Elsevier Science Home
Computer Physics Communications Program Library
Full text online from Science Direct
Programs in Physics & Physical Chemistry
CPC Home

[Licence| Download | New Version Template] aemi_v1_0.tar.gz(433 Kbytes)
Manuscript Title: SymbMat: Symbolic Computation of Quantum Transition Matrix Elements
Authors: M.F. Ciappina, T. Kirchner
Program title: SymbMat
Catalogue identifier: AEMI_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 183(2012)1832
Programming language: Mathematica.
Computer: Single machines using Linux or Windows (with cores with any clock speed, cache memory and bits in a word).
Operating system: Any OS that supports Mathematica. The notebooks have been tested under Windows and Linux and with versions 6.x, 7.x and 8.x.
Keywords: Symbolic Computation, Transition Matrices, Atomic Ionization Processes, Mathematica.
PACS: 34.10.+x, 34.50.-s, 34.50.Fa, 34.90.+q.
Classification: 2.6.

Nature of problem:
The notebooks generate analytical expressions for quantum transition matrix elements required in diverse atomic processes: ionization by ion, electron, or photon impact and ionization within the framework of strong field physics. In charged-particle collisions approaches based on perturbation theory enjoy widespread utilization. Accordingly, we have chosen the First Born Approximation and Distorted Wave theories as examples. In light-matter interactions, the main ingredient for many types of calculations is the dipole transition matrix in its different formulations, i.e. length, velocity, and acceleration gauges. In all these cases the transitions of interest occur between a bound state and a continuum state which can be described in different ways. With the notebooks developed in the present work it is possible to calculate transition matrix elements analytically for any set of quantum numbers nlm of initial hydrogenic states or Slater-Type Orbitals and for plane waves or Coulomb waves as final continuum states.

Solution method:
The notebooks employ symbolic computation to generate analytical expressions for transition matrix elements used in both collision and light-matter interaction physics.

Additional comments:
The set consists of the following 10 notebooks:
  • fba hyd.nb - This notebook computes analytical expressions for the transition matrix of collision-induced ionization in the First Born Approximation (FBA). The transitions considered are from a bound hydrogenic state with arbitrary quantum numbers nlm to a continuum state represented by a plane wave (PW) or a Coulomb wave (CW).
  • distorted hyd.nb - This notebook computes analytical expressions for the transition matrix of collision-induced ionization in Distorted Wave (DW) theories. The transitions considered are from a (distorted) bound hydrogenic state with arbitrary quantum numbers nlm to a distorted-wave continuum state. The computations are based on scalar and vectorial integrals (see the text for details).
  • dipoleLength hyd.nb - This notebook computes analytical expressions for the dipole transition matrix in length gauge. The transitions considered are from a bound hydrogenic state with arbitrary quantum numbers nlm to a continuum state represented by a PW (the Strong Field Approximation (SFA)) or a CW (the Coulomb-Volkov Approximation (CVA)).
  • dipoleVelocity hyd.nb - This notebook computes analytical expressions for the dipole transition matrix in velocity gauge. The transitions considered are from a bound hydrogenic state with arbitrary quantum nunbers nlm to a continuum state represented by a PW (the SFA) or a CW (the CVA).
  • dipoleAcceleration hyd.nb - This notebook computes analytical expressions for the dipole transition matrix in acceleration gauge. The transitions considered are from a bound hydrogenic state with arbitrary quantum numbers nlm to a continuum state represented by a PW (the SFA). For the case of the CVA we only include the transition from the 1s state to a continuum state represented by a CW.
  • fba STO.nb - This notebook computes analytical expressions for the transition matrix of collision-induced ionization in the FBA. The transitions considered are from a Slater-Type Orbital (STO) with arbitrary quantum numbers nlm to a continuum state represented by a PW or a CW.
  • distorted STO.nb - This notebook computes analytical expressions for the transition matrix of collision-induced ionization in DW theories. The transitions considered are from a (distorted) STO with arbitrary quantum numbers nlm to a distorted-wave continuum state. The computations are based on scalar and vectorial integrals (see the text for details).
  • dipoleLength STO.nb - This notebook computes analytical expressions for the dipole transition matrix in length gauge. The transitions considered are from an STO with arbitrary quantum numbers nlm to a continuum state represented by a PW (the SFA) or a CW (the CVA).
  • dipoleVelocity STO.nb - This notebook computes analytical expressions for the dipole transition matrix in velocity gauge. The transitions considered are from an STO with arbitrary quantum numbers nlm to a continuum state represented by a PW (the SFA) or a CW (the CVA).
  • dipoleAcceleration STO.nb - This notebook computes analytical expressions for the dipole transition matrix in acceleration gauge. The transitions considered are from an STO with arbitrary quantum numbers nlm to a continuum state represented by a PW (the SFA).
The symbolic expressions obtained within each notebook can be exported to standard programming languages such as Fortran or C using the Format.m package (see the text and "M. Sofroniou, The Mathematica Journal, 3 (1993) 74-80" for details).

Running time:
Computational times vary according to the transition matrix selected and quantum numbers nlm of the initial state used. The typical running time is several minutes, but it can be larger for large values of nlm.