Degenerate state

In the previous section, we saw that for a system consisting of three-state units we need a 3 x 3 matrix. It is clear that for an w-state unit, we need m m matrix. This is so because there are m x m different pairs of units, i.e., m x m different bonds. In this section we consider one example where the matrix can be reduced considerably even when there are many possible states available for each unit. 

Consider the following model as depicted in Fig. 4.3. The particles can assume m + 1 orientations with respect to the 1-D line of the system. If the particles are oriented along the line, say with ( = 0, we call their state A. All other states are referred to as B states and denoted by, 2,.. . , Altogether there are m + I states, and in general we need an (m + 1) x (m + 1) matrix to describe all possible pairs of particles. 

FIGURE 4.3. 1-D system of units having different orientations with respect to the 1-D line. 

We now assume that the interactions between A and Bi or between different j5,s are independent of the index /, i.e., 

As an example, suppose that there are two states of B, Bi and B2. Then the secular equation for such a system is 

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Mulliken symbols The designators, arising from group theory, of the electronic states of an ion in a crystal field. A and B are singly degenerate, E doubly degenerate, T triply degenerate states. Thus a D state of a free ion shows E and Tj states in an octahedral field. 

To explain the vanishing integral nile we first have to explain how we detennine the synnnetry of a product. G fold degenerate state of energy and synnnetry T, with eigenfunctions - / -fold degen... 

Figure A3.13.1 illustrates our general understanding of intramolecular energy redistribution in isolated molecules and shows how these processes are related to intemiolecular processes, which may follow any of the mechanisms discussed in the previous section. The horizontal bars represent levels of nearly degenerate states of an isolated molecule.

Figure A3.13.1. Schematic energy level diagram and relationship between mtemiolecular (collisional or radiative) and intramolecular energy transfer between states of isolated molecules. The fat horizontal bars indicate diin energy shells of nearly degenerate states.

The Role of Degenerate States in Chemistry Advances in Chemical Physics, Volume 124. Edited by Michael Baer and Gert Due Billing. Series Editors T, Prigogine and Stuart A, Rice,... 

Figure 1- Representatiori of degenerate states from nonrelativistic components, (a) Degenerate zeroth-order states at (b) Spin-orbit interaction splits 11 state, (c) With full spin-orbit...

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. 

As is well known, perturbation theory for a single state is different from that for degenerate states. The former leads to the traditional adiabatic... 

INTRODUCTION TO THE ROLE OF DEGENERATE STATES IN CHEMISTRY Kill... 

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