The Electron Correlation Problem

Martinez T J and Carter E A 1995 Pseudospectral methods applied to the electron correlation problem Modem Electronic Structure Theory yo 2, ed D R Yarkony (Singapore World Scientific) pp 1132-65... 

Any method which goes beyond SCF in attempting to treat this phenomenon properly is known as an electron correlation method (despite the fact that Hartree-Fock theory does include some correlation effects) or a post-SCT method. We will look briefly at two different approaches to the electron correlation problem in this section. 

Evaluation in the Unitary Group Approach to the Electron Correlation Problem. 

However, in the late 1960s, fir Cfzek and Josef Paldus introduced a some what different approach to the electron correlation problem instead of using a linear expansion of functions as in Eq. (13.3) they suggested an exponential ansatz of the general form [7-9]... 

The electron correlation problem remains a central research area for quantum chemists, as its solution would provide the exact energies for arbitrary systems. Today there exist many procedures for calculating the electron correlation energy (/), none of which, unfortunately, is both robust and computationally inexpensive. Configuration interaction (Cl) methods provide a conceptually simple route to correlation energies and a full Cl calculation will provide exact energies but only at prohibitive computational cost as it scales factorially with the number of basis functions, N. Truncated Cl methods such as CISD (A cost) are more computationally feasible but can still only be used for small systems and are neither size consistent nor size extensive. Coupled cluster... 

Many-body effects play a leading role in the description of chemical phenomena, and there is little point advancing detailed relativistic theories which cannot treat the electron correlation problem. A major advan-... 

The third and final approach to the electron correlation problem included briefly here is density functional theory (DFT), a review of which has been given by Kohn in his Nobel lecture [38], The Hohcnberg Kolin theorem [39] states that there is a one-to-one mapping between the potential V(r) in which the electrons in a molecule move, the associated electron density p(r), and the ground state wave function lP0. A consequence of this is that given the density p(r), the potential and wave function lf 0 are functionals of that density. An additional theorem provided by Kohn and Sham [40] states that it is possible to construct an auxiliary reference system of non-interacting... 

The electron correlation problem occurs with all polyelectronic atoms. To treat these systems using the quantum mechanical model, we must make approximations. The simplest approximation involves treating each electron as if it were moving in a field of charge that is the net result of the nuclear attraction and the average repulsions of all the other electrons. To see how this is done, let s compare the neutral helium atom and the He+ ion ... 

However, even though it is formulated rather easily, this problem cannot be solved exactly. The difficulty is the same as that encountered in dealing with polyelectronic atoms—the electron correlation problem. Since we cannot account for the details of the electron movements, we cannot deal with the electron-electron interactions in a specific way. We need to make approximations that allow the solution of the problem but that do not destroy the model s physical integrity. The success of these approximations can be measured only by comparing predictions from the theory with experimental observations. In this case we will see that the simplified model works well. 

The MOLPRO quantum chemistry package has an emphasis is on highly accurate computations, with extensive treatment of the electron correlation problem . It is maintained by H.-J. Werner and P. J. Knowles, and contains contributions from a number of authors ... 

Of course, orbital models, such as the widely used Hartree-Fock approximation neglect the effects of electron correlation. One approximation which forms the basis of a computationally tractable approach to the electron correlation problem in atoms and molecules is the many-body perturbation... 

For treating the electron correlation problem many alternative approaches have appeared in the literature, but basically three main streams (and variants of them) can be distinguished. 

Perturbative approaches to the electron correlation problem have proved to be successful even when calculating second-order corrections only but finer results require fourth order energy corrections, as we will see later in this text. The reliability of a perturbation expansion greatly depends on the partitioning of the exact Hamiltonian H = H0 + W to an unperturbed part H0 and a perturbation W. Good quality approximations are to be expected if the N-particle operator H0 is chosen as the sum of equivalent one-particle Fock operators... 

Diagrammatic methods are well established in handling the electron correlation problem which arising in the description of molecular structure within the Bom-Oppenheimer approximation. In fact, for the relativistic electronic structure problem... 

A. J. W. Thom and A. Alavi (2005) A combinatorial approach to the electron correlation problem. J. Chem. Phys. in print... 

For nondegenerate systems, single reference coupled cluster (CC) methods have proven to be quite successful in their description of the electron correlation problem (1-3). This is primarily due to their (size) extensivity and ability to efficiently incorporate higher-order correlation effects—CC models including triple excitation effects are generally within 1-2 kcal mol 1 of full Cl (3). 

The CURES-EC method was named because it CURES the ELECTRON CORRELATION problem in semi-empirical calculations. This is accomplished by adding multiconfiguration configuration interaction (MCCI) to the wave functions. The acronym CURES-EC stands for configuration interaction or unrestricted orbitals to relate experimental electron affinities to self-consistent field values by estimating electron correlation. Electron affinities of radicals can be obtained from experimental gas phase acidities by calculating C—H bond dissociation... 

A different approach to the solution of the electron correlation problem comes from density functional theory (see Chapter 4). We hasten to add that in a certain approximation of relativistic density functional theory, which is also reviewed in this book, exchange and correlation functionals are taken to replace Dirac-Fock potentials in the SCF equations. Another approach, which we will not discuss here, is the direct perturbation method as developed by Rutkowski, Schwarz and Kutzelnigg (Kutzel-nigg 1989, 1990 Rutkowski 1986a,b,c Rutkowski and Schwarz 1990 Schwarz et al. 1991). 

Several reviews mainly devoted to the methodological and computational aspects of the electron correlation problem have appeared in recent years. An attempt to complete the picture by systematically analyzing the validity and significance of that problem was made during discussions among several quantum chemists. Their remarks and questions, together with the work and the references they presented to substantiate them, are reported here. 

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