Hartree-Fock self-consistent field procedure

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

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.)... 

In this procedure Hartree-Fock self-consistent field (S.C.F.) or other calculations are carried out on the molecular ground state and on the relevant ionic states, and the calculated energy differences are compared with experimental ionization energies. 

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. 

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. 

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. 

SCF (self-consistent field) procedure for solving the Hartree-Fock equations SCI-PCM (self-consistent isosurface-polarized continuum method) an ah initio solvation method... 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Roothaan s Self-Consistent-Field Procedure.—While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by Roothaan.81 In Roothaan s approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. 

The fastest scaling for an ab initio method is Hartree Fock theory utilizing the self-consistent-field procedure (HF/SCF), which for a given basis set scales as the number of electrons N4. In other words, double the size of the calculation and it will take... 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

The Dirac-Hartree-Fock iterative process can be interpreted as a method of seeking cancellations of certain one- and two-body diagrams.33,124 The self-consistent field procedure can be regarded as a sequence of rotations of the trial orbital basis into the final Dirac-Hartree-Fock orbital set, each set in this sequence forming a basis for the Furry bound-state interaction picture of quantum electrodynamics. The self-consistent field potential involves contributions from the negative energy states of the unscreened spectrum so that the Dirac-Hartree-Fock method defines a stationary point in the space of possible configurations, rather that a variational minimum, as is the case in non-relativistic theory. 

As a matter of fact, as in the Hartree-Fock (HF) scheme, the KS equation is a pseudo-eigenvalue problem and has to be solved iteratively through a self-consistent field procedure to determine the charge density p(r) that corresponds to the lowest energy. The self-consistent solutions 4>ia resemble those of the HF equations. Still, one should keep in mind that these orbitals have no physical significance other than in allowing one to constitute the charge density. We want to stress that the DFT wavefunction is not a Slater determinant of spin orbitals. In fact, in a strict sense there is no A -electron wavefunction available in DFT. ... 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

The last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. 

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