The integral formulae for the variational solution of the molecular many-electron wave equation in terms of Gaussian functions with direct electronic correlation

Abstract

It is shown that for wave functions of the form $\Psi = \Sigma_k C_k exp (-Q_k),$ where Q$_k$ is any positive definite quadratic form in the Cartesian co-ordinates of n particles and C$_k$ a constant, all integrals required for the calculation of the electronic energy of molecules by the variation method can be readily evaluated. The potential energy integrals are reduced to quadratures and the other integrals are expressed in closed form. In the general case the quadratic forms Q$_k$ determine many-electron functions formed from correlated electron orbitals of ellipsoidal symmetry and with variable centres. The results are easily extended to wave functions of the form $\Psi = \Sigma_k P_k \exp (-Q_k),$ where P$_k$ is a polynomial.