Function symmetry

The second approach typically involves expanding the wave functions in terms of atomic or atomic-like orbitals. Frequently s- and p-symmetry functions suffice for silicon and impurities up through the third period. A minimal basis set for silicon would consist of four basis functions, one 5-function and three p-functions on each atom. Some approaches supplement the minimal basis either with more atomic-like functions or with additional types of functions, such as plane waves. Some calculations use only plane waves for the basis. 

Symmetry relationship of cross peak locations has been used for improving the quality of correlation spectra for quite a long time in NMR spectroscopy both for diagonally [21] and laterally symmetric spectra [22]. Such data processing procedures have their advantage but may introduce artifacts and remove real information and therefore should be used with caution [10]. 2Q-HoMQC spectra can be symmetrized directly using the appropriate symmetry function [33], but most commercial software do not provide such option. Also, fine structure of the direct correlation peaks in 2Q-H0MQC spectra is antisymmetric in the SQ dimension which requires extra attention. 

To obtain the symmetry functions in terms of HLSP functions we can transform the standard tableaux functions using the methods of Chapter 5. The transformation matrix is given in Eq. (5.128) ... 

Here again, the second of these is not obviously a symmetry function. 

A full MCVB calculation on BeH with the above basis yields 504 doublet standard tableaux functions, and these combine into 344 symmetry functions. In Table 10.7 we give some details of the results with experimental values for comparison. The calculated is within 0.1 eV of the experimental value, the values of... 

Table 11.2. Number of symmetry functions for 6-3IG and 6-3IG basis sets.

G The VB stmcture basis is a full valance set augmented by stractmes involving a single excitation from one valence orbital to one virtual orbital, using all possible combinations of the excitation (outside the s shells). Table 11.2 shows the number of symmetry functions (the dimension of the H and S matrices) for each case. 

The Num. row gives the number of tableau functions in a symmetry function. Thus the 2 for column one indicates that the tableau function below it is the first of a two function sum that has the correct iS+ symmetiy. 

The layout of Tables 12.3 and 12.4 is similar to that of Tables 11.5 and 11.6 described in Section 11.3.1. There is, nevertheless, one point concerning the Num. row that merits further coimnent. In Chapter 6 we discussed how the symmetric group projections interact with spatial syimnetiy projections. Functions 1, 2, and 4 are members of one constellation, and the corresponding coefficients may not be entirely independent. There are three linearly independent E+ symmetry functions from the five standard tableaux of this configuration. The 1, 2, and 4 coefficients are thus possibly partly independent and partly coimected by group theory. In none... 

Table 2A.N2 EGSO weights (standard tableaux functions) for spherical AOs, upper group, and s-p hybrids, lower group. These are weights for whole symmetry functions rather than individual tableaux. It should be recalled from Chapter 6 that the detailed forms of symmetry functions are dependent on the particular arrangement of the orbitals in the tableaux and are frequently nonintuitive.

Table 12.10. BF EGSO weights (standard tableaux functions) for spherical AOs, upper group, and hybrid AOs, lower group. These are weights for whole symmetry functions.

The number of basis functions involved in E+ or S+ symmetiy functions. The number of symmetry functions supported. 

The number of terms in the symmetry function that is generated from the tableau shown. (See text.)... 

With six electrons and six orbitals in a full valence calculation we expect 189 standard tableaux functions. These support 51 symmetry functions that, however, involve a total of only 97 standard tableaux functions out of the possible 189. Table 13.3 shows the principal terms in the wave function for the equilibrium geometry. 

A full valence orbital VB calculation in this basis involves 784 standard tableaux functions, of which only 364 are involved in 68 2" symmetry functions. For CH3 we present the results in terms of sp hybrids. This has no effect on the energy, of course. We show the principal standard tableaux functions in Table 13.5. The molecule is oriented with the C3 -axis along the z-axis and one of the H atoms on the x-axis. The three trigonal hybrids are oriented towards the H atoms. The x subscript on the orbital S5unbols in Table 13.5 indicates the functions on the x-axis, the a subscript those 120° from the first set, and the 6 subscript those 240° from the first set. 

A full valence calculation on CH4 gives 1764 standard tableaux functions, and all of these are involved in the 164 A1 sjmimetry functions. The second and fourth tableaux are also present in the principal constellation and, as with the earlier cases, these are not simple symmetry functions alone. The third tableau is ionic with the negative charge at the C atom. As before, this contributes to the relative polarity of the C—H bonds. 

After our discussion ofthe ST03Gresults we, in this section, compare some ofthese obtained with a 6-3IG basis arranged as described in Chapter 9. As before, we find that the larger basis gives more accurate results, but the minimal basis yields more useful qualitative information concerning the states of the atoms involved and the bonding. The statistics on the number of symmetry functions and standard tableaux functions for the various calculations are given in Table 13.9. 

Table 14.1. Number of symmetry functions of three types for H-ring calculations of ( [2)2 and (112)3.

In this ease all of the terms in a symmetry function have the same sign as well as magnitude for the eoeffieient. 

The Weyl dimension formula (Eq. (5.115)) tells us that six electrons in six orbitals in a singlet state yield 175 basis functions. These may be combined into 22 Aig symmetry functions. Table 15.1 shows the important HLSP functions for a rr-only calculation of benzene for the SCF optimum geometry in the same basis. The a orbitals are all treated in the core , as described in Chapter 9, and the tt electrons are subjected to its SEP. We discuss the nature of this potential farther in the next section. The functions numbered in the first row of Table 15.1 have the following characteristics. 

When there is a relatively high degree of S3mimetry as in benzene, the interpretation of the parts of the wave function must be carried out with some care. This arises from an apparent enhancement of the magnitude of the coefficient of a stmcture in the wave function when whole symmetry functions are used. Let us consider the... 

The apparent enhancement we are discussing here is more pronounced, in general, the greater the number of terms in the S5munetry function. We now consider the third sort of function from Table 15.1. These are the 12 short-bond singly ionic functions, and in this case the enhancement of the coefficient is a factor of 5.0685, i.e., the reciprocal of the normalization constant for the symmetry function that is the sum of the individually normalized HLSP functions. The resulting coefficient would then be 0.261 637, a number essentially the same as the coefficient of the Kekule symmetry function. 

This gives 3235 tableaux functions formed into 545 symmetry functions. 

We now consider naphthalene, which possesses 42 covalent Rumer diagrams. Many of these, however, will have long bonds between the two rings and are probably not very important. To the author s knowledge no systematic ab initio study has been made of this question. The molecule has D21, symmetry, and these 42 covalent functions are combined into only 16 4 symmetry functions. 

The resultant family of six electronic states can be described in terms of the six configuration state functions (CSFs) that arise when one occupies the pair of bonding o and antibonding o molecular orbitals with two electrons. The CSFs are combinations of Slater determinants formed to generate proper spin- and spatial symmetry- functions. 

By Proposition 10.7, the physical symmetry function S must be the identity. Hence 5 = [T]. 

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