Secular equations, derivation

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

Let us now turn our interest to the excited states. The energies Ev E2,. .. of these levels are given by the higher roots to the secular equation (Eq. III.21) based on a complete set, and one can, of course, expect to get at least approximate energy values by means of a truncated set. In order to derive upper and lower bounds for the eigenvalues, we will consider the operator... 

Let us now consider the possibilities for deriving an eigenfunction for a particular excited state. The straightforward application of the variation principle (Eq. II.7) is complicated by the additional requirement that the wave function Wk for the state k must be orthogonal to the exact eigenfunctions W0, Wv for all the lower states although these are not usually known. One must therefore try to proceed by way of the secular equation (Eq. III.21). A well-known theorem15 25 says that, if a truncated... 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

We begin with the derivative of the secular equation with respect to energy eigenvalues. For some background on matrix differential calculus, see the Refs. 116 and 117. 

In the derivation of normal modes of vibration we started with a set of displacements of individual atoms. By determining the reducible representation Ltot and decomposing it, we calculated the number of normal modes of each symmetry species. We could determine what these modes are by solving a secular equation. We could alternatively have used projection operators to determine the symmetry-adapted combinations. 

The band structure of nonmagnetic fee and bcc iron is shown in Fig. 7.5, being computed from the hybrid NFE-TB secular equation with resonant parameters Ed = 0.540 Ry and = 0.088 Ry. The NFE pseudopotential matrix elements were chosen by fitting the first principles band structure derived by Wood (1962) at the pure p states Nv (tiuo = 0.040 Ry), L2> ( U1 = 0.039 Ry), and X (t 200 = 0.034 Ry). Comparing the band structure of iron in the 100> and 111> directions with the canonical d bands in Fig. 

H is the Hamiltonian operator and the numbering of the CSFs is arbitrary, but for convenience we will take I l = I hf and then all singly excited determinants, all doubly excited, etc. Solving the secular equation is equivalent to diagonalizing H, and permits determination of the CI coefficients associated with each energy. While this is presented without derivation, the formalism is entirely analogous to that used to develop Eq. (4.21). 

A secular equation such as 7.1-15 is derived from an array of the individual... 

The energies of the (/-orbitals for the system , , i = 1-5, are then obtained by diagonalization of the real symmetrical matrix Zfy, i = dzi...dy2. The real (/-orbital linear combinations which correspond to these energies are then obtained by substituting the solutions, into the sets of simultaneous equations derived from the secular determinant. 

The phase integral s(r) has the derivative p(r) and it satisfies the secular equation... 

In order to find a dispersion relation co = co(k), a system of equations for the unknown amplitudes UA and UB is derived. The secular equation leads immediately to the dispersion relation (see 8-9>)... 

The m — 6 system will again be used as an example. The guest molecules cause the mixing of the lowest (r = 0) wave function with three other wave functions derived from p = 1, p = 2, and p — 3, as described in the secular equation (14). If cx,. . ., c5 are the coefficients of the basis functions in order of increasing energy in the perturbed lowest state, we have, by perturbation theory for small a,... 

We have already seen examples of semiempirical methods, in Chapter 4 the simple Hiickel method (SHM, Erich Hiickel, ca. 1931) and the extended Hiickel method (EHM, Roald Hoffmann, 1963). These are semiempirical ( semi-experimental ) because they combine physical theory with experiment. Both methods start with the Schrodinger equation (theory) and derive from this a set of secular equations which may be solved for energy levels and molecular orbital coefficients (most efficiently... 

Differentiating Equation (2.29) and zeroing each partial derivative of E with respect to cv we obtain the secular equations ... 

Diamond137 derives the secular equations in symmetrized form. 

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