Electron Correlation---Post-Hartree-Fock Methods

2 ELECTRON CORRELATION—POST-HARTREE-FOCK METHODS 

The HF method ignores instantaneous electron-electron repulsion, also known as electron correlation. The electron correlation energy is defined as the difference between the exact energy and the energy at the HF limit 

How can we include electron correlation Suppose the total electron wavefunc-tion is composed of a linear combination of functions that depend on all n electrons 

We can then solve the Schrodinger equation with the full Hamiltonian (Eq. (1.5)) by varying the coefficients c, so as to minimize the energy. If the summation is over an infinite set of these N-electron functions, i, , we will obtain the exact energy. If, as is more practical, some finite set of functions is used, the variational principle tells us that the energy so computed will be above the exact energy. 

The HF wavefunction is an A-electron function (itself composed of one-electron functions—the MOs). It seems reasonable to generate a set of functions from the HF wavefunction sometimes called the reference configuration. 

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The application of density functional theory (DFT) to the study of the structure and reactivity of some molecules with unpaired electrons (radicals) performed by our group is presented. The results describe the application of LSD, gradient corrected and hybrid DFT methods to several small molecules. On average the results are as good as highly-correlated post-Hartree-Fock methods, but still some problems remain to be solved... 

The Amsterdam Density Functional package (ADF) is software for first-principles electronic structure calculations (quantum chemistry). ADF is often used in the research areas of catalysis, inorganic and heavy-element chemistry, biochemistry, and various types of spectroscopy. ADF is based on density functional theory (DFT) (see Chapter 2.39), which has dominated quantum chemistry applications since the early 1990s. DFT gives superior accuracy to Hartree-Fock theory and semi-empirical approaches, especially for transition-metal compounds. In contrast to conventional correlated post-Hartree-Fock methods, it enables accurate treatment of systems with several hundreds of atoms (or several thousands with QM/MM)." ... 

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

Although HF theory is useful in its own right for many kinds of investigations, there are some applications for which the neglect of electron correlation or the assumption that the error is constant (and so will cancel) is not warranted. Post-Hartree-Fock methods seek to improve the description of the electron-electron interactions using HF theory as a reference point. Improvements to HF theory can be made in a variety of ways, including the method of configuration interaction (Cl) and by use of many-body perturbation theory (MBPT). It is beyond the scope of this text to treat Cl and MBPT methods in any but the most cursory manner. However, both methods can be introduced from aspects of the theory already discussed. 

Keywords many-electron correlation problem, post-Hartree-Fock methods, coupled cluster approaches, configuration interaction, externally corrected coupled cluster methods, reduced multireference coupled cluster method... 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

SM calculations are broadly based on either the (i) Hartree-Fock method (ii) Post-Hartree-Fock methods like the Mpller-Plesset level of theory (MP), configuration interaction (Cl), complete active space self-consistent field (CASSCF), coupled cluster singles and doubles (CCSD) or (iii) methods based on DFT [24-27]. Since the inclusion of electron correlation is vital to obtain an accurate description of nearly all the calculated properties, it is desirable that SM calculations are carried out at either the second-order Mpller-Plesset (MP2) or the coupled cluster with single, double, and perturbative triple substitutions (CCSD(T)) levels using basis sets composed of both diffuse and polarization functions. 

This is not the place for a full overview of the wave function based post Hartree-Fock methods currently applied for the calculation of intermolecular interactions and in particular molecule/surface interactions. Table 3 contains a brief characterization of the most widely applied schemes. The two most popular methods are MP2 (second order Moller-Plesset perturbation theory), because it covers large part of electronic correlation at comparably low ex-... 

The application of ab initio methods in the calculation of harmonic force fields of transition metal complexes has been hampered by the size of these systems and the need to employ costly post-Hartree-Fock methods, in which electron correlation is taken into account. Thus, the fruitful symbiosis between ab initio theory and experiment, to determine empirically scaled quantum mechanical force fields, has been virtually absent in studies of transition metal complexes. 

Theoretical description of the noble gas interaction requires quite advanced computational techniques, because here the binding effect comes from the dispersion interaction, which represents an electronic correlation effect. Such an effect is inaccessible in Hartree-Fock calculations. Some very expensive post-Hartree-Fock methods have to be used. The larger the number of electrons N), the more expensive the calculations quickly become as N increases (as we have seen in Chapter 10) proportionally to for the MP2 method, and even as N for the CCSD(T) method. Therefore, whereas He2 CCSD(T) calculations would take a minute, similar Xe2 calculations would take about = 26 minutes, i.e. about 3000 years. No wonder, the xenon atom has 54 electrons, and in a system of 108 electrons there are plenty of events to correlate, but because of the 3000 years this... 

Electron Correlations in Molecules Post-Hartree-Fock Methods... 

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