Foldy higher-order

The problem of nonrelativistic limit description for fundamental particles and their interactions may be solved in different ways. Although in all methods of nonrelativistic expansion the first terms of the Hamiltonians coincide, however the difference begins to arise at transition to the higher orders of expansion. The method of Foldy-Wouthuysen... 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

Higher-Order Foldy-Wouthuysen Transformations and Their Pathologies... 

For the two-electron terms we need to identify the spin-orbit operators in the transformed Hamiltonian. Again, we only consider the free-particle Foldy-Wouthuysen transformation corrections from higher-order transformations are likely to be very small. The transformed Coulomb interaction was written in (16.63) as... 

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