Nonperturbative Improvements of the ZORA Equation

The perturbation series discussed so far start from the ZORA Hamiltonian as the zeroth-order approximation. We already know that this Hamiltonian has relativistie eorreetions in it but is missing some terms of 0 c ). These terms in faet eome from the correction to the metric, as we saw in the previous section. Is there a way to obtain an improved zeroth-order approximation In this section, we consider two possible improvements. 

Our line of development here is to return to the exact Foldy-Wouthuysen transformation, presented in section 16.1 and used above to develop a perturbation series. There, we chose ll(2mc -V) as the perturbation and expanded both the square root and the inverse powers. Here, we choose the perturbation parameter as El 2mc - V) 

The operator in the second bracket is the ZORA Hamiltonian, and it is sandwiched by normalization operators. If we expand these operators as we did above, we get the FORA Hamiltonian as the first term. The higher terms differ, however, because the final energy in the previous series must be the Dirac energy, whereas here it is the energy for the approximate Hamiltonian. Inclusion of the normalization terms corresponds to a resummation of certain parts of the ZORA perturbation series to infinite order, and the name coined by Dyall and van Lenthe (1999) is lORA—infinite-order regular approximation. 

The lORA Hamiltonian still presents some problems, because there are momentum operators in the inverse square roots of the normalization operators. Defining a modified wave function, 

The spectrum of the lORA Hamiltonian can be derived by considering the asymptotic region for large r, where V - 0 and the lORA equation goes to 

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