Spin properties of determinants

Having considered in Section 2.3 the spin-tensor properties of the basic operators in second quantization, we now turn our attention to the determinants. In particular, we shall establish under what 

At this point, a comment on terminology is in order. In Chapter 1, we used the term ON vector for second-quantization vectors that correspond to Slater determinants in first quantization. The unusual term ON vector was employed in order to make a clear distinction between the first-and second-quantization representations of the same object and to emphasize the separate and independent structure of second quantization. We shall from now on use the conventional term Slater determinant or simply determinant for ON vectors, bearing in mind that determinants in second quantization are just vectors with elements representing spin-orbital occupations. 

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The Hamiltonian H therefore possesses a common set of eigenfunctions with and Sz and we shall assume that the exact wave function (2.4.1) is a spin eigenfunction with quantum numbers S and M, respectively. Consequently, when calculating an approximation to this state, we shall often find it convenient to restrict the optimization to the part of the Fock space that is spanned by spin eigenfunctions with quantum numbers S and M. It is therefore important to examine the spin properties of determinants. 

We should therefore be able to establish the general spin properties of determinants by examining the commutators between the spin operators and the spin-orbital ON operators. We note that the spin-orbital ON operators commute with the spin-projection operator ... 

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