Spin-Dependent Terms

If V is the total Coulombic potential between all the nuclei and electrons in the system, then, in the absence of any spin-dependent terms, the electronic Hamiltonian is given by... 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

The basis functions are most commonly chosen such that the spin-function is either a pure spin-up function a(cr) or a pure spin-down function )S(cr). They are defined such that a( ) = / ( j) = 1 and zero for any other argument cr. Since the BO and, consequently, the Fock operator do not contain any spin-dependent terms, the HF equations divide into spin-up and spin-down equations ... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

For the spin-dependent term, we have an equivalent expression... 

The spin-dependent terms of the photoionization process are those associated with the spin-dependent Stokes parameters of the detector. There it can be seen that... 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

For the evaluation of probabilities for spin-forbidden electric dipole transitions, the length form is appropriate. The velocity form can be made equivalent by adding spin-dependent terms to the momentum operator. A sum-over-states expansion is slowly convergent and ought to be avoided, if possible. Variational perturbation theory and the use of spin-orbit Cl expansions are conventional alternatives to elegant and more recent response theory approaches. 

RP reactions have been found to be well described by the application of a spin Hamiltonian, " a common approach used in the field of magnetic resonance, which reduces the full Hamiltonian to one that contains only spin-dependent terms. The interactions capable of influencing spin-state mixing processes in RPs are concisely introduced in the expression for the spin Hamiltonian of a RP, which can be written as a sum of interradical, intraradical, and external interactions. 

Figure 1. Spin-dependent terms Cq, —7, in Legendre expansion coefficients as functions of the interfragment distance R dashed curves from Asymptotic Theory (AT) solid curves from AT corrected to fit ah initio data.

Because H is dynamically dependent on spin and space variables, the expression in parentheses in the r.h.s. of Eq. (3) involving integration over the latter defines a spin operator. This is just the effective Hamiltonian of interest to us. By virtue of point (iii), when the integrations are to be performed for the H" term in the Hamiltonian, only the unit operator in A need to be retained. The resulting expression will thus have the form (Ap H"l ). If one takes into account that the space state 1 ) is a product (or a combination of products, see above) of localized, one-particle states, one can immediately see that upon integrating over the spatial variables r , n= 1,2,...,AI, the spatial parts of the individual spin-dependent terms will be replaced by the corresponding quantum mechanical averages. Thus, for the entire expression in Eq. (3) is none other than one of the matrix element of the standard NMR Hamiltonian, Wnmr, between two spin-product basis states,... 

Finally, with regard to (iii), after the discovery of the spin-1/2 character of the electron and the associated magnetic moment, it was obvious that spin-dependent terms must be added to U]j, corresponding to the interaction of the magnetic field produced by "1" with the magnetic moment of "2" and vice versa. This led to the now familiar spin-other-orbit and spin-spin potentials and gave... 

The majority of calculations of molecular properties that take relativistic effects into account are at the time of writing performed with 1-component relativistic methods. The corresponding Hamiltonians are obtained from 2-component relativistic Hamiltonians by the deletion of all spin-dependent terms. 

Since a is a scalar operator, even this simple energy-dependent elimination of the small component permits an exact separation of the spin-free and spin-dependent terms of the Dirac Hamiltonian by applying Dirac s relation... 

Our results indicate clearly that many-electron systems are insufficiently described by a scalar-relativistic implementation for one-electron terms of the DK transformation only. The larger the number of electrons and the heavier the nucleus of an atom the larger is the error in total energies obtained with DKH4. These deficiencies are clearly due to the missing DK transformation of the two-electron terms and the neglect of the spin-dependent terms. It is thus highly desirable to clarify the influence of these two points beyond the known implementations [59,60]. 

This factor 2 arises in the Pauli equation, only if one derives the latter either from the DE or the LLE. However, if one formulates the Pauli equation as the SE with an additional spin-dependent term, without any reference to the DE or the LLE, the gyromagnetic ratio 2 must be postulated in an ad hoc way. A direct derivation of a Galilei invariant theory for spin- particles in terms of two-component spinors does not appear to be possible [16]. This does require four-component spinors. Slight deviations from g = 2 are caused by QED effects (radiative corrections), that are outside the scope of this chapter. 

Since the B operator can by means of the Dirac relation (55) be separated into scalar relativistic and spin-orbit terms, even better approximations to the lORA (MIORA-2) and ERA (MERA-2) equations that do not have any spin-dependent terms in the metric can be obtained by only omitting the spin-orbit contribution to the B operator in the metric. 

A spin-free formulation is usually preferred where the spin has been integrated out of the above equations. If the Hamiltonian does not contain any spin-dependent terms, we can always write the density matrix as... 

Let us consider a system consisting of electrons and nuclei. Let rel denote the set of all orbital coordinates of the electrons with respect to a space-fixed (sf) frame, and let sel be the corresponding spin coordinates. Similarly, let rnu and snu denote the orbital and spin nuclear coordinates. We will consider in this chapter systems for which the electronuclear Hamiltonian fl does not contain spin-dependent terms. Generalization to spin-dependent Hamiltonians does not present a major conceptual problem, but escapes our present objectives. For the present systems, H can be written as... 

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