Invariance Property of the Brillouin-Wigner Perturbation Series

The complete perturbation series for the energy is invariant under the operation of adding a velocity-dependent interaction to the zeroth order Hamiltonian and subtracting the same quantity from the perturbation operator. The same invariance property appear to hold also for an optimum formulation of the terminated energy series generated by the $n\mathrm{th}$ order approximation to the wave function. An explicit proof is given for the first- and second-order wave functions and also for the complete energy series. Variational procedures for determining (a) the optimum velocity dependence of the zeroth order Hamiltonian and (b) the optimum uniform displacement of the zeroth order energy spectrum are discussed in relation to the invariant formulations.