1 ¼

In the case of the 0-0 transition, FCF looks like 2  N g N e X ¥ νk=2 Γðλ2 þ k=2~rÞ  g  0 0 H ð0Þ λ    ae 1 k¼0 k! k ðΔ0 νe Þðλ1þk=2~rÞ 

F ¼ s þ ~rs0 þ ~rk=2

ð48bÞ

z1 ¼ 1 þ m - 2s - F - ~rl

ð48cÞ

ð50Þ

i¼1

where Ji(t) is the individual DMTCF for the ith mode provided in the above sections. In this case the absorption spectrum for a V-mode system is given by convolving all the individual mode spectra Z Z Z IðωÞ ¼ dω1 dω2 ::: dωV -1 I1 ðω - ω1 Þ  I2 ðω1 - ω2 Þ ::: IV ðωV -1 Þ

ð48aÞ

ð49aÞ

Duschinsky mixing of modes bearing the same symmetry is essential should it prove needed, depending on the experimental conditions, and the treatment is extremely complicated. The Duschinsky effect is an orthogonal transformation that involves expressing the excited-state coordinates xei in a space in terms rotated ground-state coordinates xgi through some rotation matrix B as such xei = Bjixgi ; simply put, the normal coordinates of the ground and excited states are quite different. We shall not pursue it here as all the forthcoming calculations will assume negligible contribution of Duschinsky effect. For a V-mode system the total DMTCF reads, assuming no Duschinsky mixing, V Y Ji ðtÞ ð51Þ JðtÞ ¼

where ~k ¼ -N g N e 0 ð2νg Þs ð2Δ0 rÞs0 =ag m m

ð48dÞ

The DMTCF J(t) will result upon inserting eq 48 in eq 45. | cm0 m|2 in eq 45 are FCFs of the above system. Hole-burning, or line-narrowing fluorescence for that matter, spectroscopy of condensed-phase samples are commonly probed at low temperatures at which only cold transitions are significant and hot transitions/bands are negligible or simply do not arise. ~ s Z g N0 Nme 0 ¥ ð2νg u1=r Þ -νg u1=r~ 0 e cm 0 ¼ ae 0 u

ð52Þ

where Ii(ω) is the homogeneous absorption spectrum for mode i given by eq 36.

VII. DISCUSSION AND CALCULATIONS Three forms of anharmonic modeling of the ground and excited electronic states have been discussed in the above 5127

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Figure 1. Single-site linear absorption spectrum of a system whose ground and excited states are harmonic and Morse potentials, respectively. The spectrum was calculated at different temperatures with the following parameters: ωg = 40 cm-1, ωe = 30 cm-1, Δ = 1.4, γ = 5 cm-1, and χ = 0.0125.

sections. Studying spectral line shapes in which the initial and final states are harmonic and anharmonic, respectively, provides an idea as to what type of changes emerge as one goes from the former state to the latter. For this reason, monitoring what happens when a harmonic ground state becomes anharmonic upon optical excitation is essential, although the literature typically handles the case where both the ground and excited electronic (linearly displaced with or without curvature change) states are represented by Morse potential. While only groups of Mukamel reported numerical calculation of line shape functions reflecting vibronic transitions from a harmonic initial state to an anharmonic final state using LGF approach14,15 and

