A Gaussian Wave Packet Propagation Approach to Vibrationally

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A Gaussian Wave Packet Propagation Approach to Vibrationally Resolved Optical Spectra at Non Zero Temperatures Chadagonda Sridhar Reddy, and Mallampalli Durga Prasad J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b00308 • Publication Date (Web): 01 Apr 2016 Downloaded from http://pubs.acs.org on April 1, 2016

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The Journal of Physical Chemistry

A Gaussian Wave Packet Propagation Approach to Vibrationally Resolved Optical Spectra at Non Zero Temperatures Ch. Sridhar Reddy and M. Durga Prasad∗ School of Chemistry, University of Hyderabad, Hyderabad 500 046, India E-mail: [email protected]

∗

To whom correspondence should be addressed

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Abstract An effective time dependent approach based on a method that is similar to the gaussian wave packet propagation(GWP) technique of Heller is developed for the computation of vibrationally resolved electronic spectra at finite temperatures in the harmonic, Franck-Condon / Hertzberg-Teller approximations. Since the vibrational thermal density matrix of the ground electronic surface and the time evolution operator on that surface commute, it is possible to write the spectrum generating correlation function as a trace of the time evolved doorway state. In the stated approximations, the doorway state is a superposition of the harmonic oscillator zero and one quantum eigenfunctions, and thus can be propagated by the GWP. The algorithm has an O(N 3 ) dependence on the number of vibrational modes. An application to pyrene absorption spectrum at two temperatures is presented as a proof of the concept.

Keywords vibronic spectra, finite temperature, electronic absorption and fluorescence.

1.

Introduction

Simulation and analysis of molecular spectra from first principles is a central challenge in theoretical molecular spectroscopy. In particular, electronic spectroscopy can provide valuable information on the structure and dynamics of molecules in the electronically excited states when properly analyzed. In view of its importance, several approaches have been developed for calculating the high resolution spectra of electronic transitions. 1–60 The computation of the vibrational fine structure of electronic spectra involves three steps. In the first step, one invokes the Born-Oppenheimer approximation and constructs the potential energy surfaces(PES) associated with the two(or more) electronic states involved in the transition. In the second step, the eigenstates of the two vibrational hamiltonians 2


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are constructed. Finally, the matrix elements of the dipole operator between the vibrational states of the two surfaces are calculated. These are then use to construct the stick spectrum. Given the complexity of the problem, it is inevitable that a large number of approximations are made to simplify the tasks that are part of the algorithm. Born-Oppenheimer approximation itself is the first of these approximations, and is valid as long as the PES of the two states under consideration are well separated from other surfaces in the relevant region of the nuclear coordinate space(Franck-Condon zone). The vibrational hamiltonians are generally anharmonic. To reduce the computational requirements, harmonic approximation is invoked. This is a reasonable approximation as long as the displacements in the nuclear coordinates, as a consequence of electronic excitation, are not too large. Similarly, the dipole operator is approximated by a few terms in its Taylor series in terms of the normal coordinates of the molecules. Even with these approximations, the calculation of the electronic spectrum is a non trivial exercise. While the quadratic hamiltonians are analytically soluble in terms of the normal coordinates, the two surfaces do not have the same set of normal coordinates. This is because the electronic charge distribution in the molecule changes during an electronic transition. As a result, the potential energy terms associated with different pairs of nuclei change non uniformly. This leads to changes in the hessian matrix in a non trivial manner. Thus the normal coordinates of the two surfaces do not coincide, an effect that is termed as the Dushinsky rotation. 61 Computation of the dipole matrix element between the vibrational states of the two electronic surfaces now becomes a many body problem as a consequence of such Dushinsky rotation. Given the importance of the problem, several approaches have been developed over time to carry out such calculations. Algorithms were developed to calculate the overlaps of vibrational eigenstates of two surfaces, termed the Franck-Condon factors(FCF), in the early days. These FCF are adequate to calculate the spectrum if the dipole moment matrix elements can be approximated by the zeroth order(constant) term in the Taylor series of the dipole operator. 1–14 This approximation is often termed as Franck-Condon(FC) approximation. It

