Virial Theorem, the Variational Principle, and Nonlinear Parameters in Quantum Mechanics: Application to the 31D and 33D States of the Helium Isoelectronic Series

The connection between the virial theorem and the variational principle with respect to nonlinear parameters has been investigated for a system of $n$ particles interacting through Coulomb forces. It is found that if the nonlinear parameters are chosen to be $n$ scale parameters and considered as the components of a position vector in an n‐dimensional Euclidean space, the locus of points for which the virial theorem is satisfied is a path of steepest descent on the energy surface. It is thus possible to minimize the energy along a path of steepest descent without the necessity of calculating derivatives. As an example, variational wavefunctions and energies are obtained for the $3^{1}D$ and $3^{3}D$ states of the helium isoelectronic series through $\text{Z\hspace{0.17em}=\hspace{0.17em}30,}$ with the trial function a 39‐term configuration‐interaction expansion in members of a denumerably complete basis set. The calculated energies for He and Li⁺ agree with observed values to within 0.000007 a.u. The agreement with energies obtained from Hylleraas‐type expansions for $\text{Z\hspace{0.17em}≤\hspace{0.17em}10}$ is of the same order, and in some cases it is the configuration‐interaction energy which is lower.