Perturbation-Theoretic Approach To Atoms and Molecules

A general perturbation-theoretic procedure for obtaining accurate wave functions for atoms and molecules is developed in detail for an arbitrary separable $H_0$, and the explicit equations satisfied by the various parts of the first- and second-order perturbed wave functions are given. It is shown how the solutions for the $n\mathrm{th}$-order perturbed function can be found in terms of solutions to partial differential equations in only $n+1\mathrm{}$ variables for atoms, and in terms of solutions to sets of coupled equations in the same $n+1\mathrm{}$ variables for molecules. It turns out that for atoms the number of pair equations that need be solved to give ${\Psi }_1$ and the major part of ${\Psi }_2$ is roughly proportional to $N$, the number of electrons; while for molecules of no symmetry, with paired electrons, it is proportional to $\frac{N(N+2\mathrm{})\mathrm{}}{8}$ as distinguished from the $\frac{N^2}{4}$ expected previously. This reduction in the number of necessary pairs may be of some practical use if the pair equations are integrated numerically. Various Hamiltonians are discussed, and an "exchangeless" Hamiltonian is introduced, which can serve to simplify the computations relative to the usual Hartree-Fock procedure, and might serve to increase the rate of convergence. Numerical calculations are performed for the helium-atom ground state in order to illustrate the procedure and to examine the rates of convergence of the expansions based on several different $H_0\text{'}\mathrm{s}$. Results differing from the exact result by 2-9 × ${10}^{-4\mathrm{}}$ are obtained for the different calculations through third order, and this error cannot be reduced by changing $H_0$. The angular expansion and the numerical integration technique used on the two-dimensional equations are sufficiently accurate that the third-order energy agrees to within 7 × ${10}^{-6\mathrm{}}$ of Knight and Scherr's value.