Tensor operator singlet

A tensor operator with S=0 is called a singlet operator, an operator with S... 

The arbitrary scaling factors on the above tensor operators are again chosen in accordance with the general use. It is seen that the triplet operator is defined also for the case where i and j are identical. The singlet operator, Sjj(0,0) is proportional to the operator E j, obtained from considering spin... 

The condition j + j > 1 for a matrix element of a first rank tensor operator implies, e.g., that there is no first-order SOC of singlet wave functions. Two doublet spin wave functions may interact via SOC, but the selection rule /+ / > 2 for i (2)(Eq. [171]) tells us that electronic spin-spin interaction does not contribute to their fine-structure splitting in first order. 

Table 11 Irreps of Singlet (S) and Triplet (T) Spin Functions, the Angular Momentum Operators (A and S), an Irreducible Second-Rank Tensor Operator 2, and the Position Operators St, 9,3C in C2v Symmetry...

As defined in Equation 3.4, the spin part of i is a traceless symmetric tensor operator that can have nonvanishing matrix elements between states of spin multiplicity different by up to four, such as singlet and quintet or triplet and septet, etc., but in first order we now consider only its elements in the basis of the three sublevels of the T state. JZ is diagonalized in this subspace by rotation of the coordinate system into the principal system of axes x, y, and z in the molecular frame. In principle, the molecular magnetic axes defined in this fashion are different for every triplet state of the molecule. In the presence of symmetry, some or all of them are constrained to the molecular symmetry axes. 

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

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

Therefore, only those terms in f and T2 that transform as singlet curators should be retained in the cluster operator. From the discussion of tensor operators in Section 2.3, it follows that the singlet cluster operators should satisfy (2.3.1) and (2.3.2) ... 

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. 

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) ... 

For strings containing two or more elementary operators, it is possible to constmct more than one tensor operator. We shall in Section 2.6.7 present a general method for the construction of tensor operators from strings of elementary operators. At present, we note that, by coupling the two doublet operators and it is possible to generate a singlet two-body creation... 

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. 

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

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)... 

The electron-phonon operator is a tensor product between the electronic dipole and the nuclear dipole operators. A mixing between the AA and BB via the singlet-spin diradical AB state is possible now. A linear superposition of identical vibration states in AA and BB is performed by the excited state diradical AB. If the system started at cis state, after coupling may decohere by emission of a vibration photon in the trans state furthermore, relaxation to the trans... 

The evaluation of a spin operator times an occupation number vector is faciliated by noting that the core is a singlet spin tensor, since ata ajjj is a singlet spin operator (Eq. 5.25). The action of Sz on I na np> becomes... 

The definitions of the perturbation corrections to the interaction energy, as given by Eqs. (8), (9), and (11), involve nonsymmetric operators, like H0, V, and Rq. These operators do not include all electrons in a fully symmetric way, so H0, V, and Rq must act in a larger space them the dimer Hilbert space Hab adapted to a specific irrep of 4. To use these operators we have to consider the space Ha Hb, the tensor product of Hilbert spaces Ha and Hb for the monomers A and B. For the interaction of two closed-shell two-electron systems this tensor product space should be adapted to the irreps of the 2 2 group. In most of the quantum chemical applications the Hilbert spaces Hx, X = A and B, are constructed from one-electron spaces of finite dimension Cx - In the particular case of two-electron systems in singlet states we have Hx — x x, where the symbol denotes the symmetrized tensor product. We define the one-electron space as the space spanned by the union of two atomic bases associated with the monomers in the dimer. The basis of the space includes functions centered on all atoms in the dimer and, consequently, will be referred to as the dimer-centered basis. We assume that the same one-electron space is used to construct the Hilbert spaces Hx 1 X = A and B, i.e., Hx = In such case Ha )Hb can be represented as a direct sum of Hilbert spaces H adapted to the irreps entering the induced product [2] [2] 4, i.e., Ha ( Hb = H[2i] f[3i] f[4]- This means that every function from H , v = [22], [31], or [4], can be expanded in terms of functions from Ha Hb-... 

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