Singular integrals and their application to a hypervirial theorem

Hypervirial theorems provide relationships exact electronic wave functions must satisfy, and the extent to which this is the case is a measure (additional to the energy) of wave function quality. The hypervirial relation known as the Vinti equation has been proposed for this purpose, but its application has been hampered by the absence of analytical formulas for the singular integrals occurring therein. The authors’ methods for singular integrals arising in atomic computations [J. Chem. Phys. 121, 6323 (2004)] resolve this bottleneck; quality assessments based on the Vinti equation are provided here for a number of wave functions of varying complexity describing the He isoelectronic series (from ${\mathrm{H}}^{-}$ through ${\mathrm{Ne}}^{8+}$).