Matrix elements in the Ŝz and L̂z-adapted spaces
Calculation of matrix elements in the Ŝ z -adapted space will serve as a simple application of the previous paragraph’s general formalism. It will also serve as a preparation for more complex applications discussed in the next sections. In Ŝ z -adapted case ❘L, ∑〉 in Eq (2.37) represents a determinant, ❘L〉 designating the orbital product and ❘∑〉 the product of α,β spin functions corresponding to these orbitals. The orbital configurations are represented by the three-slope graph (cf Fig 7, 8, 32) and the primitive spin functions by the M-diagram (Fig 9). Alternatively, the ❘L,∑〉 states are represented by non-fagot graphs (cf Fig 4–8). The elements 〈L̂∑❘Ô❘RΘ〉 may be obtained directly from these graphs (Duch 1985c) analyzing the paths corresponding to ❘L, ∑〉 and ❘R,Θ〉. Let us start from analysis of the three-slope graph and the associated M-diagrams. Purely graphical method is presented first, with a more direct and computationally attractive approach evolving from it. The use of the four-slope and other non-fagot graphs is discussed later in this section. Because the L̂ z and (L̂ z ,Ŝ z -adapted spaces have also determinantal bases the same techniques as for Ŝ z eigenfunctions are applicable.
KeywordsMatrix Element Shift Operator Orbital Configuration Occupied Orbital Loop Body
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