Wave function theory

The description of electron motion and electronic states that is at the heart of all of chemistry is included in wave function theory, which is also referred to as self-consistent-field (SCF) or, by honouring its originators, Hartree-Fock (HF) theory [7]. In principle, this theory also includes density functional theory (DFT) approaches if one uses densities derived from SCF densities, which is common but not a precondition [2] therefore, we treat density functional theory in a separate section. Many approaches based on wave function theory date back to when desktop supercomputers were not available and scientists had to reduce the computational effort by approximating the underlying equations with data from experiment. This approach and its application to the elucidation of reaction mechanisms are outlined in Section 7.2.3. 

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Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

The wave function theory can be reduced to knowing about only one and two particles at a time (we only consider the ground state in the following), since the energy is given by... 

The basic idea for including electron correlation in wave function theory is to introduce additional functions orthogonal to the occupied orbitals in HF theory, and... 

If the electron density function concept is mathematically and conceptually simpler than the wavefunction concept, why did DFT come later than wave-function theory ... 

Koch H, Jdrgen H, Jensen A, Jorgensen P, Helgaker T, Scuseria GE, Schaefer III HF (1990) Coupled cluster energy derivatives. Analytic Hessian for the closed-shell coupled cluster singles and doubles wave function Theory and applications. J Chem Phys 92 4924-4940... 

A. C. Scheiner, G. E. Scuseria, J. E. Rice, T. J. Lee, and H. F. Schaefer, /. Chem. Phys., 87, 5361 (1987). Analytic Evaluation of Energy Gradients for the Single and Double Excitation Coupled Cluster (CCSD) Wave Function Theory and Application. 

The accuracy obtained in all cases depends on the details of the method used. The most accurate calculations are those obtained by HF methods with full correlation energy corrections. (The correlation energy is defined as the difference between the HF energy and the exact energy.) But these are only practical for very small values of N, since computer times now scale as N. The best DFT methods available at present are equal to the best practical HF-based methods available that is, there is some correlation energy included. At the same time the computer time required is 10 to 100 times less for DFT calculations, depending on N. It is hard to avoid the conclusion that density functional theory will almost completely replace wave function theory in the area of ab-initio calculations on molecules. 

It is an attractive feature of ab initio wave function theory that there is a clear hierarchy of methods leading from Hartree-Fock to the exact solution of the Schrodinger equation. Post-Hartree-Fock methods can be divided into three main categories [88]. The first is based on (Mqller-Plesset) perturbation theory [89] and referred to as MPn where n is the order of the perturbation. MPn is excellent when Hartree-Fock already is giving a reasonable description, as is often the case for complexes involving Ad and 5d elements. Otherwise, it fails or might only converge slowly with the order n. MP2 can be used for medium size systems of 100-200 atoms. 

Murphy RB, Pollard WT, Friesner RA. Pseudospectral localized generalized Moller-Plesset methods with a generalized valence bond reference wave function theory and calculation of conformational energies. J Chem Phys 1997 106 5073-5084. 

We have demonstrated that within a finite orbital basis, the EOM-Green s function and conventional wave function theories yield the same values for ionization potentials, electron affinities, and excitation energies. This important theorem enables the careful study of approximations to the complete solution of the EOM equations through comparison with accurate results obtained via the more fully understood wave function techniques. 

The EOM-Green s function theories differ fundamentally from conventional methods in that they are based op a Liouville operator formalism, whereas the wave function theories deal with the Hamiltonian operator. We have discussed in detail what constitutes a complete basis set for these many-body methods and have proved that they do not suffer from the problem encountered by more naive Liouville operator formalisms. 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

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