Born-Oppenheimer group- approximation

It is well known that the Born-Oppenheimer adiabatic approximation establishes geometrical shape of a molecule. The atoms (atomic groups) constituting a molecule are imagined to be placed in the vertices of certain three-dimensional (3D) shape as illustrated in Figure 9.1. 

Note that the potential matrix V( (R) is a diagonal matrix by definition, in contrast to W(s)(R). Again, in analogy to the common treatment in Sec. 3.1, we call (23) the group-Born Oppenheimer adiabatic approximation or briefly the group-adiabatic approximation. This approximation assumes that the states within the manifold g are much stronger coupled to each other — e.g. via the presence of a conical intersection of the potential surfaces — than to the rest of the electronic space. 

Deutsche and Moscowitz, a number erf semiempirical models have been proposed to explain features in experimental spectra. Although none has enjoyed any but the most modest successes, experience is gradually accumulating on the limits of applicability. Some of these are reviewed below. It is readily shown that within the Born-Oppenheimer (BO) approximation, the electronic contribution to the magnetic dipole transition moment associated with a vibrational transition of a molecule in its ground electronic state vanishes. A non-BO theory of VCD intensities was independently developed by several groups.2° 2" ... 

Having developed the mathematics of group theory, we now apply it to molecular quantum mechanics. As usual, we use the Born-Oppenheimer approximation. 

III. Born-Oppenheimer Approximations and Point-Group Symmetry. 8... 

III. BORN-OPPENHEIMER APPROXIMATIONS AND POINT-GROUP SYMMETRY... 

Point-group symmetry exists only within a particular Born-Oppen-heimer approximation. Though point-group symmetry often has little to do with spin conservation, it will be found in Section VIII that spin concepts and point-group symmetry are intermingled when a Hamiltonian involving spin interactions is considered. Also, we will find that Born-Oppenheimer approximations are important in Franck-Condon factors Franck-Condon factors are, in turn, critical in determining transition probabilities for a number of spin-forbidden processes. 

The Smith group has also developed the methodology for making high precision calculations for small systems without invoking the Born-Oppenheimer approximation and have made calculations for two-electron atomic ions, small muonic molecules, and potentials of the screened Coulomb form. Their method for determining nonlinear parameters is now referred to as random tempering.169... 

We consider two molecules, X-H and X-D, where X is a heavy group of atoms, which in the following is considered as a point mass. The two molecules have, according to the Born-Oppenheimer approximation, the same potential. The unimolecular bond breakage is described within the framework of the RRKM theory. 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

This article is not intended as a systematic review of the theory and applications of the CSA. In writing it I rather hope just to alert the chemical community to the growing potential, variety of concepts, and the promise of the current CSA. In this outlook we survey the recently developed concepts with applications, selected mainly from works carried out in our group in Cracow, only touched upon and serving as an illustration of the specificity of the CS description of the. classical chemical reactivity problems. We have limited the scope of this analysis to the CS defined within the fixed external potential (Born-Oppenheimer) approximation. A special emphasis is placed upon the concepts and quantities of already demonstrated or potential applicability in the theory... 

In the absence of external potentials, the electrostatic potential energy of nuclei and electrons can be represented by the Coulombic interactions among the electrons and nuclei. There are three groups of electrostatic interactions interactions between nuclei, interactions between electrons and nuclei, and interactions between electrons. Following the Born-Oppenheimer approximation, we neglect nuclei interactions in our DG-based model. Using Coulomb s law, the repulsive interaction between electrons can be expressed as the Hartree term ... 

Gauge Invariance of Group-Born-Oppenheimer Approximation... 

The superscript g denotes the truncated quantities both matrices and vectors now refer only to the group g of electronic states. Due to the above considerations, we call this useful result the group-Born-Oppenheimer approximation. It should be clear that truncating Eq. (10) is equivalent to truncating Eq. (7a). 

It is beyond the scope of this work to enter the subject of gauge theory deeply. We shall only illustrate some results relevant to the group-Born-Oppenheimer approximation. For more details we refer to Ref. 5 and references therein. 

In other words, the group-Born-Oppenheimer is a gauge invariant approximation. In the above equation, the dressed potential transforms as... 

The gauge invariance of the group-Born-Oppenheimer approximation provides a good starting point to discuss diabatic states. In contrast to Eq. (21a), where this approximation is formulated in the adiabatic electronic basis, Eq. (26) is expressed in an arbitrary basis. Elimination of the derivative couplings appearing in the latter equation amounts to setting to zero the left hand side of Eq. (27b) ... 

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