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Heller used the same model to study the Tully-Preston surface hopping dynamics,59 all other groups calculated FCF expressions for Morse-to-Morse transitions analytically and numerically using different levels of theory, rigor, and approximation.73-83 Despite the variation of the level of treatment of the resultant FCF expressions, they are approximate and reported in an open form. At this point running some model calculations should illustratively be helpful. A homogeneous (single-site) linear absorption line shape in Figure 1 comprising transitions from a harmonic ground state to an anharmonic final state was calculated at different temperatures using eqs 32, 35, and 36. The mass and p have been set equal to unity in all calculations. The parameters used in calculating Figure 1 were ωg = 40 cm-1, ωe = 30 cm-1, Δ = 1.4, and χ = 0.0125, which is a large anharmonicity. The same parameters were used to calculate Figure 2, which illustrates transitions from a Morse ground state to another Morse excited state. Figure 2 was calculated by Fourier transforming eq 45. The vibrational structure (progression members) that comprises the phonon-side band (PSB) in Figure 1 spectra are more pronounced than those in Figure 2. Figure 2 has more intense ZPL than Figure 1, which means weaker PSB, and thereby different pure dephasing (ZPL profile). FCF is about the extent of vibrational function overlap of which the maxima coincides at the ctps. The PSB and ZPL difference in the two sets of spectra is in part related to the ctps in both states, each of which is represented by a different nuclear Hamiltonian, whereas both states in Figure 2 have Morse nuclear Hamiltonians, leading to different overlap of vibrational wave functions, thereby rendering the alluded to difference. This disparity in vibrational wave functions overlap gives rise to different vibrational structure (PSB) stimulation in both figures upon excitation as the vertical transition cuts through the vibarational levels in the final states. Although the overall spectra in Figures 1 and 2 may appear similar, they exhibit a couple differences with respect to dynamical information that can be obtained, especially on the nature and strength of electron-phonon coupling. Electronphonon coupling plays an important role in determining FCF, and hence, electronic dephasing, and vibrational relaxation; therefore, each figure spectra exhibit different electron-phonon coupling. Interestingly, the same calculations were carried out using weak electron-phonon coupling, of which spectra (not shown) are not as significantly affected by anharmonicity in the low temperature limit. As such, one will have to be cautious as to what scheme should be employed for probing electronic dephasing. Surely, the above analysis is more readily observed in the 1 K spectra than those at higher temperatures. Note that there are three different types of electron-phonon coupling that contribute to the spectra in Figure 1, namely linear (bond length change), quadratic (frequency change), and anharmonic (a parabolic potential turning into Morse). Looking at Figure 1 one can infer that the spectrum at 1 K is very close to that of only a linearly coupled system, because at this low temperature the vibrations affected by both quadratic and anharmonic coupling are not stimulated and therefore do not show up in the spectrum. These coupling effects start to come in as the temperature rises. For this reason, the linear displacement Δ was set equal to 1.4, so that the Huang-Rhys factor S = d2/2 ≈ 1, which within the FCF of a linearly coupled system1,5 results in 1 r 0 transition and the ZPL having, approximately, an equal integrated absorption intensity. This is exactly what is 5128

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observed in Figure 1, top panel. Another drawback that may be inferred by inspecting both Figures 1 and 2 is the ZPL and PSB bands have the same width (γ = 5 cm-1), which is unphysical.5,7,13,65,66 This is tantamount to saying that vibrational and electronic relaxations occur on the same time scales. Part 2 of this work looks at this problem and suggests a solution using eqs 33 and 50.

VIII. CONCLUDING REMARKS Molecular vibrations are more realistically modeled by anharmonic oscillators such as Morse, Kratzer,84,85 or Rosen-Morse oscillators,23 all of which render sound treatment of anharmonic molecules. (The Kratzer oscillator possesses an incomplete set of unevenly spaced bound states that Morse oscillator fails to have. Besides, in some cases the Kratzer oscillator outperforms the Morse oscillator in modeling anaharmonicity.)85,86 We were able to derive and explore three different schemes of homogeneous line shape functions, and their corresponding FCF, using eigenstate representation to account for all possible molecular geometry changes upon optical excitation. The first case addressed a transition of a harmonic molecule becoming anharmonic upon electronic excitation. An interesting complication arose during the course of evaluation, namely FCF exhibited divergence due to nonuniform convergence, which was circumvented by resorting to utilizing the ctps of the harmonic state. This type of scheme offers the advantage of observing ensuing geometrical changes as a harmonic molecule makes its way to become anharmonic. (This scheme has only been handled by the Mukamel group numerically using the LGF approach; vide supra.)14,15 The second scheme treats an anharmonic molecule which undergoes a bond length and curvature change upon excitation. Although the FCF of this scheme has been extensively treated by different schools using different techniques, as elucidated above, this is the first time a closed-form expression is reported, of which the tremendous advantage is avoiding dealing with GLP.87 Finally, the third scheme is most general whereby bond length and molecular geometry change are captured. The ZPL profile was procured in all three cases. Although FCF will fall out of our derivation, the motivation of this work is not calculating FCF so much as deriving electronic DMTCFs, thereby extracting dynamical and structural information from the system of interest. Mixed quantum classical dynamics (MQCD) formalism of Kaparal and Cicicotti88 may well be utilized for evaluating DMTCF in a mixed quantum classical environment,89-92 whereby the effects of the environment (dissipation, relaxation, and depahsing) can reasonably be accounted for, whereupon the Wigner transform of the Morse potential is required for accurate evaluation. The Wigner distribution function of the Morse oscillator has been evaluated in the semiclassical limit to calculate autocorrelation functions starting with a frozen Gaussian wavepacket in phase space in a Morse potential from which propagation ensues,58,79,80 yielding accurate representation of the involved dynamics. Tannor93 and Pollak58,94 independently utilized the framework of initial value representation to calculate the DMTCF phase using classical trajectory information, rendering very sound results. The simplest way to formally incorporate friction is by employing MQCD, as was lucidly done in refs.75-78,90-92 In light of the similarities between the initial value representation and MQCD formalisms, combining them may impart useful dynamical information in dissipative environments. Future work

