Only Spin-Orbit Hamiltonians

The computational effort of a molecular calculation can be reduced significantly, if only a few electrons are taken into account explicitly and the interaction with the rest is approximated by means of effective Hamiltonians. The first step in the hierarchy of approximations is the so-called frozen-core approximation. 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

Let us first become familiar with the spin-free (sf) case we have a pair of Slater determinants interacting via (i.e., a sum of spin-independent one-and two-electron operators, h and 12). Their matrix element is given by (Sla-ter-Condon rules)  

The summation indices i and j run over all occupied spin orbitals in the determinant f . 

Single excitations and T] differ by one pair of spin orbitals i,j. 

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In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

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. 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

For F+H2 and its isotopomers the molecular beam, scattering studies mentioned above [10, 12] indicate that the reactivity of the excited spin-orbit state is below the sensitivity level of these experiments at collision energies below 2 kcaEmol. (The barrier on the SW PES with the full spin-orbit Hamiltonian included is = 1.9 kcaEmol [25]). In subsequent molecular-beam experiments on the F+HD reaction Liu and coworkers [13] also found that the reactivity of the spin-orbit excited state was only a few percent of that of the ground state. 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as... 

Matrix elements of the total spin-orbit Hamiltonian between basis states differing by A A = 1, AE = +1, for a given signed value of fl, may be calculated using only the s) or l s part of Hso. Again, it is unnecessary to use symmetrized basis functions (except when one of the states involved is a Eq state) ... 

Except in the highest symmetry cases, ab initio calculation of zero-field splitting in organic molecules requires the use of H , an operator that has only a two-electron part. Then the heavy computation involving two-electron terms cannot be avoided regardless of what spin-orbit Hamiltonian is used. These calculations are difficult because the correlation of the electrons has to be described very well before the zero-field splitting parameters are calculated accurately. Of the... 

Summarising, the valence-only spin-orbit CGWB-AIMP embedded cluster Hamiltonian, which results from choosing the CGWB-AIMP for the isolated cluster, reads ... 

The spin-orbit Hamiltonian of Eq. (31) is correct only for a bare nucleus. In the case of many-electron atoms where the nucleus is surrounded by a "core of electrons, the electrostatic potential, U (r), changes more rapidly with r because of the rapid change in shielding by the core as we... 

Use the angular momentum commutation rules L, L ] = ifiL, [L LJ = ihLx, [Lj, Lx] = ihLy, [L, l ] = 0, along with the analogous rules for the components of S and J, to prove the commutation relationships (2.45) that hold in the presence of spin-orbit coupling. Take the spin-orbit Hamiltonian to be Hgo = /( )L S, with f r) a function of the radial coordinates only. 

In these approaches, the spin-orbit part can be done either at the variation-perturbation level where only a few spin-free states are included in the spin-orbit Hamiltonian, or in a spin-orbit Cl, typically including all single excitations from the reference states as in the spin-orbit Cl method EPCISO [36].Fromager etflZ. [37] has shown that the two methods are essentially equivalent provided that the orbitals are relaxed separately in all of the spin-free reference configurations. 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

As illustrated above, any p2 configuration gives rise to iD , and levels which contain nine, five, and one state respectively. The use of L and S angular momentum algebra tools allows one to identify the wavefunctions corresponding to these states. As shown in detail in Appendix G, in the event that spin-orbit coupling causes the Hamiltonian, H, not to commute with L or with S but only with their vector sum J= L +... 

This is the most general form of a spin orbital, but if the Hamiltonian does not contain the spin explicitly, it may be more convenient to try to introduce simplified spin orbitals which contain only one nonvanishing component and hence are of either pure a or character. Corresponding to the idea of the doubly occupied orbitals, the spin orbitals are often constructed in pairs simply by multiplying the same orbital tp(r) with a and ft, respectively. 

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