Angular momenta operator table

The orbital angular momentum operations needed to calculate integrals for other orbitals are summarized in Table 4.2. 

Table 4.2 [E] Angular momentum operations on the real p and d orbitals...

Since the matrix elements of the angular momentum operator have already been determined in a simple form, and the symmetry adaptation coefficients are also known, we can proceed in the transformation to the basis set of CFTs. This work is presented in Table 8. 

Here A2 symbolizes a pseudo-scalar of A2 symmetry, normalized to unity. The actual form of this pseudoscalar need not bother us. The only property we will have to use later on is that even powers of A2 are equal to +1. Now we can proceed by defining rotation generators f x, y,t 2 in the standard way, as indicated in Table 1 [10]. Note that primed symbols are used here to distinguish the pseudo-operators from their true counterparts in real coordinate space. Evidently the action of the true angular momentum operators t y, (z on the basis functions is ill defined since these functions contain small ligand terms. 

Table 1. Effects of pseudo-angular momentum operator on real t-orbitals...

Table 7. Effects of angular momentum operator on real 4-orbitals (in units of h)...

Table D.l. Reduced matrix elements of the angular momentum operator...

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. 

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...

Table 4.11 Angular momentum operators Lx, Ly, acting on cartesian angular momentum harmonic eigenfunctions in first column produce eigenfunctions of the same type with 1 = 2...

Therefore, instead of the irreducible tensor components, a proper combination of the angular momentum operators (Jz, J+, J ) is applicable. However, the expansion coefficients (the potential constants) need to be redefined to account for the proportionality factor a (k,j). A set of equivalent operators is compiled in Table 8.17. It is quite practical to handle the equivalent operators since they can be easily constructed by matrix multiplications with the help of computers. 

To predict this property, group theory is applicable (Table 8.32). The angular momentum operator La (a = x, y, z) has the same transformation properties as the rotation operator Ra in the point group characterising the symmetry of the molecule. It transforms as the irreducible representation ra of this group. The above requirement is equivalent to the condition that the direct product of the irreducible representations rt of the wave function (which is a reducible representation Tr) contains the irreducible representation of the angular momentum (or rotation) operator ra, viz. 

These three functions are eigenfunctions of the angular momentum operator Li with eigenvalues wj = 0, 1. The normalization factor N is (3/4ir). Under the symmetry operations of Eq. 7.10(b) is totally symmetric, thus it is the 2po orbital that transforms like S" " in the site symmetry (Table 7.1). The functions 0(2p+i) and 0(2p i) are partners in the ir representation of C , and give rise to molecular functions of... 

Table 2 Useful relationships for the use of spherical tensors in the calculation of matrix elements of angular momentum operators (courtesy of Prof. JM Brown, PTCL, Oxford University)...

The analysis of the rotational spectrum of an asymmetric molecule in the vibrational state ui,... vj,... v u-6 normally allows the determination of the constants listed in this table. All rotating molecules show the influence of molecular deformation (centrifugal distortion, c.d.) in their spectra. The theory of centrifugal distortion was first developed by Kivelson and Wilson [52Kiv]. The rotational Hamiltonian in cylindrical tensor form has been given by Watson [77 Wat] in terms of the angular momentum operators J, J/and as follows ... 

In Table I these functions and operators are defined explicitly in terms of the spatial derivatives of the electrostatic potential, the angular momentum operators and the nuclear quadrupole moment Q. 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

The SH Hs acts only on the spin kets S,Ms) yielding eigenvalues that are identical with those produced by the perturbation operator H acting on the full set of spin-orbit variables a,L,Mi,S,M ). This situation is explained in Table 1 the truncated SH matrix involves integrals over the angular momentum via the perturbation theory. 

The angular momentum components span a definite irreducible representation (IR) of the given point group (Table 9), and thus its matrix element vanishes unless the direct product of the IRs for the bra kets contains the IR of the Ifl-operator hence... 

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