Factoring the Secular Equation

Before extending our treatment to two coupled nuclei, it is helpful to consider some general conditions that cause zero elements to appear in the secular equation. With this knowledge, we can avoid the effort of calculating many of the elements specifically for each case we study furthermore, the presence of zero elements usually results in the secular determinant being factored into several equations of much smaller order, the solution of which is simpler than that of a high order equation. 

Suppose the basis functions used to construct the secular equation, that is, the f , are eigenfunctions of some operator F. Consider two of these functions, f)m and / , with eigenvalues/, and f , respectively. Then 

Suppose further that the operator F commutes with the Hamiltonian  

From these premises it is shown in standard texts on quantum mechanics that 

Thus if/ —/ 0, Wmn = 0 that is, if j m and f have different eigenvalues of F, the matrix element connecting them in the secular equation must be zero. (Iffm = f , nothing can be said from Eq. 6.20 about the value of 3fm .) 

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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... 

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... 

It seems desirable to use this term for all the coordinates which factor the secular equation fully and to add special de.signations for the special types described later in this section. 

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. 

Spectra and Bonding in Metal Carbonyls The secular equations in the energy factored approximation are, for M(CO)5L,... 

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. 

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... 

As an illustration of the factors which determine the separability of such group modes, we will consider the stretching vibrations of a linear triatomic molecule, XYZ. Such a molecule has 3x3 — 5 = 4 internal vibrational modes, of which two involve stretching of the X—Y and Y—Z bonds and the other two (which are degenerate, i. e., of the same frequency) bending of the XYZ angle. By the method outlined in section I. 1. we can obtain the frequencies and forms of the stretching modes when the masses and force constants are specified. The secular equation is... 

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. 

We are now in position to complete our calculation of the AB system in general, with no restrictions whatsoever regarding the magnitude of the coupling constant Jab- By virtue of the factoring due to Fz, the secular equation is... 

We know from Section 6.7 that the true wave function i/q,. . . , t/r4 are linear combinations of the basis functions. If we begin with symmetrized functions, such as and then each of the ip s can be formed exclusively from symmetric functions or exclusively from antisymmetric functions. Stated another way, functions of different symmetry do not mix. The result is that, like the situation with Fz, many off-diagonal elements of the secular equation must be zero, and the equation factors into several equations of lower order. We shall study an example of this factoring in Section 6.13, when we consider the A2B system. 

As in the ABC case, the basis functions divide into four sets according to fz with 1,3, 3, and 1 functions in each set. However, of the three functions in the set with fz = % or — V2 two are symmetric and one antisymmetric. Hence each of the two 3X3 blocks of the secular equation factors into a 2 X 2 block and 1X1 block. Algebraic solutions are thus possible. Furthermore, the presence of symmetry reduces the number of allowed transitions from 15 to 9, because no transitions are allowed between states of different symmetry. (One of the nine is of extremely low intensity and is not observed.) Thus the A2B system provides a good example of the importance of symmetry in determining the structure of NMR spectra. 

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... 

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