Wave function for the groundstate of lithium

Abstract

A variational calculation for the groundstate of the lithium atom is carried out with an anti-symmetrical wave function built up out of the core function exp[−K/2(r₁+r ₂] (1+br ₁₂) and the valence electron function$\exp \left[ { - \frac{K}{2}\left( {r_1 + r_2 } \right)} \right]\left( {1 + br_{12} } \right)$ and the valence electron function$r_3 e^{ - \frac{a}{2}r_3 }$. The resulting improvement in the total energy of the atom is considerable. But contrary to expectation the calculated value of the ionisation potential is much inferior to those previously obtained with simple hydrogen-like core functions. The numerous integrals occurring in the problem are discussed from a uniform viewpoint. A few simple integrals involving either one or two of the inter-electronic distances are first evaluated with the help of two general formulas (D) and (E) of Section 4, and from them the remaining ones are obtained by repeated differentiation with respect to parameters occurring in the exponential part of the integrand. Considerable simplification is achieved thereby and possibilities of errors are also largely eliminated.

Incidentally, a new method of successive approximations has been developed for minimising the energy expression.