The 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. 

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) ... 

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. 

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

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

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 ... 

In the Born-Oppenheimer approximation nuclei move on the single potential energy surface created by the faster moving electrons. This approximation works so well that is at the heart of the way we think about nuclear motion. Processes in which the Born-Oppenheimer approximation breaks down are known as electronically nonadiabatic processes. Despite the reverence duly accorded the Born-Oppenheimer approximation, electronically nonadiabatic processes are ubiquitous. Indeed the study of nonadiabatic processes goes back almost far as the Born-Oppenheimer approximation itself. It is useful to group nonadiabatic processes into... 

Let us introduce a CD2 group as a defect in n-alkanes (CH3-(CH>)nCH3) or n-alkyl chain (CH3(CH2)xX or in any polymethylene chain. This is a typical case of mass defect which leaves the force constant matrix (Eq. 3-3)) unchanged because of Born-Oppenheimer approximation. The site where the CD2 group is located can be changed along the chain. Chemically, this can be easily achieved and selectively deuterated n-alkane or alkyl derivatives are either commercially available or can be (and have been) easily synthesized. 

A set of standard type approximations (Born Oppenheimer approximation, group partition, simple Hartree factorization between the groups, single determinant SCF approximation i.e. 

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