Perturbative Corrections to the ZORA Hamiltonian

There are several ways in which we can develop a perturbation series for the ZORA equation. The first is simply to ignore the normalization—a perfectly valid procedure since the wave function is only defined up to a multiplicative constant. This we will do later in the present section. The second is to follow the same procedure as in the development of the Pauli Hamiltonian in chapter 17, and the third is to start from the Foldy-Wouthuysen transformation, as in the development of chapter 16. The last two of these both explicitly involve the normalization. We will commence here with the procedure used in chapter 17. 

Just as in the Pauli approximation, we will expand the normalization operator in a series. The normalized wave function is given by (17.7), and the normalization operator from (17.10) expressed in the new expansion parameter is 

The expansion of this operator is again a double series, one for the square root and one for the inverse square. Before we proceed, we must consider what we are to use as a perturbation parameter. The power series expansion of the inverse is in El(2mc - V), and if we neglected the normalization we would simply multiply this term by a formal perturbation parameter. Here however we want to include the normalization and eliminate the energy dependence, so we are really considering an expansion in 1 /(2wc - V), of which the terms up to order 1 comprise the zeroth-order Hamiltonian. The expansion of the normalization operator (and its inverse) therefore gives us 

Following the Pauli development from the elimination of the small component, we may write 

The energy-dependent term on the left cancels with the corresponding term on the right, and we end up with 

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