Angular momenta operator matrix elements

The exploitation of the community of the transformation properties of irreducible tensors and wave functions gives us the opportunity to deduce new relationships between the quantities considered, to further simplify the operators, already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements. Indeed, it is possible to show that the action of angular momentum operator Lf on the wave function, considered as irreducible tensor tp, may be represented in the form [86] ... 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

The matrix elements of the angular momentum operators in Cartesian directions a = x,y, z form (complex) matrices La. Now we can proceed with an evaluation of the /1-tensor components in three steps. 

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. 

We shall use the T-/ -isomorphism that allows us to consider the orbital triplet T2 as a state possessing the fictitious orbital angular momentum L = 1, keeping in mind that the matrix elements of the angular momentum operator L within T2 and P bases are of the opposite signs, L(T2) = —L(P) [2]. As it was shown in our recent paper [10] this approach provides both an efficient computational tool and a clear insight on the magnetic anisotropy of the system that appears due to the orbital contributions. Within T-P formalism the spin-orbital and Zeeman terms can be represented as ... 

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

For the determination of matrix elements, it is often more convenient to use linear combinations of the Cartesian components of the angular momentum operator instead of the Cartesian components themselves. In the literature, two different kinds of operators are employed. The first type is defined by... 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

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. 

We see from the way in which the effective rotational Hamiltonian is constructed that it is naturally expressed in terms of the angular momentum operator N. In the scientific literature, however, it is frequently written in terms of the vector R (which represents the rotational angular momentum of the nuclei) rather than N. While R = N — L occurs in the fundamental Hamiltonian (7.71), its use in the effective Hamiltonian is not satisfactory because R has matrix elements (due to L) which connect different electronic states and so is not block diagonal in the electronic states. In practice, authors who claim to be using R in their formulations usually ignore any matrix elements which they find inconvenient such as those of Lx and Ly. We shall return to this point in more detail later in this chapter. 

Failure to recognise this leads to the introduction of extra terms in the Hamiltonian To summarise, a Hamiltonian formulated in terms of N2 has the highly desirable characteristic that each term is the product of a determinable parameter which governs the magnitude of the interaction and an angular momentum operator whose matrix elements are fully defined. The difficulties associated with the effect of the L2 terms are confined to the interpretation of parameters in the effective Hamiltonian and their comparison with ab initio calculations. 

Inserting the LCAO expansions for %x and of the form of (10) in (11), the matrix element of the angular momentum operators, and hence the matrix element of hso on the left hand side of (11) is easily seen to be a linear combination of terms of the form... 

The resulting SOC for the states is negligible, since both the n- and the tT--orbital are not centered at the metal (Rule D). Moreover, and even more strictly, these matrix elements of the angular momentum operator vanish according to Rule E. 

Here, we discuss how Rule E is related to the fact that angular momentum operators are purely imaginary. It further depends on the fact that, without magnetic field, the spatial orbitals are real. It is claimed that the diagonal matrix elements of the angular momentum operators vanish. For instance, in the case of lz, we have for any real orbital X = X(f)... 

Kronecker s delta X is the spin-orbit coupling constant and Ag j is defined in terms of the matrix elements of the orbital angular momentum operator L by... 

The hybrids are then written in terms of atomic states, as in Eq. (3-1), and familiar properties of the angular momentum operators are used to evaluate the intra-atomic matrix elements. In terms of our notation in Eq. (3-1), (Py = ih, but... 

The operator f a g contains the cross-terms that give rise to the Coriolis coupling that mixes states with different which is the quantum number of the projection of the total angular momentum operator J on the intermolecular axis. This term contains first derivative operators in y, and its matrix elements change on application of (18) according to... 

In equation (3) I/ matrix elements involving the angular momentum operator, L, are one centre in character and are given in units of h/i. 

The work [5], to be reviewed in this section, makes use of the asymmetric distribution Hamiltonian, as well as the cartesian component, ladder and square of the angular momentum operators and their actions on the chosen spherical harmonic basis lnij, for (L/,k) = cyc x,y,z). Here, we start from its Eqs. (26-28) for the matrix elements of H in the alternative bases ... 

It is highly useful to employ symmetry relations and selection rules of angular momentum operators for SOC matrix elements [108, 109], The Wigner-Eckart theorem (WET) allows calculations of just a few matrix elements of manifold S. M. S, M in order to obtain all other matrix elements. The WET states that the dependence of the matrix elements on the M, M quantum numbers can be entirely... 

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