Schrodinger equation finite-difference methods

T. E. Simos and P. S. Williams, On finite difference methods for the solution of the Schrodinger equation, Comput. Chem., 1999, 23, 513-554. 

Numerical Illustrations for Exponentially-Fitted Methods and Phase Fitted Methods. - In this section we test several finite difference methods with coefficients dependent on the frequency of the problem to the numerical solution of resonance and eigenvalue problems of the Schrodinger equations in order to examine their efficiency. First, we examine the accuracy of exponentially-fitted methods, phase fitted methods and Bessel and Neumann fitted methods. We note here that Bessel and Neumann fitted methods will also be examined as a part of the variable-step procedure. We also note that Bessel and Neumann fitted methods have a large penalty in a constant step procedure (it is known that the coefficients of the Bessel and Neumann fitted methods are position dependent, i.e. they are required to be recalculated at every step). 

In ref. 151 the author studies the piecewise perturbation methods to solve the Schrodinger equation and the two form of this approach, i.e. the LP and CP methods. On each stepsize the potential is numerically approximated by a constant (in the case of CP) or by a linear function (in the case of LP). After that the deviation of the true potential from this numerical approximation is obtained by the perturbation theory. The main idea of the author is that an LP algorithm can be made computationally more efficient if expressed in a CP-like form. The author produces a version of order 12 whose the two main parts are a new set of formulae for the computation of the zeroth-order solution which replaces the use of the Airy functions, and an efficient way of obtained the formulae for the perturbation corrections. The main remark for this paper is that from our experience for these methods the computational cost is considerably higher than for the finite difference methods. 

T. E. Simos, An accurate finite difference method for the numerical solution of the Schrodinger equation. Journal of Computational and Applied Mathematics, 1998, 91(1), 47-61. 

Much information of interest for atomic and molecular systems involves properties other than energy, usually observed via the energy shifts generated by coupling to some external field. The desired property is then the derivative of the energy with respect to the external field, which may be obtained by two different approaches. The finite-field method solves the Schrodinger equation in the presence of the external field, yielding... 

The first group of methods then manipulate a very small subset of vector elements Vi at a time, and a direct method continually updates the affected elements rj. Such methods are collectively known as relaxation methods, and they are primarily used in situations where, for each elements >< to be changed, the set of affected rjt and the matrix elements Ay, are immediately known. This applies in particular to difference approximations and also to the so-called Finite Element Method for obtaining tabular descriptions of the wave function, i.e. a list of values of the wave function at a set of electron positions. (Far some reason, such a description is commonly referred to as a numerical solution to the Schrodinger equation). Relaxation methods have also been applied to the Cl problem in the past, but due to their slow convergence they have been replaced by analytical methods. Even for numerical problems, the relaxation methods are slowly yielding to analytical methods. 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

It is also worth mentioning that numerical solutions of the Schrodinger equation frequently enclose the atom in a spherical box of finite radius for example, discrete variable methods, finite elements methods and variational methods which employ expansions in terms of functions of finite support, such as -splines, all assume that the wave function vanishes for r > R, which is exactly the situation we deal with here. For such solutions to give an accurate description of the unconfined system it is, of course, necessary to choose R sufficiently large that there is negligible difference between the confined and unconfined atoms. 

The finite element method (FEM). The general idea of FEM applied to solve the Schrodinger equation is to change over from the integration to a summation over many subdomains called elements [1, 86]. On each element the wavefunction is approximated by a parametrised function u. The simplest choice are polynomials of different degrees, e.g. in two dimensions... 

We should not finish this discussion without mentioning the basis-set-fi-ee method of Becke. The grids used in this approach are the same as those described in Section 4.1. The grids were designed to accurately describe orbitals and densities in the neighborhood of each nucleus. A finite-difference approximation (in spherical polar coordinates) is used to solve Poisson s and Schrodinger s equations. The accuracy obtained with this basis-set-fi-ee approach for all-electron calculations is impressive, and, with the techniques described in Sections 3, 4.1 and 4,2, an 0(N) implementation is feasible. Delley s DMol program also uses a related approach. 

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