Degeneracy index

Fermi and Dirac realized that most of the NA electrons do not contribute to Cyetal We must, instead, consider the electron gas as an FD system, with most probable occupation, Eq. (5.2.11) with slight changes in notation, and ignoring the degeneracy index g, ... 

This shorthand notation implies that the summation runs over F = (a, r) where a is the type of representation and r is the degeneracy index =... 

A continuum eigenstate of the total Hamiltonian h with eigenvalue E and degeneracy index jS will then be written as a superposition of the unperturbed eigenstates... 

Liu and Klein introduced another vertex invariant defined on the basis of matrix spectral theory. Denote by X an eigenvalue of the adjacency matrix A and by jA.) the corresponding set of orthonormal eigenvectors, where x is a degeneracy index with values from 1 to the degeneracy g(X) of the eigenvalue X. Then... 

An index with extremely low degeneracy (i.e., where no degenerate structures have been observed yet) [6], the Charge-related Topological Index (CTI), has been proposed by one of the authors (l.B.) (Figure 6-1). It has a potential-like form (Eq. (4)). 

We make a small digression and note that the spin-degeneracy problem we have alluded to before is evident in Eq. (5.102). It will be observed that / = 1,..., /x in the index of e s pnp these functions are linearly independent since the efj are. There are, thus, fi linearly independent spin eigenfunctions of eigenvalue S(S + 1). Each of these has a full complement of Ms values, of course. In view of Eq. (5.40) the number of spin functions increases rapidly with the number of electrons. Ultimately, however, the dynamics of a system governs if many or few of these are important. 

The smallest two alkanes having degenerate J values have 22 vertices, where the total number of isomers (not counting stereoisomers) is 2,278,658. It will be seen, however, from Table 1 (see Sect. 4.7.) that the degeneracy for J is much lower than for any other single index. 

Considering the very low degeneracy of index J (the last one in Table 1) in comparison with the other topological indices, a fourth approach seems possible, namely the summation of only two topological indices, J and %, which provides complete discrimination of all known structures. 

The convention is that r ranges from - 7 to 7 as E increases. The index t is a bookkeeping number, rather than a true quantum number. The degeneracy of the asymmetric-top energy levels is 27+ 1, corresponding to the 27 +1 values of M, which do not affect the energy. Each asymmetric-top wave function is a linear combination of the 27+1 symmetric-top wave functions with the same value of 7 and of M as i Let us consider some examples. For 7 = 0, the only value of K is 0, and the secular equation (5.78) is... 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

In Equation 6.67, n is the refractive index, gi and g2 are, respectively, the degeneracies of the ground state and excited state, and the term in angle brackets is the reciprocal of... 

In these expressions, gi is the degeneracy of the initial state, n is the refractive index, and xl is the local correction to the electric field, accounting for the ion being less polarizable than the bulk medium. The local electric field correction may be taken as,... 

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