Modelling atomic processes in intense laser fields often relies on solving the time-dependent Schrödinger equation (TDSE). For processes involving ionisation, such as above-threshold ionisation (ATI) and high-harmonic generation (HHG), this is a formidable task even if only one electron is active. Several powerful ideas for efficient implementation of atomic TDSE were introduced by H.G. Muller some time ago (Muller, 1999), including: separation of Hamiltonian terms into tri-diagonal parts; implicit representation of the spatial derivatives; and use of a rotating reference frame. Here, we extend these techniques to allow for non-uniform radial grids, arbitrary laser field polarisation, and non-Hermitian terms in the Hamiltonian due to the implicit form of the derivatives (previously neglected). We implement the resulting propagator in a parallel Fortran program, adapted for multi-core execution. Cost of TDSE propagation scales linearly with the problem size, enabling full-dimensional calculations of strong-field ATI and HHG spectra for arbitrary field polarisations on a standard desktop PC.
Program summary
Program title: SCID-TDSE: Time-dependent solution of 1-electron atomic Schrödinger equation in strong laser fields.
Catalogue identifier: AEYM_v1_0
Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYM_v1_0.html
Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland
Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html
No. of lines in distributed program, including test data, etc.: 334254
No. of bytes in distributed program, including test data, etc.: 20005596
Distribution format: tar.gz
Programming language: Fortran-2003 with OpenMP extensions.
Computer: Portable code. Tested on x86_64 Linux.
Operating system: Portable code. Tested on x86_64 Linux.
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 g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001046551500394X&_mathId=si49.gif&_user=111111111&_pii=S001046551500394X&_rdoc=1&_issn=00104655&md5=67dd46866682884f88be0a7f0d8c6fe1" title="Click to view the MathML source">O(d4) for a uniform grid), combined with variable grid spacing.
Restrictions: 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.
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