The Secular Matrix

By the criterion of Exercise 2-9, is an eigenvalue of the matrix in a and p. There are two secular equations in two unknowns for ethylene. For a system with n conjugated sp carbon atoms, there will be n secular equations leading to n eigenvalues . The family of , values is sometimes called the spectrum of energies. Each secular equation yields a new eigenvalue and a new eigenvector (see Chapter 7). 

If we divide each element of the secular matrix by p and perform the substitution 

For the equation set to be linearly dependent, the secular determinant must be zero 

There are n = 2 roott of the polynomial, one for each eigenvalue in the E, spectrum. 

We ai e free to pick a tefei ence poitit of energy once, but otily otice, for each system, l,et us choose the reference point t.. We have obtained the energy eigenvalues of the x bond in ethylene as one [f greater than y. 011)11 bunding) and one p lower than y ( bunding) (Fig, 6-3), 

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Later in this book, we shall need to find the roots of the secular matrix... 

This equation is a quadratic and has two roots. For quantum mechanical reasons, we are interested only in the lower root. By inspection, x = 0 leads to a large number on the left of Eq. (1-10). Letting x = leads to a smaller number on the left of Eq. (1-10), but it is still greater than zero. Evidently, increasing a approaches a solution of Eq. (1-10), that is, a value of a for which both sides are equal. By systematically increasing a beyond 1, we will approach one of the roots of the secular matrix. Negative values of x cause the left side of Eq. (1-10) to increase without limit hence the root we are approaching must be the lower root. 

Write the secular matrix, compute the eigenvalues, and draw the energy level diagram for fulvcnc. 

One restriction imposed by Huckel theory that is rather easy to release is that of zero overlap for nearest-neighbor interactions. One can retain a — as the diagonal elements in the secular matrix and replace p by p — EjS as nearest-neighbor elements where S is the overlap integral. Now,... 

Hoffman s extended Huckel theory, EHT (Hoffman, 1963), includes all bonding orbitals in the secular matrix rather than just all n bonding orbitals. This inclusion increases the complexity of the calculations so that they are not practical without a computer. The basis set is a linear combination that includes only valence orbitals... 

We fill the secular matrix H with elements Hy over the entire set of valence orbitals. The diagonal elements are... 

The various basis sets used in a calculation of the H and S integrals for a system are attempts to obtain a basis set that is as close as possible to a complete set but to stay within practical limits set by the speed and memory of contemporary computers. One immediately notices that the enterprise is directly dependent on the capabilities of available computers, which have become more powerful over the past several decades. The size and complexity of basis sets in common use have increased accordingly. Whatever basis set we choose, however, we are attempting to strike a balance. If the basis set is too small, it is inaeeurate if it is too large, it exceeds the capabilities of our computer. Whether our basis set is large or small, if we attempt to calculate all the H and S integrals in the secular matrix without any infusion of empirical information, the procedure is described as ab initio. 

The local mode Hamiltonian (5.16) includes only the operator C[2 that is, interactions of the Casimir type between bonds 1 and 2. One may wish, in some cases, to include also interactions of this type between bonds 1 and 3, C13, and 2 and 3, C23. These can be included by diagonalizing the secular matrix obtained by evaluating the matrix elements of C13 and C23 in the basis (5.4). These matrix elements are given by (5.15a) and (5.15b). 

This operator is used only for the construction of the d matrix. The gradient terms in the secular matrix are, of course, computed exactly. 

In fact the relative coefficients within a set of symmetrically equivalent atoms, such as Cl, C3, C5, and Cg in para-benzosemiquinone, can be determined by group theory alone. The appropriate set of symmetrized orbitals, also listed in Table 1, can be obtained by use of character tables and procedures described in Refs. 3 to 6. In matrix notation, the symmetrized combinations are <1> = Up, where <1) and p are column vectors and U is the transformation matrix giving the relations shown in part (c) of Table 1. The transformation U and its transpose U can be used to simplify the solution of the secular matrix X since the matrix multiplication UXU gives the block diagonal form shown in part (6) of Table 1. 

Application of the variational method leads the secular matrix equation... 

One sees from the functional dependence of the energy, eq (44), on trigonometric functions that the energies of the bands are of the standard form. Because of the one-dimensional nature of the problem and the 2x2 dimensions of the secular matrix, it is possible simply to include account of overlap automatically without specifically using methods of orthonormalization, such as Lowdin s method [42-44]. 

In constrast to the matrix H - E0 in (1.19), the secular matrix has a complicated, nonlinear energy dependence, and the one-electron energies E must be found individually by tracing the roots of the determinant of as a 10... 

It may even be improved by extending the internal L" summation in the three-centre term beyond max thus including the tails of the higher- partial waves without increasing the dimension of the secular matrix. 

The mixing coefficients depend on A and are obtained from the eigenvectors of the secular matrix. Similar equations can be written for all of the mixed vibrational states. 

In the applications, we are interested in individual eigenstates mostly in the low-energy part of the spectrum, while for higher energies it is more the spectral envelope that matters. For both of these pieces of information, the Lanczos algorithm converges particularly fast. In representative applications typically 10 -10 Lanczos steps are found to be sufficient, even though the dimension of the secular matrix may be 10 or more. 

In traditional quantum mechanical calculations the inclusion of rotational motion, particularly to high orders, has proved difficult because of the consequent increase in the size of the secular matrix. For Van der Waals systems, the so-called helicity approximation has often been invoked. This approximation assumes that the projection, k, of the total angular momentum,. J, onto some body-fixed z-axis is a constant of motion for the system. The approximation is often good for energy levels but is less reliable for properties such as transition intensities. 

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