Hartree-Fock method Slater-type orbitals

PDDO PRDDO RHF SAMO SCF SOGI STO STO-nG UA UHF VB VIP Projectors of Diatomic Differential Overlap Partial Retention of Diatomic Differential Overlap Restricted Hartree-Fock Simulated ab initio Method Self Consistent Field Spin Optimized GVB method Slater Type Orbital Slater Type Orbital expanded in terms of nGTO United Atom Unrestricted Hartree-Fock Valence Bond Vertical Ionization Potential... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

The corresponding estimate for the second eigenvalue (2s orbital energy) is —0.1789. These results are in good agreement with the actual HF/STO-3G ( Hartree-Fock method with a variational basis set of three-term Gaussians for each Slater-type orbital 10) eigenvalues eis = —2.3692 and e2s = —0.1801. 

Unfortunately, the Slater-type orbitals become increasingly less reliable for the heavier elements, including to some extent the first transition series these limitations are described in a recent review by Craig and Nyholm (5 ). The most accurate wave functions to use in these calculations would be the SCF functions obtained by the Hartree-Fock procedure outlined above, but this method leads to purely numerical radial functions. However, Craig and Nyholm (5S) have drawn attention to relatively good fits obtained by Richardson (59) to SCF 3d functions by means of two-parameter orbitals of the type... 

The extension of the matrix solution of section 4.3 for one-electron bound states to the Hartree—Fock problem has many advantages. It results in radial orbitals specified as linear combinations of analytic functions, usually normalised Slater-type orbitals (4.38). This is a very convenient form for the computation of potential matrix elements in reaction theory. The method has been described by Roothaan (1960) for a closed-shell or single-open-shell structure. 

The calculation of Mitroy started by calculating the Hartree—Fock approximation to the ground state 3s where we denote the states by the orbitals of the two active electrons in the configuration with the largest coefficient, in addition to the symmetry notation. The calculation used the analytic method with the basis set of Clementi and Roetti (1974) augmented by further Slater-type orbitals in order to give flexibility for the description of unoccupied orbitals. The total energy calculated by this method was —199.614 61, which should be compared with the result of a numerical Hartree—Fock calculation, —199.614 64. 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

QSAR - Quantitative Structure-Activity (or Property) Relation RHF - Restricted Hartree Fock SCF - Self-Consistent Field Method STO - Slater-type Orbital... 

The momentum wave functions in various atomic models are calculated for arbitrary atomic orbitals. The nonrelativistic hydrogenic, the Hartree-Fock, the relativistic hydrogenic, and the Dirac-Fock models are considered. The momentum wave functions are obtained as a Fourier transform of the wave function in the position space. The Hartree-Fock and the Dirac-Fock wave functions in atoms are given in terms of Slater-type orbitals (STO s), i.e. the Hartree-Fock-Roothaan (HFR) method and the relativistic HFR (RHFR) method. All the wave functions in the momentum space can be expressed analytically in terms of hypergeometric functions. 

The more accurate Hartree-Fock method approximates the wave function as an antisymmetrized product (Slater determinant or determinants) of one-electron spin-orbitals and finds the best possible forms for the spatial orbitals in the spin-orbitals. Hartree-Fock calculations are usually done by expanding each orbital as a linear combination of basis functions and iteratively solving the Hartree-Fock equations (11.12). The Slater-type orbitals (11.14) are often used as the basis functions in atomic calculations. The difference between the exact nonrelativistic energy and the Hartree-Fock energy is the correlation energy of the atom (or molecule). 

Kim has formulated a relativistic Hartree-Fock-Roothaan equation for the ground states of closed-shell atoms using Slater-type orbitals. Relativistic effects in atoms have been reviewed by Grant. Malli and coworkers have formulated a relativistic SCF method for molecules. In this method, four-component spinor wavefunctions are obtained variationally in a self-consistent scheme using Gaussian basis sets. 

Only basis orbitals of the same symmetry can be included in any one LCAO molecular orbital if it is to be an eigenfunction of the symmetry operators. The a basis function can combine with the Is, 2s, and 2p functions on the oxygen. The 2 basis function can combine with the 2py function on the oxygen, and the,2px function on the oxygen cannot combine with any of the other basis functions. Table 21.2 contains the values of the coefficients determined by the Hartree-Fock-Roothaan method for the seven canonical molecular orbitals, using Slater-type orbitals (STOs) as basis functions, with... 

The Slater-Condon theory and Hartree-Fock method with relativistic corrections were combined In [6] for evaluating the ground configuration in Ru XXX, Slater and spin-orbit parameters and transition probabilities (E2 and Ml type). 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

While in principle all of the methods discussed here are Hartree-Fock, that name is commonly reserved for specific techniques that are based on quantum-chemical approaches and involve a finite cluster of atoms. Typically one uses a standard technique such as GAUSSIAN-82 (Binkley et al., 1982). In its simplest form GAUSSIAN-82 utilizes single Slater determinants. A basis set of LCAO-MOs is used, which for computational purposes is expanded in Gaussian orbitals about each atom. Exchange and Coulomb integrals are treated exactly. In practice the quality of the atomic basis sets may be varied, in some cases even including d-type orbitals. Core states are included explicitly in these calculations. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

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