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Multilayer multiconfiguration time-dependent Hartree method: Implementation and applications to a Henon–Heiles Hamiltonian and to pyrazine
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10.1063/1.3535541
/content/aip/journal/jcp/134/4/10.1063/1.3535541
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/4/10.1063/1.3535541

Figures

Image of FIG. 1.
FIG. 1.

Tree structures for the MCTDH and ML-MCTDH wavefunctions of the 6D Henon–Heiles simulations. (a) MCTDH wavefunction tree, in which the coordinates are combined in groups of two. refers in this particular case to the number of primitive basis functions or grid points. (b) ML-MCTDH wavefunction tree. This tree is similar to the MCTDH tree, but the three combined modes have been separated by adding an extra layer. For the ML-MCTDH wavefunction corresponds now the number of primitive basis functions and for (and same ) the ML-MCTDH case becomes numerically identical to the MCTDH one.

Image of FIG. 2.
FIG. 2.

(a) Absolute value of the autocorrelation function for various MCTDH and ML-MCTDH 6D Henon–Heiles simulations with a coupling parameter . The numbers in parentheses correspond to () in Fig. 1. In the key, plain numbers indicate MCTDH calculations and the symbol ML designates ML-MCTDH calculations. (b) Difference between the autocorrelations of the ML-MCTDH simulations and the (50,24) MCTDH calculation.

Image of FIG. 3.
FIG. 3.

(a) Absolute value of the autocorrelation function for various MCTDH and ML-MCTDH 6D Henon–Heiles simulations with a coupling parameter . The numbers in parentheses correspond to () in Fig. 1. (b) Difference between the autocorrelations of the ML-MCTDH simulations and the (50,24) MCTDH calculation. In the key, ML is for ML-MCTDH.

Image of FIG. 4.
FIG. 4.

Tree structures for the MCTDH and ML-MCTDH wavefunctions of the 18D Henon–Heiles simulations. The MCTDH wavefunction (a) consists of six combined modes, each of them grouping three primitive coordinates. The ML-MCTDH wavefunction (b) starts dividing the system in three logical coordinates, and each of them is further divided in three combined modes. The resulting tree has three layers of TD coefficients.

Image of FIG. 5.
FIG. 5.

(a) Absolute value of the autocorrelation function for various MCTDH and ML-MCTDH 18D Henon–Heiles simulations with a coupling parameter . For the MCTDH cases the number in parentheses refers to in Fig. 4(a). For the ML-MCTDH cases the number in parentheses refers to and in Fig. 4(b), which are set equal in the reported simulations. (b) Detailed view of the last recurrence structure between 55 and 60 time units. In the key, ML is for ML-MCTDH.

Image of FIG. 6.
FIG. 6.

(a) Absolute value of the autocorrelation function for various MCTDH and ML-MCTDH 18D Henon–Heiles simulations with a coupling parameter . For the MCTDH cases the number in parentheses refers to in Fig. 4(a). For the ML-MCTDH cases the number in parentheses refers to and in Fig. 4(b), which are equal in the reported simulations. (b) Detailed view of the recurrence structures between 10 and 30 time units. In the key, ML is for ML-MCTDH.

Image of FIG. 7.
FIG. 7.

(a) Spectra for various MCTDH and ML-MCTDH 18D Henon–Heiles simulations with a coupling parameter of . The numbers in parentheses have the same meanings as in Fig. 6. (b) Detailed view of the three first peaks between 6.2 and 7.9 energy units. In the key, ML is for ML-MCTDH. The spectra are obtained by a Fourier-transform of the autocorrelation function using a filter (Refs. 3 and 5) to minimize spurious effects known as Gibbs phenomenon.

Image of FIG. 8.
FIG. 8.

Absolute value of the autocorrelation of the ML-MCTDH 1458D Henon–Heiles simulations for time units 30 to 60. (a) Reference results without coupling term between the two displaced coordinates for case (1) (red dashed), in which the two initially displaced degrees of freedom belong to different logical coordinates at all layers, and case (2) (blue line), in which the two initially displaced degrees of freedom belong to the same logical coordinate at all layers. (b) For case (2), homogeneous wavefunction (red dashed) and nonhomogeneous wavefunction with more SPFs at low layers (blue line). (c) For case (1), homogeneous wavefunction (red dashed) and nonhomogeneous wavefunction (blue line). The pairs of autocorrelation functions presented in plots (b) and (c) are almost identical until about 15 time units, but the autocorrelation functions of case (1) and (2) start to differ already after 5 time units. These differences increase with time and can be inspected for the time interval 30 to 60 by comparing (b) with (c). (d) Spectra of the calculations with the nonhomogeneous wavefunctions for cases (1) and (2). The spectra are obtained by a Fourier-transform of the autocorrelation function using a filter (Refs. 3 and 5) to minimize spurious effects known as Gibbs phenomenon.

