Seniority and the Sp(2j + 1) Group

In this section we come back to consider the seniority scheme in jⁿ configurations from another point of view. The remarkable relation (20.14) is the simple expression for certain eigenvalues of Hamiltoni-ans which are diagonal in the seniority scheme. All matrix elements in jⁿ configurations can be expressed as linear combinations of two body matrix elements V_J = 〈j ² JM|V|j ² JM〉. Yet, the interaction in states with lowest seniorities, ν = 0 and ν = 1, is a linear combination of only two parameters. These are V ₀ and the linear combination V ¯ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq2389.tif"/> defined by (20.7). This property is a special case of a more general feature of the seniority scheme, namely (25.1) T h e     a v e r a g e     i n t e r a c t i o n     e n e r g y     o f     a     s e t     o f     s t a t e s w i t h     g i v e n     s e n i o r i t y     v     i n     t h e     j n     c o n f i g u r a t i o n     i s a     l i n e a r     c o m b i n a t i o n     o f     o n l y     V 0     a n d     V ¯ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq2390.tif"/>