Submatrix Ranking Operators

Submatrix ranking operators are belonging to the class of image-space operators (Haberacker [42]) in contrast to Fourier-space operators. 

Based on the ranked list DI defines three primitive submatrix ranking operators. They determine, which value is put in the center of the submatrix ... 

Thus, utilizing the concept of irreducible tensorial sets, one is in a position to develop a new method of calculating matrix (submatrix) elements, alternative to the standard way described in many papers [9-11, 14, 18, 21-23]. Indeed, the submatrix element of the irreducible tensorial operator can be expressed in terms of a zero-rank double tensorial (scalar) product of the corresponding operators (for simplicity we omit additional quantum numbers a, a1) ... 

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

It has been shown in the previous section how the submatrix elements of irreducible tensorial products of creation and annihilation operators can be expressed in terms of pertinent one-shell submatrix elements. The submatrix elements of the operators G1-G7 are also defined in terms of the same quantities. Since the quasispin ranks from different shells that... 

Concrete irreducible forms of physical operators follow from (1) by defining ranks ki, k2, cq, a2, k and specifying the two-electron submatrix element... 

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