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Manuscript Title: Simple, accurate, and efficient implementation of 1-electron atomic time-dependent Schrödinger equation in spherical coordinates.
Authors: Serguei Patchkovskii, H. G. Muller
Program title: SCID-TDSE: Time-dependent solution of 1-electron atomic Schrödinger equation in strong laser fields.
Catalogue identifier: AEYM_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 199(2016)153
Programming language: Fortran-2003 with OpenMP extensions.
Computer: Portable code. Tested on x86 64 Linux.
Operating system: Portable code. Tested on x86 64 Linux.
RAM: Memory requirements depend on the input parameters. Time propagation of a given initial state requires O (32 * NR * (LMAX + 1) * (MMAX - MMIN + 1)) bytes of memory (double precision), where NR is the number of the radial grid points; LMAX is the maximum desired angular momentum; MMIN and MMAX are respectively the smallest and largest desired angular momentum projections. Preparation of the initial atomic states and analysis of the final wavefunction in terms of the field-free atomic states may require O (64 * NR2 * NCPU bytes of RAM (double precision), where NCPU is the number of processing threads used.
Keywords: Time-dependent Schrödinger equation, Implicit derivative operators, Muller's split propagator, Non-Hermitian representation, Strong laser fields, Linear scaling.
Classification: 2.5, 4.3.

External routines: BLAS and LAPACK (required); libhugetlbfs (optional), DGEFA and DGEDI (LINPACK); routines included with the code.

Nature of problem:
Time propagation of non-relativistic 1-electron Schrödinger equation for a central potential, under the influence of a long-wavelength laser field treated in the velocity-gauge dipole approximation.

Solution method:
The propagator is constructed by separating the Hamiltonian into a large number of non-commuting terms, where each term can be handled simply and computationally efficiently (linear scaling). Time-reversibility of the propagator is ensured by combining the individual terms symmetrically around time midpoint (See ref. [1] and the text). The numerical accuracy is achieved through implicit representation of derivatives (accurate to O (d4) for a uniform grid), combined with variable grid spacing.

Ill-conditioned Hamiltonians can occur for some choices or radial grids. The propagator is only approximately norm-conserving; small time steps may be necessary to achieve stable propagation.

Unusual features:
Due to the implicit representation of the spatial derivative operators, the overall Hamiltonian is not Hermitian. As the result, the left wavefunction is no longer given by a complex conjugate of the right wavefunction, and must be propagated explicitly.
The code makes no assumptions on the accuracy of numerical types, and can be built for any real or integer kinds supported by the compiler. Detailed instructions for building the code in single, double, and quadruple precision are included.

Running time:
Running time is input dependent. Time propagation of the Schrödinger equation scales as O (NR * (LMAX + 1) * (MMAX - MMIN + 1)) per time step.
Preparation of the initial atomic state and analysis of the results in terms of the field-free atomic states may require O (NR3 and O (NR3 * (LMAX + 1) work, respectively.
On a 3.6 GHz core i7 desktop with 4 CPU cores available, run times are from 2 seconds (probability of a perturbative bound-to-bound transition in hydrogen atom) to 1 hour (high-harmonic spectrum of a hydrogen atom driven by intense elliptically-polarised IR field).

[1] H.G. Muller, Laser Physics 9 (1999), 138-148.