Self-consistent field Hartree-Fock limit

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

The Hiickel molecular orbital (HMO) model of pi electrons goes back to the early days of quantum mechanics [7], and is a standard tool of the organic chemist for predicting orbital symmetries and degeneracies, chemical reactivity, and rough energetics. It represents the ultimate uncorrelated picture of electrons in that electron-electron repulsion is not explicitly included at all, not even in an average way as in the Hartree Fock self consistent field method. As a result, each electron moves independently in a fully delocalized molecular orbital, subject only to the Pauli Exclusion Principle limitation to one electron of each spin in each molecular orbital. 

Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called Hartree-Fock limit represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation (5) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces. 

The problem is now solved again by an iterative process, which starts with a choice of the x set and the expansion (6.58). The Hartree-Fock operator F and the matrix representation Fx are calculated, (6.64) is solved for the orbital energies, and these are used to compute a new set of coefficients in (6.63). If these are different from the starting set, the cycle is repeated until the self-consistent-field limit is reached. The total electronic energy is obtained by adding the SCF energy to the core energy for the lowest occupied n/2 levels ... 

The best possible wavefunction of the form of (10) is called the Hartree-Fock wavefunction. For molecules it is difficult to solve (11) numerically. The most widely used procedure was proposed by Roothaan.28 This involves expressing the molecular orbitals t/> (.x) as a linear combination of basis functions (normally atomic orbitals) and varying the coefficients in this expansion so as to find the best possible solutions to (11) within the limits of a given basis set. This procedure is called the self-consistent field (SCF) method. As the size and flexibility of the basis set is increased the SCF orbitals and energy approach the true Hartree-Fock ones. 

For a long time, the Hartree-Fock (HF) theory combined with the self-consistent field (SCF) procedure proved arguably to be the most useful method for computational chemists despite its well-known limitations. Within its framework, orbitals can be constmcted to reflect paired or unpaired electrons. If the molecular system has a singlet spin state, then the same orbital spatial function can be used for both the Gland (3-spin electrons in each pair. This assumption is called the restricted Hartree-Fock method (RHF). 

Assuming that an ab initio or semiempirical technique has been used to obtain p(r), we address the important question of how the calculated electrostatic potential depends on the nature of the wavefunction used for computing p(r). Historically, and today as well, most ab initio calculations of V(r) for reasonably sized molecules have been based on self-consistent-field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation. Whereas the availability of supercomputers has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms, there is reason to believe that such computational levels are usually not necessary and not warranted. The M0l er-Plesset theorem states that properties computed from Hartree-Fock wavefunctions using one-electron operators, as is V(r), are correct through first order " any errors are no more than second-order effects. Whereas second-order corrections may not always be insignificant, several studies have shown that near-Hartree-Fock electron densities are affected to only a minor extent by the inclusion of correlation.The limited evidence available suggests that the same is true of V(r), ° ° as is indicated also by the following example. 

In this substection we will shortly discuss the computational methods used for calculation of the spin-spin coupling constants. Two main approaches available are ab initio theory from Hartree-Fock (or self-consistent field SCF) technique to its correlated extensions, and density function theory (DFT), where the electron density, instead of the wave function, is the fundamental quantity. The discussion here is limited to the methods actually used for calculation of the intermolecular spin-spin coupling constants, i. e. multiconfigurational self consistent field (MCSCF) theory, coupled cluster (CC) theory and density functional theory (DFT). For example, the second order polarization propagator method (SOPPA) approach is not... 

It was decided to improve these calculations by using better electronic wavefunctions 0. Single configuration molecular orbital wavefunctions were still used. However, the molecular orbitals were expressed in terms of a so-called extended basis set of gaussian atomic orbitals (for details see reference (3)). The Hartree-Fock-self-consistent-field (HFSCF) procedure was carried out with the digital computer program POLYATOM, The quality of the wavefunctions is not quite what would be called Hartree-Fock limit wavefunctions. Calculations were carried out at several intemuclear distances and C was calculated with the inclusion of the factor A correctly calculated. The calculations were extended to include the ground states of several ions and also of HCl. 

Hartree-Fock limit, where the basis set is essentially complete, and use the term self-consistent-field (SCF) solution for one obtained with a finite, possibly small, basis set. We use the terms Hartree-Fock and SCF interchangeably, however, and specifically refer to the Hartree-Fock limit when necessary. The SCF procedure is as follows ... 

Although orbital wave functions, such as Hartree-Fock, generalized valence bond, or valence-orbital complete active space self-consistent field wave functions, provide a semi-quantitative description of the electronic structure of molecules, accurate predictions of molecular properties cannot be made without explicit inclusion of the effects of dynamical electron correlation. The accuracy of correlated molecular wave functions is determined by two inter-related expansions the many-electron expansion in terms of antisymmetrized products of molecular orbitals that defines the form of the wave function, and the basis set used to expand the one-electron molecular orbitals. The error associated with the first expansion is the electronic structure method error the error associated with the second expansion is the basis set error. Only by eliminating the basis set error, i.e., by approaching the complete basis set (CBS) limit, can the intrinsic accuracy of the electronic structure method be determined. 

Density functional theory based methods are now a very popular and rather inexpensive alternative to conventional correlated ab initio methods. However, none of the available DFT methods covers the dispersion energy" which limits their use for interactions of biomolecules. An other limitation to the application of DFT procedures in the realm of biomolecules stems from the fact that the charge transfer interactions (which probably play an important role in the "function of biosystems) are mostly strongly overestimated, though the very good performance of some so-called hybrid methods provides a large improvement. For more details concerning DFT techniques see Density Functional Applications Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field and Density Functional Theory Applications to Transition Metal Problems. The application of DFT to DNA base pairs is evaluated in Section 3.2.3. 

---