The Secular Equation

All that being said, let us now turn to evaluating the energy of our guess wave function. From Eqs. (4.15) and (4.17) we have 

it is useful to keep in mind our objective. The variational principle instructs us that as we get closer and closer to the true one-electron ground-state wave function, we will obtain lower and lower energies from our guess. Thus, once wc have selected a basis set, we would like to choose the coefficients a, so as to minimize the energy for all possible linear combinations of our basis functions. From calculus, we know that a necessary condition for a function (i.e., the energy) to be at its minimum is that its derivatives with respect to all of its free variables (i.e., the coefficients a,) are zero. Notationally, that is 

This set of N equations (running over k) involves N unknowns (the individual a,). From linear algebra, we know drat a set of N equations in N unknowns has a non-trivial solution if and only if the determinant formed from the coefficients of die unknowns (in this case the coefficients are the various quantities H i — ES i) is equal to zero. Notationally again. 

Equation (4.21) is called a secular equation. In general, there will be N roots E which permit the secular equation to be true. That is, there will be N energies ) (some of which may be equal to one another, in which case we say the roots are degenerate ) where each value of Ej will give rise to a different set of coefficients, a/y, which can be found by solving the set of linear Eqs. (4.20) using Ej, and these coefficients will define an optimal wave function pj within the given basis set, i.e.. 

in a nutshell, to find die optimal one-electron wave functions for a molecular system, we  

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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. 

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 ... 

This is a pair of simultaneous equations in Ai and A2 called the secular equations... 

In what immediately follows, we will obtain eigenvalues i and 2 for //v / = Ei ) from the simultaneous equation set (6-38). Each eigenvalue gives a n-election energy for the model we used to generate the secular equation set. In the next chapter, we shall apply an additional equation of constr aint on the minimization parameters ai, 2 so as to obtain their unique solution set. 

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... 

With the approximation that 5 = 0, the secular equations of Equation (7.45) become... 

Equation (3-111) is the secular equation for this problem. [Pauling and Wilson discuss the meaning of the term secular in this context.] Expansion of the determinant gives a polynomial in X that is called the characteristic equation ... 

The roots of the characteristic equation are called the eigenvalues of the secular equation. 

For Scheme XVIII, find the secular equation, the characteristic equation, and the eigenvalues. 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

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. 

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... 

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. 

In order to solve the secular equation, we will now use the partitioning method discussed in Section III.D(lc). Choosing the subset (a) to contain only the SCF-determinant and using Eq. III.49, we obtain... 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

These rules follow directly from the quantum-mechanical theory of perturbations and the resolution of the secular equations for the orbital interaction problem. The (small) interaction between orbitals of significantly different energ is the familiar second order type interaction, where the interaction energy is small relative to the difference between EA and EB. The (large) interaction between orbitals of same energy is the familiar first order type interaction between degenerate or nearly degenerate levels. 

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