The tensor virial theorem in quantum mechanics

Abstract

A quantum mechanical generalization of the scalar virial theorem is derived and specialized to atoms and molecules in the Born–Oppenheimer approximation. The theorem is the quantum mechanical counterpart to Chandrasekhar’s classical tensor virial theorem. The usual scalar virial equation follows by tensor contraction. One possible application is the introduction of more than one scale factor in a trial wavefunction. The scaling method proposed involves different stretchings for the different spatial coordinates. This is in contrast to the standard method of using the scalar virial theorem where the stretching is the same in all directions. An example is given where the introduction of multiple scale factors and the imposition of the tensor virial theorem yields a better result than the usual procedure of subjecting the wavefunction to a single scale transformation and imposing the scalar virial theorem.