New eigenvalues

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). 

By way of illustration it will be assumed that Ek is non-degenerate and AH is small enough to ensure that the perturbed energy level k is closer to Ek than to any other unperturbed level. The new eigenvalue problem is... 

An important feature of the solutions of eq. (5-6) is that each new eigenvalue, En, is located between a pair of zero-order, Born-Oppenheimer... 

Since the transformed operator H is no longer self-adjoint, one may expect that it may have also complex eigenvalues E which lack counterparts in the spectrum of the original Hamiltonian and which may hence be described as new eigenvalues. In such a case, we will consider the transformation... 

Even if the conditions for persistent, lost, and new eigenvalues are completely clear for the exact eigenvalue problems to the operators T and Tt, it is considerably more difficult to translate them to the approximate eigenvalue problems associated with the application of the bi-variational principle for the operators T and Tt to truncated basis sets. In this connection, the relations (A. 1.40-1.49) may turn out to be useful in formulating the problem. Some of the computational aspects, particularly the choice of the dual basis sets, are further discussed in reference A. 

Similarly, if the approximate eigenvalue Iu would correspond to a new eigenvalue of the operator Tu, one would expect that the associated Slater determinant Cua - as well as Dua -would belong to L2, whereas this would not be the case for the transformed determinant Ca = U 1Cua. In such a case, at least one of the functions y must be situated in the complement to... 

We thus transform Eq. (33) with an operator eTt. We multiply both sides of Eq. (33) on the left by eT, insert e r eT = 1 into the operator product HnRk and replace ) by e T 4>) (this is possible since T and its positive powers annihilate 4>)). As a result, we obtain a new eigenvalue problem to deal with, namely,... 

Written in matrix-form, the new eigenvalue equation has a simple structure, with all r-dependent parts in the off-diagonal of the matrix ... 

We have obtained a new eigenvalue equation with a simpler matrix. The price we paid is that the new matrix depends on the old eigenvalue e, which has to be determined. The number e now appears as a parameter in the matrix that has to be diagonalized. We have to find those values of the parameter, for which the equation (123) admits square-integrable solutions. 

Note that the new eigenvalues are additive in the sense that... 

To test the accuracy of the eigenvalues found, we can repeat the calculation with half as large and see if the new eigenvalues differ significantly from those found with the larger s. Also, we can start further into the classically forbidden region. 

The multiplication constant and perturbation. The new eigenvalue quantity, here called the multiplication constant k, is often called the effective multiplication constant and is called the static criticality factor, (7, by Weinberg and Wigner. It is the largest real eigenvalue of the criticality equation... 

Assuming the new eigenvalue is arbitrarily selected as -2, gives the new characteristic equation... 

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