Infinite-order perturbation method

The second general approach to correlation theory, also based on perturbation theory, is the coupled-cluster method, which can be thought of as an infinite-order perturbation method. The coupled-cluster wave function T cc is expressed as a power series,... 

If a single-configuration, zero-order wavefunction Po is a poor approximation to the correct wavefunction, order-by-order MBPT may converge very slowly, and going to high orders of perturbation theory may not be practical. In an effort to overcome this problem, the so-called coupled-cluster (CC) approach was developed. This method, which was reviewed in detail in Volume 5 of this series, can be thought of as an infinite-order perturbation method. 

After the lOTC method has been formulated, the so called infinite order DKHn method has been also defined [11-13]. The DKHn approximation is the generalization of the original DKH theory which enables to achieve the higher orders of the Hamiltonian. Unfortunately, the necessity to formulate the analytical form of the Hamiltonian in each order of perturbation V is still the basis of the DKHn method. The order of the DKHn approximation must be defined prior to any quantum-chemical calculations. The DKHn method is very well defined but it is only the approximation of the lOTC method which is exact. 

Perturbation methods add all types of corrections (S, D, T, Q etc.) to the reference wave function to a given order (2, 3, 4 etc.). The idea in Coupled Cluster (CC) methods is to include all corrections of a given type to infinite order. The (intermediate normalized) coupled cluster wave function is written as... 

A comparison between perturbation theoretic and truncated Cl methods is difficult, because the latter include some terms effectively to infinite order, but obviously omit some terms in lower orders. A pragmatist who prefers to do perturbation theory until it converges is likely to prefer infinite-order schemes, while a purist may prefer to obtain a result that is exact through a finite order in perturbation theory. 

It is appropriate at this point to compare some formal properties of the three general approaches to dynamical correlation that we have introduced configuration interaction, perturbation theory, and the coupled-cluster approach. First, we note that taken fax enough (all degrees of excitation in Cl and CC, infinite order of perturbation theory) all three approaches will give the same answer. Indeed, in a complete one-paxticle basis all three will then give the exact answer. We axe concerned in this section with the properties of truncated Cl and CC methods and finite-order perturbation theory. 

One can avoid these problems by using Coupled Cluster (CC) theory107, which contains infinite-order effects and therefore does not lead to the oscillatory behaviour of properties calculated with MPn108. Homoaromatic stabilization energies have been calculated for smaller molecules with CCSD(T) or QCISD(T)54 56. These are CC methods, which cover S and D excitations and, in addition, include T effects in a perturbational way109110. They represent some of the most accurate single determinant ab initio methods available today that can be applied in a routine way. 

The relationship between the coupled-electron pair approximation (c.e.p.a.) and the many-body perturbation theory has been discussed in detail by Ahl-richs.149 All of the methods denoted by c.e.p.a. (x) (x=0, 1, 2, 3) may be related to the summation of certain classes of diagrams in the many-body perturbation theory to infinite order. For example, c.e.p.a. (0), which is Cizek s linear approximation or Hurley s c.p.a. (0) ansatz150 is equivalent to the summation of all double-excitation linked diagrams in the perturbation series. This is also denoted d.e.m.b.p.t. (double excitation many-body perturbation theory) by some workers.151 168... 

Various semi-empirical methods have been compared for all properties in an important review by Klopman and O Leary, whilst Adams et < /. have compared the finite field, the variation, and the second-order infinite sum methods for the calculation of a in DNA bases. They find that the variation-perturbation method gives the most reliable results, but as their calculations were at the iterative extended Hiickel level there is no guarantee as to the generality of their conclusions. 

It is well known that electron correlation plays a key role in understanding the most interesting phenomena in molecules. It has been the focal point of atomic and molecular theory for many years [1] and various correlated methods have been developed [2]. Among them are many-body perturbation theory [3] (MBPT) and its infinite-order generalization, coupled cluster (CC) theory [4,5], which provides a systematic way to obtain the essential effects of correlation. Propagator [6-9] or Green s function methods (GFM) [10-14] provide another correlated tool to calculate the electron correlation corrections to ionization potentials (IPs), electron affinites (EAs), and electronic excitations. 

Foundations to the CC methods were laid by Coester and Kuemmel,1 Cizek,2 Hubbard,3 Sinanoglu,4 and Primas,5 while Cizek2 first presented the CC equations in explicit form. Also Hubbard3 called attention to the equivalence of CC methods and infinite-order many-body perturbation theory (MBPT) methods. From this latter viewpoint, the CC method is a device to sum to infinity certain classes of MBPT diagrams or all possible MBPT diagrams when the full set of coupled-cluster equations is solved. The latter possibility would require solving a series of coupled equations involving up to IV-fold excitations for N electrons. Practical applications require the truncation of the cluster operators to low N values. 

Thus the double perturbation theory enables to discuss the effects of electron correlation on molecular properties systematically, but there are not many numerical calculations proceeding along these lines instead of calculating the effect of /I2 by veuriation methods and subsequently dealing with d 1 by ordinary perturbation theory. The former procedure has some advantages, for, in principle, it should include ZI2 to infinite order (in other words, A1 and A 2 might enter on different levels of the perturbation treatment). 

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