Degeneracy simple problem

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. 

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

An important problem concerns the nature of the first excited state actually involved in the photodimerization reaction (the lowest excited singlet or triplet states). The simple 77-HMO method does not distinguish between the excited singlet and triplet state. Its results concern the first excited state and cannot be correlated precisely with the different multiplicities. For such a correlation, the more refined approximations of the molecular orbital method, which eliminate spin degeneracy, must be used. 

In practice we eliminate the problem of the possible degeneracy of one-electron orbitals with different angular-momentum projections by summing over the angular-momentum states to isolate the reduced matrix elements. This is simple in relevant special cases. 

The derivation of RRKM theory given here is also deficient because it has assumed that there is a single unique reaction path between the reactants and the products. In fact there may be several such paths related by symmetry, each with its own transition state. For example in the dissociation of CH4 to CH3 + H there are obviously four equivalent paths. The correct way to allow for this reaction path degeneracy has been the subject of much controversy [25,27,29-34]. However this was finally resolved by Pechukas [35], and the resolution of the problem is relatively simple. 

There exists a simple 2x2x2 array which shows degeneracy problems and often serves as a benchmark data set to test algorithms [Kruskal et al. 1983, ten Berge Kiers 1988], This array X has frontal slabs Xi and X2 ... 

ESEEM spectra at exact cancellation are close approximations to the spectra because the terms Hu are retained. It follows from this simple illustration that, although one must force a near-crossing condition in order to optimize the modulation effect, this near degeneracy does not preclude the interpretation of the exact cancellation spectra by using the conventional spin-Hamiltonian terms. In other words, cw- and FT-ENDOR are indeed analogous the ENDOR crossing problem has been examined in depdi by Schweiger et al. (1979). 

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