Degeneracies

In the statistical-mechanical evaluation of the molecular partition function of an ideal gas, the translational energy levels of each gas molecule are taken to be the levels of a particle in a three-dimensional rectangular box see Levine, Physical Chemistry, Sections 22.6 and 22.7. 

Let us prove an important theorem about the wave functions of an n-fold degenerate energy level. We have n independent wave functions ij/i, i/ 2, . ch having the same energy. Let w be the energy of the degenerate level  

In the free-electron theory of metals, the valence electrons of a nontransition metal are treated as noninteracting particles in a box, the sides of the box being the surfaces of the metal. This approximation, though crude, gives fairly good results for some properties of metals. 

For an n-fold degenerate energy level, there are n independent wave functions each having the same energy eigenvalue w  

We wish to prove the following important theorem Every linear combination 

This product of three integrals is relatively easy to evaluate, despite its length. The X and z integrals are exactly the same as the one-dimensional particle-in-a-box wavefimctions being evaluated from one end of the box to the other, and they are normalized. Therefore, the first and the third integrals are each 1. The expression becomes 

For the y part, evaluation of the derivative part of the operator is straightforward, and rewriting the integral, bringing all constants outside the integral sign, yields 

Using the integral table in Appendix 1, we find that this integral is exactly zero. Therefore, 

The final result should not he too much of a stuprise. Although the particle certainly has momentum at any given moment, it will have one of two opposite momentum vectors exactly half the time. Because the opposing momentum vectors cancel each other out, the average value of die momentum is zero. 

The above example illustrates that although the triple integral may look difficult, it separates into more manageable parts. This separability of the integral is directly related to our assumption that the wavefunction itself is separable. Without separability of T, we would have to solve a triple integral in three variables simultaneously—a formidable task We will see other examples of how separability of F makes things easier for us. Ultimately, the issue of separability is paramount in the application of the Schrodinger equation to real systems. 

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Two other examples will sufhce. Methane physisorbs on NaCl(lOO) and an early study showed that the symmetrical, IR-inactive v mode could now be observed [97]. In more recent work, polarized FTIR rehection spectroscopy was used to determine that on being adsorbed, the three-fold degeneracies of the vs and v modes were partially removed [98]. This hnding allowed consideration of possible adsorbate-adsorbent geometries one was that of a tripod with three of the methane hydrogens on the surface. The systems were at between 4 and 40 K so that the equilibrium pressure was very low, about 10 atm. 

Hydrogen atoms chemisorbed on a metal surface may be bonded to just one metal atom or may be bonded to two atoms in a symmetrical bridge. In each case, there are three normal modes. Sketch what these are, and indicate any degeneracies (assume the metal atoms to be infinitely heavy). 

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. 

Thus many aspects of statistical mechanics involve techniques appropriate to systems with large N. In this respect, even the non-interacting systems are instructive and lead to non-trivial calculations. The degeneracy fiinction that is considered in this subsection is an essential ingredient of the fonnal and general methods of statistical mechanics. The degeneracy fiinction is often referred to as the density of states. 

Wlien = N/2, the value of g is decreased by a factor of e from its maximum atm = 0. Thus the fractional widtii of the distribution is AOr/A i M/jV)7 For A 10 the fractional width is of the order of 10 It is the sharply peaked behaviour of the degeneracy fiinctions that leads to the prediction that the thennodynamic properties of macroscopic systems are well defined. 

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... 

The rotational states are characterized by a quantum number J = 0, 1, 2,. .. are degenerate with degeneracy (2J + 1) and have energy t r = ) where 1 is the molecular moment of inertia. Thus... 

Here the levels consist of several states, is the reactant level degeneracy and is the collision wavenumber (see equation (A3.4.73)). 

In this equation as well as in the succeeding discussions, we have suppressed, for notational simplicity, the pemuitation or degeneracy factor of two, required for SFG. 

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

Dunlap B I 1987 Symmetry and degeneracy in Xa and density functional theory Advances in Chemicai Physics vol LXIX, ed K P Lawley (New York Wiley-Interscience) pp 287-318... 

Wall M R, Dieckmann T, Feigon J and Neuhauser D 1998 Two-dimensional filter-diagonalization spectral inversion of 2D NMR time-correlation signals including degeneracies Chem. Phys. Lett. 291 465... 

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