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THE JOURNAL OF CHEMICAL PHYSICS 125, 124313 共2006兲

Matching-pursuit/split-operator-Fourier-transform simulations of excited-state nonadiabatic quantum dynamics in pyrazine Xin Chen and Victor S. Batistaa兲 Department of Chemistry, Yale University, New Haven, Connecticut 06520-8107

共Received 20 July 2006; accepted 25 August 2006; published online 28 September 2006兲 A simple approach for numerically exact simulations of nonadiabatic quantum dynamics in multidimensional systems is introduced and applied to the description of the photoabsorption spectroscopy of pyrazine. The propagation scheme generalizes the recently developed matching-pursuit/split-operator-Fourier-transform 共MP/SOFT兲 method 关Y. Wu and V. S. Batista, J. Chem. Phys. 121, 1676 共2004兲兴 to simulations of nonadiabatic quantum dynamics. The time-evolution operator is applied, as defined by the Trotter expansion to second order accuracy, in dynamically adaptive coherent-state expansions. These representations are obtained by combining the matching-pursuit algorithm with a gradient-based optimization method. The accuracy and efficiency of the resulting computational approach are demonstrated in calculations of time-dependent survival amplitudes and photoabsorption cross sections, using a model Hamiltonian that allows for direct comparisons with benchmark calculations. Simulations in full-dimensional potential energy surfaces involve the propagation of a 24-dimensional wave packet to describe the S1 / S2 interconversion of pyrazine after S0 → S2 photoexcitation. The reported results show that the generalized MP/SOFT method is a practical and accurate approach to model nonadiabatic reaction dynamics in polyatomic systems. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2356477兴 I. INTRODUCTION

Nonadiabatic quantum dynamics is a ubiquitous phenomenon common to a wide range of physical, chemical, and biological processes.1,2 Within the context of photochemical reactions in polyatomic systems, ultrafast nonadiabatic dynamics often determines the underlying molecular relaxation after UV-visible photoexcitation. These processes produce broad and structureless photoabsorption spectra, masking the spectroscopic features that could provide a structural and dynamical description of the system at the molecular level. Therefore, rigorous interpretations of spectroscopy often require detailed investigation using computational methods where quantum effects are explicitly considered. This paper introduces one such method as a generalization of the recently developed matching-pursuit/split-operator-Fouriertransform 共MP/SOFT兲 approach3–8 and its first application to a realistic molecular problem, the S1 / S2 interconversion dynamics of pyrazine after S0 → S2 photoexcitation. The photophysics of pyrazine provides a standard platform for studying the case of interest, ultrafast internal conversion responsible for a broadband photoabsorption spectrum with a rather diffuse superimposed structure. The underlying excited-state nonadiabatic dynamics, following S0 → S1共␲ , ␲*兲, S2共n , ␲*兲 photoexcitation, is also ideally suited to benchmark the capabilities of new theoretical methods, since it has been extensively investigated both experimentally9,10 and theoretically.11–17 Ab initio calculations have characterized the existence of a conical intersection between the S1 and S2 states, leading to ultrafast ina兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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tramolecular energy transfer.11 The experimental S2 photoabsorption spectrum has been reproduced by using a four-mode model Hamiltonian, after convoluting the resulting spectrum with an experimental resolution function,18 or explicitly including the effect of the remaining vibrational modes as a weakly coupled harmonic bath.12 These two models have allowed for direct comparisons between benchmark calculations and state-of-the-art semiclassical and quantum mechanical methods, including the multiconfigurational time dependent Hartree 共MCTDH兲 approach,12 the Herman-Kluk semiclassical initial value representation 共SCIVR兲 method,14 the multiple spawning quantum approach,15 the time dependent Gauss-Hermite discrete value representation 共TDGH-DVR兲 method,16 and the coupled coherent-state 共CCS兲 technique.17 Here, the capabilities of the generalized MP/SOFT approach are evaluated as applied to the description of the photoabsorption spectroscopy of pyrazine. Describing electronic nonadiabaticity in excited electronic states of polyatomic systems is a challenging problem. At the most fundamental level of theory, it requires solving the system of coupled differential equations defined by the multichannel time-dependent Schrödinger equation. This problem can be rigorously solved by applying numerically exact methods for wave-packet propagation,19–31 for example, approaches based on the split-operator-Fouriertransform 共SOFT兲 method,32–34 the Chebyshev expansion,35 and the short iterative Lanczos algorithms.36 These rigorous approaches demand storage space and computational effort that scale exponentially with the number of coupled degrees of freedom in the system. Their applicability is therefore limited to rather small systems 共e.g, molecular systems with less than three or four atoms兲. For this reason, studies of

