^{1,a)}, K. Giri

^{2}and G. A. Worth

^{2,a)}

### Abstract

The Gaussian-based multiconfiguration time-dependent Hartree (G-MCTDH) method is applied to calculate the absorptionspectrum of the pyrazine molecule, whose diffuse structure results from the ultrafast nonadiabatic dynamics at the conical intersection. The 24-mode second-order vibronic-coupling model of Raab *et al.* [J. Chem. Phys.110, 936 (1999)] is employed, along with 4-mode and 10-mode reduced-dimensional variants of this model. G-MCTDH can be used either as an all-Gaussian approach or else as a hybrid method using a partitioning into primary modes, treated by conventional MCTDH basis functions, and secondary modes described by Gaussian particles. Comparison with standard MCTDH calculations shows that the method converges to the exact result. The variational, nonclassical evolution of the moving Gaussian basis is a key element in obtaining convergence. For high-dimensional systems, convergence is significantly accelerated if the method is employed as a hybrid scheme.

This work was supported by ANR Project No. ANR-NT05-3-42315 and by the EPSRC through CCP6 (Collaborative Computational Project on Molecular Quantum Dynamics).

I. INTRODUCTION

II. THEORY

A. G-MCTDH: Wave function ansatz

B. Dynamical equations

1. Expansion coefficients

2. Primary-mode single-particle functions

3. Secondary-mode parametrized functions

C. Structure of the dynamical equations

D. Hamiltonian matrix elements

E. Multiconfigurational variational Gaussian wavepacket dynamics

F. Relation to other time-dependent Gaussian basis set methods

III. NUMERICAL ASPECTS

A. Initial conditions

B. Regularization schemes

C. The constant mean-field integration scheme

IV. NONADIABATIC DYNAMICS AT THE PYRAZINE CONICAL INTERSECTION

A. vibronic-coupling model

B. Four-mode model

C. Ten-mode model

D. 24-mode model

V. CONCLUSIONS

### Key Topics

- Subspaces
- 15.0
- Wave functions
- 12.0
- Absorption spectra
- 9.0
- Basis sets
- 9.0
- Equations of motion
- 9.0

## Figures

For the four-mode model (Sec. IV B), the figure illustrates the convergence properties of the all-Gaussian (vMCG) method and the comparison to classically evolving GWP particles. (a) vMCG calculations with 40 (dashed trace), 80 (full thin trace), and 200 (full bold trace) 4D GWPs (i.e., 20, 40, and 100 GWPs per state). (b) For comparison, analogous calculations with 200 4D (dashed trace), 3368 1D (full thin trace), and 1D (full bold trace) classically evolving GWPs are shown.

For the four-mode model (Sec. IV B), the figure illustrates the convergence properties of the all-Gaussian (vMCG) method and the comparison to classically evolving GWP particles. (a) vMCG calculations with 40 (dashed trace), 80 (full thin trace), and 200 (full bold trace) 4D GWPs (i.e., 20, 40, and 100 GWPs per state). (b) For comparison, analogous calculations with 200 4D (dashed trace), 3368 1D (full thin trace), and 1D (full bold trace) classically evolving GWPs are shown.

Autocorrelation functions of the 10-mode model of Sec. IV B (black trace) as compared with the full 24-mode model of Sec. IV C [gray (red) dashed trace] using standard MCTDH calculations. The structure of the first two recurrences is reproduced by the reduced-dimensional ten-mode model.

Autocorrelation functions of the 10-mode model of Sec. IV B (black trace) as compared with the full 24-mode model of Sec. IV C [gray (red) dashed trace] using standard MCTDH calculations. The structure of the first two recurrences is reproduced by the reduced-dimensional ten-mode model.

Convergence properties of the -mode G-MCTDH hybrid calculations for the 10-mode model of Sec. IV B. (a) The results of two G-MCTDH calculations are compared with the standard MCTDH (exact) result. The first calculation (dashed line), i.e., G-MCTDH I of Table II, employs the same number of particles as compared with the standard MCTDH calculation (thin solid line). The second calculation (bold solid line) has an augmented GWP basis in the secondary subspace (i.e., G-MCTDH II of Table II) and can be considered converged. (b) For comparison, a vMCG calculation (dashed line) with 20 GWPs per particle is shown (see vMCG of Table II). The G-MCTDH II calculation (bold solid line) and reference MCTDH calculation (thin solid line) are also reproduced. The vMCG calculation is significantly harder to converge than the G-MCTDH hybrid calculations.

Convergence properties of the -mode G-MCTDH hybrid calculations for the 10-mode model of Sec. IV B. (a) The results of two G-MCTDH calculations are compared with the standard MCTDH (exact) result. The first calculation (dashed line), i.e., G-MCTDH I of Table II, employs the same number of particles as compared with the standard MCTDH calculation (thin solid line). The second calculation (bold solid line) has an augmented GWP basis in the secondary subspace (i.e., G-MCTDH II of Table II) and can be considered converged. (b) For comparison, a vMCG calculation (dashed line) with 20 GWPs per particle is shown (see vMCG of Table II). The G-MCTDH II calculation (bold solid line) and reference MCTDH calculation (thin solid line) are also reproduced. The vMCG calculation is significantly harder to converge than the G-MCTDH hybrid calculations.

