Anharmonic ElectronâPhonon Coupling in Condensed Media: 2

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J. Phys. Chem. C 2010, 114, 20764–20774

Anharmonic Electron-Phonon Coupling in Condensed Media: 2. Application to Electronic Dephasing, Hole-Burning, and Photon Echo† Mohamad Toutounji* College of Science, Department of Chemistry, United Arab Emirates UniVersity, Al-Ain, United Arab Emirates ReceiVed: July 26, 2010; ReVised Manuscript ReceiVed: September 18, 2010

While different expressions of homogeneous linear electronic dipole moment time correlation functions and their respective absorption lineshapes of anharmonic systems were derived and discussed at length in part I [Toutounji, M. J. Phys. Chem. B 2010, in press] of this study, the electronic dephasing (zero-phonon line width), caused by pseudolocal phonons, and vibrational relaxation time scales were treated equally, hence, unphysical. In this study electronic dephasing and vibrational structure are correctly accounted for in harmonic and anharmonic multimode systems. Model calculations are presented. Anharmonic hole-burned spectra are calculated using linear anharmonic absorption lineshapes. An analytical expression of hole-burned absorption line shape in terms of the linear dipole moment time correlation function, leading to a hole-burned absorption line shape expressed in time-domain, is reported. An elementary method is utilized to treat nonlinear spectra in anharmonic molecules. The link between a hole-burned spectrum and a 2-pulse photon echo profile is established by combining the theoretical framework of Mukamel and that of Hays-Small, thereby calculating photon echo signals of anharmonic systems. Model photon echo calculations of a solvable harmonic model and an anharmonic system are reported. Applications to Al-phthaolcyanine tetrasulphonate (APT) in hyperquenched glassy ethanol and special pair bacterial reaction center are presented and discussed in detail. I. Introduction Optical electronic coherence loss (dephasing) caused by coupling to optical, acoustic, or pseudolocalized phonons in condensed systems is of central importance to extracting structural and dynamical information in molecular spectroscopy. While vibrationally resolved spectra may impart a wealth of information about the geometry in the ground and excited electronic states, vibronic intensities can reveal the molecular geometry and symmetry upon optical excitation. These vibronic intensities are governed by Franck-Condon (FC) factors (FCF) which are determined by the nature of electron-phonon coupling; this includes linear,1 diagonal quadratic (exchange coupling),2-6 anharmonic coupling,7-14 Duschinsky rotation (offdiagonal quadratic coupling in the context of Sturge-McCumber mechanism),15-19 Fermi resonance, and Hertzberg-Teller coupling.19,20 A comprehensive theory was developed by Skinner and Hsu21 showing how quadratic coupling of phonons to the electronic degrees of freedom is responsible for dephasing. Linearly coupled modes reflect only bond length change; quadratic coupling causes frequency shift of active modes; Duschinsky mixing effect and Hertzberg-Teller coupling reveal modes mixing of nontotally symmetric vibrations; and anharmonic coupling shows possible bond angles and geometrical distortion of molecules upon electronic excitation. Linear and nonlinear optical electronic dipole moment time correlation functions (DMTCF) are key quantities for linking theory to experiment. Toutounji et al.4 derived a DMTCF with linear and quadratic coupling included which accounted correctly for electronic dephasing. In this study the DMTCF applicability will be extended to anharmonic electron-phonon coupling whereby accounting physically and distinctively for electronic †

Part of the “Mark A. Ratner Festschrift”. * E-mail: [email protected].

dephasing and vibrational relaxation. Section III presents the DMTCF of anharmonic polyatomic molecules in which the absorption line shape exhibits a zero-phonon line profile with distinctive dephasing time and a phonon-sideband with vibrational relaxation. Additionally, an example of harmonically and anharmonically coupled modes is provided with correct inclusion of electronic dephasing. Mukamel successfully laid out a solid theoretical framework utilizing optical response theory to calculate nonlinear DMTCF in the context of his multimode Brownian oscillator (MBO) model, which is only operative in the harmonic linear coupling limit.1 Evaluating quantum mechanical nonlinear DMTCF beyond linear electron-phonon coupling22 is a formidable task. An even more formidably challenging task is dealing analytically and numerically with anharmonic oscillators, e.g. Morse, Kratzer, or Pochl-Teller oscillators, all of which accurately model diatomic and polyatomic molecular vibrations, especially when negative anharmonicities arise which can only be accounted for by employing Pochl-Teller oscillator. Toutounji successfully managed to analytically derive closed-form expressions for the quantum mechanical partition functions of all the aforementioned oscillators.23 Toutounji24-26 also carried out analytical and numerical evaluation of the linear and nonlinear DMTCF whereupon both linear and quadratic electron-phonon coupling is accounted for by employing the formalism of mixed quantum-classical dynamics (MQCD) of Kapral,27,28 leading to sound results. Additionally, MQCD formalism was also utilized to calculate spectral signals in dissipative media.29 Toutounji further derived an exact quantum mechanical expression of DMTCF for linearly and quadratically (diagonal) coupled vibrational modes utilizing Lie algebra and differential operators techniques.30 The linear absorption spectra of both linearly and quadratically (diagonal) coupled modes calculated therein had shown that quadratic coupling predominates the absorption

10.1021/jp1069913  2010 American Chemical Society Published on Web 11/03/2010

Anharmonic Electron-Phonon Coupling 2

J. Phys. Chem. C, Vol. 114, No. 48, 2010 20765

profile at elevated temperatures, thereby affecting dephasing, whereas linear coupling showed weak temperature (T) dependence, as expected. The invaluable homogeneous structural and dynamical information are often obscured by the variation of local environments and energy shift around the guest molecule in solid state giving rise to inhomogeneous broadening. The magnitude of this inhomogeneous broadening ranges from ∼40 to 500 cm-1, as in glassy hosts. 4-Wave mixing experiments1 such as photon echo, hole-burning, and pump-probe signals are effective tools to eliminate this inhomogeneous broadening, thereby unmasking useful information about molecular motion, homogeneous dephasing, and relaxation dynamics of the system of interest. Since the primary purpose of doing nonlinear spectroscopy is eliminating inhomogeneous broadening in order probe electronic dephasing and vibrational structure, this work would be the perfect juncture to present as to how one can employ nonlinear spectroscopy to study anharmonic dynamics molecules in condensed media, as the anharmonicity becomes more important as T rises which in turn will increase the dephasing.3,31-34 While part 1 (ref 14) of this study focuses on deriving only linear DMTCFs and the corresponding lineshapes, part 2 makes use of them in accounting for electronic dephasing and calaculating nonlinear signals, such as hole-burning and photon echo. Since the MBO model is not poised to handle any type of electron-phonon coupling beyond the harmonic linear coupling regime, it is necessary to come up with a strategy that helps circumvent this disadvantage and hence applies to all types of models. This paper addresses a novel way of extracting important nonlinear DMTCFs whereby many spectral and temporal nonlinear signals, such as photon echo, pump-probe, and hole burning, may be calculated without recourse to quantum path integral to calculate 4-point DMTCFs.9-12,35-38 An elementary method is proposed to treat nonlinear spectra in anharmonic,9 and harmonic for that matter, molecules. This method combines Hays-Small theory39 and the framework of nonlinear response function approach1 as shown in section IV. The advantage of this strategy is that the 4-point DMTCF of anharmonic systems can be obtained without having to evaluate them directly using nonlinear DMTCF formalism of Mukamel,1 which is an extremely difficult task to do. Hole-burning and 2-pulse photon echo signals are derived using the proposed method. Further, expressions for hole-burning (cf eq 35) and 2-pulse photon echo (cf eq 40) signals of solvable models using this approach are derived. Interestingly, the consequent zerophonon hole profile falls out of eq 35, which is consistent with literature. Section V applies the above developed techniques to chromophores in glassy ethanol and special pair bacterial reaction center at different temperatures to see if our theory would bridge the gap between previous models and experiment. II. Brief Dipole Moment Correlation Function Background Consider a system with two electronic states, a ground state |g〉 and an excited state |e〉 coupled to some nuclear degrees of freedom denoted by qj which may be viewed as primary coordinates, or pseudolocal phonons for that matter. The adiabatic electronic Hamiltonian is described by

