Spin-orbit interaction coordinates

The distinction in standard non-relativistic theory between spin-orbit interaction as relativistic on the one hand and other spin interactions as non-relativistic on the other hand does lead to some inconsistencies. Consider, for instance, a hydrogen-like atom where the coordinate system is shifted from the... 

X 10 s from the position of the observed dispersion. Such a correlation time is expected to be dominated by the electron relaxation time (see Section II.E.2). No field dependence for Tig has been observed up to 50 MHz. In the same coordination environment, electron relaxation in VO-proteins is about one order of magnitude lower than in the Cu-proteins, due to the stronger spin-orbit interaction of the latter ion. 

Using these coordinate values we may now evaluate the matrix elements of Eq. (14.12) by substituting for the dipole moments the dipole matrix elements between the initial and final states. This procedure yields explicitly time dependent matrix elements VAB(r). It is particularly interesting to consider the (0,0) resonances, for two reasons. First, the (0,0) resonances have no further splitting due to the spin orbit interaction and are therefore good candidates for detailed experimental study. Second, since these resonances only involve the matrix... 

Here Pa(a = 6, e) is the momentum conjugate to Qa. In the absence of spin-orbit interaction, the e vibration does not mix the orbital components of the 4T2 g and we have vibrational potential energy surfaces consisting of three separate ( disjoint ) paraboloids in the two-dimensional (2D) space of the Qe and Qe coordinates of the e vibration. The Jahn-Teller coupling leads only to a uniform shift (—ZsPJX = — V2/2fia>2 = —Sha>) of all vibronic levels. 

Two ions are well-known for their highly anisotropic properties. Firstly, in the rare-earth family, Dy3+ which has a 6H15/2 ground state. The spin-orbit interaction is stronger than the crystal field effects. The ratio J /J can be of the order of 100 (Jj. = 0), gjj = 20 and gi = 0 this is practically an ideal case. Secondly, in the transition element series, the ion Co2+ is also characterized by anisotropic interactions (either in the tetrahedral or octahedral coordination), the anisotropy being however lower than in the case of Dy3+. J /J is about 0.5 for this ion. Some Fe2+ compounds also display a behavior approximating to the Ising model. 

The comparison of the calculated spectra of the free ions and the ones in the crystal is not straightforward. Indeed, in the crystal, the presence of the first coordination shell increases the number of electrons and basis functions in the calculations, resulting in a blow-up of the Cl expansion, mainly due to the generated doubly-excited configurations. One should bare in mind that this increase is about six time as fast in double group symmetries as in the non-relativistic symmetry. In a non effective Hamiltonian method, the only way to keep the size of the DGCI matrix to an affordable size of few million configurations, is to cut down the number of correlated electrons. This may essentially deteriorate the quality of electron correlation as the contributions of the spin-orbit interaction... 

Ln = Pr, Nd, Pm, Sm, Dy, Ho, Er, and Tm X = F, Cl, Br, and I. Ground electronic states for all trihalides were established, assuming that the molecular symmetry was planar (Dsa) rather than pyramidal (Csv). Spin-orbit interaction was ignored. Comparison of calculated Ln-X bond lengths with experimental data showed that description of dynamic electron correlation was absolutely necessary for correct results. These studies on lanthanide systems were later extended to hydration models of trivalent rare-earth ions [253] for Y +, La +, Gd +, and Lu + geometry optimization was carried out at the MP2 level for hydrates containing from one to ten water molecules. In addition, ab initio molecular dynamics simulations (by following the dynamical reaction coordinate) for the systems with more water molecules, [M(H20)24] (M = Y, La) and [La(H20)64] ", were done and both radial distribution function and coordination number were obtained. 

Under the assumption that the emission center is Cr +, the two broad bands in the excitation spectmm are due to transitions " A2—>" Ti (410 nm) and " A2 " T2 (590 nm). These assignments enable the evaluation of the crystal field strength Dq=1700 cm which is very close to previously published data (Platonov et al. 1998). Such a Dq corresponds well to a strong crystal field site for d elements in octahedral coordination. The narrow lines observed in the excitation spectrum are characteristic for many chromium-bearing crystals. They are attributed to Cr " " transitions from A2 to the excited states E, T1 and T2. Spin-orbit interactions admix T 1 and " T2 states having the same spin multiplicity as the ground state to the states E, 1, and T2- In absorption spectra this causes an intensification of the respective absorption lines due to partial relaxation of the spin-multiplicity rule for... 

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