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© 2006 American Institute of Physics

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X. Chen and V. S. Batista

nonadiabatic dynamics in polyatomic systems are usually based on approximate methods built around semiclassical and mixed quantum-classical treatments.37–51 However practical, these popular approaches rely upon ad hoc approximations whose consequences are often difficult to quantify in applications to complex 共i.e., nonintegrable兲 dynamics. It is, therefore, an important challenge to develop practical 共yet rigorous兲 methods for validating approximate approaches as well as for providing new insights into the nature of quantum dynamics at the most fundamental level of theory.3–8,12,14–17 The recently developed MP/SOFT method3–8 is a simple time-dependent propagation scheme for simulations of quantum processes in multidimensional systems. The approach is based on coherent-state expansions of the time-dependent wave packet generated according to the matching-pursuit algorithm.52 These compact representations allow for the efficient propagation of multidimensional wave packets by analytically applying the time-evolution operator as defined by the Trotter expansion to second order accuracy. The method thus bypasses the exponential scaling problem of the standard grid-based SOFT approach,32–34 a method that is usually limited by the capabilities of the fast-Fouriertransform 共FFT兲 algorithm.53 When compared to alternative time-dependent methods based on coherent-state expansions,15,54–68 the MP/SOFT method is usually easier to implement since it has the advantage of avoiding the need of propagating time-dependent expansion coefficients, a task that would require solving a coupled system of differential equations. Furthermore, the MP/SOFT method overcomes the truncation problem 共natural to propagation schemes where the basis set, or grid base, is defined a priori兲 by dynamically adapting the coherent-state expansion according to the desired propagation accuracy. As a technical matter, the MP/SOFT method also overcomes the usual numerical difficulties introduced by overcomplete basis sets,23 simply by implementing the matching-pursuit successive orthogonal decomposition scheme to reduce the underlying redundancy of the overcomplete representation. The main drawback of the MP/SOFT method is that it requires generating a new coherent-state expansion for each propagation step. The underlying computational task, however, can be trivially parallelized. In applications to multidimensional problems, the resulting algorithm overcomes the otherwise limitations of memory/storage bandwidth in terms of readily available processing power. The capabilities of the MP/SOFT method have been recently demonstrated in several simulations of adiabatic quantum dynamics, including tunneling in model systems with up to 20 coupled degrees of freedom5 and the excited-state intramolecular proton transfer in 2-共2⬘-hydroxyphenyl兲-oxazole as modeled by the propagation of a 35-dimensional wave packet.8 The MP/SOFT approach has also been applied to the description of thermalequilibrium density matrices, finite-temperature timedependent expectation values, and time-correlation functions.6 Furthermore, MP/SOFT has been applied to the description of nonadiabatic quantum dynamics in conjunction with a perturbational description of the nonadiabatic couplings.7 However, there remains the nontrivial question

as to whether the MP/SOFT method can be rigorously implemented to model nonadiabatic processes without relying on any perturbational approximation. This paper shows that indeed MP/SOFT can be effectively applied for rigorous simulations of nonadiabatic dynamics according to a numerically exact algorithm. The paper is organized as follows. Section II describes the generalization of the MP/SOFT method to simulations of nonadiabatic quantum dynamics, the calculation of photoabsorption spectra, and the model Hamiltonian. An alternative implementation of the method is described in the Appendix, where both the diabatic propagation and the coupling between wave-packet components are combined in a single matching-pursuit expansion, rather than in separate propagation steps. Section III presents the results for the reduced four-mode model Hamiltonian of pyrazine and the comparison to the corresponding results obtained according to benchmark grid-based SOFT calculations. Furthermore, Sec. III presents full-dimensional simulations of S1 / S2 internal conversion, as described by the nonadiabatic propagation of the 24-dimensional wave packet. Finally, Sec. IV summarizes and concludes. II. METHODOLOGY A. Generalized MP/SOFT method

The generalization of the MP/SOFT method, introduced in this paper, is based on the recursive application of the Trotter expansion to second order accuracy, ˆ

ˆ

ˆ

ˆ

e−iH2␶ ⬇ e−iH0共xˆ 兲␶e−iVc2␶e−iH0共xˆ 兲␶ ,

共1兲

where ␶ is a sufficiently short time increment. To keep the notation as simple as possible, all expressions are written in atomic units, so ប = 1. The system evolves according to the Hamiltonian ˆ =H ˆ + Vˆ , H 0 c

共2兲

where n

2 ˆ = pˆ + 兺 V 共x兲兩k典具k兩 H 0 k 2m k=1

共3兲

describes the propagation of individual wave-packet components evolving on the diabatic potential energy surfaces Vk. In addition, n