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is adequate when the transition under consideration is strongly allowed. In contrast, when the dipole matrix elements are taken to the first order(including the linear terms in the normal coordinates), the approximation is called the Hertzberg-Teller(HT) approximation. It is necessary when the transition is week. 14,29,62–64 Even with analytical expressions for the FCF, the computation of the spectrum is resource hungry because the number of FCF increases exponentially with the number of vibrational degrees of freedom. Several prescreening methods have been developed to reduce the computational effort in such calculations. An alternative approach to tackle this problem was developed in the context of condensed phase spectroscopy. 15,16 Using the identity of delta function as a Fourier integral, these approaches rewrite the spectrum as the Fourier transform of the dipole dipole auto correlation function. The method was converted into a practical computational tool by Heller 17 using his gaussian wave packet propagation(GWP) technique. 18,19 Since then, several other time dependent approaches have been developed for the vibronic spectral calculations. 20–27 More recently, several authors have discussed methods to calculate the vibrational fine structure of electronic transitions at non zero temperatures. These approaches have also taken two lines, the time independent and time dependent routes. The time independent approaches follow the 0 K formulation closely, with Boltzmann averaging over the vibrational states of the initial surface. 28–35 Most of the time dependent approaches construct the correlation functions at finite temperatures directly. 39–60 The goal of the present work is to develop a time dependent approach for the calculation of the vibrationally resolved electronic spectra at non zero temperatures with in the framework of harmonic and FC/HT approximations using a GWP inspired technique that uses a more general ansatz than a simple gaussian, a superposition of the ground and first excited states of a multidimensional harmonic oscillator. It is based on the observation that the thermal density matrix of a harmonic oscillator is a gaussian when the cartesian normal coordinates are chosen as the basis vectors for the coordinate space. The doorway state,

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density matrix weighted dipole operator, thus becomes a linear polynomial in the normal coordinates multiplied by a gaussian in the Hertzberg-Teller approximation. Thus, it is possible to rewrite the spectrum generating correlation functions of the electronic transitions as the trace of a superposition of the time evolved harmonic oscillator eigenfunctions and the spectrum can be obtained as its Fourier transform. This methodology presented in section 2. Some numerical studies are presented in section 3 to illustrate the utility of the theory. Section 4 contains a summery of the present work along with a discussion on some possible directions to go beyond the harmonic approximation.

2.

Theory

We now derive working equations for the calculation of vibrational resolved absorption spectra at finite temperatures using a method that is similar to the gaussian wave packet propagation technique of Heller. Our starting point is the time dependent expression for the molar absorptivity coefficient, ǫ 51,60

ǫ = CA.ω.

Z

C(t) exp(iωt) dt ,

(1)

where CA is a constant 51 and the dipole-dipole correlation function C is defined as

C(t) = T r. dN exp(−iHe t) dN exp(+iHg t) ρg , ρg = exp(−βHg ) .

(2a) (2b)

Here β = 1/kβ T , is the inverse temperature, He and Hg are the vibrational hamiltonians of the excited and ground electronic states, and, dN is the nuclear dipole operator is defined as

dN = hΦe | d |Φg iel ,

5


(3)


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where Φe and Φg are the excited and ground electronic wave functions respectively, d is the full dipole operator and the integration is carried out over the electronic coordinates. Up to this point, we have only assumed that the Born-Oppenheimer approximation is valid. We now note that the time evolution operator and the thermal density operator on the ground electronic surface commute with each other, since both are generated by Hg . Moving the exp(+iHg t) to the right and explicitly showing the coordinate dependence, we arrive at ′

′

C(t) = T r. dN (q) exp(−iHe (q)t) dN (q) ρg (q, q ) exp(+iHg (q )t).

(4)

′

Here, ρg (q, q ) is the coordinate space representation of the thermal density operator ρg . Eq 4 can be compactly rewritten as, ′

′

C(t) = T r.dN (q) exp[−iH(q, q )t]ρD (q, q ) ,

(5)

where, ′

′

(6)

′

(7)

ρD (q, q ) = dN (q) ρg (q, q ) , and, ′

H(q, q ) = He (q) − Hg (q ) .

