Matrices secular equation

The variational coefficients, C,j, are obtained from the matrix secular equation... 

The eigenvalues E of It can be detemiined from the Hamiltonian matrix by solving the secular equation... 

In solving the secular equation it is important to know which of the off-diagonal matrix elements " I wanish since this will enable us to simplify the equation. 

The value of detennines how much computer time and memory is needed to solve the -dimensional Sj HjjCj= E Cj secular problem in the Cl and MCSCF metiiods. Solution of tliese matrix eigenvalue equations requires computer time that scales as (if few eigenvalues are computed) to A, (if most eigenvalues are... 

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... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

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). 

The matrix elements ot — Ej and p are not variables in the minimization procedure they are constants of the secular equations with units of energy. Note that all elements in the matrix and vector are real numbers. The vector is the set of coefficients for one eigenfunction corresponding to one eigenvalue, Ej. From Eq. (7-24),... 

These are just the secular equations shown in equation set (7-2) with F in place of H and the stacked matrix Eq. (7-6) of eigenvectors in place of a single eigenvector. In matrix notation... 

As there is one equation (4.5) for each i, the variational problem is transformed into solving a set of Cl secular equations. Introducing the notation Hjj = ( H ) the matrix equation becomes... 

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

This proof shows that any approximate wave function will have an energy above or equal to the exact ground-state energy. There is a related theorem, known as MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a Cl matrix) is an upper limit to the n — l)th excited exact state, within the given symmetry subclass. In other words, the lowest root obtained by diagonalizing a Cl matrix is an upper limit to the lowest exact wave functions, the 2nd root is an upper limit to the exact energy of the first excited state, the 3rd root is an upper limit to the exact second excited state and so on. 

There seems to be a certain complementarity between the degree of difficulty in evaluating HKL for various one-electron sets y)k and the order of the secular equation needed to obtain a certain accuracy in the result. The work carried out in getting extensive tables of molecular integrals has also been of essential value for facilitating the calculation of the matrix elements HKL. 

This is an eigenvalue problem of the form of Eq. III.45 referring to the truncated basis only, and the influence of the remainder set is seen by the additional term in the energy matrix. The relation III.48 corresponds to a solution of the secular equation by means of a modified perturbation theory,19 and the problem is complicated by the fact that the extra term in Eq. III.48 contains the energy parameter E, which leads to an iteration procedure. So far no one has investigated the remainder problem in detail, but Eq. III.48 certainly provides a good starting point. 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

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... 

SCF, see Self-consistent field treatment (SCF) Schroedinger equation, 2,4,74 Secular equations, 6,10, 52 solution by matrix diagonalization, 11 computer program for, 31-33 Self-consistent field treatment (SCF), of molecular orbitals, 28 Serine, structure of, 110 Serine proteases, 170-188. See also Subtilisin Trypsin enzyme family comparison of mechanisms for, 182-184, 183... 

The benzene molecule can now be treated very simply by the Slater method, with the help of the rules formulated by one of us4 for finding the matrix elements occurring in the secular equation. The bonds between the six eigenfunctions can be drawn so as to give the independent canonical structures shown in Fig. 1. Any other... 

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