Particle-hole formulation of the unitary group approach to the many-electron correlation problem. II. Matrix element evaluation

Graphical methods of spin algebras are used to derive the expressions for the matrix elements of the total particle-number-conserving operators in the basis of hole-particle states, adapted to the chain $\mathrm{U}(n^{\prime }+n^{\prime \prime })\supset \mathrm{U}\left(n^{\prime }\right)\otimes \mathrm{U}\left(n^{\prime \prime }\right)$, where $n^{\prime }$ and $n^{\prime \prime }$ designate dimensions of particle and hole subspaces, respectively. The matrix elements are expressed as a product of segment values, each associated with one orbital level as in the particle formalism, and of an additional segment value, representing a linkage of the hole and particle subspaces. It is shown that the particle formalism segment values can be used throughout except for the link segment, whose possible values are derived. An example of hole-particle bases and of their graphical representations is given and the advantages of the hole-particle formalism in shell-model calculations are outlined. An extension of this formalism to particle-number-nonconserving operators, needed in applications involving the $m\mathrm{p}-n\mathrm{h}$ propagators with $m\ne n$, $m$, $n=0\mathrm{}$, 1, and 2, is discussed.