Hartree-Fock self-consistent field orbitals

Solving the Schroedinger equation for an atom with N electrons is a formidable computational task because of the numerous electron-electron repulsion terms, Vry. In order to calculate the electron repulsion of one electron, the wavefunctions for the other electrons must be known and vice-versa. The best atomic orbitals are obtained by a numerical solution of the Schroedinger equation. The procedure first introduced by D.R. Hartree is called self-consistent field (SCF). The procedure was further improved by including electron exchange by V. Fock and J.C. Slater. The orbitals obtained by a combination of these procedures are called Hartree-Fock self-consistent field orbitals. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

Thus, the orbitals uk and vk satisfy Hartree-Fock equations which are identical in form and differ only in the numerical values of the constants X/Jt and Ajk respectively. But since the latter are unknowns in the equation, and since 7(p) is itself invariant as shown in Eq. (21), we can say that the Hartree-Fock self-consistent-field equations are invariant under the orbital transformation given by Eqs. (5) and (6). This means in effect, that the energy integral ( H "X11/0 is minimized by the vk s as well as by the uk s — a circumstance which is in agreement with the invariance of and ( 1 under the transformation (5). 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

The Section on More Quantitive Aspects of Electronic Structure Calculations introduces many of the computational chemistry methods that are used to quantitatively evaluate molecular orbital and configuration mixing amplitudes. The Hartree-Fock self-consistent field (SCF), configuration interaction (Cl), multiconfigurational SCF (MCSCF), many-body and Mpller-Plesset perturbation theories,... 

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. 

The choice of orbital exponent (a) to use for a particular atomic Slater orbital has been the subject of several investigations. Originally, Slater (9) proposed a set of empirical rules for choosing exponents however, these are not used frequently in modern calculations. Hartree-Fock self-consistent-field (SCF)... 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

Ab-initio molecular orbital calculations which use the Hartree-Fock self-consistent field theory with one-electron molecular orbitals. This method is based on the variation theorem to seek the nuclear geometry of the molecule or hydrogen-bonded complex with lowest energy [248-253]. It uses no experimental data. 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

It is obvious by symmetry that the coefficients are related ca = cb, a = /b and Ca = =teB, but what about the ratios of ca to a to epP. I ll just mention for now that there is a systematic procedure called the Hartree-Fock self consistent field method for solving this problem. In the special case of the hydrogen molecular ion, which only has a single electron, we can calculate the variational integral and find the LCAO expansion coefficients by requiring that the variational integral is a minimum. Dickinson (1933) first did this calculation using Is and 2porbital exponents to be is = 1.246 and 2pa = 2.965 (See Table 3.2.)... 

The first of a series of more extensive molecular-orbital calculations used Hartree-Fock self-consistent field calculations, thereby deriving a value of... 

THE HALL-ROOTHAAN EQUATIONS, THE ORBITAL APPROXIMATION AND THE MODERN HARTREE-FOCK SELF-CONSISTENT FIELD METHOD... 

We performed Hartree-Fock self-consistent-field (HF-SCF) calculation and obtained PES s corresponding to T, Csv and Did deformation modes of the cluster as shown in Fig. l(a)-(c).Douhle-zeta basis functions, which express each valence orbital of the atom with two functions, are employed with two d functions for silicon and p functions for the negative ion state of oxygen. We assume no electron orbitals around the point charges. 

Hartree incorporated the Pauli principle by allowing no more than two electrons to be present in each orbital, but the wavefunctions that he used did not involve spin, and were not antisymmetric with respect to interchange of electrons. In 1930, V. Fock modified Hartree s approach by using fully antisymmetric spin orbitals that did not distinguish between electrons. This improved way of calculating atomic orbitals is known as the Hartree Fock self-consistent field (SCF) method. Nowadays, fast computers are used and procedures are followed which allow the one-electron wave equations to be solved simultaneously. 

Hartree-Fock self-consistent field calculations indicate that the energy of an electron in the 4s orbital of vanadium lies above that of the 3d orbital in the ground state configuration, [Ar]3dMs. Explain why [Ar]3d 4s and [Ar]3d are less stable configurations than the ground state. 

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. 

In order to extend these methods to make them feasible for the study dynamical chemical processes in biopolymers, simplifying assumptions are necessary. The most obvious choice is the use of semi-empirical techniques within the Hartree Fock, linear combination of atomic orbitals framework. These methods can achieve speedups on the order of 1000 over typical ab initio calculations using split valence basis sets within the Hartree Fock approximation. Often greater accuracy can be achieved as well because of the parameterization inherent in the semi-empirical approaches. One semi-empirical approach which has proven successful in representing many chemically interesting processes is the AMI and MNDO Hartree Fock Self-Consistent Field methods developed and paramerterized by Dewar and coworkers [46]. These methods have recently been implemented in a mixed quantum/ classical methodology for the study of chemical and biochemical processes by Field et al. [47]. 

The next step is to make the Hartree-Fock self-consistent field (HF-SCF) approximation as described previously for a multi-electron atom in Section 8.4. The Hartree-Fock approximation results in separation of the electron motions resulting (along with the Pauli principle) in the ordering of the electrons into the molecular orbitals as shown in Figure 9-5 for carbon monoxide. Hence, the many-electron wavefunction i for an N-electron molecule is written in terms of one-electron space wavefunctions,/, and spin functions, a or p, like what was done for complex atoms in Section 8.4. At this stage it is assumed that the N-electron molecule is a closed-shell molecule (all the electrons are paired in the occupied molecular orbitals). How molecules with open shells are represented will be discussed later in this Section. 

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