Operator diamagnetic spin-orbit

As mentioned above there are four main contributions to the nuclear spin-spin coupling constants the Fermi contact (FC), the paramagnetic spin-orbit (PSO), the spin-dipolar (SD) and the diamagnetic spin-orbit (DSO) contributions. The Fermi contact term is usually the most important of these and also the most sensitive to geometry changes [8]. The Fermi contact contribution arises from the interactions between the terms containing S(riM) and < (riN) in the operators Hon for nuclei N and M (see Eqn. (12)). 

The V2AA term in eq. (10.68) gives a Diamagnetic Spin-Orbit operator, which is an operator of order c. Although we otherwise only consider terms up to order c, the nuclear spin-spin coupling constant only contains terms of order c, which is why we need to include... 

The diamagnetic spin-orbit operator describes the direct interaction between the two orbital magnetic moments induced in the electron density by the presence of the magnetic moments of nuclei K and L. As for the diamagnetic shielding operator, this is a 3 x 3 nonsymmetric tensor operator, with in general nine independent elements. 

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

The dominant diamagnetic Hamiltonian term is a simple one-electron operator and its expectation value, when the ground-state determinantal wave function is constructed from the set of occupied molecular spin-orbitals, is... 

If the Hartree-Fock equations associated with the valence pseudo-Hamiltonian (167) are solved with extended basis sets, then all the above F are almost basis-set-independent. At the present time, and for practical reasons, most of the ab initio valence-only molecular calculations use coreless pseudo-orbitals. The reliability of this approach is still a matter of discussion. Obviously the nodal structure is important for computing observable quantities such as the diamagnetic susceptibility which implies an operator proportional to 1/r. From the computational point of view, it is always easy to recover the nodal structure of coreless valence pseudo-orbitals by orthogonalizing the valence molecular orbitals to the core orbitals. This procedure has led to very accurate results for several internal observables in comparison with all-electron results. The problem of the shape of the pseudoorbitals in the core region is also important in relativity. For heavy atoms, the valence electrons possess high instantaneous velocities near the nuclei. Schwarz has recently investigated the compatibility between the internal structure of valence orbitals and the representations of operators such as the spin-orbit which vary as 1/r near the nucleus. ... 

There are electron spin-independent and spin-dependent paramagnetic terms (linear in the vector potential), and a diamagnetic spin-independent term that (quadratic in the vector potential). By substituting (12.12a) or (12.12b) in the spin-free linear terms, one obtains the ZORA form of the Orbital Zeeman (OZ) operator... 

The latter is often called the orbital diamagnetic (OD) or diamagnetic nuclear spin-electron orbit operator (DSO), whereas the first could be called the... 

Combined with the paramagnetic term in Eq. (6.53) this reformulated diamagnetic contribution replaces in the paramagnetic term the dependence of the orbital angular momentum operator on the nuclear centre of mass Rcm by a dependence on the position of nucleus K. An alternative expression for the spin-rotation tensor is thus... 

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