Spin-Free Terms

All of the magnetic and electric hyperfine matrix elements were derived in our discussion of the LiO spectrum. We now use the symbol m to denote nuclear spin-free terms, as listed above in equations (8.406), and denote the hyperfine terms, previously given in equations (8.384) and (8.385), by the symbol hf. Since the 14N nucleus has spin 7=1, each J level is split into three hyperfine levels, characterised by / =./,. / 1, except for the J = 1 /2 level which has only two hyperfine components, with F = 1 /2 and 3 /2. Consequently if we neglect matrix elements off-diagonal in J for the moment, each characteristic set of J, F levels is described by a 4 x 4 matrix, or two 2x2 matrices, as follows. 

The magnetic hyperfine interactions were discussed in chapter 8, where we followed rather closely the analysis of Jette and Cahill [ 187] we will come to these a little later, but first consider briefly the nuclear spin-free terms. 

This operator looks a little unsymmetrical due to the term involving the magnetic field. The lack of symmetry is apparent rather than real, because the unmodified operator had no such asymmetry. The spin-free term in W is also unsymmetrical in form, if we apply the momentum operator to the potential ... 

We will evaluate the numbers of integrals required for a calculation with the unmodified Dirac Hamiltonian, and compare them with the number of integrals required for a calculation with the spin-free modified Dirac Hamiltonian, and with the number required for a nonrelativistic calculation. The spin-free Hamiltonian is formed by summing all the spin-free terms defined above, but we will consider the Coulomb term and the Gaunt and Breit terms separately. For the purpose of this evaluation, we make the following assumptions and definitions ... 

The terms of leading order 1 / mc) comprise a Darwin-like term and a spin-orbit term, and reduce to the Pauh expressions in the limit 1. These terms are therefore regularized, just like the nuclear potential terms. The leading spin-free term is a term in V IT, which vanishes for external electric fields, just as it does in the Pauli Hamiltonian. 

These operators will be considered again later, when all the spin-free terms will be accumulated. We note here only that the Darwin term from the Gaunt interaction... 

The contributions to the two-electron operator from the gauge term of the Breit interaction may be developed in the same manner, using the representation for the modified Dirac operator. The derivation is more complicated, and details may be found in appendix I. The only term that contributes to 0(c ) is a spin-free term ... 

We assume that we somehow know how to handle the spin-free terms of this equation, and that they form a suitable Ho. The spin-orbit part may be divided into a one-electron and a two-electron operator. 

The simple perturbative treatment of spin-free terms from the Pauh Hamiltonian does reasonably well for the two light members of the series, but less so for AuH. As this approximation is extremely easy to program for SCF calculations and requires almost no extra computational effort, it appears as an attractive qualitative approach to relativity in medium heavy species. Only the Dirac-Coulomb (DC) results in table 22.3 account for spin-orbit interaction. The closeness of DKH and RECP results to the DC values indicate that spin-orbit effects are of minor importance, something we would expect in closed-shell molecules, where the bonding is dominated by s orbitals. Under these conditions, the two approximate spin-free methods can compete with the full DC operator. The agreement between the results from these three schemes also indicate that the discrepancy between the calculated and experimental values is due to insufficient description of the correlation. This observation is in line with the common experience that MP2 calculations on transition-metal compounds frequently yield somewhat short bond lengths. 

This complicates the term scheme considerably. For a triplet system, for instance, we must ima e that all terms in Fig. 1 (except for those with I = 0) become three-fold multiple. With respect to the intensities, we must note that we deal — just os in III — with a spin-free term with the azimuthal quantum number I and a partition z, and so all rules derived in III remain intact. Since, meanwhile, we obtain in the case of a triplet system, for instance, six or seven lines from one, all of which lie very close together, the multiplet fine structure of these bands will be difiicult to analyse. They can be derived easily with the help of theory, if we keep in mind what has been said above, that we obtain a simple multiplet for each of the terms indicated in Fig. 1. In this way we also obtain the formulae for the intensities. 

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

The first simplification in the TDAN model is to consider only a few electronic orbitals on the scattered atom. For many applications, it is sufficient to consider one only, that from which, or into which, an electron is transferred. Let the ket 10 > denote the spatial part of the orbital. When far from the surface, suppose its energy is So> let Uq be the Coulomb repulsion integral associated with the energy change when it is occupied by two electrons of opposite spin. In terms of creation and annihilation operators and Co for 0>, with ff( = aorfi)a spin index, that part ofJt which refers to the free atom is... 

For the illustrative calculations shown here, the spin-free wave functions, 4, for the H/ isotopomers were obtained as 50-term expansions in a basis of FSECG s gi(r) ... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

The easiest and, in many respects, the most satisfactory way is to consider only the totally symmetric tensor components (i.e., the spin-free density matrices) and to define the spin-free cumulants in terms of these [17, 30]. This corresponds to replacing the considered state by an Ms-averaged ensemble. 

Such localized states as under discussion here may arise in a system with local permutational symmetries [Aa] and [AB], If [Aa] + [S] and [Ab] = [5], the outer direct product [Aa] 0 [AB] gives rise to a number of different Pauli-allowed [A], If the A and B subsystems interact only weakly, these different spin-free [A] levels will be closely spaced in energy. The extent of mixing of these closely spaced spin-free states under the full Hamiltonian, H = HSF + f2, may then be large. Thus, systems which admit a description in terms of local permutational symmetries may in some cases readily undergo spin-forbidden processes, such as intersystem crossing. 

Level-6. The most complete treatment utilizes the basis set of all free-atom terms v,L,Ml,S,Ms) for the given electronic configuration dn, and the calculation of energy levels is performed by involving the operators of the electron repulsion, CF, spin-orbit interaction, orbital-Zeeman and spin-Zeeman terms ... 

The primitive VB model is defined in terms of overlap and Hamiltonian matrix elements over the basis states of eqn. (2.1.3). For fixed there are 2N possible spin-product functions so that this gives the dimension of the model s space. Indeed (though not originally formulated in this manner) the model may be mathematically represented entirely in spin space, despite the fundamental spin-free nature of the interactions. One may introduce a spin-space overlap operator by integrating out the spin-free coordinates... 

It is well known that two main contributions determine the overall isotropic hyperfine coupling of a given atom together with small spin-orbit terms, which are, however, negligible for organic free radicals ... 

In this section, a new function, called paired-permanent-determinant (PPD), is introduced, which is an algebrant. An overlap matrix element in the spin-free VB method may be obtained by evaluating a corresponding PPD, while the Hamiltonian matrix element is expressed in terms of the products of electronic integrals and sub-PPDs. 

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