Vˆc = 兺

n

兺 Vkj共x兲兩j典具k兩

共4兲

k=1 j⫽k

represents the couplings between diabatic states. The paramˆ and Vˆ is described in Sec. II C. etrization of H 0 c In order to analyze the nonadiabatic dynamics of pyrazine, after S0 → S2 photoexcitation, we focus on the two-level 共n = 2兲 time-dependent wave packet, 兩⌿共x;t兲典 = ␸1共x;t兲兩1典 + ␸2共x;t兲兩2典,

共5兲

where ␸1共x ; t兲 and ␸2共x ; t兲 are the nuclear wave-packet components associated with the S1 and S2 states 兩1典 and 兩2典, respectively.

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Nonadiabatic quantum dynamics in pyrazine

The MP/SOFT implementation of Eq. 共1兲, as applied to the propagation of the initial state introduced by Eq. 共5兲, involves the following steps. 共I兲

Propagate ␸1共x ; t兲 and ␸2共x ; t兲 diabatically for time ␶,

冉

␸1⬘共x;t + ␶兲 ␸2⬘共x;t + ␶兲

冊冉 =

e−i关pˆ

⫻ 共II兲

冉

2/2m+V 共xˆ 兲兴␶ 1

0

0 e

冊

−i关pˆ 2/2m+V2共xˆ 兲兴␶

冊

␸1共x;t兲 . ␸2共x;t兲

共6兲

␸1⬙共x;t + ␶兲 ␸2⬙共x;t + ␶兲

with M⬅

␸l⬙共x;t兲 = 兺 c共l兲 ␹ j典, j 具x兩˜

冉

冊 冉 =M

␸1⬘共x;t + ␶兲 ␸2⬘共x;t + ␶兲

冊

where N

共7兲

,

具x兩˜␹ j典 ⬅ 兿 A j共k兲 k=1

cos共2Vc共xˆ 兲␶兲

− i sin共2Vc共xˆ 兲␶兲

− i sin共2Vc共xˆ 兲␶兲

cos共2Vc共xˆ 兲␶兲

冊

冊冉

−

冊

2

0 ␸1共x;t + 2␶兲 e−i关pˆ /2m+V1共xˆ 兲兴␶ = 2/2m+V 共xˆ 兲兴␶ ˆ −i关p ␸2共x;t + 2␶兲 2 0 e ⫻

冉

␸1⬙共x;t + ␶兲 ␸2⬙共x;t + ␶兲

冊

共9兲

.

Step 共III兲 can be combined with step 共I兲 of the next propagation time slice for all but the last propagation time increment. The diabatic propagation of individual wave-packet components 关steps 共I兲 and 共III兲兴 is based on the MP/SOFT implementation of the Trotter expansion, 2/2m+V 共xˆ 兲兴␶ k

⬇ e−iVk共xˆ 兲␶/2e−ipˆ

2/共2m兲␶ −iV 共xˆ 兲␶/2 k

e

,

共10兲

as described in Refs. 5 and 7. The underlying propagation scheme can be outlined as follows.

␸l⬘共x ; t兲 Decompose the target functions ⬅ e−iVl共xˆ 兲␶/2␸l共x ; t兲 into matching-pursuit coherent-state expansions, n

␸l⬘共x;t兲 ⬇ 兺 c共l兲 j 具x兩␹ j典.

共11兲

j=1

Here, 具x 兩 ␹ j典 are N-dimensional coherent states, N

冋

具x兩␹ j典 ⬅ 兿 A j共k兲exp − k=1

␥ j共k兲 共x共k兲 − x j共k兲兲2 + ip j共k兲 2

册

⫻共x共k兲 − x j共k兲兲 ,

冑

m m + i␶␥ j共k兲

共p j共k兲/␥ j共k兲 − i关x j共k兲 − x共k兲兴兲2 2/␥ j共k兲 + 2i␶/m

冊

p j共k兲 . 2␥ j共k兲

共14兲

The time-evolved wave-function components are thus n

−iV共x兲␶/2 ␸l共x;t + ␶兲 = 兺 c共l兲 具x兩˜␹ j典, j e

共15兲

j=1

.

e−i关pˆ

冉

⫻exp

共8兲

.