Figure 2. Single-site linear absorption spectrum of a system whose ground and excited states are Morse potentials. Same parameters as in Figure 1 were used. The difference in ZPL and PSB (vibrational structure) in Figures 1 and 2 spectra may be attributed to the respective ctps. While the ground-state nuclear Hamiltonian in Figure 1 is parabolic, the ground-state nuclear Hamiltonian in Figure 2 is Morse potential; different ground-state nuclear Hamiltonians result in different vibrational wave function overlap due to the variation in the corresponding ctps.

will involve developing mixed quantum-classical linear and nonlinear DMTCF expression where the quantum subsystem will be a two-level system coupled to classical anharmonic vibrations via mixed quantum classical formalism. In part 2 of this work, we extend our approach to encompass nonlinear spectroscopic techniques such as photon echo, pump-probe, and hole-burning, all of which will build on the herein derived DMTCF expressions. Moreover, we explore how pure dephasing may be incorporated in this work. Additionally, we project out our future work directions and the respective motivations, some of which are already underway. 5129

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’ APPENDIX A: EVALUATING THE INTEGRATION IN EQ 31 In this appendix we evaluate the integral in eq 35, which will yield the result in eq 36. The integral in its general from may be recast as Z 0 2 ðA1Þ I ¼ Hn ðRxÞe-R x -εx dx using Hn ðRxÞ ¼

½n=2 X

ð-1Þn

p¼0

n! ð2RxÞn-2p p!ðn - 2pÞ!

where R is just a constant. Using eq A2 in eq A1 yields ½n=2 n-2p Z X 0 2 n n!ð2RÞ ðxÞn-2p e-R x -εx dx ð-1Þ I ¼ p!ðn 2pÞ! p¼0

ðA2Þ

ðA3Þ

with R0  R2/2 and ε  aΞ. Upon completing the square and changing the variable from x to u = x þ ε/2R0 followed by utilizing the binomial theorem, the integral in (A3) reads Z n-2p X 0 2 2 0 xn-2p e-R x -εx dx ¼ eε =4R ð-ε=2R0 Þn-2p v¼0



n - 2p v

!Z

k -R0 u2

ue

du

ðA4Þ

where k = n - 2p - v. The integral Z

0 2

uk e-R u

  kþ1 0 2 ;R u Γ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi du ¼ 2 R0 ðkþ1Þ

ðA5Þ

xR

0 2

xn-2p e-R x -εx dx

xL

¼

n-2p X

ð-ε=2R0 Þn-2p

v¼0

n - 2p vj

’ APPENDIX B: EVALUATING EQ 28 WITH RESPECT TO MORSE COORDINATE Alternatively, one can proceed on to evaluate the expansion coefficient in eq 29 by carrying out the overlap integral with respect to Morse coordinate as follows. The expansion coefficient reads Z ¥ 2 2 -aðx-xo Þ ðHn ðRxÞe-R x =2 ð2νe-aðx-xo Þ Þs e-νe cmn ¼ Nn Am -aðx-xo Þ L2s ÞÞ dx m ð2νe

ðB1Þ Letting y = 2Δνe-ax yields Z ¥ 1 cmn ¼ Nn Am ay 0     R -ð1=2Þ½ðR=aÞlnð2ΔνyÞ2 -y=2 s 2s lnð2ΔνyÞ e  Hn e y Lm ðyÞ dy a ðB2Þ where s = ν - (m þ 1/2). One can learn more about implicit summation for Hermite functions and related polynomials from refs 95 and 96. The above is quite challenging because the use of different Hamiltonians for the ground and excited states invokes two quite different arguments of the corresponding wave functions. To evaluate this formidable integral, one may write

!

 ! eε =4R kþ1 0 ε 2 ;R xL þ 0  pffiffiffiffiffiffiffiffiffiffiffiffiffi Γ 2 2R 2 R0 ðkþ1Þ !!   kþ1 0 ε 2 ;R xR þ 0 -Γ 2 2R 2

where R0  R2/2 ! ½n=2 n-2p n-2p X X n n!ð2RÞ 2 n-2p n - 2p I ¼ ð-1Þ ð-ε=R Þ v p!ðn-2pÞ! v¼0 p¼0 !   2 2 eε =2R k þ 1 R2 ε 2  ðkþ1Þ pffiffiffi Γ ; xL þ 2 2 2 R R 2 !!   k þ 1 R2 ε 2 ; xR þ 2 -Γ ðA9Þ 2 2 R

-¥

where Γ(...) is the upper incomplete Gamma function. Thus,  ! kþ1 0 ε 2 ;R x þ 0 Γ Z 2 2R k -R0 u2 pffiffiffiffiffiffiffiffiffiffiffiffiffi du ¼ ue 2 R0 ðkþ1Þ   !x 1 kþ1 0 ε 2  R ;R x þ 0 ¼ - pffiffiffiffiffiffiffiffiffiffiffiffiffi Γ  x 2 2R 2 R0 ðkþ1Þ L !  2 1 kþ1 0 ε ;R xL þ 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi Γ ðkþ1Þ 2 2R 0 2 R   !! kþ1 0 ε 2 ;R xR þ 0 -Γ ðA6Þ 2 2R Z

