Kramers-unrestricted configuration interaction

One solution to avoid these complications is to take a brute-force approach and not distinguish between barred and unbarred spinors at all. The advantage is that such a formalism will also be applicable in conjunction with Kramers-unrestricted Hartree-Fock procedures. 

In order to make such brute-force procedures feasible one has to make the computation of coupling coefficients very efficient. It is thereby very instructive to visualize this process using graphical representations [34] of the Cl spaces. As an example of such a graphical representation of a Cl space we consider the case with 3 electrons in 6 spinors. 

All possible determinants with 3 electrons in 6 one-electron spinors can be depicted as paths in a graph (Fig. 1). An occupied spinor P is represented by a diagonal arc connecting vertex (P-1, n-1 to vertex P,n) while unoccupied spinors are represented by vertical arcs. The determinant 246 is thus represented by the thick line in the graph. 

This graphical representation provides a prescription for the enumeration of the determinants via the one-to-one mapping of paths in the graphs to determinants. We define a vertex weight fV P, n) as the number of different paths that lead from the top of the graph to this vertex. It is easy to see that this weight obeys the recursion formula 

All vertical arcs (connecting vertices with the same number of electrons) have zero weight. 

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