共13兲

j=1

Propagate ␸1⬙共x ; t + ␶兲 and ␸2⬙共x ; t + ␶兲 diabatically for time ␶,

冉

共1兲

n

Couple the two wave-packet components ␸1⬘共x ; t + ␶兲 and ␸2⬘共x ; t + ␶兲,

冉

共III兲

共2兲

共l兲 Eq. 共11兲, are defined as follows: c共l兲 1 ⬅ 具␹1 兩 ␸l⬘典 and c j j−1 共l兲 ⬅ 具␹ j 兩 ␸l⬘典 − 兺k=1ck 具␹ j 兩 ␹k典 for j = 2 − N. Apply the kinetic energy part of the Trotter expansion to ␸l⬘共x ; t兲 by first Fourier transforming the coherentstate expansion of ␸l⬘共x ; t兲 to the momentum representation, multiplying it by exp关−i共p2 / 2m兲␶兴 and finally computing the inverse Fourier transform of the product to obtain

共12兲

where A j共k兲 are normalization factors and ␥ j共k兲, x j共k兲, and p j共k兲 are complex-valued parameters selected according to the matching-pursuit algorithm, as described below. The expansion coefficients c共l兲 j , introduced by

which can be reexpanded in coherent states as in step 共1兲. In the present study, step 共II兲 can be performed analytically, since the wave-packet components ␸1⬘ and ␸2⬘ are represented as coherent-state expansions 关see Eq. 共11兲兴 and the couplings Vc共x兲 are expanded to second order accuracy in powers of x 共see Sec. II C兲. Therefore, the underlying computational task necessary for nonadiabatic propagation is completely reduced to generating the coherent-state expansions defined by Eq. 共11兲. In general, however, step 共II兲 requires a matching-pursuit coherent-state reexpansion of the target states ␸1⬙ and ␸2⬙. As described in the Appendix, such a reexpansion can be efficiently combined with the diabatic propagation in order to avoid an extra computational overhead. Coherent-state expansions are obtained by combining the matching-pursuit algorithm with a gradient-based optimization technique as follows. 共a兲

Starting from an initial trial coherent state 兩␹ j典, optimize the parameters x j共k兲, p j共k兲, and ␥ j共k兲 so that they locally maximize the overlap with the target state 兩␸l⬘共t兲典. In the present study, initial guess parameters ␥ j共k兲, x j共k兲, and p j共k兲 are chosen as defined by the basis elements of previous wave-packet representation 共or initial state兲 and they are locally optimized according to a steepest descent optimization procedure. The first basis function is the state 兩␹1典 that locally maximizes the overlap with the target state, 兩␸l⬘共t兲典 = c1兩␹1典 + 兩⑀1典,

共16兲

where c1 ⬅ 具␹1 兩 ␸l⬘共t兲典 and 兩⑀1典 is the residual state after projecting out 兩␹1典 from 兩␸l⬘共t兲典. Note that 兩⑀1典 is or-

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共b兲

J. Chem. Phys. 125, 124313 共2006兲

X. Chen and V. S. Batista

thogonal to 兩␹1典, due to the definition of c1. Therefore, 兩⑀1兩 艋 兩␸l⬘共t兲兩. Subdecompose 兩⑀1典, analogously, as follows; 兩⑀1典 = c2兩␹2典 + 兩⑀2典,

共17兲

where c2 ⬅ 具␹2 兩 ⑀1典. Note that 兩⑀2 兩 艋 兩⑀1兩, since 兩␹2典 is orthogonal to 兩⑀1典. Step 共b兲 is repeated each time on the following residue. After n successive orthogonal projections, the norm of the resulting residue 兩⑀n兩 is smaller than a desired precision ␧, 兩 ⑀ n兩 =

冑

of vibrational modes explicitly considered in the model 共i.e., N = 4 in the reduced model system and N = 24 in the fulldimensional model兲. Earlier calculations have simulated the experimental broadening of the spectra, due to finite resolution of the spectrometer, by convoluting the photoabsorption spectra with a suitable experimental resolution function. In order to allow for direct comparisons with these earlier theoretical studies, the survival amplitude is damped with a single exponential,12

冉 冊

h共t兲 = exp −

n

1 − 兺 c2j 艋 ␧,

共18兲

j=1

defining the truncation of the matching-pursuit expansion. The resulting nonadiabatic propagation scheme is a natural generalization of the MP/SOFT method described in Ref. 5. In fact, in the limiting case of adiabatic quantum dynamics 共i.e., V jk共x兲 = 0兲, the two methods are identical. The generalized MP/SOFT method also reduces to the perturbational approach introduced in Ref. 7 when the coupling matrix elements, introduced by Eq. 共7兲, are expanded to first or second order accuracy in powers of the propagation time increment ␶. In contrast to such a perturbational scheme, however, the novel approach is rigorous and unitary.

