Complex virial theorem and complex scaling

We present the simple generalization to complex energies of the normal global real scaling used for bound-state calculations to produce a variational energy which satisfies the virial theorem. We show that in two limiting cases, one or the other of which is almost always satisfied in all calculations, the virially stabilized complex energy is sensitive to only the real part or the imaginary part of the complex virial expression. We then compute the virial expression for a number of wave functions for the $1s2s^2{}_{}{}^2{}_{}{}^{}S {\mathrm{He}}^{-}$, $1s2s2p{}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{} {\mathrm{He}}^{-}$, and $1s^{2}2s^2\mathrm{kp}{}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{} {\mathrm{Be}}^{-}$ resonances and the corresponding virially stabilized resonance energies. In all calculations one of the limiting cases was applicable.