Electronic-field representation

In the previous section, we discussed how radiative transitions are manifested by curve crossings in the diabatic electronic-field representation, so that our understanding of surface crossings in the usual field-free situation can be directly applied to the problem of collision-induced radiative processes. Moreover, by changing the radiation frequency the position of the crossing point is changed. 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

Figure 13. Left ligand field energy-level diagram calculated for plastocyanin. Center contains energies and wavefunctions of the copper site. Energy levels determined after removing the rhombic distortions to give and C symmetries are shown in the left and right columns, respectively (from Ref. 11). Right electronic structural representation of the plastocyanin active site derived from ligand field calculations (from Ref. 11).

On the right in Fig. 32 is an electron-domain representation of Lin-nett s model of an Octet-Rule satisfying atom in field-free space. For domains of (i) fixed size and distribution of charge, (ii) fixed distances from the nucleus, and (iii) fixed tetrahedral disposition with respect to other domains of the same spin-set, three of the four contributions to the total energies of the two structures in Fig. 32 are identical, namely the energies arising from (i) electronic motion, (ii) nuclear-electron attractions, and (iii) electron-electron repulsions between electrons of the same... 

Fig. 32. Electron-domain representation of (left) strong-field and (right) weak-field models of an octet, after Linnett 12 6>...

In 1996 a force field model was proposed in which geometry-dependent charges are calculated by EEM. This model, called consistent implementation of the electronegativity equalization method (CIEEM), combines a force field representation for the PES (e.g., Eq. [19]) with the EEM equation for the electronic energy (Eq. [7]) in such a way that the atomic terms of Eq. [7] explicitly enter the expression for the potential energy. Thus, within the CIEEM, the expression for the steric energy in the MM force field model will read... 

In order to derive the PPP-hamiltonian, it is assumed that the nuclear framework of the molecular system is invariant under the point group Cg, which contains the operations of identity and a reflection. This is not always true for all molecules to which the model is applied, and then the magnitude of the perturbation caused by the noninvariant part of the nuclear potential must be examined. The electron field operators t) are expressed as the sum of two components, each transforming according to an irreducible representation of C, ... 

Experimental information alone, apart from extreme cases, is insufficient to uniquely determine even the harmonic force field of polyatomic molecules. Severe difficulties are encountered if vibrational anharmonicity is considered. Since accurate anharmonic force field representations of PESs are of considerable interest in many branches of chemical physics, independent information obtainable from electronic structure calculations on the force fields is of considerable practical importance. 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Figure Bl.15.8. (A) Left side energy levels for an electron spin coupled to one nuclear spin in a magnetic field, S= I =, gj >0, a<0, and a l 2h)<(a. Right side schematic representation of the four energy levels with )= Mg= , Mj= ). +-)=1, ++)=2, -)=3 and -+)=4. The possible relaxation paths are characterized by the respective relaxation rates W. The energy levels are separated horizontally to distinguish between the two electron spin transitions. Bottom ENDOR spectra shown when a /(21j)< ca (B) and when co < a /(2fj) (C).

Figure Bl.22.4. Differential IR absorption spectra from a metal-oxide silicon field-effect transistor (MOSFET) as a fiinction of gate voltage (or inversion layer density, n, which is the parameter reported in the figure). Clear peaks are seen in these spectra for the 0-1, 0-2 and 0-3 inter-electric-field subband transitions that develop for charge carriers when confined to a narrow (<100 A) region near the oxide-semiconductor interface. The inset shows a schematic representation of the attenuated total reflection (ATR) arrangement used in these experiments. These data provide an example of the use of ATR IR spectroscopy for the probing of electronic states in semiconductor surfaces [44]-...

In Chapter IV, Englman and Yahalom summarize studies of the last 15 years related to the Yang-Mills (YM) field that represents the interaction between a set of nuclear states in a molecular system as have been discussed in a series of articles and reviews by theoretical chemists and particle physicists. They then take as their starting point the theorem that when the electronic set is complete so that the Yang-Mills field intensity tensor vanishes and the field is a pure gauge, and extend it to obtain some new results. These studies throw light on the nature of the Yang-Mills fields in the molecular and other contexts, and on the interplay between diabatic and adiabatic representations. 

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