Spin tensor state

These are the defining equations for a spin tensor state of rank S. The tensor state is obviously an eigenfunction of the projected spin (2.3.5). To determine the effect of the total-spin operator on the tensor state, we combine relations (2.3.4) and (2.3.5) with the expression (2.2.38) for the spin... 

It is also possible to employ highly correlated reference states as an alternative to methods that employ Hartree-Fock orbitals. Multiconfigu-rational, spin-tensor, electron propagator theory adopts multiconfigura-tional, self-consistent-field reference states [37], Perturbative corrections to these reference states have been introduced recently [38],... 

The eigenvalue of Qz is N — 2 for a state of gN. We can now consider that the identical components of g1 and g together form a quasi-spin tensor of rank 1/ 2, whose array of ranks we can now indicate by writing G(l - - -]. The e, operators can be broken down into parts that have well-defined quasi-spin ranks however, it turns out that e2 is a quasi-spin scalar, which can be used to explain some similar matrix elements of e2 in g 2 and g 4 [10]. 

Given a molecule that possesses C2p symmetry, let us try to figure out how to calculate ( Ai ffsol Bi) from wave functions with Ms = 1. The coupling of an Ai and a B state requires a spatial angular momentum operator of B2 symmetry. From Table 11, we read that this is just the x component of It. A direct computation of (3A2, Ms = 1 t x spin-orbit Hamiltonian with x symmetry and So correspondingly for the zero-component of the spin tensor. This is the only nonzero matrix element for the given wave functions. 

The derivation above is not the only way to obtain the required result, but it is straightforward, if somewhat tedious. The reduced matrix element of the fourth-rank spin tensor, T4, S. S, S), which can arise in the analysis of higher spin states, is obtained by further use of the recursion relationship given by Edmonds [80], See also the general expression given in equation (5.134) of chapter 5. 

The equations for other solution averaging conditions are given in references (18) and (19). In the special case of a spin-only state with an isotropic g-tensor (i.e. gy = gi and g n = gsi = 2) equation (9) reduces... 

We conclude that T - lvac) - provided that it does not vanish - represents a tenscH- state with spin eigenvalues 5 and M. Because of the close relationship between spin tensw operators and spin eigenfunctions, the terminology for spin functions is often used for spin tensor operators as well. Thus, a spin tensor operator with 5 = 0 is referred to as a singlet operator, S — gives a doublet operator, 5 = 1 a triplet, and so on. 

Spin tensor operators play an important role in the second-quantization treatment of electronic systems since they may be used to generate states with definite spin properties. In the remainder... 

The form of ir in (10.10.4) is useful for discussing the differences between RHF and UHF theory and, in particular, for examining what happens when an already optimized RHF state is reoptimized in the full set of symmetry-breaking variational parameters. For the direct optimization of the UHF wave function (10.10.3) itself, it is more convenient to work with excitation operators that are not spin tensor operators. Decomposing the singlet and triplet excitation operators in alpha and beta parts (see Section 2.3.4)... 

Chapters 1-3 introduce second quantization, emphasizing those aspects of the theory that are useful for molecular electronic-structure theory. In Chapter 1, second quantization is introduced in the spin-orbital basis, and we show how first-quantization operators and states are represented in the language of second quantization. Next, in Chapter 2, we make spin adaptations of such operators and states, introducing spin tensor operators and configuration state functions. Finally, in Chapter 3, we discuss unitary transformations and, in particular, their nonredundant formulation in terms of exponentials of matrices and operators. Of particular importance is the exponential parametrization of unitary orbital transformations, used in the subsequent chapters of the book. 

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