Factoring of the Secular Equation

Hoffmann R, Lipscomb WN. Theory of polyhedral molecules. I. Physical factorizations of the secular equation. J Chem Phys 1962 36 2179-89. 

Here the Fjt are again force constants but pertain to vibrations described by the symmetry coordinates Sr Sh and so on. From the standpoint of physical insight, it is the fik that have meaning for us, whereas mathematically the Ffi and the associated symmetry coordinates provide the easiest route to calculations because of symmetry factorization of the secular equation. Clearly, if we could express the Ffs in terms of the fik s we would have an optimum situation. The following considerations will show how to do this. 

A second variant of the ABC system occurs when the chemical shift of one nucleus differs substantially from that of the other two — an ABX system. The presence of one nucleus only weakly coupled to the others permits factoring of the secular equation so that algebraic solutions are possible. The basis functions for the ABX system are just those shown in Table 6.3 for the general three-spin system. However, because (vA — vx) and (vB — vx) are much larger than Jax and /BX, we can define an Fz for the AB nuclei separately from Fz for the... 

We discussed the ABX system briefly in Section 6.13. Here we provide additional details on the form of the matrix elements, the factoring of the secular equation, and the expressions for transition frequencies and intensities. In addition, we describe in some detail the use of the resultant algebraic expressions to analyze an experimental ABX spectrum. Although such analysis for a specific case can be carried out by computer spectral simulation, it is instructive to see the steps used in the general algebraic procedure, which is analogous to that used in Section 6.8 but is more tedious. 

Problem 30-2. Investigate the factorization of the secular equation for np3, using the results of Problem 30-1, and list terms which belong to this configuration. 

Since these four functions are constructed from only three linearly independent functions II, III, and IV, they cannot be linearly independent in fact, it is seen that A + B 4- C = 0. The factoring of the secular equation will be found to occur when it is set up in terms of D and any two of the functions A, B, and C. The energy of the quartet level can be obtained from either of the linear factors it is given by the relation ... 

R. Hoffmann and W. N. Lipscomb, J. Chem. Phys., 36, 2179 (1962). Theory of Polyhedral Molecules. I. Physical Factorizations of the Secular Equation. R. Hoffmann,/. Chem. Phys., 39, 1397 (1963). An Extended Huckel Theory. 1. Hydrocarbons. 

One of the most powerful tools in simplifying the treatment of larger molecules is the use of molecular symmetry in factoring the secular equation. This will be developed at length in the next three chapters. The methods of the present chapter are presented in a form suitable for application to the molecule as a whole, but it will be seen later that they can also be applied to the separate factors of the secular equation which can be obtained when there is symmetry. It is the combination of the developments of the present chapter with the symmetry considerations to be introducicd later which provides the most effective approach now available. ... 

Sans serif type is used to indicate coordinates, potential cneigy constants, etc., when refeired to the basis which accomplishes the maximum factoring of the secular equation po.saible by symmetry. 

By using these results to eliminate the ratios of the vanishing frequencies, (5) can be used even for those factors of the secular equation which contain translations and/or rotations. In particular, it can be used for the whole secular equation, i.e., for all the frequencies. It then has the form... 

When the product rule is applied to a single factor of the secular equation, the character tables may be used to indicate how many powers of or which ratios TUlx), if any, will appear, thus if Tt and appear in the given symmetry species, M /M) I Jh) will appear in the product rule expression. If a doubly degenerate species is under consideration and Rx and 74 are involved, (I JIx) = would be used... 

When every molecule has the same symmetry, as in (18), then the sum rule applies separatel to each factor of the secular equation. But when the isotopic molecules passcss different symmetry, the possibility of... 

In order to carry out in praciticc the factoring of the secular equation described in the previous section, it is first necessary to construet the. symmetry coordinates. ... 

Factoring of the Secular Equation. These considerations also lead to the conclusion that the secular equation of such molecules can often be factored further than would appear from the ordinary point group of the whole molecule, for any orientation of the tops. Thus in nitromethane (CII3NO2) the mo.st favorable orientation leads to a point group 6,., which would yield only two factors, of degrees 9 and 0. However, with the i])proximations used above it will be found that the secular equation ac-iually factors into throe parl.s, of degrees. 5, 5, and 4. 

The above checks test only the transformation to symmetry coordinates and can be used on the G matrix as well as on F. They do not reveal errors in the original G elements. When factors of the secular equation are expanded, further tests are available. 

Polyhedral Molecules. 1. Physical Factorizations of the Secular Equation. 

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