The Linear Variational Problem

Given a trial function 0 , which depends on a set of parameters, then the expectation value 0. 0 will be a function of these parameters. In general, it will be such a complicated function that there is no simple way 

To show this we assume that the basis functions are real and orthonormal 

The case where the basis functions are complex and not orthogonal will be considered in Chapter 3. The matrix representation of the Hamiltonian operator in the basis is an N x N matrix H with elements 

Since the Hamiltonian is Hermitian and the basis is real, H is symmetric, i.e., Hij = Hji. The trial function is normalized, so that 

Our problem is to find the set of parameters for which 0. 0 is a minimum. Unfortunately, we cannot simply solve the equations 

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

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... 

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

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. 

Caleulations that employ the linear variational prineiple ean be viewed as those that obtain the exaet solution to an approximate problem. The problem is approximate beeause the basis neeessarily ehosen for praetieal ealeulations is not suffieiently flexible to deseribe the exaet states of the quantnm-meehanieal system. Nevertheless, within this finite basis, the problem is indeed solved exaetly the variational prineiple provides a reeipe to obtain the best possible solution in the space spanned by the basis functions. In this seetion, a somewhat different approaeh is taken for obtaining approximate solutions to the Selirodinger equation. 

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.

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

Among the classes of the trial wave functions, those employing the form of the linear combination of the functions taken from some predefined basis set lead to the most powerful technique known as the linear variational method. It is constructed as follows. First a set of M normalized functions dy, each satisfying the boundary conditions of the problem, is selected. The functions dy are called the basis functions of the problem. They must be chosen to be linearly independent. However we do not assume that the set of fdy is complete so that any T can be exactly represented as an expansion over it (in contrast with exact expansion eq. (1.36)) neither is it assumed that the functions of the basis set are orthogonal. A priori they do not have any relation to the Hamiltonian under study - only boundary conditions must be fulfilled. Then the trial wave function (D is taken as a linear combination of the basis functions dyy... 

The most direct way to represent the electronic structure is to refer to the electronic wave function dependent on the coordinates and spin projections of N electrons. To apply the linear variational method in this context one has to introduce the complete set of basis functions k for this problem. The complication is to guarantee the necessary symmetry properties (antisymmetry under transpositions of the sets of coordinates referring to any two electrons). This is done as follows. 

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... 

No matter how defined, restricting our A a-like solutions to be the lowest-energy solutions, FONs never occur. Instead, for the case of near-degeneracy, such solutions will involve orbitals that are linear combinations of symmetry-distinct orbitals with complex coefficients. With such extensions we can use integral occupation numbers and employ finite differences to evaluate derivatives in Parr s exact HK-based thermodynamic theory of the electron gas. The HK variational problem... 

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. 

The linear variational method is a useful and accurate tool for solving problems in different fields, such as quantum chemistry, molecular physics and solid state physics, among others [200]. 

Most of the studies cited are based on the linearity or the availability of analytical solutions for the systems considered. Many engineering problems, mostly in the chemical engineering field, are non-linear and results must be obtained through computations. Several numerical studies have been reported in the literature on distributed systems. One of the early computational results on distributed optimization was given by Denn ei al. (1966). The solution of the linearized variational equations by Green s function that leads to both the necessary conditions and computational scheme based on the method of steepest ascent, was obtained. The computational difficulties due to the discretization scheme of the catalyst decay problem was overcome by Jackson (1967). Computational results based on steepest ascent for the optimal control for a non-linear distributed systems were also reported by Denn (1966). 

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]. 

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 ... 

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. 

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