Operator effective spin

Any set of energetically well-isolated levels can be described by an effective spin Hamiltonian operator by choosing S to match the corresponding number of levels. This can be just one isolated Kramers doublet of a high-spin multiplet if the... 

A special focus will be on phenomenological Hamiltonians involving electronic spin interactions. For this it is necessary to define atomic surrogate spin operators—so-called local spin operators—that may be directly related to the effective spins in... 

D. Maynau, P. Durand, J. P. Duadey, and J. P. Malrieu, Phys. Rep. A, 28, 3193 (1983). Direct Determination of Effective-Hamiltonians by Wave-Operator Methods. 2. Application to Effective-Spin Interactions in -Electron Systems. P. Durand and J. P. Malrieu, in Advances in Chemical Physics (Ah Initio Methods in Quantum Chemistry—I), K. P. Lawley, Ed., Wiley, New York, 1987, Vol. 67, pp. 321-412. Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling. 

It took several decades for the effective Hamiltonian to evolve to its modem form. It will come as no surprise to learn that Van Vleck played an important part in this development for example, he was the first to describe the form of the operator for a polyatomic molecule with quantised orbital angular momentum [2], The present formulation owes much to the derivation of the effective spin Hamiltonian by Pryce [3] and Griffith [4], Miller published a pivotal paper in 1969 [5] in which he built on these ideas to show how a general effective Hamiltonian for a diatomic molecule can be constructed. He has applied his approach in a number of specific situations, for example, to the description of N2 in its A 3 + state [6], described in chapter 8. In this book, we follow the treatment of Brown, Colbourn, Watson and Wayne [7], except that we incorporate spherical tensor methods where advantageous. It is a strange fact that the standard form of the effective Hamiltonian for a polyatomic molecule [2] was established many years before that for a diatomic molecule [7]. 

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

The operators ajj ) make it possible to introduce the so-called pseudospin formalism that features the effective spin-1/2 operators... 

Equations for the Fock space coupled cluster method, including all single, double, and triple excitations (FSCCSDT) for ionization potentials [(0,1) sector], are presented in both operator and spin orbital form. Two approximations to the full FSCCSDT equations are described, one being the simplest perturbative inclusion of triple excitation effects, FSCCSD+T(3), and a second that indirectly incorporates certain higher-order effects, FSCCSD+T (3). 

One of our main motivations for pursuing the development of a density functional response theory for open-shell systems has been to calculate spln-Hamiltonian parameters which are fundamental to experimental magnetic resonance spectroscopy. It is only within the context of a state with well-defined spin we can speak of effective spin Hamiltonians. The relationship between microscopic and effective Hamiltonians rely on the Wigner-Eckart theorem for tensor operators of a specific rank and states which transform according to their irreducible representations [45]. 

The detailed structure of the Mossbauer and ESR spectra for the iron transport compounds can be described in terms of a spin Hamiltonian with effective spin S= 5/2 for the high spin ferric ion. These parameters can give information about the site symmetry of the iron. Since there is no orbital angular momentum in the 6S state, the effective spin is the same as the real spin of the iron ion. There is spherical symmetry in the 6S state to first order, but spin-orbit coupling to excited (non-spherical) orbital states gives rise to asymmetries about the iron site which are reflected in the spin Hamiltonian. The general form of the spin Hamiltonian which we will use here is a quantum mechanical operator which acts on the electronic states ms> and nuclear states mf>. 

However, relativistic effects have a much more profound influence on a material s properties. Thus, the existence of a spin quantum number allows for the existence of magnetism. Moreover, as shown in most standard textbooks on physical chemistry, the phenomenon of phosphorescence can only be explained through the existence of relativistic effects. Phosphorescence involves transitions between, for example, singlet and triplet states which are only possible if some spin-operating effects exist, e.g. spin-orbit couplings. Furthermore, several experimental techniques are indirectly based on exploiting relativistic effects. These include, for example, electron-spin and NMR spectroscopies. 

Judd, Crosswhite, and Crosswhite (10) added relativistic effects to the scheme by considering the Breit operator and thereby produced effective spin-spin and spin-other-orbit interaction Hamiltonians. The reduced matrix elements may be expressed as a linear combination of the Marvin integrals,... 

An even simpler but less well-justified approximation avoids the calculation of the matrix elements of the two-electron part of the operator altogether. Only the matrix elements of the one-electron part of are computed, and in the sum over nuclei a in Equation 3.3, contributions from each atom are not multiplied by Z but by the effective spin-orbit coupling nuclear charge of atom a, which has been optimized empirically to represent the partial compensation of the one-electron part by the two-electron part of the operator. Recommended values of for atoms of main-group and transition metal elements are listed in Table 3.1. This method is generally acceptable in molecules containing heavy atoms but is not very accurate in those composed of light atoms only. 

The interaction of the spin magnetic moment with the orbital magnetic moment of the unpaired electron leads to an orientationally dependent shift of the resonance frequency. This effect is normally described by an effective spin operator S and an anisotropic g matrix. The quantimi-mechanical Hamilton operator for the interaction of the electron spin with the external magnetic field Zeeman interaction) can, therefore, be described by... 

The effective part of this operator interchanges spins, that is,... 

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