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Figure 10.2 Diagram of Hamiltonian block structure for N odd. The blocks are ordered by Mk
Na Np Mk 
We now consider the composition of the Hamiltonian matrix elements Hpq between determinants P and Q on each block diagonal. For convenience we will simply refer to these as diagonals. For the main diagonal, \MK(P) — MK(Q)\ = 0, and there must be the same number of A and B spinors. The matrix elements consist of one-electron integrals oftheclass htu, and two-electron integrals of the classes (tu\vw) and (tu\vw). These classes include all integrals that are related by time-reversal symmetry. For the first off-diagonal, \MK(P) — MK(Q)\ = 1, and one determinant must have an A spinor where the other has a B spinor. The matrix elements are therefore composed of integral classes htu and (tu\vW). For the second off-diagonal, \MK(P) — MK(Q)\ =2, and one determinant must have two A spinors where the other has two B spinors. The matrix elements are therefore composed of the integral class (tu\vW).
The classes of two-electron integrals required for the construction of the matrix elements are therefore different for each block diagonal, and the use of time-reversal properties to classify the integrals provides an efficiency in the construction of the Hamiltonian matrix.
To proceed with the analysis of the structure of H and the relation between time-reversal and point-group symmetry, we consider the case of D2h and subgroups, the so-called "binary" groups. There are two good reasons for this. First, these groups are particularly convenient for computational implementations. As all the operations are twofold, only a parity factor (or sign bit) is required to describe the effect of an operation on a function. In addition, each group contains only one of the three types of irreps described previously, and it is therefore customary to classify these groups accordingly:
• The groups D2h, D2, and C2v, whose fermion irreps are all real, will be called real groups.
• The groups C2h, C2, and Cs, whose fermion irreps are all complex, will be called complex groups.
• The groups Q and Ci, whose fermion irreps are all quaternion, will be called quaternion groups.
The advantage of having to deal with only one type of fermion irrep will become apparent below.
With the restriction to binary group symmetry, we realize that the integrals on the first off-diagonal are only nonzero for the quaternion groups. For the two other types of binary groups, a spinor and its time-reversed partner belong either to different one-dimensional irreps (complex groups) or to different rows of the same two-dimensional irrep (real groups), and therefore integrals of the types htu and (tu\vW) must vanish. The elements on the first off-diagonal are represented by the grey shading in the diagrams, and we see that when these disappear for the real and complex groups, the diagonal and second off-diagonal blocks partition into two disjoint, interleaving groups, represented by the striped white and grey shading. As a consequence the determinants may be partitioned into two sets. The relation between these sets depends on the number of electrons:
• For N odd, the N-electron wave function must transform under one of the fermion irreps. In this case time reversal of one set produces the other set. We see this from figure 10.2 by inversion through the center of the diagram.
The two sets of determinants therefore form a basis for the doubly degenerate Kramers pairs of the N-electron states. For the fermion irreps of the binary groups, the only other element of symmetry that can be exploited is inversion, which may be handled in the same way as in nonrelativistic CI theory.
• For N even, the overall symmetry of the wave function must be that of a boson irrep. The application of time reversal maps each set of determinants onto itself. The two sets form bases for states of different boson symmetry, and the set with Mk even contains the basis for the totally symmetric irrep. For the complex binary groups C2 and Cs there are only two boson irreps, and so the Hamiltonian matrix is fully blocked by the use of Kramers pairs. Thus selection of the parity of Mk determines the state symmetry. The last of the complex groups, C2h, has inversion symmetry, which gives rise to two more boson irreps, but again inversion may be treated by nonrelativistic methods.
Thus, apart from inversion, time-reversal symmetry provides for full symmetry blocking of the Hamiltonian matrix for an odd number of electrons in complex or real groups and for an even number of electrons in complex groups.
For an even number of electrons in real groups, the remaining element of symmetry is introduced by forming a real basis in the manner described in (9.58). From the consideration of the Hamiltonian matrix in this real basis in (9.60), it may be seen that the elements of H in the real basis that connect the positive and negative combinations come from the imaginary part of the elements in the determinant basis. However, for the real groups, the Hamiltonian must be real because all the integrals from which it is constructed are real. Therefore, these blocks must be zero, and the Hamiltonian is partitioned into four blocks, defined by the parity of MK and the sign of the combination of determinants in the real basis. The symmetry of each set of such configuration state functions (CSFs) may be deduced from the discussion of the symmetry of two-particle states above.
The matrix elements between the CSFs in the real basis involve two Hamiltonian matrix elements between determinants, Hpq and Hpq . However, Hpq is only nonzero for \Mk(P)-Mk(Q)\ < 2, because otherwise the excitation between P and Q is more than a two-particle excitation. Thus it is only in the center blocks of the Hamiltonian that the linear combination of matrix elements needs to be taken.
The only remaining case to consider is that of the quaternion groups with N odd. The Hamiltonian matrix may be blocked by the quaternion transformation given previously,
producing two quaternion matrices of half the rank, with matrix elements
As for the real groups for N even, the blocks for which both Hpq and Hpq are nonzero occur only for \MK(P) — MK(Q)\ < 2. Although the amount of work forming the Hamiltonian or forming its product with a vector would be the same without the transformation, the indexing work is halved, and therefore it may be useful to perform the transformation.
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