兩t兩 , ␶

共22兲

leading to a Lorentzian broadening of the spectrum with full width at half maximum 共FWHM兲 ⌫ = 2 / ␶. The experimental broadening is reproduced by using ␶ = 150 fs for the 24dimensional model, and ␶ ⬇ 30 fs for the reduced fourdimensional system. Furthermore, in order to reduce the effects of truncation due to the finite-time propagation, the calculated spectrum is convoluted with a sinc function which has a FWHM= 2.5/ T, where T = 150 fs is the finite propagation time. The convolution is implemented by smoothly zeroing the survival amplitude at t = T with the damping function12

冉 冊

g共t兲 = cos

␲t . 2T

共23兲

B. Calculation of photoabsorption spectra

The S2 photoabsorption spectrum I共␻兲 is computed as the Fourier transform of the survival amplitude C共t兲, I共␻兲 ⬀ ␻

冕

⬁

dtC共t兲ei共␻+⑀0兲t ,

共19兲

⬁

where ⑀0 denotes the energy of the ground vibrational state of pyrazine and ␻ = 2␲c / ␭. The time-dependent survival amplitude, ˆ

C共t兲 = 具⌿共0兲兩e−iHt兩⌿共0兲典 =

冕

C. Model Hamiltonian

A detailed analysis of the model Hamiltonian, used for the reported MP/SOFT simulations, is readily available in Ref. 12. Here, the model is described only briefly. The Hamiltonian, introduced by Eq. 共2兲, is defined as follows: 2 ˆ = 兺 − 1 ⳵ 共兩1典具1兩 + 兩2典具2兩兲 + 兺 1 m ␻2Q2共兩1典 H 0 j j j 2m j ⳵Q2j j j 2

⫻具1兩 + 兩2典具2兩兲 + ⌬共兩2典具2兩 − 兩1典具1兩兲 +

dx⌿쐓共x;− t/2兲⌿共x;t/2兲, 共20兲

is obtained by overlapping the time-evolved states ˆ ˆ ⌿共x ; t / 2兲 = e−iHt/2⌿共x ; 0兲 and ⌿共x ; −t / 2兲 = eiHt/2⌿共x ; 0兲, after propagating the initial state ⌿共x ; 0兲 forward and backward in time on the nonadiabatically coupled excited electronic states. Here, ⌿共x ; 0兲 is the initial ground-state wave function multiplied by the transition dipole moment which is assumed to be constant as a function of nuclear coordinates 共Condon approximation兲. The initial state is defined according to Eq. 共5兲 in terms of the two ground vibrational state wave-packet components,

␸1共x;0兲 = 0, N

␸2共x;0兲 = 兿 j=1

冉冊 1 ␲

共21兲 1/4

2

e−x j /2 ,

associated with the S1 and S2 states, respectively. Here, N is the dimensionality of the system, as defined by the number

兺

共ai兩1典具1兩

i苸G1

+ bi兩2典具2兩兲Qi +

兺

共i,j兲苸G2

共ai,j兩1典具1兩 + bi,j兩2典具2兩兲QiQ j 共24兲

and Vˆc =

兺

ci共兩1典具2兩 + 兩2典具1兩兲Qi +

i苸G3

⫻具1兩兲QiQ j .

兺

共i,j兲苸G4

ci,j共兩1典具2兩 + 兩2典 共25兲

The parameters introduced by these equations have been obtained at the complete-active-space self-consistent-field 共CASSCF兲 ab initio level,18 including a total of 102 coupling constants ai, bi, ci, ai,j, bi,j, and ci,j, explicitly describing the 24 vibrational modes of pyrazine.12 It is important to note that these parameters have been kindly supplied by Raab et al. and are slightly different from those reported in Ref. 12. In addition, to facilitate the comparison with experimental data, the 24-dimensional potential energy surfaces were shifted in energy by 4.06 eV.

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124313-5

Nonadiabatic quantum dynamics in pyrazine

The first and second terms, introduced by Eq. 共24兲, define the harmonic expansion of the diabatic surfaces, where ␻ j are the experimental ground-state vibrational frequencies eq −1/2 are the dimenand m j = ␻−1 j . Further, Q j = 共x j − x j 兲 / 共m j␻ j兲 sionless normal-mode coordinates. The third term in Eq. 共24兲 introduces the couplings between the S1 and S2 potential energy surfaces at the ground-state equilibrium configuration 共Q = 0兲. The fourth and fifth terms in Eq. 共24兲 include the linear and quadratic contributions to the diabatic-state expansions, where G1 and G2 indicate the set of modes having Ag and B2g symmetries, respectively. The nonadiabatic couplings are described to second order accuracy, as given by Eq. 共25兲, where G3 represents the modes with symmetry B1g that linearly couple the S1 and S2 states and G4 is the set of all pairs of modes the product of which has B1g symmetry, including the combinations Ag ⫻ B1g, B2g ⫻ B3g, Au ⫻ B1u, and B2u ⫻ B3u. The reduced four-mode model is constructed by following earlier work,12 including only the vibronic coupling mode ␯10a and the three totally symmetric modes with the strongest linear coupling parameters, ␯6a, ␯1, and ␯9a. In addition, to facilitate the comparison with experimental data, the four-dimensional potential energy surfaces were shifted in energy by 3.94 eV. III. RESULTS