Hel ) Hg |g〉〈g| + He |e〉〈e|

(1)

where the harmonic nuclear Hamiltonians Hg and He are given by

Hg )

[ ∑[ ∑ j

He )

j

]

pj2 1 + µjωg,j2qj2 , 2µj 2 2

pj 1 + µjωe,j2(qj - dj)2 2µj 2

]

(2)

where µ is the reduced mass, dj is the final state linear displacement, and ωg,j and ωe,j are the ground and excited state frequencies for mode j. In case of anharmonicity, the anharmonic nuclear Hamiltonians Hg and He are given by

Hg )

[ ∑[ ∑ j

He )

j

]

pj2 + De,j(e-2ajqj - 2e-ajqj) 2µj 2

]

pj + D'e,j(e-2a'j(qj-qo,j) - 2e-a'j(qj-qo,j)) 2µj

(3)

where De,j and D′e,j and a and a′ are the dissociation energies and Morse parameters in the initial and final states, respectively. q0,j is the linear displacement (bond length) in the upper state We shall assume no symmetry change upon electronic excitation (neither Duschinsky mixing effect nor Hertzberg-teller coupling will be taken into account this study). The primary coordinates qj may be coupled to other modes via some system-bath coupling Hamiltonian, whereby decoherence, dephasing, and other dissipative dynamical processes can be accounted for.1,26,27,35-38,40-42 The linear electronic DMTCF is defined as ˆ

J(t) )

ˆ

ˆ

Tr(eiHt/pµˆ e-iHt/pµˆ e-βH) ˆ

Tr(µˆ 2e-βH)

(4)

where β ) 1/kT, Tr(...) denotes the trace over the entire system (electronic and nuclear degrees of freedom) and, µˆ is the transition electronic dipole moment operator given by

µˆ ) µˆ ge |g〉〈e| + µˆ eg |e〉〈g|

(5)

In the Born-Oppenheimer approximation, µˆ eg is a matrix element in the electronic subspace and a nuclear operator in the nuclear subspace. The linear DMTCF serves as a key quantity in linking experiment to theory for understanding linear spectra. Nonlinear DMTCFs are central to understanding nonlinear optical spectroscopy, especially 4-wave mixing experiments. Section IV briefly sheds some light on nonlinear DMTCFs and their role in calculating photon echo and holeburning signals. III. Electronic Dephasing and Linear Absorption Lineshapes Generally speaking, optical coherence loss (dephasing) in condensed-phase systems is determined by the nature of electron-phonon coupling. In case of harmonic phonons, two types of coupling may emerge: linear and quadratic, or both, electron-phonon coupling.4,15-17,28 While linear electron-phonon coupling takes place when the final electronic state is linearly displaced with respect to the initial state, quadratic coupling arises when there is a force constant change upon electronic excitation. In case of anharmonic phonons, anharmonic electron-phonon coupling must be considered in order to

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Toutounji

accurately probe dephasing. The homogeneous broadening of the zero-phonon line (ZPL) is supposed to signify electronic dephasing in this study. The inclusion of a damping constant in linear DMTCF that reads both electronic and vibrational relaxation times equally is unphysical. We shall proceed along the same guidelines provided in ref4 to correctly account for the ZPL width. In that study4 the DMTCFs of all modes take on the form in which the damping constant is that of vibrational relaxation only, precluding any electronic dephasing contribution and the ZPL profile is simply a Delta function. This technique is particularly helpful when having a system with different groups of modes each of which has different characteristics. This is important, because harmonic phonons can dephase electronic coherence linearly, quadratically, or both. If the system of interest consists of linearly coupled harmonic modes, quadratically coupled harmonic modes, and anharmonic modes, the total DMTCF of an N-mode system representing all the important j modes will read

Sj[4ω ¯ jγjsin(βpγj /2) + 4ω ¯ j2sinh(βpω ¯ j)] Ξj ≡ 4ω ¯ jωj[cos(βpγj /2) - cosh(βpω ¯ j)]

(One can show that the infinite sum contribution in the above equation becomes negligible at high T and considerable in the low T limit.)44 This type of methodology has been used before to emulate experimental nonlinear signals such as hole-burning, photon echo, and pump-probe in glassy ethanol and water, leading to very meaningful results.4,45 The linear DMTCFs exhibiting linear, quadratic, and anharmonic electron-phonon coupling are4

Jl,j(t) ) exp{-Sj[coth(βpωj /2) e-γj|t|/2[coth(βpωj /2) cos(ωjt) - i sin(ωjt)]]}, (9)

(6)

Jel(t) ) exp[-γel |t|/2]

(6a)

j

j

Mg

∏ Jl,j(t)

(10)

1 - e-βpωg-i(ωe-ωg)t-γj|t|/2 j

j

The anharmonic DMTCF derived in ref 14 reads (assuming no molecular dissociation)

N

Jl(t) )

1 - e-βpωg

Jq,j(t) )

J(t) ) Jel(t)Jl(t)Jq(t)Janh(t)

(8a)

-1

Janh(t) ) Q

(6b)

M'e

∑ ∑ |cm'm|2e-βE +iE t/p-iE ,t/p-γ/t/2 g m

g m

g m

m)0 m')0

j

(11) N

Jq(t) )

∏ Jq,j(t)

(6c)

with

j

N

Janh(t) )

∏ Janh,j(t)

|

(

e Nmg Nm' m + 2s (∆′r)s' m a

(6d)

where DMTCFs Jel(t), Jl(t), Jq(t), and Janh(t) represent the electronic, linear, quadratic, and anharmonic coupling, with γel being the electronic dephasing. Alternatively, the electronic contribution to the linear response function in the context of the MBO model may be written as34,43-45

