Symmetry configuration

Spatial function Spin function Symmetry Configuration... 

As an effect of the linear and quadratic vibronic integrals the adiabatic potential surface stays no longer paraboloid-shaped. It exhibits an additional warping with several local minima and saddle points out of the reference high-symmetry configuration Q0. 

There are several obvious mechanisms that lower both the internal energy and symmetry. These are first of all electron-nuclear bonding and interatomic and intermolecular interactions. They underlie the formation of condensed matter (atoms, molecules, and solids) by cooling which takes place in a series of typical SB. Beside these cases there are many SB that are at first sight not associated with bonding, but with spontaneous distortions of high-symmetry configurations (which... 

Fig. 1. The adiabatic potentials (AP) in the Jahn-Teller (a), Renner-Teller (b), and pseudo-Jahn-Teller (c) effects for systems with a double degenerate electronic term interacting with one coordinate Q (E bi problem in the JT case). In all three cases the ground state is unstable in the high-symmetry configuration <2 = 0, while the stable configurations at Q0 are at lower symmetry. The differences between these cases are in the behavior of the AP at Q = 0.

An important question is whether the JT vibronic coupling (JT, RT, and PJT) mechanism of SB is unique, or it is applicable to a limited number of special cases and hence there may be other mechanisms of SB that are in principle different from the JT one. The answer is that the JT mechanism of spontaneous distortions of high-symmetry configurations (chemically stable systems and transition states of chemical reactions) that leads to SB is unique, and there is no other in principle different mechanism that produces such distortions [1,2,5,11], It was also shown that in ensembles of systems (e.g. local centers in crystals), just the interaction between them (e.g. mutual polarization) cannot produce SB without local (JT) distortions of each system [12]. The JT effects provide thus the necessary and sufficient condition of SB in the systems under consideration. 

However, so far all the applications of the JT effect theory were realized only for chemically bonded systems in their high-symmetry configuration and transition states of chemical reactions. We show here that this is an unnecessary restriction the JT type instability is inherent to all the cases of degeneracy or pseudodegeneracy in molecular systems and condensed matter including nonbonded states in molecule formation from atoms, intermolecular interaction, and chemical reactions. 

The necessary and sufficient condition of instability (lack of minimum of the AP) of high-symmetry configurations of any polyatomic system is the presence of two or more electronic states, degenerate (except 2-fold spin degeneracy) or pseudodegenerate, which interact sufficiently strong under the nuclear displacements in the direction of instability . 

An interesting case of SB is presented by enantiomer formation. In recent papers [15] it was shown that enantiomers can be presented as the low symmetry, PJT distorted configurations of a hypothetical high-symmetry structure, and as such their interaction in the liquid phase via collisions under special conditions may lead to some kind of cooperativity and phase transition (SB) resulting in singleenantiomer broken symmetry configuration. 

The final wavefunction for BH3 can thus be said to consist of two fully-symmetric configurations, one of them essentially a double excitation out of the other, and a symmetry-adapted linear combination of six equivalent low-symmetry configurations. The latter can be viewed as distortions of the main configuration. The wavefunction therefore includes only three truly independent configurations, and is thus readily amenable to physical interpretation, while achieving an accuracy that vouches for the significance of such interpretations. 

The present review has been written with the aim of presenting the various aspects of molecular geometry relevant to stereoisomerism, namely symmetry, configurational and conformational isomerism, enantiomerism and diastereoisomerism. It certainly was apparent to the reader that the examples were never examined for the full spectrum of their stereochemical features, but only for that particular aspect under... 

Somewhat paradoxically, symmetry is seen to play an important role in the understanding of the Jahn-Teller effect, the very nature of which is symmetry destruction [52], In a recent review the original paper published by Jahn and Teller [53] was called one of the most seminal papers in chemical physics [54], Only a brief discussion of this effect will be given here for more detail we refer the reader to References [55-59], Bersuker says that all structural instabilities and distortions of high-symmetry configurations of polyatomic systems are of Jahn-Teller origin (here he also refers to other related effects, such as the Renner-Teller effect and the pseudo-Jahn-Teller effect—they will be mentioned later). Bersuker likes to call this the... 

Potluri S, Yan AK, Chou JJ, Donald BR, Bailey-Kellogg C. Structure determination of symmetric homo-oligomers by a complete search of symmetry configuration space, using NMR restraints and van der Waals packing. Proteins 2006 65 203-219. 

The multiconfiguration Hartree—Fock procedure is concerned with a particular symmetry manifold /j. It is therefore necessary to specify an eigenstate only by the principal quantum number n. The eigenstate ) is expanded in a set of Nr symmetry configurations r) that belong to the same manifold. That is they are eigenstates of parity and total angular momentum with quantum numbers /,y,m. 

The symmetry configurations are linear combinations of single determinant configurations p) that have the required quantum numbers. They are formed by coupling the angular momenta of the configurations. Normally all symmetry configurations are formed from a common set of orbitals. 

If the states i ) and i) belong to different symmetry manifolds, characterised by the quantum numbers, j, then the Hamiltonian matrix element is zero. It is economical to consider the diagonalisation in a particular symmetry manifold and we will begin our discussion in this way. The basis states fk) are now symmetry configurations consisting of linear combinations of configurations which have the symmetry /j of the manifold. 

In one sense the hydrogen atom is a trivial case since the symmetry configurations are one-orbital determinants and in any case the exact eigenstates are known. However, we use it to illustrate the answer to a nontrivial question. How well can the lower-energy eigenstates of an atomic system be represented by an M/-dimensional square-integrable basis for each symmetry manifold We remember that a complete set of atomic states includes the ionisation continuum. 

It is sensible to choose the basis orbitals a) that form the basis configurations in an optimal way. The symmetry configuration that most-closely approximates the lowest-energy state of the Xj manifold is the Hartree—Fock configuration ro), in which the lowest-energy orbitals are occupied. The basis orbitals include ones that are unoccupied in ro). They may be calculated as eigenstates of the Hartree—Fock equation (5.26). 

The configuration-interaction representation of the lower-energy states of an atom is the IV-electron analogue of the Sturmians in the hydrogen-atom problem. We choose an orbital basis of dimension P, form from them a subset of all possible A/ -electron determinants pk),k = 0,Mp, and use these determinants as a basis for diagonalising the IV-electron Hamiltonian. It may be convenient first to form symmetry configurations kfe) from the pfe). 

In general the observed ion states can be grouped into symmetry manifolds, characterised by the quantum numbers /,/ We consider each symmetry manifold separately. The configuration-interaction basis for the target consists of symmetry configurations r), which are linear combinations, with symmetry /,y, of determinants formed from the set of orbitals a). 

The angular distributions of recoiling iodine atoms were also measured for all four alkyl iodides studied. As is discussed elsewhere, such distributions provide information about the symmetry, configuration, and lifetime of the parent dissociative excited state. The details of these results will be presented in a future paper. Briefly, the angular distributions show that the transition dipole moment lies along the C—I bond and that the excited state breaks up on a time-scale short compared toarotational period. 

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