Simplification of the Secular Equation

The secular equation (9.83) is easier to solve if some of the off-diagonal elements of the secular determinant are zero. In the most favorable case, all the off-diagonal elements are zero, and 

Now we want to find the correct zeroth-order wave functions. We shall assume that the roots (9.89) are all different. For the root = H[i, the system of equations (9.81) is 

When the secular determinant is in diagonal form, the initially assumed wave functions 

The converse is also true. If the initially assumed functions are the correct zeroth-order functions, then the secular determinant is in diagonal form. This is seen as follows. From we know that the coefficients in the expansion are 

Applying the same reasoning to the remaining functions we conclude that = 0 for i m. Hence, use of the correct zeroth-order functions makes the secular determinant diagonal. Note also that the first-order corrections to the energy can be found by averaging the perturbation over the correct zeroth-order wave functions  

Using procedures similar to those for the nondegenerate case, one can now find the first-order corrections to the correct zeroth-order wave functions and the second-order energy corrections. For the results, see Bates, Volume I, pages 197-198 Hameka, pages 230-231. 

As an example, consider the effect of a perturbation H on the lowest degenerate energy level of a particle in a cubic box. We have three states corresponding to this level Hiese unperturbed wave functions are orthonormal, and the 

Solving this equation, we find the first-order energy corrections  

The triply degenerate unperturbed level is split into three levels of energies (through first order) (6h /Sma ) +, 6f /8ma ) + E. Using each of the 

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An alternative method of simplification of the secular equation consists in rejecting the less probable structures. If in benzene, for example, only the Kekul structures are considered,... 

Since Hij = Sij = 0 if classification results in a considerable simplification of the secular equation. Further simplification is possible if the are combined into linear combinations which are eigenfunctions of as well as of Sg. 

In both cases, if Xi is a root of the secular equation Pg[x) =0, —Xi is a root too. There is at least one zero in the spectrum for odd N. Thus in order to construct the whole set of MOs it is sufficient to find only the positive (or negative) eigenvalues, and this can be used for simplification of the MO calculations 77>. 

In order to describe quantitatively the Q-branch transformation at this stage, one should use the secular simplification of the problem [133, 257]. It neglects completely the off-diagonal elements of the kernel of integral operator (6.9), i.e. terms taking into account transfer from the other branches in equation (6.4) ... 

The secular equations which must be solved for the values of W are frequently of very high degree, even after all of the simplifications arising from considerations of symmetry have been introduced. In such cases we have reduced... 

In order to make clear the manner in which we have simplified the treatment of the various radicals, we give in Table IV a statement of the value and significance of the coefficients occurring in the function ip in the individual cases. It will not be necessary to give a similar table for the functions p since, except in the cases of the a and the /3-naphthyldiphenylmethyl radicals which have been discussed above, no simplifications were necessary for them, and the secular equations were solved rigorously. 

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. 

Formulas (78) and (79) allow the simplification of the expression of secular equation elements corresponding to orthogonal orbitals ... 

Now consider the HMO treatment of naphthalene. For butadiene and benzene, we set up the secular equation without bothering with the intermediate step of constructing symmetry orbitals from the Ipir AOs. For these molecules, the secular equation was easy enough to solve without the simplifications introduced by symmetry orbitals. For naphthalene the 10 X 10 secular determinant is difficult to deal with, and we first find symmetry orbitals. The point group of naphthalene (Fig. 17.6) is 2h-... 

Simplification of secular equations. Because the Hamiltonian is totally symmetric - that is, for a molecule of C2v symmetry such as H2O, of symmetry species Ai - the matrix elements Hij = ipi, Ti. ipj) as well as the overlap integrals Sij = (tpi, ipj) will be equal to zero unless the direct product representation r. contains Ai. This is the basis for the assertion that states of different symmetry do not mix. ... 

Since only the topology of the carbon framework is of significance in the HMO method, the full symmetry of s-traru-butadiene need not be used to get the maximiun simplification possible in the HMO method instead, it is sufficient to use only the C2 axis, (a) Write down the two possible symmetry species for the group (b) Construct rr-electron symmetry orbitals for butadiene, classifying them according to the symmetry species of %. (c) Set up and solve the two Hiickel secular equations for butadiene using the symmetry orbitals of (b) as basis functions. 

Further simplification of (9.23) can be carried out in different ways. In one, the quantities and Q are determined by a calculation similar to the one used in Section 9.4 to compute the -factor for the H + Ig reaction, leading to an expression involving masses, moments of inertia, and vibrational frequencies. The other method depends upon the solution of the vibrational secular equation in the reactant valley and at the saddle point. 2 The result, for a linear transition state, is... 

The last step in the equation above is known as secular approximation and it is a generally appropriate simplification valid as consequence of the much larger magnitude of the Zeeman interaction with the external magnetic field as compared to the chemical shift one [5]. 

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