νg ≡

√2µDe , pa

2s ) 2νg - 2m - 1,

[(

∑ (ω n)1

4γju2j /β 2 j

+ νn2)2 - (γjνn)2

) ] |t|/2

2γjuj2 /βνn

∑ (ω 2 + ν 2)2 - (γ ν )2

n)1

j

n

j n

}

√2µD'e

(11b)

pa'

2s' ) 2νe - 2m' - 1

r ≡ νe /νg

(11d)

and

(8)

δ ) -∆′(1 + r)

(11e)

However, eq 11 is unphysical as it assigns one relaxation time constant for both electronic and vibrational decay. To correct this behavior, we propose the following one mode anharmonic DMTCF

[

1 41+s+s'a2ss'(∆′r)2s'Γ2(s + s') -γel|t|/2 e + Q Γ(2ν )Γ(2ν )(1 + ∆′r)2(s'+s) g e Mg

M'e

∑ ∑ |cm'm|2e-βE +iE t/p-iE t/p-(γ +γ )|t|/2 g m

where

(11a)

(7)

|〈0|0〉| 2 ) Re exp -iSjγj /ωj - Ξj ∞

νe ≡

∆′ ≡ ea'qo,

where β ) (kT)-1, Sj is the Huang-Rhys factor, γj is vibration damping of mode uj ) ωj3/2djj/p1/2, the positive bosonic Matj j )(ωj2 - (γj/2)2)1/2. subara frequencies νn ) (2π/pβ)n, and ω 43,45 The corresponding FCF of this ZPL is

{

)|

(11c)

2sin(βpγj /2) + (γj /ω¯j)sinh(βpω¯j) cosh(βpω¯j) - cos(βpγj /2) ∞

)

(

j

JMBO (t) ) exp - Sjωj el

)(

m' + 2s' Γ(s + s') × m' δs+s' 2 1 ∆′r F2 s + s', -m, -m';2s + 1, 2s' + 1; , δ δ

|cm'm | 2 )

m)1 m')1

g m

e m'

el

j

]

(12)



In order to use eq 12 in eqs 6 in the case of an anharmonic multimode system, this form must be used

Janh,j(t) )

[

1 41+s+s'a2ss'(∆′r)2s'Γ2(s + s') + Q Γ(2ν )Γ(2ν )(1 + ∆′r)2(s'+s) g e Mg

M'e

∑ ∑ |cm'm|2e-βE +iE t/p-iE t/p-γ |t|/2 g m

m)1 m')1

g m

e m'

j

]

(13)

In eqs 11-13, Q is the canonical partition function, Γ is the Gamma function, m and m′ are the initial and final quantum states, respectively, Mg and M′e are the finite number of quantum levels in Morse potential, and γj is vibration damping of mode j. It is important here to assume that molecule of interest is not dissociating.14 Each J(t) yields a ZPL width ) γel and progression members of width ) (γel + γj) upon taking Fourier transform (FT). Therefore, the utility of eqs 6-11 comes in handy when information about electronic dephasing needs to be found out. The dependence of γel on T has to do with the nature of the ensuing dephasing mechanism, e.g., for chromophores in glassy ethanol and water Jacson-Silbey mechanism46 was adopted to explain electronic dephasing. The prime over “M′e” in eqs 12 and 13 signals the preclusion of 0 r 1 transition at high temperature. Note that the FT of eq 12 yields an anharmonic homogeneous absorption line shape distinctively exhibiting a sharp ZPL peak and progression members on the high energy side for one mode. In the case of a many anharmonic mode system, one will have to use the product of eqs 6a and 13. Equation 6a accounts for the electronic relaxation, whereas the first term and second term in eq 13 represents the FCF for the ZPL and vibarational progression profiles, respectively. It is noteworthy here that, although numerical (certainly not analytical) anharmonic homogeneous absorption lineshapes might have been calculated before,6,7 this work is a first-time report of an analytical anharmonic line shape which accounts for electronic dephasing correctly in a one- and multimode system. We shall calculate anharmonic homogeneous linear absorption lineshapes at different temperatures distinctively showing different ZPL profiles having different homogeneous broadening (dephasing), which is independent of progression members widths which comprise the phonon-sideband (PSB), as physically should be the case. Figure 1 shows single-site (homogeneous) absorption spectra of an anharmonically coupled normal mode calculated by taking the FT of the product of eq 12 at the indicated T. This anharmonic pseudolocal mode is characterized by ground state frequency ωg ) 40 cm-1, excited state frequency ωe ) 30 cm-1, linear displacement d ) 1.40, anharmonicity χ ) 0.0125, vibrational damping γanh ) 5 cm-1, and γel ) 2 cm-1. While the ZPL homogeneous width is being held constant over the shown T range, it has lost intensity to the PSB wing. (The ZPL width is being held constant in Figure 1 to better show the ZPL profile and independence of the vibrational structure as T changes, although in reality it should narrow as T decreases and broaden as T increases, as illustrated in Figure 2.) Figure 2 shows the same set of spectra but with different T-dependent ZPL width (using Jackson-Silbey formula)46 of γel(T) ) 2nj(ωg) cm-1 and vibrational damping of γanh ) 5(nj(ωg) + 1) cm-1, with nj(ωg) being the thermal occupation number. One observes the progressive ZPL homogeneous broadening as T increases and that the ZPL width approaches 0 as T f 0. Figure 3 shows an interesting case of a 3-mode model system, one of which is an anharmonic. The system consists of a linearly coupled mode

Figure 1. Linear homogeneous absorption spectra of an anharmonically coupled normal mode of a model system calculated by taking FT of the product of eqs 6a and 11 at the indicated Ts. The parameters used to calculate the spectra were ωg ) 40 cm-1, ωe ) 30 cm-1, d ) 1.40, χ ) 0.0125, γanh ) 5 cm-1, and γel ) 2 cm-1.

with ωl ) 200 cm-1, γl ) 5 cm-1, and Huang-Rhys factor Sl ) 0.7; quadratically coupled mode with ωg ) 40 cm-1, ωe ) 30 cm-1, and γq ) 3 cm-1; and an anharmonic mode with ωg ) 40 cm-1, ωe ) 30 cm-1, d ) 1.40, χ ) 0.0125, γanh ) 8 cm-1, and γel ) 2 cm-1. The insets in the figure show better resolved vibrational structure on different scales.47 IV. Nonlinear Spectroscopy Framework We shall adopt the formalism of Mukame1 to lay out some fundamentals of nonlinear spectroscopy using nonlinear response theory. However, the nonlinear response functions of the MBO model are not poised to treat electron-phonon coupling beyond the linear coupling regime. The general set up of 4-wave mixing experiments would be envisaging three laser pulses, each of which is represented by an electric field E(t), successively interacting with the molecular sample of interest at different time intervals. The result of these interactions with the sample is a fourth polarization signal given by expanding quantum Liouville equation to the third order in the electric field yielding

p(3)(t) )

∫0∞ dt1 ∫0∞ dt2 ∫0∞ dt3 S(3)(t3, t2, t31)E(t - t3)

E(t - t3 - t2)E(t - t3 - t2 - t1)

×

(14)

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Toutounji

Figure 3. A homogeneous absorption spectrum of a 3-mode system which consists of a linearly coupled mode with ωl ) 200 cm-1, γl ) 5 cm-1, and Huang-Rhys factor Sl ) 0.7; quadratically coupled mode with ωg ) 40 cm-1, ωe ) 30 cm-1, and γq ) 3 cm-1; and an anharmonic mode with ωg ) 40 cm-1, ωe ) 30 cm-1, d ) 1.40, χ ) 0.0125, γanh ) 8 cm-1, and γel ) 2 cm-1. The insets are to show better resolved vibrational structure on different scales.

