Bracketing theorem

The secular problem, in either form, has as many eigenvalues Ei and eigenvectors Cij as the dimension of the Hu matrix as . It can also be shown that between successive pairs of the eigenvalues obtained by solving the secular problem at least one exact eigenvalue must occur (i.e., Ei+i > Egxact > Ei, for all i). This observation is referred to as the bracketing theorem. ... 

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). 

Illustrate the bracketing theorem mentioned in connection with the augmented Hessian method, by solving equation (4 31) for the energy E in the form E = f(E). Plot both the functions E and f(E) and show that the crossing points (the eigenvalues Ej) satisfies the betweenness condition. 

The discussion of the various aspects of the MCSCF method requires the discussion of some background material. In this section, some of the elementary concepts of linear algebra are introduced. These concepts, which include the bracketing theorem for matrix eigenvalue equations, are used to define the MCSCF method and to discuss the MCSCF model for ground states and excited states. The details of V-electron expansion space represent-... 

An important application of the bracketing theorem is the case where the M matrix corresponds to a matrix representation of the Hamiltonian operator in... 

Fig. 1. Graphical example of ihe bracketing theorem. The vertical asymptotes are the eigenvalues of the matrix M. The horizontal asymptote is the diagonal element B of the full matrix. The intersections of the branches of the function L(A) with the straight line R(A) are the eigenvalues of the full matrix. These intersections satisfy the bracketing theorem relations...

Fig. 2. Schematic representation of the lowest eigenvalues of the Hamiltonian matrix as a function of the wavefunction expansion length. As shown in Fig. 1, the eigenvalues for two successive dimensions satisfy the bracketing theorem ordering, given by ( + 1) ( l (> + 11 ( ). .. j"+ j+V < ...

CSFs into the wavefunction expansion. Although unattainable in molecular calculations, the second limiting case, corresponding to full Cl for a complete orbital set, is called the complete Cl expansion s. The eigenvalues of the complete Cl expansion are the exact energies within the clamped-atomic-nucleus Born-Oppenheimer approximation. A correspondence may then be established with the bracketing theorem between the lowest eigenvalues of a limited CSF expansion and those of the exact complete Cl expansion. This is illustrated schematically in Fig. 2. 

These bracketing theorem considerations allow an ideal MCSCF method to be defined for excited states. This ideal method is first to define the lower-energy states to be optimal as determined by separate MCSCF calculation on these states, and then to define the energy of the excited state to be the appropriate Hamiltonian matrix eigenvalue obtained from a wavefunction expansion that includes these lower-energy states as expansion terms. 

The basic concepts necessary to the development of the MCSCF method have been introduced. These concepts include the bracketing theorem, which... 

The solution of the Schrddinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an altemative insight in the mathematical structure of perturbation theory, particularly with respect to the bracketing theorem and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately. 

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