Hamiltonian spin-free modified Dirac

The spin-free modified Dirac Hamiltonian is obtained by simply omitting the second term on the right-hand side of (15.11) ... 

We will evaluate the numbers of integrals required for a calculation with the unmodified Dirac Hamiltonian, and compare them with the number of integrals required for a calculation with the spin-free modified Dirac Hamiltonian, and with the number required for a nonrelativistic calculation. The spin-free Hamiltonian is formed by summing all the spin-free terms defined above, but we will consider the Coulomb term and the Gaunt and Breit terms separately. For the purpose of this evaluation, we make the following assumptions and definitions ... 

From (15.8) we can extract a modified Dirac Hamiltonian that consists of a spin-free and a spin-dependent term. 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

This equation is exact just as the modified Dirac equation of chapter 15 is exact. It can also be separated into spin-free and spin-dependent terms, but now the separation must be done in both the Hamiltonian and the metric. Visscher and van Lenthe (1999) have shown that the spin separation gives different results for the two modified equations, and therefore the spin separation is not unique. This regular modified Dirac equation can be used in renormalization perturbation theory, with ZORA as the zeroth-order Hamiltonian. 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

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