Linear variational problem

Figure 1.2. A qualitative graph showing schematically the interleaving of the eigenvalues for a series of linear variation problems for n = 1,..., 5. The ordinate is energy.

NB We assumed this not to happen in our discussion above of the convergence in the linear variation problem. 

In its simplest form the energy optimization is a linear variation problem. 

In linear variational problems, one way of satisfying Hurley s conditions is to make the basis set closed with respect to the differential operators d/dp. Such a basis set is in principle infinite. Practically, however, the Hellmann-Feynman theorem will be approximately satisfied if, for each significantly populated basis function x, its derivatives with respect to the orbital centers, X, x are included in the basis set (Pulay, 1969). The use of augmented basis sets in conjunction with the Hellmann-Feynman theorem was considered by Pulay (1969, 1977) but dismissed as expensive. Recently, Nakatsuji et al. (1982) have recommended such a procedure. However, an analysis of their procedure (Pulay, 1983c Nakatsuji et al., 1983) reveals that it is not competitive with the traditional gradient technique. Much of the error in the Hellmann-Feynman forces is due to core orbitals. Therefore, methods based on the Hellmann-Feynman theorem presumably work better for effective core... 

The requirements (26) and (27) can be met in a simple and practical way by requiring the ipp to be solutions of a linear variation problem with matrix elements determined by integration over the z alone, for each and every value assigned to q. 

Given a set of basis states, excited eigenstates can be computed varia-tionally by solving a linear variational problem, and the Metropolis method can be used to evaluate the required matrix elements. The methods involving the power method, as described above, can then be used to remove the variational bias systematically [13,19,20]. 

One seeks a solution to the linear variational problem in Eq. (3.34) in the sense that for all i the Rayleigh quotient < , >/< > is stationary with respect to variation of the coefficients d. The solution is that the matrix of coefficients d has to satisfy the following generalized eigenvalue equation ... 

However, if the component orbitals of are the SCF orbitals in the given basis they already are the best possible orbitals of a single determinant. Thus there can be no improvement of an SCF single determinant by the addition of single-excitation determinants. If we recall the form of the linear variation problem given in Chapter 1, it is clear that this result implies that all integrals of the form... 

If we take an optimum single-determinant wavefunction and remove one electron from each one of a chosen set of occupied orbitals we can generate n different (n — l)-electron single-determinant wavefunctions, and these wavefunc-tions should beat some relationship to the states of the corresponding molecular ion. In fact, they can be taken as a basis for the linear expansion of the states of the ion, and the diagonalisation of the associated linear variation problem would give the best approximation to these states available with this restricted class of function. 

This is simply a standard linear variation problem to which we have the solutions from Chapter 1 a single matrix equation involving the matrix of H the full many-electron Hamiltonian operator and the overlap matrix ... 

In this case the linear variational problem would be generated by the ratio of singlet spin functions, determining D and D in... 

This is not quite true since, when an expansion method is used, there is always an empty shell . If there were no empty shell i.e. if the number of basis functions were not greater than the number of MOs) there would be no linear variational problem to solve. 

We will illustrate the variational technique by rederiving the matrix eigenvalue equation of the linear variational problem given in Subsection 1.3.2. Given a linear variational trial wave function. 

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