Non-degenerated states

The first-order perturbation theory of the quantum mechanics (4, III) is very simple when applied to a non-degenerate state of a system that is, a state for which only one eigenfunction exists. The energy change W1 resulting from a perturbation function / is just the quantum mechanics average of / for the state in question i.e., it is... 

That is, if we are dealing with non-degenerate states. Otherwise the wave function might be a limited linear combination of Slater determinants. 

Here we ignore any possible perturbation to the site energies at the ends of the chain, n = 1 and n = m.) We apply Brillouin-Wigner perturbation theory (Ohanian 1990), whereby the eigenvalue of a non-degenerate state can be expressed as... 

To evaluate both the coupling and the magnitude of the moments, we consider first an array of one-electron atoms in non-degenerate states, and let 0 denote an atomic orbital for an electron on atom (i). Then the electron on atom (i) can overlap on to neighbouring atoms (/), so we may write its wave function in the form... 

Let us now consider a number of different situations and their experimental consequences with the aid of figure 6.26. This diagram illustrates four different situations concerning the relative populations of the two non-degenerate states. In case (i) the... 

The integrals which determine the transition probabilities between states /, m and /, m of a molecule in a non-degenerate state are, from equation (6.300),... 

For the diagonal element, the vibronic coupling constant, wF is nonzero if and only if the symmetric product of F, [F ] contains F. For a non-degenerate state, F x r = Ai, where, 4i is a totally symmetric representation. Therefore, the distortions are totally symmetric in a non-degenerate electronic state. As for a degenerate state, the symmetric product contains some non-totally symmetric representations that cause Jahn-Teller distortions. 

Non-linear polyatomic molecules are treated similarly except now there are three rotational constants in general. Thus, for non-linear XYZ (such as CINO) molecules, experimental values of Ay, By and Cy in the non-degenerate states... 

This expression for the perturbation energy can be very simply described The first-order perturbation energy for a non-degenerate state of a system is just the perturbation function averaged over the corresponding unperturbed state of the system. 

The Fermi-Dirac distribution law for the kinetic energy of the particles of a gas would be obtained by replacing p W) by the expression of Equation 49-5 for point particles (without spin) or molecules all of which are in the same non-degenerate state (aside from translation), or by this expression multiplied by the appropriate degeneracy factor, which is 2 for electrons or protons (with spin quantum number ), or in general 21 + 1 for spin quantum number I. This law can be used, for example, in discussing the behavior of a gas of electrons. The principal application which has been made of it is in the theory of metals,1 a metal being considered as a first approximation as a gas of electrons in a volume equal to the volume of the metal. 

Equations (90), (94), (106), (110) and (138), (144) strictly hold for one single transition in the spectrum, more precisely for a transition between two non-degenerate states and for one polarization orientation (p = x or y, +1 or — 1). In that case all molecules are in the ground state a, leading to the factor Na in k which is related to Ca in 8. If several transitions have the same energy, because of degeneracy or because more than one state is populated, Na remains as such but in Lambert-Beer s law, the concentration becomes Ca, the total concentration of the crystal field manifold states. Therefore, a summation over a and j must be performed leading to the expression ... 

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