GRMS or Graphical Representation of Model Spaces pp 161-178 | Cite as

# Non-fagot graphs and the Ŝ^{2}—adapted space

## Abstract

S-diagrams are obtained from M-diagrams by removing the points with negative *M* values (cf Fig **9**,**19**,**46**). Similarly the four-slope graphs G_{4}(*n*: *N*,*S*) and the two-slope graphs G_{2,2}(2*n*: *N*, *S*) representing Ŝ^{2}—adapted spaces are obtained from the corresponding graphs representing Ŝ
_{ z }-adapted spaces removing points for negative *M* (cf Fig **5c**,**7c** with Fig **20b**,**c**). It is amazing how much this small change complicates segmentation rules derived in section **2.3** for Ŝ_{ z }—adapted four-slope graphs. For single shift operators segment values (called one-body segments) instead of ±1 phase factors for determinants should be expressed using *a*_{ s }, *b*_{ s } coefficients of Eq (2.69) and *C*^{+}, *C*^{−} coefficients for segments [12] or [21] due to equation Eq (2.70). Therefore one-body segment values should be quite straightforward to obtain without graphical methods of spin algebras or general angular momentum theory. Products of many shift operators (many-body segments) in Ŝ
_{ z }
-adapted spaces are calculated using the same segmentation values, Eq (2.55). It is not so simple with the spin eigenfunction basis because the product (2.66) involves more than one intermediate state. One may artificially introduce these intermediate states and perform summations using one-body segment values (Shavitt 1978), but for products of two shift operators special tricks are possible reducing the number of intermediate summations to at most two (cf Drake and Schlessinger 1977; Paldus and Boyle 1980; Payne 1982).

## Keywords

Shift Operator Terminal Segment Triplet Pair Intermediate Spin Parent Path## Preview

Unable to display preview. Download preview PDF.