Evaluating the first electronic trace in a 2-dimensional electronic basis set yields this nuclear dynamical quantity

S(3)(t3, t2, t1) ) 〈g|Trp{µˆ G(t3)µˆ G(t2)µˆ G(t1)µˆ Fˆ }|g〉 + 〈e|Trp{µˆ G(t3)µˆ G(t2)µˆ G(t1)µˆ Fˆ }|e〉 (17) Evaluating eq 17 yields

Figure 2. Same linear homogeneous absorption spectra as those of Figure 1 but with different T-dependent ZPL width of γel(T) ) 2nj(ωg) cm-1 and vibrational damping of γanh ) 5(nj(ωg) + 1) cm-1. One observes the homogeneous broadening of the ZPL as T increases. (3)

where S (t3, t2, t1) is the third-order nonlinear response function, and physically provides the polarization response to the E field applied at different intervals. The time arguments t1, t2, and t3 represent time delays between the sequential interactions with the E field. Note that polarization has a spatial dependence in eq 14, but it is not shown as it is of no interest here. The polarization here may be an echo signal, pump-probe, or holeburning, and hence S(3), is a fundamental quantity for calculating these signals. S(3) is given by

S(3)(t3, t2, t1) ) Tr Trp{µˆ G(t3)µˆ G(t2)µˆ G(t1)µˆ Fˆ }

(15)

where the first trace is carried out over electronic basis and second one is done over nuclear degrees of freedom and “p” signifies phonons. The Green’s function G(t) (time evolution operator) and density operator Fˆ and in eq 15 are defined as ˆ

ˆ

G(t)µˆ ) e-iHt/pµˆ eiHt/p Fˆ ) |g〉Fg〈g|

(16)

S(3)(t3, t2, t1) ) 2p-3u(t1) u(t2) u(t3) Im

4

∑ RR(t3, t2, t1)

R)1

(18) Here {RR(t1,t2,t3)}4R)1 are the nonlinear correlation functions, and u(tn) is the Heaviside step function

R1(t3, t2, t1) R2(t3, t2, t1) R3(t3, t2, t1) R4(t3, t2, t1)

) ) ) )

Trp{Geg(t3)Gee(t2)Geg(t1)Fg} Trp{Geg(t3)Gee(t2)Geg(t1)Fg} Trp{Geg(t3)Ggg(t2)Gge(t1)Fg} Trp{Geg(t3)Ggg(t2)Geg(t1)Fg}

(19)

where the nuclear Green’s function is

Gnm(t)A ) e-iHnt/pAeiHmt/p

(20)

with A being some nuclear operator and n and m signify the system electronic states. Valuable information may be inferred from eq 19. For example two types of nuclear Green’s functions may be observed: diagonal Green’s function and off-diagonal Green’s function. While the former is responsible for the population free evolution, the latter is responsible for coherence evolution, for which the reason off-diagonal Gnm(t) (n * m) is called depahsing time Green’s function. The underlying physical



significance of these Green’s functions will become clear upon applying them to photon echo spectroscopy where dephasing 4 and rephrasing processes are involved. The set {RR(t1,t2,t3)}R)1 forms fundamental components in calculating any nonlinear spectroscopic signals, e.g., photon echo, hole-burning and pump-probe which may be obtained using the appropriate nonlinear optical correlation functions.1 A. Hole-Burned Spectrum from Linear DMTCF. We will rely on our linear DMTCFs derived in paper 114 of this work to calculate nonlinear signals, from which one can then extract the data to find nonlinear DMTCFs. This is to the contrary to what is commonly done when calculaing nonlinear DMTCFs; that is, they are evaluated by tracing over some nuclear basis set, such as Fock or coherent states, starting with nuclear density operator as an equilibrium ground state. Hole-burning is one of the selective spectroscopy techniques to probe electronic dephasing of a guest molecule embedded in a solid matrix. The idea of hole-burning is basically exciting a group of molecules within an inhomogeneously broad band, leaving a hole in the ground state.1 To this end, a hole-burning signal can hence be 4 as1 calculated in terms of {RR(t1,t2,t3)}R)1

( p1 ) ω ∫ 3

S(ω, ωB, τ) ) 2

L

∞

0

dt1

∫0∞ dt3 {χ(t3 + t1)eiωt +iω t 3

B1

×

[R1(t3, τ, t1) + R4(t3, τ, t1)] + χ(t3 - t1)eiωt3-iωBt1 × (R2(t3, τ, t1) + R3(t3, τ, t1))} (21) where ωB is the burn frequency and ω is light frequency. Assuming a very large inhomogeneous broadening, i.e., χ(t3 + t1)δ˜ (t3 + t1), whereupon R1(t3,τ,t1) and R4(t3,τ,t1) are eliminated as a result, leading to this hole-burning signal (neglecting some prefactors)

S(ω, ωB, τ) ) Re

∫0∞ dt1 ei(ω-ω )t (R2(t1, τ, t1) + R3(t1, τ, t1)) B 1

(22) (R2(t3,τ,t1) + R3(t3,τ,t1)) will be called photon echo response function for a reason will be clear shortly, vide infra. Note all of the dynamical contribution will come from photon echo response function. Although hole-burning is a nonlinear technique, one may still calculate it using the linear correlation function J(t) via the hole-burning theory of Hays-Small.38 The absorption spectrum following a burn time η is given by38 Iη(ω, ωB) )

∫

∞

-∞

dΩχ(Ω - νm)J(ω - Ω) exp[-θJ(ωB - Ω)η]

(23)

where Ω is the variable frequency of the zero-phonon line (ZPL) of a single absorber and ωB is the burn frequency. The absorption spectrum before burning is I0(ω) which is obtained by setting the burning time η ) 0. Thus the hole-burned spectrum is given by SHB(ω,ωB) ) Iη(ω) - I0(ω). χ(Ω - νm) is a Gaussian function, with variance ∆2 centered at νm, which governs the distribution of ZPL frequencies due to structural heterogeneity. θ is the product of three terms: the absorption cross-section, the laser burn flux and the quantum yield for holeburning. The advantage of eq 23 over eq 22 is that once the DMTCF of any type of electron-phonon coupling, be it linear coupling, both linear and quadratic, or anahrmonic coupling14 is available the corresponding hole-burning signal is readily calculated directly using eq 23. This is a tremendous simplifica-

tion to the calculational process of nonlinear signals. However, the price to be paid for this simplification is that there will be less availing information to be gathered. Equation 22 is the FT of the echo response function and thus one can obtain it via inverse FT which can in turn be used to calculate photon echo profiles, vide infra. (R2(t1, τ, t1) + R3(t1, τ, t1)) )