This section demonstrates the capabilities of the generalized MP/SOFT method, introduced in Sec. II, as applied to the description of the ultrafast internal conversion of pyrazine at the S1 / S2 conical intersection. Figure 1 shows the comparison between the real and imaginary parts of the survival amplitude C共t兲 = 具⌿0 兩 ⌿t典, during the first 150 fs of dynamics, as predicted by the fourmode model Hamiltonian introduced in Sec. II C. Figure 1 also compares the MP/SOFT results 共dots兲 with benchmark grid-based calculations 共solid lines兲. MP/SOFT calculations are converged by using dynamically adaptive coherent-state expansions with n = 200 basis functions. Grid-based SOFT calculations required representations on basis set of 1 048 576 grid points, defined in the range of normal mode coordinates 兩Q10a兩 艋 7.0 a.u., 兩Q6a兩 艋 7.0 a.u., 兩Q1兩 艋 7.0 a.u., and 兩Q9a兩 艋 7.0 a.u.. The integration time step was defined as ␶ = 0.25 fs for both grid-based and MP/SOFT calculations. Figure 2 shows the photoabsorption spectra, obtained by Fourier transform of the survival amplitudes presented in Fig. 1, as described in Sec. II B. The efficiency of the MP/ SOFT method is demonstrated in terms of the rather modest number of basis functions 共n = 200兲 required by the matching-pursuit coherent-state propagation, as compared to the rather demanding 共memory/storage兲 requirements of the corresponding grid-based calculations 共1 048 576 grid points兲. The excellent agreement between MP/SOFT and benchmark grid-based calculations, during the whole dynamical range of S1 / S2 internal conversion, demonstrates that the generalized MP/SOFT methodology introduced in this paper is a numerically exact and practical approach for the description of nonadiabaticity. The ultrafast falloff of the overlap

J. Chem. Phys. 125, 124313 共2006兲

FIG. 1. Comparison between the real part 共a兲 and the imaginary part 共b兲 of the survival amplitude C共t兲 = 具⌿0 兩 ⌿t典, during the first 150 fs of dynamics after S0 → S2 photoexcitation of pyrazine, as predicted by the four-mode model Hamiltonian described in Sec. II C. The spectra are convoluted with a survival amplitude damping factor ␶ = 30 fs, as described in Sec. II B. MP/SOFT results 共dots兲 are compared to benchmark grid-based calculations 共solid lines兲.

between the initial and the time-evolved wave functions, within the first 15 fs of dynamics 共Fig. 1兲, is caused by the wave packet moving away from its initial state and internal converting at the conical intersection between the S1 and S2 excited states. This ultrafast relaxation is responsible for the broad photoabsorption band in the frequency domain 共Fig. 2兲. All the subsequent dynamics, at longer times, determines the diffuse structure superimposed to the broadband in the frequency domain. The longer time dynamics gives rise to a first recurrence feature at t ⬇ 30 fs 共the subsequent at t ⬇ 60 fs, etc.兲 due to the partial recurrencies of the timeevolved wave function to the initial state. This phenomenon, which is common to other polyatomic systems where the

FIG. 2. Photoabsorption spectra of pyrazine predicted by the four-mode model Hamiltonian described in the text 共Sec. II C兲. Benchmark quantum calculations 共solid line兲 are compared to MP/SOFT results 共dots兲 implemented as described in Sec. II A. Both spectra are convoluted with a survival amplitude damping factor ␶ = 30 fs, as described in Sec. II B.

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X. Chen and V. S. Batista

J. Chem. Phys. 125, 124313 共2006兲

FIG. 5. Comparison between the photoabsorption spectra of pyrazine predicted by the 24-mode model 共␶ ⬇ 150 fs, n = 400兲 and the four-mode model 共␶ ⬇ 30 fs, n = 200兲 described in the text.

