Eddie Red

Poisson's equation solution of Coulomb integrals in atoms and molecules

The integral bottleneck in evaluating molecular energies arises from the two-electron contributions. These are difficult and time-consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behaviour of atomic orbitals. In this work, it is shown that the two-centre Coulomb integrals involved can be expressed as one-electron kinetic-energy-like integrals. This is accomplished using the fact that the Coulomb operator is a Greens function of the Laplacian. The ensuing integrals may be further simplified by defining Coulomb forms for the one-electron potential satisfying Poissons equation therein. A sum of overlap integrals with the atomic orbital energy eigenvalue as a factor is then obtained to give the Coulomb energy. The remaining questions of translating orbitals involved in three and four centre integrals and the evaluation of exchange energy are also briefly discussed. The summation coefficients in Coulomb forms are evaluated using the LU decomposition. This algorithm is highly parallel. The Poisson method may be used to calculate Coulomb energy integrals efficiently. For a single processor, gains of CPU time for a given chemical accuracy exceed a factor of 40. This method lends itself to evaluation on a parallel computer. †Dedicated to Professor Nicholas Handy.

Solution of Poisson's equation using spectral forms

A new technique is presented for the solution of Poissons equation in spherical coordinates. The method employs an expansion of the solution in a new set of functions defined herein for the first time, called ‘spectral forms’. The spectral forms have spherical harmonics as their angular part, but use a new set of radial functions that automatically statisfy the boundary conditions, up to a multiplicative constant, on the Poisson solution. The resultant problem reduces to a set of simultaneous equations for the expansion coefficients . The matrix A is block diagonal in the spherical harmonic indices l,m and is independent of any parameters. The simultaneous equations may be solved by LU decomposition. The LU decomposition only needs to be done once and multiple right hand sides (B-vectors) can be treated by a matrix-vector multiply. For a parallel computing platform, each such B-vector may be dealt with on a separate processor. Thus the algorithm is highly parallel. This technique may be used to calculate Coulomb energy integrals efficiently on a parallel computer.

Solution of the time-dependent Schrodinger equation using a basis in time

Abstract A computational theory for the solution of the time-dependent Schrodinger equation, wherein a basis in space and time is used, is applied to the precession of electron spin in a magnetic field. This constitutes a ‘proof of principle’ for formulating an initial value problem as a two-point boundary value problem in time with one boundary value determined by the solution of simultaneous equations. This is similar to a method introduced by Schneider et al. [Bull. Am. Phys. Soc. 45 (2000) 39], and Weatherford [Bull. Am. Phys. Soc. 46 (2001) 109]. This method allows for an effective implementation of a time-dependent exchange/correlation theory for two electrons [Computational Chemistry: Reviews of Current Trends 5 (2000)], such that ‘derivative-overlap’ integrals of one-electron orbitals, such as 〈φ j | φ k 〉, may be evaluated analytically.