Perturbation theory partitioning techniques

W Another type of energy partitioning in MO theory was proposed by Clementi 86>. This kind of partitioning called Bond Energy Analysis" (BEA) was found to be particularly useful in larger molecular clusters 87-90) Since the individual contributions of BEA are not directly related to the quantities discussed in intermolecular perturbation theory, dementi s technique will not be used here (see Chapter V). 

P. O. Lowdin. Partitioning technique, perturbation theory, and rational approximations. Intern. J. Quantum Chem., 21 69, 1982. 

The theory of the Bk method [22] is based on the partitioning technique in perturbation theory [23, 24]. Suppose the Hamiltonian matrix H of the MR-CI space is partitioned as... 

The well-established perturbation theory of intermolecular interaction [53 59] can be applied to hydrogen-bonded systems in combination with the frozen molecule approximation, when the interaction is either sufficiently weak [60 62], or when the interaction is treated at a more qualitative level. When the interaction becomes larger, structural relaxations become sizable. Then the more usual approach to treat the hydrogen-bonded complex or cluster as a supermolecule becomes more practical and also more appropriate. However, also in this case, the detailed analysis of the interaction energy is often done with the aid of different variants of energy partitioning techniques [63,64] which closely follow the lines of intermolecular perturbation theory. 

One should simply mention briefly the application of the effective Hamiltonian approaches which use them as technical tools to solve numerically complex problems. The uses of partitioning techniques and of quasidegenerate perturbation theory are especially frequent in solving the configuration-interaction (Cl) problem in molecular physics. 

Studies in Perturbation Theory. I. An Elementary Iteration-Variation Procedure for Solving the Schrodinger Equation by Partitioning technique... 

The Calculation of Upper and Lower Bounds of Energy Eigenvalues in Perturbation Theory by means of Partitioning Technique... 

Studies in Perturbation Theory. XIII. Treatment of Constants of the Motion in Resolvent Method, Partitioning Technique, and Pertubation Theory... 

An important feature of modern perturbation theory is the so-called partitioning technique. In this procedure, the functional space for the wave function is separated into two parts, a model space and an orthogonal space. 

In the second section of this chapter, we shall employ the partitioning technique to develop various types of perturbation theory, including Rayleigh-Schrddinger perturbation theory and Brillouin-Wigner perturbation theory. This involves the expansion of the inverse operators which occur in the effective Hamiltonian operator and other operators obtained by the partitioning technique. Different expansions lead to different types of perturbation theory. 

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. 

The eigenvalue problem Hit = Et in quantum theory is conveniently studied by means of the partitioning technique. It is shown that, if is a real variable, one may construct a function i = /( ) such that each pair and i always bracket at least one true eigenvalue E. If is chosen as an upper bound by means of, e.g., the variation principle, the function i is hence going to provide a lower bound. The reaction operator t associated with the perturbation problem H = Ho + V... 

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