Coupled-cluster perturbation theory

In conclusion, it is possible to restructure the M0ller-Plesset energy and wave-function corrections such that their separability and size-extensivity become apparent. However, the separability of the corrections is not obvious in the original formulation of M0ller-Plesset theory but becomes transparent only when commutators are introduced. In Section 14.3, we shall develop coupled-cluster perturbation theory, where the connected (termwise size-extensive) commutator form arises naturally, without the need to restructure the expressions by hand. 

The zero-order wave functions are the Hartree-Fock state and the excited determinants (14.3.3). In coupled-cluster perturbation theory (CCPT), we expand the full coupled-cluster wave function... 

The accurate calculation of these molecular properties requires the use of ab initio methods, which have increased enormously in accuracy and efficiency in the last three decades. Ab initio methods have developed in two directions first, the level of approximation has become increasingly sophisticated and, hence, accurate. The earliest ab initio calculations used the Hartree-Fock/self-consistent field (HF/SCF) methodology, which is the simplest to implement. Subsequently, such methods as Mpller-Plesset perturbation theory, multi-configuration self-consistent field theory (MCSCF) and coupled-cluster (CC) theory have been developed and implemented. Relatively recently, density functional theory (DFT) has become the method of choice since it yields an accuracy much greater than that of HF/SCF while requiring relatively little additional computational effort. 

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

The main problem connected with the use of MP theory (as well as with other perturbation theories) concerns the convergency of the expansion. A related problem is connected with the truncation of the MP expansion is it possible to truncate it after the fourth order or at some higher order A practical solution to this problem is the coupled-cluster (CC) theory [2]. 

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. 

The ab initio HF calculations reported below have been performed with the GAUSSIAN 76 [26] program package. The atomic basis sets applied are a minimal (STO-3G [26]) one, a split valence (6-31G [26]) one, a split-valence one plus a set of five d-functions on carbon (6-31G [26]), and one with an additional set of p-functions on hydrogen (6-31G [26]). The correlation energy has been computed using Mpller-Plesset many body perturbation theory of second order (MP2) [27], the linear approximation of Coupled Cluster Doubles theory (L-CCD)... 

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the perturbation expansion... 

The failing of MBPT is that it is basically an order-by-order perturbation approach. For difficult correlation problems it is frequently necessary to go to high orders. This will be the case particularly when the single determinant reference function offers a poor approximation for the state of interest, as illustrated by the foregoing examples at 2.0 R. A practical solution to this problem is coupled-cluster (CC) theory. In fact, CC theory simplifies the whole concept of extensive methods and the linked-diagram theorem into one very simple statement the exponential wavefunction ansatz. 

It is now well established by numerous and extensive applications that the single reference (SR) based many-body methods, viz. many-body perturbation theory (PT) [1], coupled cluster (CC) theory [2], coupled electron-pair approximations (CEPA) [3], etc. provide rather accurate descriptions of the energy in and around the equilibrium geometry of the closed-shell states. In particular, the single reference coupled cluster (SRCC)... 

They employed ab initio calculations such as the nth order many-fcody perturbation theory, abbreviated to MBPT(n), whose better known variant is the nth order Mpller-Plesset perturbation theory, MPn. The MBPT(n) procedure takes account of all the terms to a predetermined final order n for the electron correlation. In the coupled cluster (CC) theory, on the other hand, certain contributions from the MBPT formalism are summated for all orders... 

One of us [1] reviewed the situation of electron correlation a quarter of a century ago in a paper with the title electron correlation in the seventies [2]. At that time most quantum chemists did not care about electron correlation, and standard methods for the large scale treatment of electron correlation, like Mpller-Plesset (MP) perturbation theory or coupled-cluster (CC) theory were not yet available. However precursors of these methods such as lEPA (independent electron pair approximation) and CEPA (coupled-electron-pair approximation) had already been developped and were being used, mainly in research groups in Germany [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. 

Complete Active Space (CAS) 141 configuration interaction singles (CIS) 89, 93-95 coupled cluster (CC) theory 140, 142 coupled perturbed Hartree-Fock (CPHF) 19 coupled perturbed Kohn-Sham (CPKS) 19 CPCM 9... 

Today we know that the HF method gives a very precise description of the electronic structure for most closed-shell molecules in their ground electronic state. The molecular structure and physical properties can be computed with only small errors. The electron density is well described. The HF wave function is also used as a reference in treatments of electron correlation, such as perturbation theory (MP2), configuration interaction (Cl), coupled-cluster (CC) theory, etc. Many semi-empirical procedures, such as CNDO, INDO, the Pariser-Parr-Pople method for rr-eleetron systems, ete. are based on the HF method. Density functional theory (DFT) can be considered as HF theory that includes a semiempirical estimate of the correlation error. The HF theory is the basie building block in modern quantum chemistry, and the basic entity in HF theory is the moleeular orbital. 

Although a wide variety of theoretical methods is available to study weak noncovalent interactions such as hydrogen bonding or dispersion forces between molecules (and/or atoms), this chapter focuses on size consistent electronic structure techniques likely to be employed by researchers new to the field of computational chemistry. Not stuprisingly, the list of popular electronic structure techniques includes the self-consistent field (SCF) Hartree-Fock method as well as popular implementations of density functional theory (DFT). However, correlated wave function theory (WFT) methods are often required to obtain accmate structures and energetics for weakly bound clusters, and the most useful of these WFT techniques tend to be based on many-body perturbation theory (MBPT) (specifically, Moller-Plesset perturbation theory), quadratic configuration interaction (QCI) theory, and coupled-cluster (CC) theory. 

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