Note that we have made no approximations regarding either the hamiltonians or the dipole operators up to this point. We now make two further approximations. First, we assume that the two vibrational hamiltonians are harmonic. Using the dimensionless normal coordinates of the ground surface as the basis for the coordinate system, the two hamiltonians can be written as 1X ∂2 2 ωi 2 + ωi qi + Eg , Hg = − 2 i ∂qi X (1) ∂2 1X 1X (2) ωi 2 + qi Vij qj + Ee . V i qi + He = − 2 i ∂qi 2 i,j i 6


(8a) (8b)

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Here Eg and Ee are the electronic state energies of the ground and excited states at the equilibrium geometry of the ground surface. {ωi } is the set of normal mode frequencies of (1)

the ground surface. Vi

(2)

and Vij are the gradient and hessian matrix elements of the excited

surface at the ground state equilibrium geometry. The thermal density matrix ρg is now given by 65

ρ=

Y

ρk ,

(9a)

k

ρk =

mk ωk tanh(βωk )

12

exp

i −mk ωk h 2 ′2 ′ (qk + qk ) cosh(βωk ) − 2qk qk 2 sinh(βωk )

.

(9b)

Here mk = ωk−1 , since we are using dimensionless normal coordinates. Second, we assume that the dipole operator contains no more than linear terms in the normal coordinates (Hertzberg-Teller approximation),

dN = d 0 +

X

dk qk ,

(10a)

k

d0 = dN (q eq ) , ∂dN dk = . ∂qk qeq

(10b) (10c)

Using the definitions in eqs(5 − 10), the correlation function can be rewritten as ′

C(t) = T r.dN (q) .ρD (q, q , t) , ′

′

ρD (q, q , t) = exp(−iHt)ρD (q, q , 0) .

(11a) (11b)

We now note that every term in ρD is either a gaussian or a gaussian multiplied by a linear polynomial in q, that is, it is a superposition of ground and some of the first excited states of a harmonic oscillator, and, H is a quadratic Hamiltonian. As is well known, a gaussian propagated by a quadratic hamiltonian remains a gaussian. 18,19 What is not so well known, perhaps, is that all the harmonic oscillator eigenfunctions retain their structure 7



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when propagated by quadratic hamiltonians. 66 So, in the spirit of GWP, we posit the ansatz, ′ ρD (q, q , t) = d0 + dT (q − q0 ) exp i (q − q0 )T A (q − q0 ) + pT (q − q0 ) + γ .

(12)

Here A, is 2N × 2N dimensional matrix, p, (q − q0 ) and d are 2N element vectors consisting of both primed and unprimed variables. We use bold lowercase symbols to represent vectors, and, bold uppercase for matrices. Components of q0 are the displacements along the normal coordinates, {pk } are the associated momentum expectation values and γ is a complex variable containing the normalization and phase information. The evolution of A matrix reflects the effects of frequency renormalization and Dushinsky rotation during the time propagation of ρD . The governing equations for these parameters in the ansatz can be obtained in the usual way by comparing the left and right hand sides after substituting the ansatz 12 in the Schrödinger equation

′

iρ˙ D (q, q , t) = HρD .

(13)

These are, ∂H = , ∂pk p ∂H , p˙k = − ∂qk q0 2 X ∂ H 1 ˙ , Akl = −2 Akp ωp Apl − 2 ∂q ∂q 0 k l q p X Pk ωk Pk ′ iX γ˙ = − V (q0 , q0 ) + Akk ωk , 2 2 k k q˙k0

d˙0 = 0 ,

(14a) (14b) (14c) (14d) (14e)

d˙k = −2

X

Akl dl .

(14f)

l

Note that these equations are essentially the multi dimensional generalization of the equa-

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tions derived by Meyer. 66 These equations hold both for the unprimed and primed variables. The initial conditions for the variables are set from ρD at t = 0. Thus at t = 0, all the components of q0 and p are zero and the matrix elements of A are defined from eq 9b. In the case of primed variables, eqs 14(a,b) are explicitly given by ′

′

q˙k0 = pk ωk , ′

′

p˙k = −ωk qk0 , ′

′

′

′

with the initial conditions qk0 (0) = pk (0) = 0. Consequently, qk0 (t) = pk (t) = 0, for all values of t. The rest of the variables have to be integrated to obtain the time evolved ρD . As noted by Tannor and Heller earlier, 67 these equations have analytical solutions in terms of the gradient and hessian matrices for harmonic surfaces . We now turn to the evaluation of the correlation function C. This is given by