Image of FIG. 9.
FIG. 9.

Tree structure used in most of the ML-MCTDH simulations of 24D pyrazine. The maximum depth of the tree is five layers, and the first one separates the 24 vibrational coordinates and the discrete electronic degree of freedom. The number of SPFs denoted need not be necessarily the same, and in fact in some of the ML-MCTDH simulations they are chosen to be different. The same is true for the three ’s, which can be different. For the fast ML-1 calculation a sixth layer was added, see text.

Image of FIG. 10.
FIG. 10.

Absolute value of the autocorrelation function for the 24D pyrazine calculations. The numeration of the calculations is consistent with Table I. (a) Best MCTDH result and two best ML-MCTDH results from 0 to 300 fs. (b) Detailed view of the recurrences between 20 and 70 fs for all reported MCTDH simulations and the best ML-MCTDH simulation, and (c) for the two worst and two best converged ML-MCTDH runs. In the key, ML is for ML-MCTDH.

Image of FIG. 11.
FIG. 11.

(a) Spectra of the fastest ML-MCTDH 24D pyrazine calculation ( TD coeff., 7 min. of CPU) and the best reference MCTDH result ( TD coeff., 2901 h of CPU). (b) Detailed view of the energy domain between 2.1 and 2.6 eV. The spectra are obtained by a Fourier-transform of the autocorrelation function using a filter3,5 to minimize spurious effects known as Gibbs phenomenon. The numeration of the calculations is consistent with Table I. In the key, ML is for ML-MCTDH.

Image of FIG. 12.
FIG. 12.

(a) Spectra of the three reference MCTDH simulations and three of the ML-MCTDH simulations, among them the two best results, for 24D pyrazine. The spectra are obtained by a Fourier-transform of the autocorrelation function using a filter (Refs. 3 and 5) to minimize spurious effects known as Gibbs phenomenon. The numeration of the calculations is consistent with Table I. For the energy domain between 2.1 and 2.6 eV: (b) The three reference MCTDH calculations compared to the best ML-MCTDH result and (c) ML-MCTDH simulations 2, 7, and 8. In the key, ML is for ML-MCTDH.

Tables

Generic image for table
Table I.

Simulation parameters of the various MCTDH and ML-MCTDH 24D pyrazine calculations. The second column contains the wall-clock time of each simulation. ML-MCTDH calculations were run on a single CPU, so that the wall-clock time equals the CPU time. The MCTDH calculations were run on eight CPUs using shared-memory parallelization (Refs. 56 and 57). The speed-up factor of the eight-processor parallel pyrazine calculations is 2.9, 3.3, and 3.7 for the MCTDH-1, -2, and -3 cases, respectively. The wall-clock times given for the MCTDH reference results are already scaled up by the corresponding speed-up factor and therefore reflect the time that such simulation would have taken on a single processor. Therefore they can be readily compared to the ML-MCTDH values. All simulations were run on the same machine and CPU type, namely, Quad-Core AMD Opterons, processor type 2384 running at 2.7 GHz. The third column shows the total number of time-dependent coefficients propagated in each case. The fourth column contains, for ML-MCTDH simulations, the number of SPFs for each node of the tree according to the representation in Fig. 9. The parentheses indicate that different values were taken for each of the branches. The asterisk for the ML-1 case indicates that there is a further layer below the branches corresponding to (see text and Fig. 9 for details). For the MCTDH calculations, the fourth column contains in each parenthesis the number of SPFs for combined modes to for electronic states and , respectively. All reported simulations, except for ML-1, were run with a rather high integrator precision of (see text). ML-1 was run with . Reducing the high integrator precision will speed up the calculations by factors between 1.5 and 2 without introducing visible changes into the spectra.

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2011-01-28
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Multilayer multiconfiguration time-dependent Hartree method: Implementation and applications to a Henon–Heiles Hamiltonian and to pyrazine
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/4/10.1063/1.3535541
10.1063/1.3535541
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