Splitting, electron states

Results for the lowest states of Ru+ and Os+ are shown in Table 2. The table shows that, in keeping with the well-known assessment of the method, when a large basis set is employed, reasonable accuracy for excitation energies is realized at the MRSD level. As shown in the table, this accuracy carries over to the spin-orbit split electronic states derived from MRSD SOCI calculations. 

The SHG/SFG technique is not restricted to interface spectroscopy of the delocalized electronic states of solids. It is also a powerful tool for spectroscopy of electronic transitions in molecules. Figure Bl.5.13 presents such an example for a monolayer of the R-enantiomer of the molecule 2,2 -dihydroxyl-l,l -binaphthyl, (R)-BN, at the air/water interface [ ]. The spectra reveal two-photon resonance features near wavelengths of 332 and 340 mu that are assigned to the two lowest exciton-split transitions in the naphtli-2-ol... 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

The vibronic structure of a electronic state at variable strengths of the vibronic and spin-orbit coupling is presented in Figure 5. The splitting of the... 

Figure 5, Low-eriergy vibronic spectrum in a electronic state of a linear triatomic molecule. The parameter c determines the magnitude of splitting of adiabatic bending potential curves, is the spin-orbit coupling constant, which is assumed to be positive. The zero on the...

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

Figure 13. Vibrational levels for the first-excited electronic state of HD2 calculated [8] using split basis (SB) technique with A(R) = tp/2 coordinate-transformation (CT) treatment with A(R) — tp/2 Eq. (A. 14) with A (R) — y(p, 9, tp). Shown by the longer line segments are the levels assuming different values in two sets of calculations.

Electron spin resonance (or electron paramagnetic resonance) is now a well-established analytical technique, which also offers a unique probe into the details of molecular structure. The energy levels involved are very close together and reflect essentially the properties of a single electronic state split by a small perturbation. 

The reliability of this method for the evaluation of (vertical) electronic spectra has been clearly established in the present work, and further calculations on other molecules (ethylene, vinylydene... for example) have confirmed the very promising potentialities of such an approach that avoids the possible artefacts brought in by any arbitrary truncated CIs when dealing with excited states [49]. We also emphasize that this methodology is able to give reliable splittings between states ranging from 10 keal/mol to more than 10 eV. 

Instead, we believe the electronic structure changes are a collective effect of several distinct processes. For example, at surfaces the loss of the bulk symmetry will induce electronic states with different DOS compared to bulk. As the particle sizes are decreased, the contribution of these surface related states becomes more prominent. On the other hand, the decrease of the coordination number is expected to diminish the d-d and s-d hybridization and the crystal field splitting, therefore leading to narrowing of the valence d-band. At the same time, bond length contraction (i.e. a kind of reconstruction ), which was observed in small particles [89-92], should increase the overlap of the d-orbitals of the neighboring atoms, partially restoring the width of the d-band. 

An advanced subject in the theory of quadrupole splitting is the fact that the quadrupole splitting can become temperature dependent. At the heart of this effect is the change in Boltzmann populations of electronically nearly degenerate many-electron states with temperature. 

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