Douglas spin-free approximation

For systems having unpaired electrons, it is usually necessary to include spin-orbit interaction, leading automatically to a two-component treatment. This was actually the original form of the Douglas—Kroll theory—the one-component, spin-free formulation is the result of a further approximation. The mere nature of spin-orbit interaction almost invariably calls for a multistate treatment, and it would thus seem that the level of complexity is dramatically increased as a result of the configuration interaction treatment that is typically required. In all fairness, though, one must keep in mind that the spin-orbit interaction is typically of importance because questions of multistate nature are being asked, and in such cases even a nonrelativistic treatment would often require a Cl treatment. 

We could of course proceed as we did for the Douglas-Kroll transformation and use a transformation that depends only on the nuclear potential. This would remove the awkwardness of having 1 /r,y in the denominator, but we still have the product of c f(2mc — Vi) with 1 /r,-y to deal with. If we are only interested in spin-free relativistic effects, we could neglect the transformation of the electron-electron interaction, as we did in the Douglas-Kroll-Hess approximation. This approximation yields the 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... 

Spin-free relativistic effects are readily incorporated into the ab initio model potential approximation by using a one-component spin-free relativistic method for the atom, such as the Cowan-Griffin method" or the Douglas-Kroll-Hess method. 

In chapter 16, we showed how approximate relativistic spin-free Hamiltonians could be derived using unitary ttansformations. For applications to molecular systems the most frequently used model is the Douglas-Kroll-Hess model. Having successfully carried out a spin-free Douglas-Kroll-Hess calculation, we may be faced with the challenge of trying to account for the spin-orbit effects. In order to do this, we have to identify the terms that have been neglected in order to arrive at a spin-free formalism. 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

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