Theory perturbation, degenerate state

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

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... 

In this context equations (50) and (53) can be considered forming a completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [lej. 

B) How many lines are expected from this model The total number of nuclear spin states is (2 f + 1) x (2I2 + 1) x (2/3 + 1). Thus, if the model structure has six protons (I = 1/2), there should be (2 x 1/2 + l)6 = 26 = 64 nuclear spin states. If some of the nuclei are expected to be equivalent, then the number of lines will be less than the number of spin states, i.e., some of the spin states will be degenerate (to first-order in perturbation theory). Thus, if the six protons are in three groups of two, it is as if you had three spin-1 nuclei and you expect (2 x 1 + l)3 = 33 = 27 distinct lines. If there is one group of four equivalent protons and another group of two, then it is as if you had one spin-2 nucleus and one spin-1 nucleus and you expect (2x2+ 1)(2 x 1+1) =15 lines. 

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... 

These functions are all eigenfunctions of VT with the first two functions forming the ground state when Kris applied alone and the last four forming the degenerate upper state. Using first-order perturbation theory with the perturbation operator of Eq. (35), we obtain the following determinant ... 

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

Now consider perturbation theory for an unperturbed energy level that is n-fold degenerate, with the states... 

S. Wilson, K. Jankowski, and J. Paldus, Int.]. Quantum Chem., 28,525 (1985). Applicability of Non-Degenerate Many-Body Perturbation Theory to Quasi-Degenerate Electronic States. II. A Two-State Model. 

If a point-charge ionic model places the empty 3d orbitals of a degenerate manifold an energy AEP above the 0-2p orbitals and AES above the 0-2s orbitals, the antibonding d-like states may be described in second-order perturbation theory to give the ligand-field wavefunc-tions... 

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