Variance minimization. A variational principle for accurate lower and upper bounds of the eigenvalues of a selfadjoint operator, bounded below

Abstract

In connection with Temple's formula variance minimization yields accurate values especially for groundstate energies of Schrödinger operators with a discrete spectrum.

The result in a.u. for the groundstate E ₀ of the He-atom in the infinite nuclear mass approximation is −2.9037243865₅≤E ₀≤−2.9037243769₆ i.e. E ₀ is determined with an absolute error smaller than 0.0022 cm⁻¹.