Relation to the Spin Density

It is instructive to see how the nonrelativistic unrestricted KS-DFT formalism emerges from relativistic spin-DFT. This is most easily seen for the collinear approach, although the noncollinear one also reduces to the same nonrela- 

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Isotropic Hyperfine coupling constants an are related to the spin densities p(rN) at the corresponding nuclei by... 

An unambiguous identification of anomalous muonium with the bond-center site became possible based on pseudopotential-spin-density-functional calculations (Van de Walle, 1990). For an axially symmetric defect such as anomalous muonium the hyperfine tensor can be written in terms of an isotropic and an anisotropic hyperfine interaction. The isotropic part (labeled a) is related to the spin density at the nucleus, ip(0) [2 it is often compared to the corresponding value in vacuum, leading to the ratio i7s = a/Afee = j i (O) Hi/) / (O) vac- The anisotropic part (labeled b) describes the p-like contribution to the defect wave function. 

For any reservoir in equilibrium the fluctuation-dissipation theorem provides the relation between the symmetrized and antisymmetrized correlators of the noise Sx(x) = Ax(x) coth(w/2T). Yet, the temperature dependence of Sx and Ax may vary depending on the type of the environment. For an oscillator bath, Ax (also called the spectral density Jx(x)) is temperature-independent, so that Sx(x) = Jx(x)coth(x/2T). On the other hand, for a spin bath Sx is temperature-independent and is related to the spins density of states, while Ax([Pg.14]

The hyperfine coupling tensor (A) describes the interaction between the electronic spin density and the nuclear magnetic momentum, and can be split into two terms. The first term, usually referred to as Fermi contact interaction, is an isotropic contribution also known as hyperfine coupling constant (HCC), and is related to the spin density at the corresponding nucleus n by [25]... 

Free radicals are generally short-lived, highly reactive species, usually characterized experimentally by their magnetic properties only. Thus a successful theoretical approach must be able to provide at the same time reliable structural and magnetic properties. Here we have chosen as representative models the methyl, aUyl and formaldehyde cation radicals. The isotropic hyperfine coupling constant (hcc) of a magnetically active nucleus N (a(N)) is related to the spin densities at the nucleus by [69]... 

We shall demonstrate this for Arnold s EPR-spectroscopic scale (ax, Table 3) [36, 39], which relates the radical substituent constant ((7rad) to the benzylic a spin densities (p) according to Eq. (17), in which px and p refer to the substituted and unsubstituted benzyl radicals. Since the a hyperfine coupling constants (a) are linearily related to the spin densities by the McConnell equation [36,38], that is, ax oc px (cf. Section III.B), the coupling constants may be used instead of the spin densities [Eq. (18)]. According to Eq. (8) (pA = pB for the symmetrically substituted cases), the D value is proportional to the square of the spin density and a trAD scale may be defined by taking the square root of the ratio of the D values instead... 

The matrix element is easily related to the spin densities in subsystems A and B thus... 

The anisotropy of the line width is clearly manifested. When the external magnetic field is applied parallel to the stretching direction of the film, the signal intensity has a higher intensity in most fi equency regions than that in the case of the field being perpendicular to the direction. The spectra seem to be composed of more than two components. The ENDOR features at low temperatures can be interpreted as the direct evidence of the soliton like spin density by the simulation of the anisotropic spectrum. The maximum frequency of the ENDOR spectrum is related to the spin density, p(0) at the central carbon of the soliton as indicated in Fig. 7.41. 

The relation to the spin density can be made more explicit by invoking a Gordon decomposition of the current density to produce expressions for charge- and spin-related currents [392,397]. Although we have already encountered the Gordon decomposition for the 4-current in section 8.8.1, Appendix F considers explicitly the decomposition of its spatial components, that is, of the current density, in standard notation. From Appendix F, we take the result for the many-electron case,... 

Concerning the Hesns Hamiltonian, the quantities that need to be computed are the isotropic and anisotropic spin-spin coupling tensor. The isotropic contribution, the so-called Fermi contact term, is related to the spin density at the nucleus K under consideration [139],... 

For P protons Ajjj, can be related to the spin density p in the tt electron system by... 

For the calculations of the hyperfine coupling (HFC) constants the unrestricted Becke s UB3LYP hybrid functional in combination with the triple-zeta EPR-llI basis set [50] implemented in Gaussian 98 program [45] was used. Solvent effects on the hyperfine constants were estimated using the polarizable continuum model (PCM) [51] implemented in Gaussian 98 [45]. The isotropic HFC constant, a, is related to the spin density at the corresponding nucleus by ... 

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