Evaluation Of Exchange Energy Research Articles

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 Green's function of the Laplacian. The ensuing integrals may be further simplified by defining Coulomb forms for the one-electron potential satisfying Poisson's 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.

Analytical evaluation of molecular electronic integrals using Poisson's equation: Exponential‐type orbitals and atom pairs

AbstractThe 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 behavior of atomic orbitals. The two‐center two‐electron integrals are essential to describe atom pairs in molecules and distinguish those that are bound. In this work on analytical integration, it is shown that the two‐center 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 Green's function of the Laplacian. The ensuing integrals may be further simplified by defining spectral forms for the one‐electron potential satisfying Poisson's equation therein. A sum of overlap integrals with the atomic orbital energy eigenvalue as a factor is then obtained to give the Coulomb energy. This is most easily evaluated by direct integration. The orbitals involved in three and four center integrals are translated to two centers. This is discussed very briefly. The evaluation of exchange energy is a straightforward extension of this work. The summation coefficients in spectral forms are evaluated analytically from Gaunt coefficients. 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 4. This method lends itself to efficient evaluation on a parallel computer. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem 106, 2006