Spin operators, evaluation

The dimensions of the spin spaces for the active electrons in Table 2, cf. Eq. (9)) are certainly not small. It proved difficult to find a spin basis in which very few of the coefficients were large and so we adopted instead a spin correlation scheme cf. Section 4.2). In the present work, we exploited the way in which expectation values of the two-electron spin operator evaluated over the total spin eigenfunction 4, depend on the coupling of the individual spins associated with orbitals ( )/ and j. Negative values indicate singlet character and positive values triplet character. Special cases of the expectation value are ... 

The coefficients of the spin operators must be evaluated using the electron wave function, an operation that is not usually possible in practice. However, we can parameterize the problem, defining the matrix D with elements ... 

The Holstein-Primakoff transformation also preserves the commutation relations (70). Due to the square-root operators in Eqs. (78a)-(78d), however, the mutual adjointness of S+ and 5 as well as the self-adjointness of S3 is only guaranteed in the physical subspace 0),..., i- -m) of the transformation [219]. This flaw of the Holstein-Primakoff transformation outside the physical subspace does not present a problem on the quantum-mechanical level of description. This is because the physical subspace again is invariant under the action of any operator which results from the mapping (78) of an arbitrary spin operator A(5i, 2, 3). As has been discussed in Ref. 100, however, the square-root operators may cause serious problems in the semiclassical evaluation of the Holstein-Primakoff transformation. 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

We will assume that the different states are pure spin ones and thus must obey well defined relations when the Spin operators act upon them. We will see that this analysis allows us to determine the global unknowns that must be evaluated. 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

The evaluation of a spin operator times an occupation number vector is faciliated by noting that the core is a singlet spin tensor, since ata ajjj is a singlet spin operator (Eq. 5.25). The action of Sz on I na np> becomes... 

In order to evaluate integrals over spin variables let us note that Eq. (5) implies that the symmetric group total spin operators S2 and S2. Consequently, the eigenfunctions of these operators form bases for representations of [Pg.614]

In this equation, the effects of coupling are expressed only for the axis, as initially there is no magnetization in the xy plane. In Appendix 3, the effects of the spin operator I were examined and evaluated along each axis (eq. A3-8). The product-operator formalism expands upon this approach by exploring the effects of adding a time factor to the Hamiltonian. In this fashion, we can evaluate the evolution of magnetization over time. 

After evaluating the effects of the spin operators on simple spin-orbitals, Eq. (3.4.33) reduces to... 

The spin-spin operator is a scalar product of two tensorial operators of rank 2, one of which operates on spin coordinates only, the other (spherical harmonics of rank 2) of which affects only spatial coordinates. Hss can be factored into its spin and spatial parts, and the Wigner-Eckart theorem can be used to evaluate matrix elements of the spin part. Considering only diagonal matrix elements,... 

In has the value of 5-04929 x 10 ergs per gauss or 5 04929 x 10 j T i jjjg jnatrix elements are easy to evaluate using the spin operator form oi 3 — and result in eigenvalues of... 

To obtain the spin coupling function for any given vector-coupled state of the composite system AB, when A and B are remote, we use (33) to evaluate the expectation value of the total spin operator S. Thus,... 

A noteworthy point is that Eq. (25.7) is the equation for the exact solutions that are also eigenfunctions of spin operator, S. Following the precursor s procedure [21, 22], we will evaluate the isotropic HFC values from broken-symmetry (BS) DFT solutions with using the equation... 

Thus, the correct result is obtained provided the the expectation values of the product of spin operators can be evaluated. This analysis is given in the next section. 

In an isotropic liquid the mean value of is zero and the quadrupole interaction contributes only to relaxation. In an anisotropic medium, on the other hand, the mean value of is no longer zero and a quadrupole splitting appears in the NMR spectrum. While the quadrupole hamiltonian in Eq. (7.3) may be evaluated in any coordinate system it is convenient to express the spin operators in a laboratory-fixed coordinate system and the electric field gradients in a principal axes coordinate system fixed at the nucleus. It is then suitable to rewrite the hamiltonian as... 

The spin and orbital gyromagnetic ratios g, and gj are the free nucleon values Sg, and 8gi are caused by both core polarization and meson exchange. The last term, arising from the dipole-dipole interaction, is a rank-one tensor product of the spherical harmonic of order 2, Y, and the spin operator. The separate contributions of Sg, 8g(, and gp can be extracted from experimental data that include high-spin and low-spin states and can also be calculated by theory. For the rather extensive evaluation of the neighborhood of 2o pjj/i57) gystematjcg hag en closed... 

In the case of -electron systems, the spin operators have to be applied on Slater determinants or linear combinations of these. The action of the Al-electron operator 5 is most conveniently evaluated in the -electron version of Eq. 1.19 with S, S and S defined as the sum of the corresponding one-electron operators. 

This series can be evaluated either by numerical methods or by collecting the partial series of common spin operators (such as Ix, /, and so on) [5,12]. For the present case, 1 = 312, we have for a tt/2 pulse, for example ... 

Abstract An expression for the square of the spin operator expectation valne, S, is obtained for a general complex Hartree-Fock wave fnnction and decomposed into four contributions the main one whose expression is formally identical to the restricted (open-shell) Hartree-Fock expression. A spin contamination one formally analogous to that found for spin nnrestricted Hartree-Fock wave functions. A noncollinearity contribntion related to the fact that the wave fnnction is not an eigenfunction of the spin- S operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy-plane. All these contributions are evaluated and compared for the H2O+ system. The optimization of the collinearity axis is also considered. 

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