Spin tensor operators

The incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

A spin tensor operator of integral or half-integral rank S is a set of 2S - -1 operators where M runs from —S to S in unit increments and which fulfills the relations [1] 

A tensor operator working on the vacuum state generates a set of spin eigenfunctions with total and projected spins S and M (provided the tensor operator does not annihilate the vacuum state). We may prove this assertion in the following way. Since the second-quantization spin-component 

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 

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. 

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As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

So far we know the selection rules for spin-orbit coupling. Further, given a reduced matrix element (RME), we are able to calculate the matrix elements (MEs) of all multiplet components by means of the WET. What remains to be done is thus to compute RMEs. Technical procedures how this can be achieved for Cl wave functions are presented in the later section on Computational Aspects. Regarding symmetry, often a complication arises in this step Cl wave functions are usually determined only for a single spin component, mostly Ms = S. The Ms quantum numbers determine the component of the spin tensor operator for which the spin matrix element (S selection rules dictated by the spatial part of the ME. 

Z < M 5>[Cs, + tfSf] which contains the singlet-spin tensor operator. 

The nonrelativistic Hamiltonian (2.2.18) is a spin-free operator - that is, a spin tensor operator of zero rank see Section 2.3. Determinants of different spin projections therefore give vanishing Hamiltonian matrix elements and we may restrict the determinants of the Cl expansion to have the same spin projection. If the total number of electrons is N and the spin projection is M. the numbers of electrons with alpha and beta spins are given by... 

In second quantization, the singlet one- and two-electron spin tensor operators have the following representations ... 

One important observation that should be made about the spin tensor operators is that any singlet operator commutes with both the shift curators S and the spin-projection operator 5 see (2.3.1) and (2.3.2). It therefcM e follows that singlet opo tors also commute with 5 since this operator may be expressed in terms of the shift operators and the spin-piojection operator (2.2.38) ... 

The commutator of two Hermitian operators is an anti-Hermitian operator. From (2.3.2), we can therefore conclude that spin tensor operators are not in general Hermitian. Indeed, the only possible exception to this rule are the operators where M = 0, which may or may not be Hermitian. It is therefore of some interest to examine the Hermitian adjoints of the spin tensor operators. Taking the conjugate of the relations (2.3.1) and (2.3.2), we obtain ... 

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... 

Comparing (2.3.34) and (2.2.40), we note that the three components of the spin-orbit operator are treated alike in the Cartesian form (2.2.40) but differently in the spin-tensor form (2.3.34). The spin-tensor representation (2.3.34), on the other hand, separates the spin-orbit operator into three terms, each of which produces a well-defined change in the spin projection. From the discussion in this section, we see that the singlet and triplet excitation operators (in Cartesian or spin-tensor form) allow for a compact representation of the second-quantization operators in the orbital basis. The coupling of more than two elementary operators to strings or linear combinations of strings that transform as irreducible spin tensor operators is described in Section 2.6.7. 

We give in this section an introduction to the construction of CSFs and more generally to the construction of spin tensor operators. We shall employ the genealogical coupling scheme, where the final CSF for N electrons is arrived at in a sequence of N steps [2]. At each step, a new electron is introduced and coupled to those already present. We thus arrive at the final CSF through a sequence of N —I intermediate CSFs, each of which represents a spin eigenfunction. 

For ease of notation, we discard the superscript in p and consider the product (p t) with an unspecified Slater determinant. Writing the determinant as a product of creation operators (2.6.2) and the CSF by means of a spin tensor operator (2.6.3), we obtain... 

From this point of view, then, the construction of CSFs from determinants (2.6.1) may be viewed as the construction of a spin tensor operator from strings of creation operators... 

Thus, in the construction of a spin tensor operator, we employ (2.6.33) for creation operators and (2.6.34) for annihilation operators, assuming that the number and relative ordering of the creation and annihilation operators are fixed. 

Assume that are components of a normalized spin tensor operator of rank S —... 

Consider the spin tensor operator. Show that... 

Use the properties of spin tensor operators to verify the following relationship between the matrix elements of a singlet operator... 

Applying (2.3.1) twice, we find that the spin tensor operator must satisfy the condition... 

Spin-adapted rotations The spin-free nonrelativistic Hamiltonian commutes with the total and projected spin operators. We are therefore usually interested only in wave functions with well-defined spin quantum numbers. Such functions may be generated from spin tensor operators that are totally symmetric in spin space. For optimizations, we need consider only singlet opeiatois since these are the only ones that conserve the spin of the wave function. Spin poturbations, on the other hand, may mix spin eigenstates and require the inclusion also of triplet rotations. 

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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