Brillouin-Wigner BW and Rayleigh-Schrodinger RS expansions

Let us assume that zero-order Hamiltonian Ho is a Hermitian operator having a complete set of eigenfunctions 

Let us assume again that functions X a are non-degenerate, in which case they are automatically orthogonal. If one of 4 a is a reasonable approximation of the exact wave function xp for the state investigated, then this zero-order approximation is called the model function  

The purpose of PT is to find a scheme for generating a sequence of successive improvements to this zero-order approximation. Let us define 

According to these definitions, the P operator projects out of any function describing the system, the part that is proportional to the model function, and the Q operator projects the part that is orthogonal to this function. The sum of P + Q satisfies the condition 

Operating with P on the exact wave function xp, we find 

---