FIG. 3. 共a兲 Comparison between the absolute values of C共t兲, predicted by the MP/SOFT methodology during the first 150 fs of dynamics after S0 → S2 photoexcitation of pyrazine, with n = 200, 300, and 400 coherent-state basis functions. 共b兲 Absolute value of survival amplitude differences obtained with n = 300 and n = 400, illustrating the convergence rate and the quality of the calculations.

potential energy surfaces responsible for excited-state relaxation are coupled at a conical intersection,69 is quantitatively described by the MP/SOFT approach. Figures 3 and 4, respectively, report the corresponding calculations of survival amplitudes and photoabsorption spectra for the 24-dimensional model of pyrazine. Figure 3, panel 共a兲, shows the comparison of the absolute values of C共t兲 as predicted by the MP/SOFT methodology with coherent-state expansions based on n = 200, 300, and 400 coherent-state basis functions. Figure 3, panel 共b兲, illustrates the convergence rate and the quality of the calculations by showing the absolute value of the differences between survival amplitudes obtained with n = 300 and n = 400. It is shown that the differences in C共t兲 obtained by increasing the number of basis functions in the model oscillate, reaching a

plateau of maximum values at about 100 fs. The resulting changes in the spectrum, arising from these differences, are shown in Fig. 4. The comparison of the spectra and survival amplitudes, obtained with an increasing number of coherent states, indicates that the calculations scale only modestly with the dimensionality of the problem. In fact, even calculations based on 200 basis functions for the 24-dimensional model provide a very good approximation to fully converged results. These results favorably compare to the corresponding results obtained with state-of-the-art time-dependent methods, including MCTDH calculations based on 2 771 440 configurations and 502 200 configurations.12 Regarding computational expense, computations of survival amplitudes for 150 fs required 620 propagation steps. Both four-dimensional and 24-dimensional calculations, based on n = 200 coherent states, were completed within 24 h on a parallel computer with 48 Opteron processors. Each processor runs at a clock speed of 2.2 GHz. Analogous calculations for the 24-dimensional model with n = 300 and n = 400 basis functions required 80 and 160 h, respectively, on the same parallel computer. For completeness, Fig. 5 shows the calculated photoabsorption spectra, obtained for the four-mode model 共␶ ⬇ 30 fs, n = 200兲 as compared to the photoabsorption of the 24-mode model 共␶ ⬇ 150 fs, n = 400兲. This comparison indicates that, in the full-dimensional model, the remaining 20 vibrational modes influence the underlying spectroscopy of the reduced four-mode model similarly to the effect of a phenomenological broadening. IV. CONCLUDING REMARKS

FIG. 4. Comparison between the pyrazine photoabsorption spectra obtained by implementing the MP/SOFT methodology with n = 200, 300, and 400 coherent-state basis functions. The spectra are convoluted with a survival amplitude damping factor ␶ = 150 fs, as described in Sec. II B.

We have introduced a numerically exact 共nonperturbative兲 approach for simulations of nonadiabatic quantum dynamics. The resulting approach is a generalization of the recently developed MP/SOFT algorithm.3–8 It is based on the recursive application of the time-evolution operator, as defined by the Trotter expansion to second order accuracy, in dynamically adaptive coherent-state expansions generated with the matching-pursuit algorithm. The coherent-state representations allow for an analytic evaluation of the Trotter expansion, bypassing the exponential scaling problem of the grid-based SOFT method. The resulting approach is unitary

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J. Chem. Phys. 125, 124313 共2006兲

Nonadiabatic quantum dynamics in pyrazine

APPENDIX: NONADIABATIC MP/SOFT

and rigorous and becomes identical to the original MP/SOFT formulation5 in the adiabatic limit 共i.e., when the couplings are negligible兲. In addition, the method can be reduced to the recently developed perturbational scheme for simulations of nonadiabatic dynamics,7 simply by expanding the coupling propagation matrix elements to first or second order accuracy in the propagation time increment. We have demonstrated how to apply the generalized MP/ SOFT method for computations of survival amplitudes and photoabsorption spectra of pyrazine, as described by a fourmode model Hamiltonian that allows for direct comparisons to benchmark grid-based calculations and a 24-mode model that describes the nonadiabatic dynamics at the S1 / S2 conical intersection in full dimensionality. The reported computational results demonstrate that the MP/SOFT method is a practical and accurate approach to model nonadiabatic reaction dynamics in polyatomic systems, since both the accuracy and efficiency compare quite favorably with state-ofthe-art time-dependent methods.

