2-G matrix

Even for a single radical tire spectral resolution can be enlianced for disordered solid samples if the inliomogeneous linewidth is dominated by iimesolved hyperfme interactions. Whereas the hyperfme line broadening is not field dependent, tire anisotropic g-matrix contribution scales linearly with the external field. Thus, if the magnetic field is large enough, i.e. when the condition... 

If we consider G as a unitary transformation matrix that diagonalizes the g matrix and 1 is the diagonal matrix with elements toy, j =, N as... 

Now, we are in a position to present the relevant extended approximate BO equation. For this purpose, we consider the set of uncoupled equations as presented in Eq. (53) for the = 3 case. The function icq, that appears in these equations are the eigenvalues of the g matrix and these are coi = 2 (02 = —2, and CO3 = 0. In this three-state problem, the first two PESs are u and 2 as given in Eq. (6) and the third surface M3 is chosen to be similar to M2 but with D3 = 10 eV. These PESs describe a two arrangement channel system, the reagent-arrangement defined for R 00 and a product—anangement defined for R —00. 

The Two-State Case The g matrix in this case is given in the form ... 

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

Note that the factor of 1/2 has disappeared from the energy expression this is because the G matrix itself depends on P, which has to be taken into account. We write SSg in terms of the Hartree—Fock Hamiltonian matrix h, where... 

It is first transfonned to mass-dependent coordinates by a G matrix eontaining the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, this is in line with the Bom-Oppenheimer approximation that the electrons follow the nucleus). 

The above-mentioned examples are particular cases of the following G-matrix ... 

Roothaan G matrix ) and depends on the density matrices Ra(= TaTaO all shells in fact... 

The major contribution to the components of the D tensor as well as the deviations of the g values from 2.0023 arises from the mixing of ligand field states by SOC other contributions to D result from direct spin-spin coupling, which mixes states of the same spin S. The D tensor and the g matrix both carry chemical information as they are related to the strength and symmetry of the LF, which is competing and counteracting to the effects of SOC. Details on the chemical interpretation of the parameters by quantum chemical means is found in Chap. 5. 

Thus, since K is arbitrary, it is possible to modify the system poles as desired. The matrix F corresponds to the G matrix used earlier to decouple the system and, in fact, reduces to G for the case K = 0 (no state variable feedback). 

In eqn (4.1), g and A-t are 3x3 matrices representing the anisotropic Zeeman and nuclear hyperfine interactions. In general, a coordinate system can be found - the g-matrix principal axes - in which g is diagonal. If g and A, are diagonal in the same coordinate system, we say that their principal axes are coincident. 

When a radical is oriented such that the magnetic field direction is located by the polar and azimuthal angles, 6 and cp, relative to the g-matrix principal axes, the resonant field is given, to first order in perturbation theory, by 4... 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

These equations have three solutions (i) 9 = 0 (ii) 9 = 90°, q> = 0 and (iii) 9 = (p = 90°. Since 9 and cp are in the g-matrix axis system, observable features are expected for those fields corresponding to orientations along the principal axes of the g-matrix. This being the case, the principal values of the g-matrix are obtained from a straightforward application of eqn (4.10). 

The g-value of a free electron is a scalar, ge = 2.00232. In a radical species, g becomes a matrix because of the admixture of orbital angular momentum into S through spin-orbit coupling. The components of the g-matrix thus differ from ge to the extent that p-, d-, or f-orbital character has been incorporated, and they differ from one another, depending on which p-, d-, or f-orbitals are involved. 

Notice that dz2 is unique among the d-orbitals in that lz does not couple it to any other orbital. Thus if the major metal contribution to the SOMO is dzi, g2 will be close to the free electron value. Accordingly, when one g-matrix... 

Since the g-matrix has only three principal values and there are almost always many potentially interacting molecular orbitals, there is rarely sufficient information to interpret a g-matrix with complete confidence. When a well-resolved and reliably assigned optical spectrum is available, the energy differences, E0—Em, are known and can be used in eqn (4.11) to estimate the contribution of the corresponding MOs to the g-matrix. Extended Hiickel MO (EHMO) calculations can be useful (but do not trust EHMO energies ), but one is most commonly reduced to arguments designed to show that the observed g-matrix is consistent with the interpretation placed on the hyperfine matrix. 

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