Degenerate symmetry species, for linear

Handling of degenerate symmetry species for linear molecules symt- Symmetry can be exploited to improve the efficiency of ab initio quantum chemical programs. This program recognizes degenerate symmetry species for linear molecules. 

Handling of degenerate symmetry species for linear molecules symt... 

The ith element of the array invlab identifies the symmetry of the orbital i. This index is used for handling the degenerate symmetry species that occur with linear molecules. All elements of invlab are set equal to 1 if the molecule is not linear or if real spherical harmonics are not used. For linear molecules, the elements of invlab are assigned as follows ... 

The carbon atom is at the center of the molecule, and the carbon Ij and Is AOs are each sent into themselves by every symmetry operation. These AOs transform according to the totally symmetric species A]. The carbon 2p 2py, and 2p AOs are given by x, y, or z times a radial function. Their symmetry behavior is the same as that of the functions x, y, and z, respectively. From the formulas for rotation of coordinates [Eq. (15.52)], we see that any proper rotation sends each of the functions x, y, and z into some linear combination of x, y, and z. Any improper rotation is the product of some proper rotation and an inversion (Problem 12.15) the inversion simply converts each coordinate to its negative. Hence the three carbon 2p orbitals are sent into linear combinations of one another by each symmetry operation. They must therefore transform according to one of the triply degenerate symmetry species. Further investigation (which is omitted) shows the symmetry species of the 2p AOs to be T2. 

When a molecule has doubly degenerate symmetry species, three adiabatic electronic states can intersect conically. Three-state CIs can occur e.g. for E and II states of linear molecules or for non-degenerate and degenerate states in other point groups, as in the PIT effect. These multiple CIs cause a huge increase of NA effects. 

In IR spectra of spherical tops, the allowed vibrational transitions are all of a single, triply degenerate symmetry species (t2 symmetry for a tetrahedral molecule). The stmcture of a band is like that of a perpendicular band of a linear molecule, with A/ = 0, 1, giving rise to P, Q and R branches. 

For an unsymmetrical linear molecule there are N—1 non-degenerate vibrational normal coordinates belonging to the symmetry species 2+ (see Table 7-8.1) and they are of the longitudinal type and N—2 pairs of vibrational normal coordinates (doubly degenerate), each pair belonging to the II symmetry species and they are of the transverse type. 

To obtain information about the geometry of molecules, e.g., about the number of AB bonds in an AB system, the spectra have to be analyzed more carefully. In a non-linear molecular compound, the total number of vibrations is 3m- 6 (m = number of atoms). In order to discuss the geometry, one must first look at the number of stretching vibrations. This number corresponds to the number of AB bonds. As a result of the symmetry, the number of observed stretching vibrations is sometimes smaller, e.g., only two vibrations are usually observed for AB3-systems, since one of them is doubly degenerate. This is true for species with D h- and with Cj -symmetry. 

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