The Partitioning Technique

The Partitioning Technique.—Let P denote the projector onto some zero-order model wave function cP0 and Q its complement. The electronic Schrodinger equation 

Wigner, Math. u. naturw. Anz. ungar. Akad. Wiss., 1935,53,475. 

This effective hamiltonian has eigenfunctions in the model space but has the exact energy as an eigenvalue. 

Various forms of perturbation theory result from different expansions of the inverse in the effective hamiltonian using the identity 

0 denotes some zero-order hamiltonian and E0 its ground state eigenvalue then the perturbation series of Lennard-Jones,48 Brillouin,49 and Wigner50 is obtained by putting 

Abstract The partitioning technique is described in some detail. The concept of a model space is introduced and the wave operator and the reaction operator are defined. Using these ideas, both the Rayleigh-Schrodinger and the Brillouin-Wigner expansions are developed, first for the case of a single-reference function and then for the multi-reference case. 

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. 

The basic idea here is to And effective operators which act only within the 

These words were written by Lindgren and Morrison in their book Atomic Many-Body Theory [1] which was first published in 1982. 

In this section, we first consider the partitioning technique for the case of a singlereference function. We develop the basic apparatus of the partitioning technique. We define the model function and the associated projection operators. We formulate an effective Hamiltonian whose eigenfunctions lie in the model space, but whose eigenvalue is equal to that of the original Hamiltonian. We define the wave operator, which when applied to the model function yields the exact wave function, and the reaction 

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A central problem in physics and chemistry has always been the solution of the Schrodinger equation (SE) for stationary states. Such stationary states may relate to electronic structure problems, in which case one is primarily interested in bound states, or to scattering problems, in which case the stationary solutions are continuum states. In both cases, one of the most powerful tools in the theoretical arsenal for solving such problems is the partitioning technique (PT), which has been developed in a series of papers prominently by Per-Olov Lowdin [1-6] and Herman Feshbach [7-9]. 

The partitioning technique also nicely shows why a bridge, consisting of a molecule or a solvent, is a better mediator than empty space. In empty space the direct matrix element Hda is the only coupling between donor and acceptor. This matrix element decreases with P 4, which effectively forbids ET distances larger than 4-5 A. 

The partitioning technique developed by Lowdin provides a synthesis... 

The discussion of the operator - ( ) should have the preceding as a starting point, but for our purposes it is sufficient to have found an unperturbed operator with a partially discrete spectrum, with known solutions and therefore tractable by methods of the partitioning technique-... 

There exists another path of approaching the ZFS that is beyond the SH formalism but retains features of the model space. It is based upon the partitioning technique. 

This, however, is not the case when one restricts oneself to the first iteration only. The first iteration in the partitioning technique can be considered an improvement lying beyond the SH formalism. 

The differences in the energy levels of a model subspace when the partitioning technique is applied in the first iteration this involves diagonalization only inside the model space ... 

Returning to the partitioning technique formulation, see Appendices A and B, we recover the following modifications of the projection operator formulations... 

Formal expressions for the wave and reaction operators are easily derived by means of the partitioning technique,1 which leads to the formulas... 

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... 

Both alternatives can be useful depending on the particular problem, one of them can be more convenient than the other one. As we will see in Section III.D, the second version is particularly adapted to the construction of effective Hamiltonians by the partitioning technique. 

We develop the partitioning technique with the use of the iterative KAM perturbation algorithms. We derive an effective Hamiltonian of second order. The scheme we show can be easily extended to higher orders. 

In the literature a different technique has been widely used to construct effective Hamiltonians, based on the partitioning technique combined with an approximation procedure known as adiabatic elimination for the time-dependent Schrodinger equation (see Ref. 39, p. 1165). In this section we show that the effective Hamiltonian constructed by adiabatic elimination can be recovered from the above construction by choosing the reference of the energy appropriately. Moreover, our stationary formulation allows us to estimate the order of the neglected terms and to improve the approximation to higher orders in a systematic way. 

The estimates of the relative hydrophobicity of solutes obtained by measurements of their comparative solubility in water and in organic solvents are usually in agreement with those obtained by the partition technique 45,461 (see below). 

Some Aspects on the Bloch-Lindgren Equation and a Comparison with the Partitioning Technique. 

Resonances are currently observed in collision processes. We will show in this section, how they can be studied in the framework of the partitioning technique by means of the effective Hamiltonians discussed in Section 2. The theory is described below and illustrated by model applications in Section 3.2. 

Using the partition technique (see Appendix B) the transition operator can be expressed as... 

Another type of effective Hamiltonian has been proposed by Davidson and coworkers for the practical treatment of Cl problems in nearly degenerate cases, under the name shifted approximation (which refers to a version of the partitioning technique proposed by Gershgom et a/. and already discussed). To second order the effective Hamiltonian... 

The discussion may be formalized using the partitioning technique of Appendix 4. The transition state may be found by inspection of the LUMO and LUMO -E 1 in the electron case and the HOMO and HOMO - 1 in the hole case. According to Equation 10.20 MOs have the character c )i + ( )2 and c )i - (f)2 at the transition state. This form is easily checked out by inspection. For this geometry, the Fock matrix may be written as (see Appendix 4)... 

This system of equations is of infinite dimension therefore it can be solved by the partitioning technique. 

It is not the aim of this section to review scattering theory which can be found in textbooks, for instance, [1-3] and in review articles [10-12]. Here we shall only recall the expressions of the transition operator in the framework of the partition technique. In a second step we will use the model Hamiltonian of Part I to derive new exact expressions of the transition matrices. The Hilbert space is partitioned into the space of the resonances (the n-dimensional model space) and the space of the collision states (the infinite-dimensional complementary space). The projectors onto these two spaces are Pq and Qo respectively Pq + Qo = 1- The exact resolvent can be written in the form (see Part I and Ref. [13])... 

By the partition technique, the two ethylenic bonds may be regarded as independent components. Under photochemical condition we have to consider interaction of HOMO of excited part with the LUMO of unexcited part. Upper lobe of one is to overlap the lower lobe of other to give bicyclobutane by n s+n s cycloaddition. This cycloaddition is symmetry allowed under photochemical condition. 

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