Symmetry relations between the matrix elements

Having defined our starting point, the second quantized no-pair Hamiltonian, we may now take a closer look at the relations between the matrix elements. For future convenience we will also change the notation of these matrix elements slightly. Due to hermiticity of the Dirac Hamiltonian and the Coulomb-Breit operator we have 

The symmetry of the interaction with respect to interchange of integration variables makes that 

Restriction to Kramers-paired basis spinors gives 

The latter two relations were derived using the fact that the operator K, Eq. (12), commutes with full Coulomb-Breit operator. It is also possible to apply time-reversal to individual particle coordinates if we take the different transformation character of the two parts of the two-electron interaction into account. This gives in addition 

If the nuclear framework exhibits spatial symmetry the number of independent matrix elements can be reduced further. It is e.g. possible to make all matrix elements real if a mirror plane or a two-fold rotation axis is present [13]. 

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