Finite Matrix Methods for Dirac Hamiltonians

Non-relativistic quantum theory of atoms and molecules is built upon wave-functions constructed from antisymmetrized products of single particle wave-functions. The same scheme has been adopted for relativistic theories, the main difference now being that the single particle functions are 4-component spinors (bispinors). The finite matrix method approximates such 4-spinors by writing 

We now insert (96) and (98) into (97), giving a ratio of two quadratic forms 

The kinetic matrices (where T = S when T = L and vice versa) 

We can now apply standard methods of the calculus of variations to R c) this function is stationary with respect to weak variations of the coefficient vector c and cl if c is an eigenfunction of the equation 

Whilst this argument mirrors simple textbook derivations for nonrelativistic problems, for example [73, 32], such a non-rigorous treatment is not enough for relativistic calculations. We have already seen in Section 1 that early attempts to solve Dirac s equation by matrix methods encountered unexpected 

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