General Energy and Wavefunction Expressions

Given the above decomposition of //, we now express the Schrddinger equation 

Making use of the fact that = QH , which is easily seen to be valid from definition of Q, we can operate on Eq. (3.4) with the projector Q to obtain a closed expression for Q jy  

We now consider two especially relevant choices of the, in principle, arbitrary parameter If is taken to be equal to Ej, then the above perturbation series describe the Brillouin-Wigner approximations to y and Ej, The choice yields the Rayleigh-Schrodinger perturbation series. 

To express any of the above perturbation expansions in terms of creation and annihilation operators, we simply write y , H , and U in the second-quantized manner. The zeroth-order Hamiltonian is virtually always taken to be a one-electron operator 

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