Solution of the time-dependent Schrödinger equation employing a basis of explicit discrete-coordinate eigenfunctions: spherical and azimuthal symmetry, adiabaticity, and multiphoton excitation of a rotating Morse oscillator

Time dependent Schrödinger equation on arbitrary structured grids: Application to photoionization

“…It is well known that discretizing the Hamiltonian b H by taking centered differences on a Cartesian grid leads to a form of b U that is both numerically stable and unitary. However, when more complex grids are used, sometimes strict unitarity, and therefore conservation of probability, is lost [1,3,4]. This loss of unitarity is particularly troublesome when modeling photoionization, since small changes in free state probabilities are of principal interest.…”

“…The loss of strict unitarity that may occur when the TDSE is solved on a curvilinear grid results from errors in the spatial discretization of b H. This discretization is typically carried out by means of three or five point difference formulae [1,4], or by some system of quadratures [3]. This leads to matrices H and U, which serve as the discrete analogs of b H and b U.…”

“…Numerical solutions of the TDSE are discussed in many articles, e.g. [1][2][3][4][5][6]. In most cases, the Cayley form of the time translation operator b UðDtÞ ¼ 1 À 1 2 i b HDt…”