^{1,a)}, G. Jolicard

^{1}, D. Viennot

^{1}and J. P. Killingbeck

^{1,2}

### Abstract

The constrained adiabatic trajectory method (CATM) is reexamined as an integrator for the Schrödinger equation. An initial discussion places the CATM in the context of the different integrators used in the literature for time-independent or explicitly time-dependent Hamiltonians. The emphasis is put on adiabatic processes and within this adiabatic framework the interdependence between the CATM, the wave operator, the Floquet, and the (*t*, *t* ^{′}) theories is presented in detail. Two points are then more particularly analyzed and illustrated by a numerical calculation describing the ion submitted to a laser pulse. The first point is the ability of the CATM to dilate the Hamiltonian spectrum and thus to make the perturbative treatment of the equations defining the wave function possible, possibly by using a Krylov subspace approach as a complement. The second point is the ability of the CATM to handle extremely complex time-dependencies, such as those which appear when interaction representations are used to integrate the system.

We acknowledge the support of the French Agence Nationale de la Recherche (Project CoMoC).

I. INTRODUCTION

II. STATIONARY AND DYNAMIC TREATMENTS FOR ADIABATIC PROCESSES IN MOLECULAR PHYSICS

III. COMPARISONS BETWEEN THE CATM, THE SOD SCHEME, AND THE SPLIT-OPERATOR METHOD

A. Model for illuminated by intense pulses

B. Comparative integration schemes

C. Results

IV. THE INFLUENCE OF THE ABSORBING POTENTIAL AND OF A KRYLOV SUBSPACE PROCEDURE ON THE CONVERGENCE OF THE CATM

V. USE OF AN INTERACTION REPRESENTATION FOR THE HAMILTONIAN BEFORE APPLYING THE CATM

A. The Gibbs phenomenon and the CATM

B. Working in the interaction representation

C. Results

VI. CONCLUSION

### Key Topics

- Eigenvalues
- 35.0
- Wave functions
- 18.0
- Hilbert space
- 16.0
- Subspaces
- 15.0
- Operator theory
- 10.0

## Figures

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.

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.

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

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

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

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

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.

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.

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* = 10^{6} time steps (CPU time: 471.1 s), by the split-operator scheme with *N* = 10^{5} time steps (CPU time: 12.9 s), with *N* = 10^{6} time steps (CPU time: 126.7 s) or with *N* = 10^{7} (CPU time: 1244.3 s).

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* = 10^{6} time steps (CPU time: 471.1 s), by the split-operator scheme with *N* = 10^{5} time steps (CPU time: 12.9 s), with *N* = 10^{6} time steps (CPU time: 126.7 s) or with *N* = 10^{7} (CPU time: 1244.3 s).

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).

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).

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.

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.

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.

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.

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).

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).

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

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

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

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

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.

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.

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.

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.

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.

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

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

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

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.

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.

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〉.

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〉.

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

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

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.

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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