The Off-Diagonal Hypervirial Theorem

A second very useful theorem can be derived from the Ehrenfest theorem, if one considers the wavefunctions of two different stationary states and the unperturbed Hamiltonian instead of a general time-dependent wavefunction (Chen, 1964) 

Several important relations can be derived from the off-diagonal hypervirial relation (Hansen and Bouman, 1979), if one chooses an operator P that does not commute with the Hamiltonian The most important one is obtained for P = O, which is a cartesian component of the sum of the position operators of all electrons defined as  

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