C(t) =

Z

dN (q)

′

lim ρD (q, q , t) dq ′

q →q

(15)

After some tedious but straight forward algebra we obtain, "

C(t) = exp(iΦ) × ∆0

s

X πD ∂ + ∆2k,l det(B) ∂Bk,l k,l

9


s

πD det(B)

#

,

(16)


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where, T

Φ = q0u App q0u + γ + iλT Bλ ,

(17a)

B = i(Auu + Aup + Apu + App ) ,

(17b)

T

b = i[2q0u (Apu + App ) + pT u],

(17c)

1 λ = − B−1 b , 2

(17d)

T T ∆0 = D0 D1 + λT du (0)DT 2 λ + (D0 D2 + du (0)D1 )λ,

(17e)

∆2 = d(0).D2 ,

(17f)

D0 = d0 + dT (0).q0u ,

(17g)

0 D1 = d0 + dT p (t).qu ,

(17h)

D2 = du (t) + dp (t).

(17i)

Here the subscripts u and p indicate restrictions to unprimed and primed variables. Thus, Auu etc are N × N matrices while pu etc are N -element vectors. Once the correlation function is calculated, the spectrum is obtained from eq 1 by Fourier transforming the correlation function.

3.

Results

In this section we present the absorption (S1 ← S0 ) spectrum of pyrene at two different temperatures as a proof of the concept of the theory developed in the previous section. Pyrene is a polyaromatic hydrocarbon molecule that is quite popular as a probe molecule in photo physical studies. It belongs to the D2h point group and has 72 vibrational modes. The ground (S0 ) and the first excited (S1 ) states belong to the A1g and B1u irreducible representations respectively. Because of its importance, several theoretical 68,69 and experimental 70,71 studies have been carried out on it. The parameters of the two vibrational hamiltonians have been obtained by optimizing the equilibrium geometry in both the states by the Density 10


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functional/time dependent density functional theory based approach. The hybrid B3LYP functional 72,73 was used with 6-311++G(2d,2p) basis. There is some computational evidence that this basis is best suited for vibrational calculations. 74 All calculations were performed with the G09 suite. 75 The hessian matrices for both the states were extracted from the output of G09 along with the optimized geometries, and these were used to obtain the ground state vibrational frequencies and normal coordinates as well as the V (1) and V (2) matrices of the excited state. Thus our calculation corresponds to the adiabatic Hessian(AH) model of Santoro and coworkers. 33 The S1 ← S0 transition is symmetry allowed, and is quite strong. Since the FranckCondon approximation is adequate to describe such strong transitions, 29 we have carried out our calculation in the Franck-Condon approximation, i.e. with dk ≡ 0. With all the parameters in place, we integrated the working equations(eqs 14) numerically with a fourth order Runge-Kutta algorithm with a step length of 0.363 femto seconds and the correlation function was sampled after every 3.63 femto seconds for 213 steps.This gave a resolution of δω = 1.12 cm−1 in the spectrum. The correlation function was multiplied πt by the window function cos 2T and a damping function exp(−αt) with a decay constant

α = 1.12 cm−1 to avoid aliasing and other limitations of the discreet Fourier transformation. The spectrum of pyrene at 873, 423 and 0 K are presented in the three panels of Figure 1.

The two high temperature spectra are compared with the experimental spectra of Thöny et. al. 71 As can be seen, the agreement between the experimental and computed spectra is quite good. The computed peak positions(frequencies), relative intensities and even the line widths match the experimental observations quite well. A few comments on the line widths at finite temperatures are in order here. Line broadening at higher temperatures occurs via two mechanisms. The first is the system bath interaction, which enables the molecule to exchange its energy with the thermal bath with which the molecule interacts, and forms a decay channel. This effect cannot be described by any algorithm, including the current