∫

∞

-∞

d(ω - ωB)e-i(ω-ωB)t1SHB(ω, ωB)

(24)

Assuming a very large inhomogeneous broadening, χ(Ω - νm) ∼ 1, yields (R2(t1, τ, t1) + R3(t1, τ, t1)) )

∫

∞

-∞


(25)

It will prove useful to get an analytical expression for SHB(ω,ωB) ) Iη(ω) - I0(ω) under short burning time, η, approximation which allows us to expand the exponential, leading to

Iη(ω, ωB) ≈

∫-∞+∞ dΩJ(ω - Ω)(1 - FηJ(ωB - Ω)) (26)

We are interested in this integral

SHB(ω, ωB) ) Iη(ω) - I0(ω) ) θη

∫-∞+∞ dΩJ(ω - Ω)J(ωB - Ω)

(27)

This integral is very hard to do directly without resorting to integral theorems in Fourier theory. Examining the integral in eq 27 reveals that it might be identified as a convolution integral which would be easiest to evaluate using integral theorems in FT theory. As such, the above integral can efficiently be handled by utilizing Parseval’s identity from Bessels’ functions theory,48 yielding

SHB(ω, ωB) )

1 θη 2π

∫-∞+∞ dt|J(t)|2e-iωteiω t B

(28)

The time- and frequency-shifting properties in FT theory has been used to arrive at eq 28. For simplicity sake, we shall evaluate the above integral in the low T limit. Equation 28 is a first-time report of a hole-burned absorption spectrum written in terms of time-domain DMTCF. It is worth noting that the square of J(t) is what imparts twice the width of ZPL to that of the ZPH, and leads reduction of ZPH integrate intensity. Figure 4 shows an absorption (top frame) and hole-burned (bottom frame) spectra of an anharmonic mode calculated with eq 24 with ωB ) 0, ωg ) 40 cm-1, ωe ) 30 cm-1, d ) 1.40, χ ) 0.0125, γanh ) 10 cm-1, inhomogeneous broadening w (fwhm) ) 64.4 cm-1, and γel ) 3 cm-1 at T ) 1 K. Same hole-burned spectrum was calculated with eq 28, yielding identical line shape (not show) to that of Figure 4. One can see from Figure 4 the ZPH width is twice that of the ZPL. SolWable Model: Linear Electron-Phonon Coupling. For clarity and illustrative reasons it would be helpful to apply our formula to a solvable model such as a system that is characterized with linear electron-phonon coupling in which pure electronic dephasing and vibrational relaxation are distinctly and

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Toutounji where the burning frequency ωB ) 0. Evaluating the integral yields

∫

∞ SHB(ω) ) θη dt exp[-γel |t| - iωt]| exp[-Sj + Sje-γj|t|/2-iωjt]| 2 2π -∞ ∞ θη exp(-2Sj) ) dt exp[-γel |t| + -∞ 2π -γj|t|/2 cos(ωjt)] exp(-iωt) 2Sje ∞ θη exp(-2Sj) ) dt exp(-γel |t|) + -∞ 2π exp(z cos(ωjt)) exp(-iωt)

∫ ∫

(32) with z ≡ 2Sje-γj| t|/2. Using the identity of ordinary Bessel generating function Jk(y) and Legendre polynomials Pk(y)49

exp(iz cos φ) )

∞

∑

π (2k + 1)ikJk+1/2(z)Pk(cos φ) 2z k)0 (33)

Upon inserting eq 33 in eq 32 and with some further simplification and using definition of ordinary Bessel functions Jk(y)49

Jk(y) )

y 2

k

()

∞

ν

y ∑ ν!Γ(k(-1) + ν + 1) ( 2 )

2ν

(34)

ν)0

eq 32 evaluates to Figure 4. Homogeneous absorption (top panel) and hole-burned spectra (bottom panel) at T ) 1 K of a model system made up of an anharmonically coupled mode which was calculated with these parameters: ωB ) 0; ωg ) 40 cm-1; ωe ) 30 cm-1; d ) 1.40, χ ) 0.0125; γanh ) 10 cm-1; w (fwhm) ) 64.4 cm-1; and γel ) 3 cm-1. While the absorption spectrum is made up of a very sharp peak (ZPL) and cold transitions (progression members), comprising the PSB, the anharmonic hole-burned spectrum consists of the ZPH, which has twice the width of the ZPL, real PSBH, and pseudo PSBH on the high and low energy sides, respectively.

correctly accounted for, as it will allow us to test the correctness and applicability of eq 28. This hole burning signal formula is a first-time report which turns out to be spectrally informative and insightful, vide infra. The electronic DMTCF in the low T limit for this system with mode j is4

J(t) ) exp[-γel |t|/2] exp[-Sj(1 - e-γj|t|/2-iωjt)]

(29) whose linear homogeneous absorption line shape is ∞

I(ω) ) exp(-Sj)

Sjm (γel + mγj)/2π m! (ω - mω )2 + ((γ + mγ )/2)2 m)0 j el j (30)

∑

with ZPL of full width at half-maximum (fwhm) ) γel and FCF of exp (-Sj)and progression members with fwhm ) γj + γel. The corresponding hole-burned line shape is SHB(ω) )

1 θη 2π

∫

∞

-∞

∫

θη ∞ dt exp[-γel |t| - iωt] × 2π -∞ | exp[-Sj(1 - e-γj|t|/2-iωjt)]| 2 (31)

dt|J(t)| 2e-iωt )

+∞

SHB(ω) )

+∞

(-1)k+3νSk+2ν ei(k+ν)π j × ν!Γ(ν + k + 1) ν)0

∑∑

θη -2Sj e 2π k)-∞

(2γel + kγj + 2νγj) kγj + νγj (ω + kωj)2 + γel + 2

(

)

2

(35)

Once clearly observe the zero-phonon hole (ZPH) profile at k ) 0 reads

γel θη exp(-2Sj) 2 π ω + γel2

(36)

The FCF and fwhm of ZPH are exp (-2Sj) and 2γel, respectively, as expected. Two more components besides ZPH that make up a hole-burned spectrum are the real and pseudo phonon-sideband hole (PSBH) profiles which appear to the right and left of the ZPH, respectively. Examining eq 35 one can readily see that terms in the sum that carry negative “k” signify the real PSBH which appear on the high energy side, and the positive terms “k” connote the pseudo-PSBH which appear on the low energy side, as the quantum number “k” signifies the number of phonons associated with the vibronic transitions. Note that “ν” is just a dummy index and is not a quantum number. This reflects the accuracy and correct applicability of our formula in eq 28. Equation 35 is a first-time report of a full and explicitly clear profile of a hole-burned spectrum. B. Two-Pulse Photon Echo Spectroscopy from HoleBurning Through Linear DMTCF. Consider an impulsive stimulated photon echo signal in which the three infinitely short pulses are applied sequentially. The integrated intensity of the echo signal, SSPE, reads1,7


