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Constrained adiabatic trajectory method: A global integrator for explicitly time-dependent Hamiltonians
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10.1063/1.3673320
/content/aip/journal/jcp/136/1/10.1063/1.3673320
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/1/10.1063/1.3673320

Figures

Image of FIG. 1.
FIG. 1.

Adiabatic laser pulse with angular frequency ω = 0.2958678 a.u. and total duration 10 000 a.u. (0.24 ps) with a Gaussian shape.

Image of FIG. 2.
FIG. 2.

Imaginary part of the time-dependent complex absorbing potential defined in Eq. (32).

Image of FIG. 3.
FIG. 3.

Dissociation and transition probabilities for submitted to the pulse of Fig. 1. The initial state was the fundamental state v = 0.

Image of FIG. 4.
FIG. 4.

Final dissociation probability given by the CATM (squares), the SOD scheme (rounds), and the split-operator method (triangles) vs number of Fourier basis functions (CATM) or number of time steps (SOD and split-operator). The associated CPU-time are mentioned near each point. The CATM and the SOD scheme work within the H 0 eigenbasis and the split-operator works within a DVR x-basis.

Image of FIG. 5.
FIG. 5.

Time evolution of the transition probability |〈2|Ψ(t)〉|2 given by the CATM with N = 2048 Fourier functions (CPU time: 42.5 s), by the SOD method with N = 106 time steps (CPU time: 471.1 s), by the split-operator scheme with N = 105 time steps (CPU time: 12.9 s), with N = 106 time steps (CPU time: 126.7 s) or with N = 107 (CPU time: 1244.3 s).

Image of FIG. 6.
FIG. 6.

Laser pulse with angular frequency ω = 0.2958678 a.u. and total duration 212 a.u. with Gaussian shape. All results in the current section relate to this pulse (with variable intensity).

Image of FIG. 7.
FIG. 7.

Number of iterations required until convergence vs electric field amplitude E 0, with V 0 = 0, using RDWA (rounds) or RDWA+Krylov procedure (triangles). All the domain of convergence is covered. For each case, the last point (coloured in black) is the edge of the convergence domain, in the present calculation conditions.

Image of FIG. 8.
FIG. 8.

Number of iterations required until convergence vs electric field amplitude E 0, with V 0 = 0.4, using RDWA (rounds) or RDWA+Krylov procedure (triangles). All the domain of convergence is covered. For each case, the last point (coloured in black) is the edge of the convergence domain, in the present calculation conditions.

Image of FIG. 9.
FIG. 9.

Number of iterations required until convergence vs absorbing potential amplitude V 0, with E 0 = 0.5, using RDWA (rounds) or RDWA+Krylov procedure (triangles).

Image of FIG. 10.
FIG. 10.

The ε of Eq. (40) vs absorbing potential amplitude, with E 0 = 0.5, using RDWA (rounds) or RDWA+Krylov procedure (triangles).

Image of FIG. 11.
FIG. 11.

vs iteration number, with E 0 = 0.5 and V 0 = 0.4, using perturbative RDWA (rounds) or RDWA+Krylov procedure (triangles).

Image of FIG. 12.
FIG. 12.

Spatial dependence of the modulas of the eigenstates of the field-free molecular ion : first two eigenstates of the first electronic potential curve on the left: |〈x|j = 0〉| (line) and |〈x|1〉| (dashed line); first 3 pseudo-eigenstates of the second electronic potential curve : |〈x|102〉| (long dashes, in the middle) |〈x|j = 100〉| (dotted line, on the right), |〈x|101〉| (dotted-dashed line, on the right). These two last functions located at the edge of the grid correspond to eigenvalues with and are some of those which play a very minor role in the dynamics.

Image of FIG. 13.
FIG. 13.

Laser pulse with pulsation ω = 0.2958678 a.u. and total duration 254 a.u. (i.e., 6.14 fs) with Gaussian turning on and off and continuous wave during 85 a.u. All results in the current section relate to this pulse with varying intensity.

Image of FIG. 14.
FIG. 14.

Number of iterations required until convergence vs electric field amplitude E 0, with V 0 = 0.3, using the CATM and RDWA, without interaction representation (rounds) or using an interaction representation following Eqs. (43) (triangles) or Eqs. (46) (squares). All the domain of convergence is covered. For each case, the last point (coloured in black) is the edge of the convergence domain.

Tables

Generic image for table
Table I.

Matrix representation of one block t i of the absorbing potential within the bi-orthogonal eigenbasis set {|j〉} of H 0.

Generic image for table
Table II.

Comparison of the CATM final transition probabilities P(|j〉, T 0) as a function of the electric field amplitude E 0 (in units of ) with two different procedures: Simple RDWA (A), RDWA+Krylov subspace diagonalization (B). The biggest initial residue is defined in Eq. (40). The absorbing potential amplitude was V 0 = 0.4.

Generic image for table
Table III.

Computational parameters corresponding to the results of Table IV. The electric field amplitude was 0.25 and the initial state is the fundamental state |i = 0〉.

Generic image for table
Table IV.

Comparison of the final transition and dissociation probabilities to bound states with the different conditions described in Table III.

Generic image for table
Table V.

Comparison of the CATM results as a function of the electric field amplitude (1 corresponds to ) with three different procedures: Simple CATM (A), CATM+interaction representation [Eqs. (43)] (B), CATM+interaction representation with respect to real parts [Eqs. (46)] (C). The biggest initial residue ε is defined in Eq. (40). The absorbing potential amplitude is V 0 = 0.3.

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/content/aip/journal/jcp/136/1/10.1063/1.3673320
2012-01-04
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Constrained adiabatic trajectory method: A global integrator for explicitly time-dependent Hamiltonians
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/1/10.1063/1.3673320
10.1063/1.3673320
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