Tensor operator triplet

The relations of Eq. (5.17) allow the scaling of all components of a tensor operator with the same factor. The factor used above is the one commonly used. It is noted that if i and j are identical the triplet operator of Eq. (5.24) vanishes. 

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... 

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...

In low-symmetry molecules, diagonal and off-diagonal matrix elements of the electronic dipolar coupling tensor may contribute to ( h[)0 ) J ssl b ). Therefore, they are specified mostly in terms of their Cartesian components. If symmetry is C2V or higher, the off-diagonal matrix elements of the tensor operator in Eq. [163] vanish (i.e., the principal axes diagonalizing the SCC tensor coincide with the inertial axes). For triplet and higher multiplicity states, one then obtains... 

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. 

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. 

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. 

As an example of the genealogical scheme for tensor operators, we consider the construction of a one-electron excitation operator of triplet symmetry S = 1 and M = 0. Because of the requirement of zero spin projection, only two strings will contribute ... 

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 symmetry of the spin-orbit operator was derived in section 10.3. The spatial part of this operator transforms as the vector of rotations R = (Rx, Ry, Rz)- This means that the spin-orbit operator will connect states of different spatial symmetry. In C2v, for example, states of all spatial symmetries are connected by the spin-orbit operator. In D2h, the gerade states are all connected by the spin-orbit operator, and likewise the ungerade states, but there is no connection between gerade and ungerade states because the spin-orbit operator commutes with the inversion operator (it is an even operator). The spin operator transforms as a spherical tensor of rank 1 it is essentially a triplet operator. Therefore, it can connect states whose S and Ms values differ by 0 or 1. 

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