∫0∞ dt|R(t, τ, τ′)|2|χ(t - τ′)|2

SSPE(τ′, τ) )


(37)

where τ′ is the delay between the first and second pulses and τ is the delay between the second and third pulses. Here, R(t3,t2,t1) is the echo response function defined as

R(t3, t2, t1) ≡ R2(t3, t2, t1) + R3(t3, t2, t1)

(38)

The form of the inhomogeneous broadening χ(t - τ′) results in the maximum of the echo appearing along the line t ) τ′ after the interaction with the third pulse which we can now consider to have occurred at t ) 0. For t > τ′, the echo decays due to dephasing. 2-puse photon echo (PE) of very large inhomogeneously broadened system may be obtained by setting τ ) 0, leading to the impulsive integrated PE signal

∫0∞ dt|R(t, 0, τ′)|2|χ(t - τ′)|2 ∞ ) ∫0 dt|R(t, 0, τ′)δ(t - τ′)| 2

SPE(τ′) )

(39)

) |R(τ′, 0, τ′)| 2 The primary goal here is to obtain nonlinear DMTCF (echo response function) without having to evaluate a 4-point time evolution operator from which a 2-PE signal may be procured using linear DMTCF, as we did SHB(ω,ωB) above, without resorting to nonlinear DMTCF formalism as they are not available for quadratically and anharmonically coupled systems. This should be valuable thing to do as all kinds of DMTCFs have previously been derived (cf eqs 9-11). The echo response function in the impulsive and large inhomogeneous broadening limits may be obtained from eqs 25 and 39 as

|R(t1, 0, t1)| 2 )

|∫

∞

0


|

Figure 5. Integrated 2-PE signal for a linearly coupled vibrational mode calculated with with eq 37 using ωj ) 50 cm-1, Sj ) 1.0, T ) 0 K, and γj ) 5 cm-1. The inset is the corresponding absorption spectrum with S ) 1 which displays a pronounced vibrational structure of an uderdamped mode (quantum beats in time domain), which would be suppressed in case of an overdamped mode. Equation 38 produces the same PE signal, ratifying its accuracy and correct applicability; see text for details.

2

(40)

in SHB(ω,ωB) which is expressed in terms of DMTCF J(t) in eq 27. Below a PE signal of a linearly coupled system will be calculated from SHB(ω,ωB) to test the herein approach. SolWable Model: Linear Electron-Phonon Coupling. To test the applicability of our approach that leads to eq 40, we shall calculate the impulsive 2-PE signal for a solvable model in which phonons are linearly coupled to electronic transition since the solution to this model is well-known in the literature. Consider a linearly coupled system with one vibrational mode whose ωj ) 50 cm-1, Sj ) 1.0, T ) 0 K, and γj ) 5 cm-1. The integrated 2-PE signal for this system is calculated in Figure 5 (top part) with eq 38 using the linear DMTCF, J(t), of Mukamel.1 Same signal was calculated with the herein derived eq 40 resulting in a very similar echo signal. A clearer illustration of the photon echo profile for this system is the 3-dimensional surface at the bottom of Figure 5, in which one can see that the echo profile starts with intense peak classified as induction decay followed by quantum beats, which correspond to FC progressions in the frequency-domain. The reason quantum beats are very pronounced here is that because the Huang-Rhys factor S ) 1 which tends to stimulate considerable vibrational structure (beats) as dictated by FCFs for linear coupling. The signals calculated by eqs 39 and 40 are the same. The inset in Figure 5 displays the corresponding homogeneous

Figure 6. Same as in Figure 5 but with a finite inhomogeneous broadening of w ) 50 cm-1; see text for details.

linear absorption spectrum calculated with the same parameters as those of the PE signal, revealing the same FC progressions (beats in PE). Of course, if the induction decay was to be chopped off, more quantum beats with better intensity would show exactly as the absorption spectrum shows. Figure 6 exhibits a 2-pulse echo signal for the same system in Figure 5 but with a finite inhomogeneous broadening w (fwhm) ) 50 cm-1. Upon comparing Figure 5 and 6 one observes that the echo profiles forming on the diagonal line along t ) τ′ in Figure 6 has a finite width due to the finite inhomogeneous broadening whereas that in Figure 5 has much thinner beats due the Deltafunction-like inhomogeneous broadening. To our knowledge, the best experimental relation between optical line shape and

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Toutounji

Figure 7. 2-PE signal of an anharmonic system calculated with ωg ) 50 cm-1, ωe ) 50 cm-1, d ) 1.40 (S ) 0.98), γel ) 3 cm-1, γanh ) 10 cm-1, and χ ) 0.0125 at T ) 1 K using eq 38. The inset exhibits the corresponding anharmonic absorption spectrum showing the same vibrational structure which corresponds to the quantum beats in the PE signal.

photon echo at different T has been produced by Rebane et al. using single laser shot.50 Now using the same approach one can calculate the integrated 2-PE signal of anharmonic molecules. For example one can use eq 12 in eq 40 to calculate the integrated 2-PE signal for an anharmonic system with these parameters: ωg ) 50 cm-1, ωe ) 50 cm-1, d ) 1.40 (S ) 0.98), γel ) 3 cm-1, γanh ) 10 cm-1, and χ ) 0.0125 at T ) 1 K as shown in Figure 7. The inset displays the corresponding anharmonic absorption spectrum showing the same vibrational structure which corresponds to quantum beats in Figure 7. The echo signal and absorption line shape represent linear (only bond length) and anharmonic (only shape change) coupling while the curvature of the final state is kept fixedsno frequency change. The beats in the echo signal reflect that the vibrational structure representing an underdamped mode whereas the initial intense peak signifies free induction decay. The results are meaningfully sensible. V. Real Applications Our first real application would be the chromophore Alphthaolcyanine tetrasulphonate (APT) in glassy ethanol which has been studied before.3,4 This system was found out to consist of a linearly coupled mode and a quadratically coupled mode which was thought responsible for dephasing the ZPL. Homogeneous absorption spectra of APT in ethanol are calculated in Figure 8 at 15, 60, and 100 K by taking FT of eqs 6 and 13 with ωl ) 25 cm-1, γl ) 10 cm-1, and Sl ) 0.5 for the linear mode, and ωg ) 50 cm-1, ωe ) 35 cm-1, γanh(T) ) 5 (nj(ωg) + 1) cm-1, d ) 0, χ ) 0.0125, and γel(T) ) 8nj(ωg) cm-1 for the quadratic (and anharmonic) mode. This is an interesting case as it allows us to ascertain how much the spectra would change upon invoking some anharmonicity into the quadratic mode at elevated Ts, leading to a better agreement with experiment. A second, and final, application of our work is calculating the homogeneous absorption spectra of the special pair band of the bacterial reaction center which is characterized by linearly coupled modes at ωm ) 30 and ωsp ) 120 cm-1.4,38 This system has received considerable attention as its study is related to

