Electrons relaxed orbitals

The fact is that the molecular orbitals describing the resulting cation may well be quite different from those of the parent molecule. We speak of electron relaxation, and so we need to examine the problem of calculating accurate HF wavefunctions for open-shell systems. 

In this particular example, the Xa orbital energies resemble those produced from a conventional HF-LCAO calculation. It often happens that the Xa ionization energies come in a different order than HF-LCAO Koopmans-theorem ones, due to electron relaxation. 

The symmetry-breaking of the HF function occurs when the resonance between the two localized VB form A+...A and A...A+ is weaker than the electronic relaxation which one obtains by optimizing the core function in a strong static field instead of keeping it in a weak symmetrical field. If one considers for instance binding MOs between A and A they do not feel any field in the SA case and a strong one in the SB solution. The orbitals around A concentrate, those around A become more diffuse than the compromise orbitals of A+ 2 and these optimisations lower the energy of the A. A form. As a... 

Fig. 5.6 Changes in the shape of the valence contribution due to geometric and electronic relaxation in [FeF4]" . Full line [FeF4] at its equilibrium geometry, dashed line [FeF4] at its equilibrium geometry. The square of the valence orbital that mainly contributes to p(0) along the Fe-F bond (distances are in units of the Bohr radius) is also drawn (from [19])...

A fully realistic picture of solvation would recognize that there is a distribution of solvent relaxation times (for several reasons, in particular because a second dispersion is often observable in the macroscopic dielectric loss spectra [353-355], because the friction constant for various types or modes of solute motion may be quite different, and because there is a fast electronic component to the solvent response along with the slower components due to vibration and reorientation of solvent molecules) and a distribution of solute electronic relaxation times (in the orbital picture, we recognize different lowest excitation energies for different orbitals). Nevertheless we can elucidate the essential physical issues by considering the three time scales Xp, xs, and Xelec-... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

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. 

The NMRD profiles of V0(H20)5 at different temperatures are shown in Fig. 35 (58). As already seen in Section I.C.6, the first dispersion is ascribed to the contact relaxation, and is in accordance with an electron relaxation time of about 5 x 10 ° s, and the second to the dipolar relaxation, in accordance with a reorientational correlation time of about 5 x 10 s. A significant contribution for contact relaxation is actually expected because the unpaired electron occupies a orbital, which has the correct symmetry for directly overlapping the fully occupied water molecular orbitals of a type (87). The analysis was performed considering that the four water molecules in the equatorial plane are strongly coordinated, whereas the fifth axial water is weakly coordinated and exchanges much faster than the former. The fit indicates a distance of 2.6 A from the paramagnetic center for the protons in the equatorial plane, and of 2.9 A for those of the axial water, and a constant of contact interaction for the equatorial water molecules equal to 2.1 MHz. With increasing temperature, the measurements indicate that the electron relaxation time increases, whereas the reorientational time decreases. 

Koopmans s theorem is not valid for Xa calculations, but Slater s transition state concept applies. This approximation allows the interpretation of electronic transitions in terms of 1-electron orbitals and yet includes electronic relaxation effects (208). The virtual (unoccupied) orbitals of Xa theory have a physical significance and can be used to discuss excitations of the electronic system, because the same potential due to the iV - 1 other electrons affects both occupied and unoccupied... 

Mechanisms analogous to those illustrated in Fig. 3.1 apply also to electron relaxation (see below). However, electrons have other more efficient relaxation mechanisms which overcome the former ones. They are based on the presence of spin orbit coupling. Molecular motions modulate the orbital magnetic moment and then affect the electron spin. Several possible mechanisms for electron relaxation... 

Solid-state theories ascribe electron relaxation to the coupling of electronic spin transitions with transitions between lattice vibrational levels, or more generally with phonons. Disappearance (depopulation of a vibrational level) or creation (population of a vibrational level) of phonons modulate the orbital component of the electron magnetic moment. 

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