Electronic potentials hypersurfaces

Ab Initio Calculation of Adiabatic Channels on Electronic Potential Hypersurfaces... 

In addition, we assume, for the systems of interest here, that the electronic motion is fast relative to the kinetic motion of the nuclei and that the total wave functions can be separated into a product form, with one term depending on the electronic motion and parametric in the nuclear coordinates and a second term describing the nuclear motion in terms of adiabatic potential hypersurfaces. This separation, based on the relative mass and velocity of an electron as compared with the nucleus mass and velocity, is known as the Born-Oppenheimer approximation. 

Fig.4. Potential hypersurfaces of T, (3P0) and r4 (3P,) levels in the e (Q2, Q3) subspace calculated from the Fukuda coupling matrix with the vibronic coupling parameter Are t = 4x 10 9 J/m from Eq. (5) (denoted by b in [84]), the electron (exchange) repulsion parameter G and spin-orbit parameter are both set equal to 0.3 eV...

Vibronic spectra reflect changes in the electronic and vibrational state of a molecule at the same time. It is possible to calculate the geometry of the excited species and the potential hypersurface close to the equilibrium state. For this, a spectrum is required with sufficiently well resolved vibronic structure to carry... 

Fig. 8.1 Electrostatic potential for benzene (a), nitrobenzene (b) and trinitrobenzene (c) on the 0.001 electron bohr3 hypersurface with the color code from +0.075 a.u. (green) to -0.075 a.u. (red).

If it is assumed that the crossing between the different substate systems proceeds via an excited vibrational/phonon state, the intensity ratio of the delayed excitation peak relative to the fast one would for that specific vibrational satellite (and presumably all higher lying ones) differ distinctly from the ratio found for the electronic origins. However, such behavior is not observed, at least for vibrational states up to = 1500 cm above the zero-point vibrational levels of the triplet substates. Thus, it can be concluded that the relaxation from an excited vibrational state takes place by a fast process within the individual potential hypersurface of each triplet sublevel to its zero-point vibrational level (intrasubstate relaxation). Subsequently, a comparatively slow sir and/or emission occurs from that electronic state. This result is schematically depicted in Fig. 24. 

The introduction of the Born-Oppenheimer approximation is not sufficient to make the problem actually solvable. To determine the electronic wave function — a necessary step to construct the potential hypersurface W (R) — we have to resort to further approximations... 

In the great majority of atom-transfer reactions between species in their ground electronic states, even if several non-degenerate states do correlate with the separated reactants, their existence is of limited importance. Chemical reaction usually proceeds via the potential hypersurface that is at all points lowest in energy. The process is then said to be electronically adiabatic. The only effect of there being more than one overall electronic state is the introduction of a statistical factor, which will be temperature dependent if the spin-orbit terms are split by to allow for the fact that not all collisions will occur on the lowest potential. In assessing the results of dynamical calculations, one should determine whether any allowance has been made for this effect as no standard practice has been established. 

If relative energies are of interest, for example between two electronic states or between two different regions of the potential hypersurface, only the difference in the corrections is important, i.e. 

Electronic energy hypersurfaces lepiesent the potential energy surface (PES) for the motion of the nuclei. In the quantum mechanical picture on r some energies will be allowed we will have the vibrational and rotational energy levels, as for diatomics. The same energy levels corresponding to... 

The additional electron-phonon coupling constants could be calculated again with the help of quantum mechanically computed electronic wave functions of the stack (band structures) using again a potential hypersurface to determine the wave functions of these coupled vibrations. Presently, however, to obtain an orientation one can treat these coupling constants as parameters substituting different values for them. 

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