Figure 8. Linear homogeneous absorption spectra of APT in glassy ethanol exhibiting one linearly coupled mode and another anharmonically coupled mode, with d ) 0, at different Ts: ωl ) 25 cm-1; γl ) 10 cm-1; Sl ) 0.5; ωg ) 50 cm-1, ωe ) 35 cm-1, γanh(T) ) 5 (nj(ωg) + 1) cm-1, d ) 0, χ ) 0.0125, and γel(T) ) 8nj(ωg) cm-1.

proteins, algae, and photosynthetic structure.38,51 Small and his co-workers simulated this system and characterized it by using Hays-Small theory38 of hole-burning spectroscopy employing parameters extracted from their own hole-burning experiments.38,50 In their report,38,51 it was stated that their simulated spectra had shown weak T-dependence because they could only include linear electron-phonon coupling and did not have quadratic or anharmonic coupling accounted for in their model. Having run spectra with quadratic coupling in refs 24 and 28 their rational seems to be accurate and well-assessed. Furthermore, the linear


J. Phys. Chem. C, Vol. 114, No. 48, 2010 20773 VI. Concluding Remarks

Figure 9. Homogeneous absorption spectra of the special pair band of the bacterial reaction center at different Ts; see text for details.

absorption spectra of the special pair mode and the marker mode calculated in Figure 9 which accounts for linear and quadratic/ anharmonic coupling, respectively, in this system only further ratify their conclusion with respect to the broadening and shifting38 that arise due inclusion of quadratic or anharmonic coupling, as these two types of coupling are responsible for dephasing the zero-phonon transition. Figure 9 is calculated by taking FT of eq 6 made up of a linearly coupled mode with ωsp ) 120 cm-1, Ssp ) 1.5, and γsp ) 25 cm-1, and only anharmonically (no frequency change) coupled mode characterized with ωm ) 30 cm-1, γm,anh ) 15 cm-1, and Sm ) 0.98 and χ ) 0.0125. The ZPL width in this system was set γel ) 3 cm-1. Figures 2 and 9 are good representative of the overall effect of quadratic and anharmonic electron-phonon coupling on the spectra as they evidently start to dominate the absorption profiles as T rises and thereby weakening the linear coupling.

The theme of this study is 5-fold. First, employing our anharmonic DMTCF and the corresponding linear absorption line shape derived in ref14 to correctly reflect the T-dependence of homogeneous broadening of the ZPL, using Jackson-Silbey theory, and the damping of the thermally activated phonons involved in the transition. Second, applying this methodology to encompass a multimode system made up of different types of coupled modes that play a role in dephasing the zero-phonon transition. It is known that quadratic and anharmonic coupling contribute considerably to broadening and shifting the spectrum, whereas linear coupled modes weakly dephase the ZPL. Third, calculating anharmonic hole-burned spectra in which the ZPL dephasing is accounted for. Fourth, obtaining an anharmonic hole-burned line shape directly from a linear DMTCT, J(t). Fifth, calculating hole-burning and photon echo signals from the nonlinear echo response function using simple inverse FT. In this article important equations for probing homogeneous linear spectra for various types of vibrational modes taking into account electronic dephasing correctly have been reported. More importantly reported are eqs 24 and 28 which show how one can obtain the dynamical nonlinear echo response function of the homogeneous line shape. Equation 28 expresses hole-burning line shape in terms of the linear DMTC; this is a very new way of evaluating a hole-burning line shape. Equation 35 provides the correct solution for a solvable system for applicability and correctness verification. Equation 35 is a first-time report of explicitly clear profile components of a hole-burned spectrum. With the aid of eq 35, one can readily calculate the integrated intensity of an impulsive 2-pulse photon echo signal. Future Prospects. Morse potential eigenfunctions do not readily lend themselves to numerical calculations either in coordinate or momentum space; hence, tricky. Their instability stems from having to deal with the generalized Laguerre polynomials (GLP)52 which may be viewed as alternating series. Many schemes53-57 have been developed to deal with the numerical instability of Morse oscillator eigenfunctions. Therefore it is incumbent to come up with a robust, exact analytical approach that circumvents the aforementioned drawbacks by providing an alternative to dealing with the GLP. As such, the following works are in progress. 1. Future work will involve utilizing Liouville transformation to map Morse potential into a simpler solvable potential using path integrals. This will render an exact propagator of Morse potential which in turn may be employed to analytically and exactly evaluate DMTCFs, or any dynamical correlation function for that matter. The advantage of this method is that we can obtain a closed-form expression in the time-domain whereby all time- and frequency-resolved signals may be calculated using the robust optical nonlinear response function formalism of Mukamel.1 2. Coherent states of Morse oscillator (analogue of harmonic oscillator coherent states) have been constructed58-61 and used efficiently in Morse dynamics before,62 rendering very good results. This technique should be more useful than the aforementioned path integral as it turns out to be far more tractable and simpler expression to handle upon evaluating anharmonic J(t). 3. The wavelets of both the harmonic oscillator63 and Morse oscillator64 have successfully been constructed, leading to remarkable results. Equivalently, it is believed that wavelets of Morse oscillator can serve more usefully as a mathematical tool to probe its anharmonic dynamics. In fact, this can be a robust algebraic technique if one can properly invoke the translational

