### Molecular Orbital integrals using MAPLE in MATLAB

#### Jun Bai Wang

Department of Physics, University of Bergen, Bergen, Norway

Time-dependent Schrödinger equation for description of atomic collisions
The approach to atomic collisions used in this work is called semiclassical,
referring to the fact that only the electrons are treated as quantal particles,
while the atomic motion is simulated by a classical trajectory. This leads
to time-dependent Schrödinger equation for the active electrons, in
the present case one active electron. The matrix formulation of this problem
arises from expansion of the unknown electron wavefunction
in a set of basis functions
, much in analogy with Fourier series or expansions using orthogonal polynomials

The unknown quantities to be found are the expansion coefficients, which
form a vector. In this formulation, the time-dependent Schrödinger
equation

is replaced by a set of coupled differential equations, which are conveniently
expressed by matrix notation

In practical work, the most demanding part of the analysis is the evaluation
of matrix elements of the hamitonian between the basis states. This contribution
describes one particular task, related to finding instantaneous eigenstates
of the hamiltonian. The problem is close to finding the molecular orbitals
in the so called LCAO (linear combination of atomic orbitals) approach.

The method is based on earlier work of Hansen and Kocbach, using momentum
wavefunctions for hydrogen-like states on each of the two colliding nuclei
and obtaining analytic expressions for the matrix elements. These expressions
are in the form of text strings, which are directly used as programs for
numerical evaluation.

The setup of the problem is performed by a MATLAB program, which also
calls the integrated MAPLE kernel for symbolic manipulations and evaluations.
Finally, the same MATLAB program also uses the above mentioned text representations
of the analytic formulae to perform numeric evaluations. This is made possible
by MATLAB's function eval() (this takes any text and treats it as a program).
The data are then further treated by the numerical part of MATLAB.

MATLAB is first of all optimized for numerical calculations, so that
the presented approach is quite efficient with respect to the speed of
numerical calculations.

For more information junbai@kvark.fi.uib.no.