Shavitt graph

I. Shavitt, Graph Theoretical Concepts for the Unitary Group Approach to the Many-Electron Correlation Problem, Int. J. Quantum Chem. Symp. 11, 131-148 (1977). 

Wave function analysis with Shavitt graph density in the graphically contracted function method... 

Keywords Multifacet graphically contracted function method Configuration interaction Graphical unitary group approach Shavitt graph Wave function analysis Graph density Node density Arc density... 

The partial lower-walk overlap matrix measures the overlap of all those CSFs that lead from the vacuum node of the Shavitt graph to node k, while the partial upper-walk overlap matrix determines the overlap of all those CSFs that lead firom the head node of the Shavitt graph to node k. Since the CSF coefficients are expressed as products of the arc factor matrices in Eq. (3), the partial overlap matrices may be computed recursively [30]. Assuming that every node of the Shavitt graph is characterized by fmax facets, the recursive computation of all matrices yf and would scale as 0 Nrowf ),... 

Figure 10 contains a sequence of graph density plots computed from MFGCF expansions that have been converged to the full-CI limit. The order of levels in these Shavitt graphs is the same as the MO order above. Of particular interest are the node densities at the 2aig level. 

The so-called graphically contracted function scheme (GCF) introduce a completely new parameterization of the CASSCF-CI space (based on the so-called Shavitt graphs) while maintaining key qualities of the CASSCF approach. The new approach is still in a vigorous development phase and suffers from problems of conceptual importance. For example, the GCF approach is not invariant to the order of the active orbitals. However, the method exhibits enormous development potential and does not only offer a reduced parameter space but also an efficient way in which matrix elements can be computed. Recent improvement of the recursive algorithm, used in GCF, presents CASSCF calculation correlating 156 electrons in 156... 

Figure 1 Shavitt graph for a system of four orbitals and four electrons and the CI-SD expansion. A thick line corresponds to a walk for the ground state CSF. The numbers at arcs are the step numbers d...

We now show how the Shavitt graph may be used for the determination of one-electron coupling coefficients in equation (15). For the CSFs , and y, differing by occupation numbers in two or more orbitals, the Hamiltonian matrix element is vanishing, and for the diagonal matrix elements the coupling coefficients are given simply by the occupation numbers... 

A computer program requires the Shavitt graph in a digital form, the so-called distinct row table (DRT). As a matter of fact, the first achievement in the exploitation of the UGA was the Paldus tableaux for individual CSFs, The distinct rows from these were collected in the ORT and the latter was then given diagrammatic expression by Shavitt and called after him the Shavitt graph. 

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