Multiple Perturbation Theory for Many-Electron Systems and Properties

Multiple Perturbation Theory for Many-Electron Systems and Properties 

In the previous two sections, we have presented the Breit-Pauli perturbation Hamiltonian for one- and two-electron relativistic corrections of order 1/c to the nonrelativistic Hamiltonian. But there is a problem for many-electron systems. For the perturbation theory to be valid, the reference wave function must be an eigenfunction of the zeroth-order Hamiltonian. If we take this to be the nonrelativistic Hamiltonian and the perturbation parameter to be 1/c, we do not have the exact solutions of the zeroth-order equation. 

To overcome this problem, we can use double perturbation theory. The zeroth-order Hamiltonian is then normally some one-particle operator, such as the Fock operator, and the two perturbations are correlation and relativity. Formally, we write the Hamiltonian as 

In principle, we should have an infinite series in because there are relativistic corrections to the Hamiltonian at all orders in n, but since we are developing a method appropriate for the Breit—Pauli approxiination, we need only consider the lowest-order terms. Technically, we ought to write Ho as Hoo to indicate that it is of zeroth order in both perturbations, but we will omit the second zero. 

The energy and the wave function are expanded in infinite series in both perturbations. 

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