Nondegenerate

The majority of photochemistry of course deals with nondegenerate states, and here vibronic coupling effects aie also found. A classic example of non-Jahn-Teller vibronic coupling is found in the photoelection spectrum of butatiiene, formed by ejection of electrons from the electronic eigenfunctions [approximately the molecular orbitals). Bands due to the ground and first... 

Consider the function at a certain set of R s.Rfl. where c()(r, R) is nondegenerate. When the nuclei move away from that point, and approach it back via a different route, the wave function must return to its original value. However, in... 

There is a single nondegenerate zeroth-order vibronic state uj- Dr 00 +). The zeroth-, first-, and second-order formulas read... 

Zeroth-order level, corresponding to the vibronic state l)r (r Uc (c +) = Ur Ur uc Uc +) is nondegenerate. The zeroth, first- and second-order energies are... 

Suppose now that the total wave function (state) is nondegenerate, and F is half-integer. From Eq. (29), it then follows... 

Thus B is a diagonal mati ix that contains in its diagonal (complex) numbers whose norm is 1 (this derivation holds as long as the adiabatic potentials are nondegenerate along the path T). From Eq. (31), we obtain that the B-matrix hansfomis the A-matrix from its initial value to its final value while tracing a closed contour ... 

Although the models mentioned here are of a very specialized form (the non-adiabatic coupling terms have identical spatial dependence), still the fact that such contradictory results ai e obtained for the two situations could hint to the possibility that in the transition process from the nondegenerate to the degenerate situation, in Eq. (113), something is not continuous. 

A molecule has three nondegenerate conformers (Fig. 5-8). One is 450 x 10 J above the ground state, and the second high-energy conformer is 900 x 10 J above the ground state. What are the percentages of each of the three conformers relative to the total number of molecules in a sample of the normal conformational mixture at 300 K ... 

The first Hohenberg-Kohn theorem states that, for a nondegenerate ground state, there is a one-to-one mapping among p. V. and iq... 

For our veetors, the eonstraint that the nondegenerate veetor v(3) be orthogonal to the one we are trying to find v(l), amounts to... 

E. Nondegenerate Eigenveetors of Hermitian Matriees are Orthogonal If two eigenvalues are different, then... 

In the specific case considered here, X(E) = 4, X(2C3) = 1, and X(3av) = 0 (You should try this.). Notice that the contributions of any doubly occupied nondegenerate orbitals... 

The second difference is the appearance of a doubly degenerate E symmetry species whose characters are not always either the - - 1 or — 1 that we have encountered in nondegenerate point groups. 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

The vibrational term values for a polyatomic anharmonic oscillator with only nondegenerate vibrations are modified from the harmonic oscillator values of Equation (6.41) to... 

Densities less than indicate classical, nondegenerate behavior, whereas those above imply quantum, degenerate conditions, such as can be found in condensed-state plasmas (see Fig. 1). 

---