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one, in which the system bath interaction is not explicitly included in the hamiltonian. The second source of the line broadening is the appearance of hot bands at elevated temperatures. Consider, for example, a transition in a two mode system, (n , 0) ← (0 , 0) at 0 K, this transition occurs at nω1e beyond the 0-0 band with an intensity |hn , 0|0 , 0i|2 in the FC approximation. Here ω1e is the frequency of the first mode in the excited state. At an elevated temperature, β, there would be several hot bands corresponding to ((n + p) , q) ← (p , q) for all natural numbers p and q, appearing at nω1e +p(ω1e −ω1g )+q(ω2e −ω2g ) with intensity given by the associated FCF multiplied by the corresponding Boltzmann factor exp[−β(pω1g + qω2g )]. Since the vibrational frequencies ω1 and ω2 change during the transition, all these lines come at different energies, though, quite close to the 0 K transition energy of nω1e . As the temperature increases, the Boltzmann factors increase, and produce a closely packed set of hot bands. It is these closely packed hot bands that provide the second route line broadening at higher temperatures. The experimental spectrum of Thöny et. al 71 was measured in the gas phase. Consequently the line broadening seen in the experimental spectrum is predominantly due to the enhanced presence of hot bands at higher temperatures rather than the system bath interactions. As can be seen in Figure 1 the current formulation is able to estimate the temperature dependence of this hot band induced line broadening quite well. Comparison with the 0 K spectrum(Figure. 1, panel c) shows that, the small decay constant(α = 1.12 cm−1 ) that we used could not have produced the line broadening that we see at the elevated temperatures. There have been some earlier studies that indicate that the results of quantum dynamical calculations are nearly independent of the medium influence for large molecular systems. For example, Peluso and coworkers, 76–79 based on their quantum dynamical studies of the rate constants of the electron transfer reactions, noted that the electron transfer is primarily influenced by the intramolecular vibrations instead of the solvent modes for large molecular systems. In effect, their studies and ours presented here imply that for each active mode or transition, in our case, the rest of the molecular degrees of freedom act as the bath.

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4.

Conclusions

We have developed, in the present work, a GWP inspired approach to the calculation of vibronically spectra on harmonic surfaces at finite temperature spectra. Two important observations made such a development possible. The first observation is that the time evolution operator on the lower surface commutes with the thermal density operator on that surface, since both are generated by the same hamiltonian, Hg . This allowed us to rewrite the spectrum generating correlation function in terms of a time evolved density matrix weighted dipole operator, ρD , which satisfies a Schrödinger equation with an effective hamiltonian, H(eq 7 and 13). The second observation is that the density matrix for a harmonic system is a gaussian. Consequently, with in the restrictions of FC and HT approximations, ρD , that is to be propagated, is just a superposition of a harmonic oscillator ground and some of its first excited states. Since the harmonic oscillator eigenstates retain their coherence during time evolution under the influence of a general quadratic hamiltonian, it became possible to parametrize the time evolved doorway state, ρD , with an ansatz parametrized as the superposition of harmonic oscillator eigenstates with time dependent centroid (q0 ), scaling and mixing parameters (the A matrix elements) and the superposition coefficients (d0 and dk ). The advantage of this formulation is that the number of linearly independent parameters required to define the wave packet, the matrix elements of A , p , q0 , d and γ, is proportional to N 2 . Such an ansatz is capable of treating systems with fairly large number of vibrational degrees of freedom. The current method can be easily extended to the computation of fluorescence spectra and electronic circular dichroism. To obtain the fluorescence spectra for example, all that one has to do is to switch the roles of ground and excited states. 51 The essential structure of the correlation function remains unchanged. Similar arguments hold in the context of circular dichroism calculations as well. From a computational perspective the most intensive part of the present algorithm is the solution of the differential equation for A(eq 14(c)), the computation λ(eq 17(d)) and the determinant of B. All these three are O(N 3 ) processes. Consequently the overall algorithm 13



