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.
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 .
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.
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.
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].
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.
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.
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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