Variational Methods for Atoms and the Virial Theorem

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly −1, a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case.