Calculation of London—van der Waals Energies

A simple expression for the London—van der Waals interaction energies of like molecules, in the absence of retardation effects, is obtained by a perturbation theory calculation using a new approximate method. The approximate method, which is useful for certain second-order p.t. sums when the unperturbed wave-functions can be described as products of determinants, is presented with a discussion of its generally small error. This method enables one to express the London—van der Waals energy (also called ``dispersion energy'') as products of certain expectation values over the unperturbed wavefunctions, involving essentially approximate solutions, to the first-order perturbation Schrödinger equation in its differential form. In particular, the energy in the inverse sixth power of the intermolecular separation R can be expressed as the product of electric dipole polarizabilities and mean-square electric dipole moments. For two interacting 1s hydrogen atoms the energy is very easily computed and found to be EL=−e2a0[6.75(Ra0)−6 (Ra0)−8+819(Ra0)−10+···].The first two terms compare favorably with the results of variational calculations while the third does not. Using the approximate perturbed wavefunction, the expectation values of r2 and 1/r for each electron proportional to R—6 are obtained. These expectation values which contribute to the atomic magnetic susceptibility and to the magnetic shielding of the proton agree well with the variational calculations. Also obtained is the third-order p.t. energy between three like molecules which for three 1s hydrogen atoms is E3=−23.1(R12/a0)−3(R13/a0)−3(R23/a0)−3Ω,where Ω is an angular factor.