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scales cubically with the number of vibrational degrees of freedom. This cubic scaling behavior with respect to the number of vibrational modes is similar to other approaches to the calculation of the finite temperature spectra based on the correlation functions. 50,51,55 Our approach might be a little slower compared to the others since we have chosen to integrate eq 14 numerically, though, they have analytical solutions. 67 As we will discuss below, we were motivated to use the numerical integration procedure for eq 14 instead of using the analytical solutions to keep the flexibility of the program to go beyond the harmonic approximation. The algorithm is amenable to extensive parallelization, since almost all the operations are matrix multiplications. Even otherwise, the algorithm is quite fast. It required 31 minutes to calculate each of the spectra presented in Figure. 1 on a single core of an intel cpu running at 3.1GHz speed. We now compare our formulation with the other methods discussed in literature. The work that is closest in spirit to ours is due to Yan and Mukamel. 48 These authors also rewrite the dipole correlation function as the overlap of the time evolved density matrix weighted dipole operator with the dipole operator just as we did. At that point, they perform a Wigner transformation on the doorway state and propagate it in the phase space by Liouville equation. Since we work in the coordinate space through out, the working equations in the two approaches are different. In addition, they confined their attention to the FC approximation. More recently, Pollak and coworkers 58 developed a path integral based approach for the calculation Franck-Condon factors. The method was generalized by Shuai and coworkers 55 and applied to a wide class of systems. The same equations were derived by Borelli et. al 50 and Barone and coworkers 51 following alternative routes. All these approaches lead to expressions for the correlation functions that are similar to ours in that, they require matrix inversion and the computation of a determinant at each step. Both of these are O(N 3 ) processes. We now turn to possible extensions of the present approach to molecules in which the anharmonic terms in the PES make a noticeable contribution. 80–83 The molecular PES are

14


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generally better approximated in fewer terms if they are expanded in curvilinear coordinates. 80–86

The curvilinear coordinates mimic the molecular vibrations much more faithfully for

large displacements and thus require fewer terms to represent the PES. On the other hand, the kinetic energy operator is quite complex in such coordinates. 87–90 Because of the simplicity of the PES, it is tempting to use curvilinear coordinates for the computation of the vibronic spectra. Peluso and coworkers 82 compared the performance of the two coordinate systems. They found that in the quadratic approximation, the curvilinear coordinate system provided a significant improvement(for the photo electron spectrum of N H3 ) compared to the cartesian coordinate system. However, it was unable to reproduce all the features of the spectrum faithfully. On the other hand, a calculation in the cartesian coordinate system with a sixteenth order PES reproduced the spectrum to a very good extent. In view of these results, and, the relatively few systems for which comparative studies have been made, it is not clear, at this stage, which of the coordinate systems is to be preferred, particularly for large molecules such as pyrene. We now consider the possible framework for incorporating the effects of anharmonicity terms in the cartesian normal coordinate basis. The influence of anharmonic terms in the ground state hamiltonian can be incorporated in an approximate manner by constructing the thermal density matrix of the ground surface by a variational procedure 65,91–94 based on Feynman-Gibbs-Bogoliubov inequality,

F ≤ F0 + hV − V0 iβ ,

(18)

and using a gaussian ansatz for the density matrix. The effects of the anharmonic terms on the excited PES can be similarly treated approximately by time dependent variational principles. 95–97 The resulting equations look similar to eq 14. However the effective gradient and hessian(V (1) andV (2) ) are no longer constants in time. They are given by the quantum mechanical averages of the gradient and hessian elements at the current centroid position,

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making them explicitly time dependent. Consequently, analytical solutions for these equations are not possible and the wave packet no longer follows the classical trajectory. These equations then necessarily have to be numerically integrated. As noted in the earlier paragraph, the motivation to extend the current approach along these lines to incorporate the effects of anharmonicity was the reason for us to numerically integrate eq 14 rather than use the analytical solutions.

Acknowledgement We thank Prof. K. D. Sen for his kind interest in the present work. Ch. Sridhar Reddy acknowledges a sustaining fellowship from UGC and DST(JC Bose fellowship). Financial support from DST (HPCF, PURSE, IYC and FIST programs) and UGC(CAS and UPE programs) is gratefully acknowledged.

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(a)

1 0.8 0.6 0.4 0.2 1

(b)

0.8 0.6 0.4 0.2 0 1

(c)

0.8 0.6 0.4 0.2 0 -2000

-1000

0

1000

2000

3000

4000

Figure 1: Pyrene absorption (a) at 873 K (b) at 423 K and (c) at 0 K. For 423 K and 823 K experimental spectra adapted from 71 in doted line(in red) and computed spectra in continuous line (in thick black). 0 K spectrum in thin line(in blue). 17



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