(a) Autocorrelation function and (b) spectrum for the nearly converged G-MCTDH calculation (black bold solid line), i.e., G-MCTDH II of Table II, as compared with the standard MCTDH result [gray (red) dashed line]. The spectra are convoluted with a damping function with decay constant .

(a) Autocorrelation function and (b) spectrum for the nearly converged G-MCTDH calculation (black bold solid line), i.e., G-MCTDH II of Table II, as compared with the standard MCTDH result [gray (red) dashed line]. The spectra are convoluted with a damping function with decay constant .

Spectrum for a small G-MCTDH calculation [gray (red) dashed trace], i.e., G-MCTDH III of Table II, as compared with the standard MCTDH result (black trace). This illustrates that good results can be obtained even with a cheap calculation.

Spectrum for a small G-MCTDH calculation [gray (red) dashed trace], i.e., G-MCTDH III of Table II, as compared with the standard MCTDH result (black trace). This illustrates that good results can be obtained even with a cheap calculation.

Autocorrelation function for the G-MCTDH 24-mode calculation (bold black trace), i.e., G-MCTDH of Table IV, as compared with a standard MCTDH calculation [gray (red) dashed trace], i.e., MCTDH of Table IV Panels (a) and (b) show different time intervals, i.e., 150 fs vs 300 fs. Both calculations are near convergence.

Autocorrelation function for the G-MCTDH 24-mode calculation (bold black trace), i.e., G-MCTDH of Table IV, as compared with a standard MCTDH calculation [gray (red) dashed trace], i.e., MCTDH of Table IV Panels (a) and (b) show different time intervals, i.e., 150 fs vs 300 fs. Both calculations are near convergence.

Spectra obtained by Fourier transformation of the autocorrelation functions of Fig. 6, i.e., G-MCTDH 24 mode of Table IV [gray (red) dashed trace] vs MCTDH 24 mode of Table IV (black trace). The spectra are convoluted with a damping function with a decay constant of .

Spectra obtained by Fourier transformation of the autocorrelation functions of Fig. 6, i.e., G-MCTDH 24 mode of Table IV [gray (red) dashed trace] vs MCTDH 24 mode of Table IV (black trace). The spectra are convoluted with a damping function with a decay constant of .

Comparison of the G-MCTDH 24-mode calculation (solid line) with the experimental spectrum of Ref. 19 (dotted line). In panel (a), the (high-frequency side) and (low-frequency side) regions are shown, while panel (b) is centered on the region. Note that intensities in the region are not correctly reproduced since the initial condition used in the wavepacket propagation is restricted to one of the diabatic states.

Comparison of the G-MCTDH 24-mode calculation (solid line) with the experimental spectrum of Ref. 19 (dotted line). In panel (a), the (high-frequency side) and (low-frequency side) regions are shown, while panel (b) is centered on the region. Note that intensities in the region are not correctly reproduced since the initial condition used in the wavepacket propagation is restricted to one of the diabatic states.

Diabatic populations obtained for the G-MCTDH 24-mode calculation of Table IV (black trace) as compared with a standard MCTDH calculation of Table IV [gray (red) dashed trace].

Diabatic populations obtained for the G-MCTDH 24-mode calculation of Table IV (black trace) as compared with a standard MCTDH calculation of Table IV [gray (red) dashed trace].

## Tables

Computational resources required by the four-mode vMCG calculations. is the number of 4D GWPs used to describe the wavepacket in the two electronic states.

Computational resources required by the four-mode vMCG calculations. is the number of 4D GWPs used to describe the wavepacket in the two electronic states.

MCTDH vs G-MCTDH basis sets for ten-mode calculations. The round brackets denote the combination of vibrational modes and the square brackets the number of SPFs or GWPs used for the representation of the wave function in the and states. The number of modes in one combination defines the dimensionality of the corresponding SPFs. These SPFs are represented on a grid whose size is given by the product of the number of grid points used for each mode of the corresponding combination.

MCTDH vs G-MCTDH basis sets for ten-mode calculations. The round brackets denote the combination of vibrational modes and the square brackets the number of SPFs or GWPs used for the representation of the wave function in the and states. The number of modes in one combination defines the dimensionality of the corresponding SPFs. These SPFs are represented on a grid whose size is given by the product of the number of grid points used for each mode of the corresponding combination.

Computational resources required by the ten-mode calculations detailed in Table II. “Num (A-vec)” is the number of expansion coefficients, and “Num (SPFs/GWPs)” is the number of grid points and GWP parameters. Memory and CPU are the computer resources on an Intel Pentium Quad Core Q6600 (2.4 GHz, 8 Gbytes) machine.

Computational resources required by the ten-mode calculations detailed in Table II. “Num (A-vec)” is the number of expansion coefficients, and “Num (SPFs/GWPs)” is the number of grid points and GWP parameters. Memory and CPU are the computer resources on an Intel Pentium Quad Core Q6600 (2.4 GHz, 8 Gbytes) machine.

MCTDH vs G-MCTDH basis sets for 24-mode calculation. The conventions are as in Table II.

MCTDH vs G-MCTDH basis sets for 24-mode calculation. The conventions are as in Table II.

Computational resources required by the 24-mode calculations. The columns headed “ (MCTDH)” and “ (G-MCTDH)” are for the calculations detailed in Table IV. Conventions are chosen as in Table III.

Computational resources required by the 24-mode calculations. The columns headed “ (MCTDH)” and “ (G-MCTDH)” are for the calculations detailed in Table IV. Conventions are chosen as in Table III.

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