Coupled cluster method systems

The final topic we will discuss in this chapter is size-consistency, which has been mentioned several times already. A method is said to be size-consistent if the computed energy of the composite system A + B, with A and B at infinite distance from each other, yields the same energy as if the method is applied to A and B separately and the energies axe added, i.e. E(A+B)=E(A)+E(B). Some of the methods we have discussed are automatically size-consistent. This is true, for example, for the Hartree-Fock method and the complete Cl method, and it is also true for the methods discussed in the chapter on perturbation theory, such as the coupled cluster method. It is, however, not true for the SD-CI or the MR-CI method. We will in this section show that it is possible, by a slight modification of the formalism, to correct these Cl methods to be approximately size-consistent. The experience gathered over the past two decades on size-consistency corrections indicates that the calculated results axe much improved at the SD-CI level, whereas relative energies axe improved at the MR-CI level but the situation for geometries is less clear at this level. 

In order to get more detailed information about, e.g., bond strengths and equilibrium geometries in transition metal systems it is necessary to include electron correlation. This can be done either by traditional ab initio quantum chemistry models, e.g., Cl-methods and coupled cluster methods, or by density functional theory (DFT) based methods. Correlated ab initio methods are often computationally very demanding, especially in cases where multi-reference based treatments are needed. Also, the computational cost of these methods increases dramatically with the size of the system. This implies that they can only be applied to rather small systems. 

First let us review static and dynamic electron correlation. Dynamic (dynamical) electron correlation is easy to grasp, if not so easy to treat exhaustively. It is simply the adjustment by each electron, at each moment, of its motion in accordance with its interaction with each other electron in the system. Dynamic correlation and its treatment with perturbation (Mpller-Plesset), configuration interaction, and coupled cluster methods was covered in Section 5.4. 

Our present focus is on density functional theory and coupled cluster methods for describing molecular systems interacting with a structured environment, and we focus on the derivation of linear response properties and compare the expressions that we obtain for the two different electronic structure methods. Based on linear response... 

In the following we want to focus on some problems which arise if ab initio methods are used to calculate isotropic hfcc s. We will mostly concentrate on approaches where the Configuration Interaction (Cl) method is used in various versions. To illustrate the performance of other theoretical methods such as M0ller-Plesset perturbation theory (MP), Coupled Cluster methods (CC) or quadratic Cl (QCISD), the results obtained with those approaches will be compared for a few model systems. Because an understanding of the influences... 

Stanton JF, Gauss J (1994) Analytic energy gradients for the equation-of-motion coupled-cluster method Implementation and application to the HCN/HNC system. J Chem Phys 100 4695 1698. 

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

The full configuration interaction method [34-36] is exact in the sense that after choosing appropriate atomic basis functions (defining the model in this way), the resulting many-electron wavefunction is an exact eigenfunction of the model Hamiltonian, the computational effort, nevertheless, increases in an exponential manner. Truncation of the full Cl expansion (especially after single and double excitations, CI-SD) considerably reduces the necessary computational resources, but leads unfortunately to the serious problem of nonsize-consistency [37, 38] which makes the results even for medium systems unrealistic. The coupled-cluster method [39, 40] theoretically properly describes extended systems as well, but numerous experiences show the enormous increase of computational work with the size of the system. 

In the Fock space coupled cluster method, the Hartree-Fock solution for an iV-electron state, 0), is used as the vacuum. The Fock space is divided into sectors, (m,n), according to how many electrons are added to and removed from 0). Thus, the vacuum is in the (0,0) sector, single ionizations are in the (0,1) sector, one-electron attached states are in (1,0), and (1,1) are single excitations relative to 0). The orbitals are also divided into active, which can change occupation, and inactive, for which the occupation is fixed. All possible occupations of the active orbitals in all possible sectors constitute the multireference space for the system. 

Another major point in which the theoretical methods differ is the quantum chemical approach to solve the operator equation of the Hamilton operator itself. The most important schemes are Hartree-Fock self consistent field (HF-SCF), density functional theory (DFT) and multi-body second order perturbation theory (MP2). Different combinations have been established, so for instance GIAO-SCF, GlAO-DI I, (,IA()-MI>2, or DF I-IGLO. Most precise measurements on small systems were done with coupled cluster methods, as for instance GIAO-CCSDT-n. ... 

Electronic correlation in extended systems remains a central problem despite impressive progress in recent years. For small systems a number of very powerful methods have reached a high degree of accuracy thanks to a combination of formal algebraic and numerical techniques. These include configuration interaction,1-5 propagator methods,2,4 5 many-body perturbation procedures,3-5 and coupled-cluster methods.4 For extended systems density functional methods6,7 dominate the scene. Certain forms of correlation are taken into account by such methods, but how and to what extent are still unclear.8... 

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