Post-Hartree-Fock Treatments

In reality, elearons avoid each other better than can be described by a Hartree-Fock (HF) wavefunction. The ab initio total energy of an HF wavefunction is in error with respect to the true nonrelativistic energy by an amount called the correlation energy. A wavefunction that accounts for more of the elearon correlation gives a lower (better) total energy. There are two popular methods to account for electron correlation. To properly explain these requires a fuller development than is possible here. However, we wish to mention them in order to at least give a flavor of what is involved. The interested reader is direaed to excellent texts that describe details of the notation. 

The first approach is Moller-Plesset (MP) many-body perturbation theory. To the Hartree-Fock wavefunction is added a correction corresponding to exciting two electrons to higher energy Hartree-Fock MOs. Second-order, third-order, and fourth-order corrections to the Hartree-Fock total energy are designated MP2, MP3, and MP4, respectively. For double substitutions, i,j (occupied) into m,n (virtual), 

This method is size consistent, i.e., relative errors are more or less proportional to the size of the molecule. It is not variational, i.e., the computed total energy may not be an upper bound to the true energy. An MP2 treatment can double computer time requirements of a Hartree-Fock calculation and can overcorrect the correlation energy. MP3 and MP4 are several times more expensive in terms of computer time than MP2 and can correct back toward the true energy. 

This method is variational, but not size consistent. The latter means that it will not be good for computing hydrogen bond energies or other association energies. Cl is much more costly in computer time than the MP method. 

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In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

As the above narrative indicates, most of the ideas for the treatment of the many-electron problem were first developed by the nuclear and solid-state physicists. This is the case not only for perturbative methods, but also for variational ones, including the configuration interaction method, which nuclear physicists refer to as the shell model, or for the unitary group approach (see Ref. [90] for additional references see Refs. [23, 78-80]). The same applies to the CC approach [70]. For this reason, quantum chemists, who were involved in the development of post-Hartree-Fock methods, paid a close attention to these works. However, with Cizek s 1966 paper the tables were turned around, at least as far as the CC method is concerned, since a similar development of the explicit CC equations, due to Liihrmann and Kiimmel [91] had to wait till 1972, without noticing that by that time quantum chemists were busily trying to apply these equations in actual computations. 

This description of quantum mechanical methods for computing (hyper)polarizabilities demonstrates why, nowada, the determination of hyperpolarizabilities of systems containing hundreds of atoms can, at best, be achieved by adopting, for obvious computational reasons, semi-empirical schemes. In this study, the evaluation of the static and dynamic polarizabilities and first hyperpolaiizabilities was carried out at die Time-Dependent Hartree-Fock (TDOT) [39] level with the AMI [50] Hamiltonian. The dipole moments were also evaluated using the AMI scheme. The reliability of the semi-empirical AMI calculations was addressed in two ways. For small and medium-size push-pull polyenes, the TDHF/AMl approach was compared to Hartree-Fock and post Hartree-Fock [51] calculations of die static and dynamic longitudinal first hyperpolarizability. Except near resonance, the TDHF/AMl scheme was shown to perform appreciably better than the ab initio TDHF scheme. Then, the static electronic first hyperpolaiizabilities of the MNA molecule and dimer have been calculated [15] with various ab initio schemes and compared to the AMI results. In particular, the inclusion of electron correlation at the MP2 level leads to an increase of Paaa by about 50% with respect to the CPHF approach, similar to the effect calculated by Sim et al. [52] for the longitudinal p tensor component of p-nitroaniline. The use of AMI Hamiltonian predicts a p aa value that is smaller than the correlated MP2/6-31G result but larger than any of the CPHF ones, which results fi-om the implicit treatment of correlation effects, characteristic of die semi-empirical methods. This comparison confirms that a part of die electron... 

The definition of the gas-phase acidity through reaction (7.3) implies that this quantity is a thermodynamic state function. Thus, one could use quantum chemical approaches to obtain gas-phase acidities from the theoretically computed enthalpies of the species involved. However, two points must be noted before one proceeds A chemical bond is being broken and an anion is being formed. Thus, one may anticipate the need for a proper treatment of electronic correlation effects and also of basis sets flexible enough to allow the description of these effects and also of the diffuse character of the anionic species, what immediately rules out the semi-empirical approaches. Hence, our discussion will only consider ab initio (Hartree-Fock and post-Hartree-Fock) and DFT (density functional theory) calculations. 

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)." ... 

When the molecular wavefunction is decribed at a post Hartree-Fock level, the treatment is more complex, but there are several methods which can recover to a large extent the simple description summarized above. 

As already noted, HF orbitals is a usual choice of the orbital set. If some other choice is considered, it is profitable to use an orthonormal orbital set. Then the overlap matrix in equation (10) reduces to a unit matrix, and, what is more important, the evaluation of Hij is greatly simplified. On the other hand, it is well known that the use of HF orbitals brings about a slow convergence of the Cl expansion. This is mainly due to the HF virtual orbitals. When large atomic basis sets are used, and they must inevitably be used for any meaningful post-Hartree-Fock ab initio calculation, the virtual orbitals are not concentrated for their most part in the space of valence-shell orbitals, and they do not provide an efficient treatment of electron correlation. This deficiency of HF virtual orbitals may be cured by various approaches, but they do not meet with widespread use. 

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