Using eq A7 in eq A3 yields ! ½n=2 n-2p n-2p X X n n!ð2RÞ 2 n-2p n - 2p I ¼ ð-1Þ ð-ε=R Þ v p!ðn-2pÞ! v¼0 p¼0 !   2 0 eε =4R kþ1 0 ε 2 ;R xL þ 0  pffiffiffiffiffiffiffiffiffiffiffiffiffi Γ 2 2R 2 R0 ðkþ1Þ  2 !! kþ1 0 ε ;R xR þ 0 -Γ ðA8Þ 2 2R

2

e-ð1=2Þ½ðR=aÞlnð2ΔνyÞ ¼

0

ðA7Þ

¥ X ðg lnð2ΔνyÞÞσ Hσ ð0Þ σ! σ¼0

ðB3Þ

with g = (R/a)2/2. However, properties of Hermite polynomials dictate that only even terms will contribute, and as a result eq B3, with some algebra, reads pffiffiffi 2σ ¥ X σ ð 2g lnð2ΔνyÞÞ -ð1=2Þ½ðR=aÞlnð2ΔνyÞ2 ð2σ - 1Þ!! ¼ ð-1Þ e ð2σÞ! σ¼0 ðB4Þ 5130

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Using eq B4 in eq B2 yields Z ¥ X ð2σ - 1Þ!! ¥ 1 cmn ¼ Nn Am ð-1Þσ ð2σÞ! 0 ay σ¼0     pffiffiffi R 2σ -y=2 s 2s lnð2ΔνyÞ ð 2g lnð2ΔνyÞÞ e  Hn y Lm ðyÞ dy a ðB5Þ

the Research Affairs Sector at United Arab Emirates University for partial financial support of this project under grant 01-03-211/09.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

with κ  2s. Extensive use of Hermite functions and their recursion relationships renders 

Hn

   R R lnð2ΔνyÞ ¼ Hn ðln 2Δν þ ln yÞ a a 0 1 n-j   n X n  R R @ A 2 ln y ¼ ln 2Δν Hj a a j j¼1

’ REFERENCES ðB6Þ

Hence the expansion integral becomes cmn

0 1 μ ¥ n X X X n σ ð2σ - 1Þ!! ¼ Nn Am ð-1Þ Τm @ Að2R=aÞn-j að2σÞ! m¼0 j¼1 j σ¼0

Z

pffiffiffi

¥

 Hj ðln 2Δν=aÞð 2gÞ2σ

ððln yÞn-j ðlnð2ΔνyÞÞ2σ

0

 e-y=2 ysþm-1 Þ dy

ðB6aÞ

where Τm ¼ ð-1Þm

Γðμ þ k þ 1Þ ðμ - mÞ!Γðk þ m þ 1Þm!

ðB7Þ

The above integral may be recast as

0 1 μ ¥ n X X X n ð2σ 1Þ!! cmn ¼ Nn Am ð-1Þσ Τm @ Að2R=aÞn-j að2σÞ! m¼0 j¼1 j σ¼0 pffiffiffi  HM ðln 2Δν=aÞð 2gÞ2σ ! Z



2σ X 2σ ðlnð2ΔνÞÞ2σ-k k k¼0

¥

ððlnðyÞÞη e-y=2 yθ-1 Þ dy

0

ðB8Þ where η  2σk þ n - M and θ  s þ m. Evaluating the integral gives cmn ¼ Nn Am

¥ X

ð-1Þ

0 1 μ n n - 1Þ!!X X Τm @ Að2R=aÞn-j að2σÞ! m¼0 j¼1 j

σ ð2σ

σ¼0

pffiffiffi

 Hj ðln 2Δν=aÞð 2gÞ2σ

! 2σ X 2σ Dη ðlnð2ΔνÞÞ2σ-k η ð2θ ΓðθÞÞ k Dθ k¼0

ðB9Þ where Γ(θ) is the Gamma function. Equation B9 has been numerically tested and agrees well with eq 29, especially on the divergence. The rationale for this divergence should physically be more profound than just a mathematical result.

’ ACKNOWLEDGMENT I sincerely thank Larry Schulman at Clarkson University, Alan Jeffrey (editor of ref 64), and Francisco Fernandez at La Plata, Argentina, for educational and helpful discussions. I also thank

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’ NOTE ADDED AFTER ASAP PUBLICATION This paper was published ASAP on August 19, 2010. There was a minor revision to the text in the Concluding Remarks. The updated paper was reposted on October 4, 2010.

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