The CPD or ZORA Hamiltonian

In the development of the Pauli Hamiltonian in section 17.1, truncation of the power series expansion of the inverse operator after the first term yielded the nonrelativistic Hamiltonian. In (18.1), the zeroth-order term is the Hamiltonian first developed by Chang, Pelissier, and Durand (1986), often referred to as the CPD Hamiltonian. The name given by van Lenthe et al. is the zeroth-order regular approximation, ZORA, which we will adopt here. The zeroth-order Hamiltonian is 

The fact that this Hamiltonian includes relativistic corrections can be demonstrated by expanding (2mc - V) in a series, to give 

The first term is simply f, the nonrelativistic kinetic energy. Commuting a p) to the right and using the Dirac relation (4.14), the second term gives 

We could also partition t zora commuting ( r p) to the right, to give 

Here we see a modified kinetic energy term that is cut off near the nuclei, a spin-free relativistic correction, and a spin-orbit term, both of which are regularized and behave as 1/r for small r. We may compare this with the regularization of the free-particle Foldy-Wouthuysen or Douglas-Kroll transformed Hamiltonian of section 16.3. The regularization clearly corrects the overestimation of relativistic effects that plagues the Pauli Hamiltonian. There is another consequence of the small r behavior. Since the relativistic correction operator behaves like 1 jr, it ought to be possible to use T zoRA variationally—and in fact we may demonstrate that there is a variational lower bound. 

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