Reduction of the Breit Hamiltonian to non-relativistic form

The first stage in deriving a molecular Hamiltonian is to reduce the Breit equation to non-relativistic form and Chraplyvy [17] has shown how this reduction can be performed by using an extension of the Foldy-Wouthuysen transformation. First let us remind ourselves of the most important features in the transformation of the Dirac Hamiltonian. The latter was written (see (3.57) and (3.58)) as 

Now the Breit Hamiltonian for two electrons in the presence of electromagnetic fields is, as we have seen, 

By comparison with (3.228) the even-even, odd-even, even-odd and odd-odd operators are 

The Breit Hamiltonian operates on sixteen-component spinor functions which contain fourtypes of function, designated A L. 1- fiu- fu, which represent upper and lower 

As a preliminary to this it is worthwhile noting a few properties of the matrix 

The Breit Hamiltonian (3.228) can be written in a fonn analogous to (3.225), namely, 

The Breit Hamiltonian operates on sixteen-component spinor functions which contain fourtypes of function, designated V uL. fm, V i,whichrepresentupperandlower components as previously defined, the small letters u, referring to the first particle (1) and the capital letters U, L referring to the second particle (2). Our aim is to find a transformation which gives a Hamiltonian operating only on the components uu in other words, we seek a Hamiltonian which, to order c, contains only terms which are overall even-even in character. 

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