This section introduces the implementation of the MP/ SOFT approach for simulations of nonadiabatic quantum dynamics, according to a propagation scheme where both the diabatic propagation and the mixing of wave-packet components are combined in a single matching-pursuit reexpansion procedure. The advantage of this alternative approach, relative to the method introduced in Sec. II, is that a single matching-pursuit reexpansion is required per propagation step, even for problems where the coupling of wave-packet components cannot be performed analytically. The approach is based on the embedded form of the Trotter expansion, ˆ

e−iH2␶ ⬇ e−i共pˆ

共III兲

冉

ˆ

ˆ

冉

冉

冊冉

2

冉

⫻

冉

␸1⬙共x;t + ␶兲 ␸2⬙共x;t + ␶兲

␸1⬘共x;t + ␶兲 ␸2⬘共x;t + ␶兲

冊

.

ˆ

ˆ

␸1⬙共x;t + ␶兲 ␸2⬙共x;t + ␶兲

冊

=

2/2m兲␶

0

0 e

−i共pˆ 2/2m兲␶

冊冉 冊

冊 冉 =M

␸1⬘共x;t + ␶兲 ␸2⬘共x;t + ␶兲

冊

,

␸1共x;t兲 . ␸2共x;t兲

共A3兲

共A4兲

.

e−iVc2␶ = D† with D = D† ⬅

共A5兲

In practice, however, step 共III兲 is combined with step 共I兲 of the next propagation time slice for all but the last propagation time increment. Note that Eq. 共A3兲 results, in analogy to Eq. 共7兲, from considering that

冊冉

e−i共pˆ

with

cos共2Vc共xˆ 兲␶兲e−iV2共xˆ 兲2␶

冊

,

Couple the two wave-packet components ␸1⬘共x ; t + ␶兲 and ␸2⬘共x ; t + ␶兲

共II兲

Analytically propagate ␸1⬙共x ; t + ␶兲 and ␸2⬙共x ; t + ␶兲 for time ␶, according to the free-particle propagator, 0 ␸1共x;t + 2␶兲 e−i共pˆ /2m兲␶ = 2 ␸2共x;t + 2␶兲 0 e−i共pˆ /2m兲␶

e

共A2兲

ˆ

ˆ

e

Analytically propagate ␸1共x ; t兲 and ␸2共x ; t兲 for time ␶, according to the free-particle propagator,

共I兲

− i sin共2Vc共xˆ 兲␶兲e−i共V1共xˆ 兲+V2共xˆ 兲兲␶

− i sin共2Vc共xˆ 兲␶兲e−i共V1共xˆ 兲+V2共xˆ 兲兲␶

e

ˆ = pˆ2 / 共2m兲 + Vˆ + Vˆ , with Vˆ = V 共x兲兩1典具1兩 + V 共x兲兩2典 where H 0 c 0 1 2 ⫻具2兩 and Vˆc = Vc共x兲兩1典具2兩 + Vc共x兲兩2典具1兩. The MP/SOFT implementation of Eq. 共A1兲, as applied to the propagation of the initial state introduced by Eq. 共5兲, involves the following steps.

One of the authors 共V.S.B.兲 acknowledges a generous allocation of supercomputer time from the National Energy Research Scientific Computing 共NERSC兲 center and financial support from Research Corporation, Research Innovation Award No. RI0702, a Petroleum Research Fund Award from the American Chemical Society PRF No. 37789-G6, a junior faculty award from the F. Warren Hellman Family, the National Science Foundation 共NSF兲 Career Program Award CHE No. 0345984, the NSF Nanoscale Exploratory Research 共NER兲 Award ECS No. 0404191, the Alfred P. Sloan Fellowship 共2005-2006兲, a Camille Dreyfus Teacher-Scholar Award for 2005, and a Yale Junior Faculty Fellowship in the Natural Sciences 共2005-2006兲.

cos共2Vc共xˆ 兲␶兲e−iV1共xˆ 兲2␶

e

共A1兲

ACKNOWLEDGMENTS

M⬅

ˆ 共x兲␶ −iVˆ 共x兲2␶ −iVˆ 共x兲␶ −i共pˆ2/2m兲␶ 2/2m兲␶ −iV 0 c 0

冉

冉

eiVc共x兲2␶ 0

0 e

−iVc共x兲2␶

− 1/冑2 1/冑2 1/冑2

1/冑2

冊

冊

D,

,

共A6兲

共A7兲

since e−iVc2␶ = 1 + 共− iVc2␶兲 +

1 共− iVc2␶兲2 + ¯ 2!

共A8兲

and

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124313-8

Vc ⬅

J. Chem. Phys. 125, 124313 共2006兲

X. Chen and V. S. Batista

冉

0

Vc共x兲

Vc共x兲

0

冊 冉 = D†

− Vc共x兲

0

0

Vc共x兲

冊

32

D,

共A9兲

with DD† = 1. 1

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