Ground state degeneracy

Figure 21-14 Two energy levels with different degeneracies. Ground-state atoms can absorb light to be promoted to the excited state. Excited-state atoms can emit light to return to the ground state.

Orbital energy Designation Degeneracy Ground-state occupancy ... 

The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

Figure 20, The potential surface near the degeneracy point of a degenerate E state that distorts along two coordinates and Q. The parameter is the stabilization energy of the ground state (the depth of the moat ), [Adapted from [70]].

The energy differences among conforiiiers relative to the ground state are 0.0, 0.85, 1.62, and 3.32 kcal mol . The relative populations of the states, judged by the number of times they were found in a random search or 50 trials, are 0.16, 0.21, 0.15, and 0.08 when degeneracy is taken into account. In the limit of ver y many runs, a Boltzmann distr ibution would lead us to expect a ground state that is much more populous than the output indicates, but this sample is much too small for a statistical law to be valid. 

Figure 10.17 The effect of degeneracy on a Schottky heat capacity from a single excited state 30 cm-1 above the ground state, (a), o= i = l (b). o = 3. gi = 1 and (c), go — Ug =3.

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

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