Time Reversal and Symmetry in the Many-Electron Hamiltonian

Time Reversal and Symmetry in the Many-Electron Hamiltonian 

We have seen how time-reversal symmetry and double-group symmetry are intimately connected in the matrices of one- and two-electron operators. These two symmetries are just as intimately connected in the many-electron Hamiltonian matrix. 

As for the one-electron matrices, we must choose a representation for the basis with which the matrices are represented. We take as our A-particle basis the determinant basis introduced in chapter 9, given in terms of A and B strings, and consider all possible determinants that may be constructed from a given set of Kramers pairs. We group these determinants into subsets with a given value of Na and Nb, characterized by a pseudo-quantum number Mk, 

In the nonrelativistic case, with a and p strings. Mg = Ms- Because the nonrelativistic operators are spin-independent, the Hamiltonian matrix is blocked by Ms- This block diagonalization of the Hamiltonian matrix does not persist in the relativistic case, and in the absence of any point group symmetry, the N-particle basis extends to all Mg values. 

After dividing the determinants into subsets defined by their Mg values, we order the subsets from highest to lowest Mg value. The subsets define a partitioning of the Hamiltonian matrix H into blocks. Determinants from sets whose Mg values differ by more than two have zero Hamiltonian matrix elements between them, because the excitation between them is more than a two-electron excitation, and the Hamiltonian contains at most two-electron operators. With the arrangement of the determinants in Mg blocks from highest to lowest, H is therefore block pentadiagonal. This structure is shown in figures 10.1 and 10.2 for an even and an odd number of electrons. 

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