Matrix Elements and Symmetry

Having discussed the symmetry of the spinors and products of spinors, we are now in a position to discuss the symmetries of the one- and two-electron integrals. For computational applications, the integrals over molecular orbitals are often divided into symmetry classes for convenient handling of symmetry. In addition, the consideration of symmetry may produce some simplification in the expressions for the many-electron Hamiltonian. In the relativistic case we must use both point-group and time-reversal symmetry. 

The Kramers-restricted relativistic case is a little more complicated. First, because the basis is complex, we must use the conjugate representation for the bra in any integral. In the notation used above, we can express this as 

So if r] , and are from the same row of the same irrep, the product transforms as r p)r(q). It is this product that contains the symmetric irrep. Thus, for 

Here we refer to the orbitals or spinors by their indices. 

The symmetry reduction for the one-electron integrals follows from the consideration of time-reversal symmetry  

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