20774


and squeezing operators65 on the initial electronic state to account for the molecular geometry change upon optical excitation. Translational and squeezing operators are supposed to displace and distort the shape of the final state with respect to initial state. One can easily draw the connection between wavelets and affine coherent states of the system at hand.66 As such, wavelets67 may be utilized in that sense to see what type of novel dynamical and structural information may be extracted, as new dynamics may emerge. 4. One can evaluate linear and nonlinear optical response functions for anharmonic molecules, using MQCD of Kapral25,26,41 for linear response function.22-24,44 However, MQCD formalism is not poised to handle nonlinear correlation functions, and one will have to resort to the method developed by Toutounji24 for calculating nonlinear anharmonic response functions, which starts out with the four-point correlation function of Mukamel1 and then utilizes some Wigner transform identities as they appear in refs 68-70. The results can then be used to calculate all types of nonlinear signals of anharmonic molecules, e.g., holeburning, pump-probe, photon echo, etc. References and Notes (1) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University: New York, 1995. (2) Reinot, T.; Kim, W.-H.; Hayes, J. M.; Small, G. J. J. Chem. Phys. 1996, 104, 793. (3) Reinot, T.; Kim, W.-H.; Hayes, J. M.; Small, G. J. J. Chem. Phys. 1997, 106, 457. (4) Toutounji, M.; Small, G. J.; Mukamel, S. J. Chem. Phys. 1998, 109, 7949. (5) Osadko, I.,S. AdV. Polym. Sci. 1994, 114, 125. (6) Yan, Y.; Mukamel, S. J. Chem. Phys. 1986, 85, 5908. (7) Grad, J.; Yan, Y.; Mukamel, S. J. Chem. Phys. 1986, 86, 3441. (8) Yan, Y.; Mukamel, S. J. Chem. Phys. 1988, 88, 5735. (9) Tanimura, Y.; Ishaizaki, A. Acc. Chem. Res. 2009, 42, 1270, and references therein. (10) Suzuki, Y.; Tanimura, Y. Phys. ReV. E 1999, 59, 1475, and references therein. (11) Tanimura, Y.; Maruyama, Y. J. Chem. Phys. 1997, 107, 1779. (12) Maruyama, Y.; Tanimura, Y. Chem. Phys. Lett. 1998, 292, 28, and references therein. (13) Kallaush, S.; Segev, B.; Sergeev, A. V.; Heller, E. J. J. Phys. Chem. A 2002, 106, 6006. (14) Toutounji, M. J. Phys. Chem. B 2010, DOI: 10.1031/jp104731s. (15) Yan, Y.; Mukamel, S. J. Chem. Phys. 1986, 85, 5908. (16) Wadi, H.; Pollak, E. J. Chem. Phys. 1999, 110, 11890. (17) Ianconescu, R.; Pollak, E. J. Phys. Chem. A 2004, 108, 7778. (18) Banerjee, S.; Gangopadhyay, G. J. Chem. Phys. 2005, 123, 114304. (19) Santoro, F.; Lami, A.; Improta, R.; Bloino, J.; Barone, V. J. Chem. Phys. 2006, 128, 224311. (20) Shiu, Y.,J.; Hayashi, M.; Mebel, A.,M.; Chen, Y.-T.; Lin, S. H. J. Chem. Phys. 2001, 115, 4080. (21) Hsu, D.; Skinner, J. L. J. Chem. Phys. 1984, 81, 1604. Skinner, J. L.; Hsu, D. AdV. Chem. Phys. 1986, 65, 1. (22) Mukamel, S. Annu. ReV. Phys. Chem. 1990, 41, 647. (23) Toutounji, M. Int. J. Quantum Chem. 2010, in press. (24) Toutounji, M. J. Chem. Phys. 2002, 117, 3848. (25) Toutounji, M. Chem. Phys. 2003, 293, 311. (26) Toutounji, M. J. Chem. Phys. 2004, 121, 2228. (27) Kapral, R.; Ciccotti, G. J. Chem. Phys. 1999, 110, 8919. (28) Toutounji, M.; Kapral, R. Chem. Phys. 2001, 268, 279. (29) Toutounji, M. J. Chem. Phys. 2006, 125, 194520.

Toutounji (30) Toutounji, M. J. Chem. Phys. 2008, 128, 164103. (31) Nagasawa, Y.; Passino, S. A.; Joo, T.; Fleming, G. R. 1997, 106, 4840. (32) Bardeen, C. J.; Cerullo, G.; Shank, C. V. Chem. Phys. Lett. 1997, 280, 127. (33) Knox, R.; Small, G J.; Mukamel, S. Chem. Phys. 2002, 281, 1. (34) Toutounji, M. Int. J. Quantum Chem. 2009, 109, 3399. (35) Tanimura, Y.; Mukamel, S. J. Phys. Chem. 1993, 97, 12596. (36) Tanimura, Y.; Mukamel, S. Ultrafast Dynamics of Chemical Systems; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994. (37) Ashizaki, A.; Tanimura, Y. J. Chem. Phys. 2005, 123, 014503. (38) Hays, J.; Lyle, P. A.; Small, G. J. J. Phys. Chem. 1994, 98, 7337, and references therein. (39) Tanimura, Y. Phys. ReV. A 1990, 41, 6676. (40) Weiss, U. Quantum DissipatiVe Systems; World Scientific: Singapore, 1999. (41) Kapral, R. J. Phys. Chem. A 2001, 105, 2885. (42) Shiokawa, K.; Kapral, R. J. Chem. Phys. 2002, 117, 7852. (43) Toutounji, M.; Small, G. J. J. Chem. Phys. 2002, 117, 3848. (44) Toutounji, M. Chem. Phys. 2003, 293, 311. (45) Toutounji, M. J. Chem. Phys. 2009, 130, 094501. (46) Jackson, B.; Silbey, R. Chem. Phys. Lett. 1983, 99, 331. (47) In the case of only harmonically linear coupling, the use of the MBO model ZPL has a superior advantage over the method laid out in ref 4 as the MBO model has the capability of incorporating non-Markovian dynamics as well sketched out by Knox and coworkers in ref 33. (48) Walker, J. Fourier Analysis; Oxford University Press: New York, 1988. (49) Gradsteyn, I. S.; Ryzhik, I. M. Tables of Integrals, Series and Products; Academic Press: New York, 2007. (50) Rebane, A.; Gallus, J.; Ollikainen, O. Laser Physics 2002, 12, 1126. (51) Lyle, P. A.; Kolackzkowski, S. V.; Small, G. J. J. Phys. Chem. 1993, 97, 6933, and references therein. (52) GLPs, as may be seen from their definition in ref 49 are alternating series and thereby dealing with them computationally can become difficult and hence high precision will be required. This difficulty increases as the number of Morse bound states increases, as a substantial loss of significant figures will take place during evaluation; in fact quadruple precision and beyond may be needed in some cases. Therefore finding a technique which avoids the use of GLP in probing Morse or Kratzer dynamics certainly deems necessary. (53) Vasan, V. S.; Cross, R. J. J. Chem. Phys. 1983, 78, 3869. (54) Karlsson, F.; Jedrzejek, J. Chem. Phys. 1987, 86, 3532. (55) Ratner, M.; Buch, V.; Gerber, R. Mol. Phys. 1980, 53, 345. (56) Dahl, J.; Springborg, M. J. Chem. Phys. 1988, 88, 4835. (57) Levine, R. D. Chem. Phys. Lett. 1983, 99, 27. (58) Molnar, B.; Foldi, P.; Benedict, M. G.; Bartha, F. Europhys. Lett. 2003, 61, 445. (59) Foldi, P.; Benedict, M. G.; Czirjak, A.; Molnar, B. Fortschr. Phys. 2003, 51, 122. (60) Popov, D. Phys. Lett. A 2003, 316, 369. (61) Nieto, M. M.; Simmons, L. M. Phys. ReV. A 1979, 19, 438. (62) Wu, J.; Cao, J. J. Chem. Phys. 2001, 115, 5381. (63) Dai, D.-Q. J. Math. Phys. 2000, 41, 3086. (64) Olhede, S.; Walden, A. T. IEEE Trans. Signal Process. 2002, 50, 2661. (65) Fernadez, F. M.; Castro, E. A. Algebraic Methods in Quantum Chemistry and Physics; CRC Press: Boca Raton, FL, 1995. (66) Calais, J.-L. Int. J. Quantum Chem. 1996, 58, 541. (67) Modisette, J. P.; Nordlander, P.; Kinsey, J. L.; Johnson, B. Chem. Phys. Lett. 1996, 250, 485. (68) Wigner, E. P. Phys. ReV. 1932, 40, 749. ¨ zizmir, E.; Rosenbaum, M.; Zwiefel, P. F. J. Math. Phys (69) Imre, K.; O 1967, 5, 1097. (70) Wang, Z.; Heller, E. J. Phys. A: Math. Theor. 2009, 42, 285304, and references therein.

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