Factorization of secular equation

As the ground CFT is orbitally triply degenerate, it does not fulfill the conditions for a direct application of the SH formalism. By using the T-p isomorphism, the 9 x 9 interaction matrix in the basis set of transformed octahedral kets (which accounts for the Cl of the 3 -terms) is of the form that can be factored into secular equations of lower dimensions as already presented in Table 13 the eigenvalues can be written in a closed form. 

The process by which we have factored the secular equation into two linear factors and a quadratic corresponds to using the real functions pi Pip, and fap, for the ipu s instead of the set fa faPl, faPl, and fap, (see Sec. 185). In terms of the real set the secular equation has the form... 

It is to be noted that in place of fap, and fap, any linear combinations of these might have been used in setting up the secular equation 24-25, without changing the factoring of that equation, so that these linear combinations would also be satisfactory zeroth-order wave functions for this perturbation. 

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. ... 

In other words, in order to factor the secular equation, the coordinates are formed into linear combinations such that each combination (or new coordinate) belongs to one of the symmetry species of the molecular point group. In the water illustration, Si and S2 are of species A, while S3 is Bi. When only real one-dimensional species occur, the proof is immediate that no cross terms will occur in cither the kinetic or potential energies between two coordinates, S<" and S<" say, of different symmetry species r( i ) and FThere will always be some operation B of the group for which... 

In Chap. 6 it was asserted that the introduction of symmetry coordinates would factor the secular equation. A general proof will now be given. 

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

In the example considered above, Arj - A/s is the only symmetry coordinate of species B2. Thus, it results in a factor of degree one in the completely reduced secular determinant It is therefore a normal coordinate. On the other hand, the two normal coordinates of species Ai are linear combinations of the symmetry coordinates Acr and Arj + Ar2. They can only be found by solution of the secular equations. 

Unfortunately, the secular equation doesn t factor and the energies must be computed numerically. A plot of the computed energies is shown in Figure 6.4 as a function of magnetic field. 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

The secular determinant is in block-diagonal form and factors into three determinants. Two of the roots of the secular equation are... 

The wave functions have the form (5.54), but since Pc does not commute with H, we cannot separate out a chi factor the Schrodinger equation is not separable, and we will try another method of dealing with the problem. We saw in Section 2.3 that the eigenvalues of an operator H can be found by expanding the unknown eigenfunctions in terms of some known complete orthonormal set [Pg.361]

This is simpler than 7.1-15, in that many of the terms in the lOth-order polynomial equation which will result on expanding the determinant will now be equal to zero. Nevertheless, the basic, awkward fact is that a lOth-order equation still has to be solved. This is not a task to be confronted with pleasurable anticipation without the use of a digital computer it would be a protracted, tedious job. Fortunately, in this case and all others in which the molecule possesses symmetry, the secular equation can be factored—that is, reduced to a collection of smaller equations—by using the symmetry properties in the right way. The method of symmetry factoring will now be explained and illustrated. 

Even with the simplifications that result from a drastic approximation such as the Hiickel approximation, the secular equation for the MOs of an n-atomic molecule will, in general, involve at least an unfactored nth-order determinant, as just illustrated in the case of naphthalene. It is clearly desirable to factor such determinants, and symmetry considerations provide a systematic and rigorous means of doing this. 

As a very persuasive illustration of the effectiveness of symmetry factorization in reducing a computational task that would be entirely impractical without a digital computer to one that is a straightforward pencil-and-paper operation, we shall again consider the naphthalene molecule. It has been shown in Section 7.1 that the secular equation for the n MOs is the 10 x 10 determinantal equation, 7.1-15, if the set of 10pn orbitals is used directly for constructing LCAO-MOs. 

It follows from the properties of determinants that, if the entire determinant is to have the value zero, each block factor separately must equal zero. Thus the 10 x 10 determinantal equation has been reduced to two 2x2 and two 3x3 secular equations. For example, the energies of the two MOs of Au symmetry are given by the simple secular equation... 

We shall return later (page 172) to the symmetry-factored secular equation for the n MOs of naphthalene and solve for the energies, LCAO-MO coefficients, and other useful results. 

Figure 7.3 Symmetry-factored form of the secular equation for the n orbitals of naphthalene.

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. 

The problem of expressing, for the H20 molecule, the relationship between the frequencies of the fundamental modes and a set of force constants has now been solved in such a way that the equations are as simple as symmetry will permit them to be. In this case, the secular equation is factored into one of second order for the two A, vibrations and one of first order for the sole Bt vibration. The explicit forms of these separate equations are... 

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