Hartree-Fock LCAO calculation

Ci44= -3.2, and Ci55= -3.2 [9]. Treatment of exchange interactions in Hartree-Fock LCAO calculations on the STO-2G level shows that for a-BN a cut-off radius of 10 a.u. is sufficient to reduce the error in total energy to less than 1 xIO"" a.u. per cell [10]. 

For the variational procedure two sets of Bloch functions were used (both obtained with the help of CRYSTAL code [23]). The first set (Si) corresponds to the full electron Hartree-Fock LCAO calculations, the second one (52) to the pseudopotential... 

The approximation of the one-electron Hamiltonian is the next step in the framework of the one-electron approximation - the electron-electron interactions are excluded from the Hamiltonian. In solid-state theory the LCAO one-electron Hamiltonian approximation is known as the tight binding method. In molecular quantum chemistry the one-electron Hamiltonians of Huckel or Mulhken-Rtidenberg tjqses (see Chap. 6) were popular in the 1950s and the beginning of the 1960s when the first-principles, Hartree-Fock LCAO calculations were practically impossible. 

The study of the approximate density matrix properties allowed the implementation of the cyclic cluster model in the Hartree- Fock LCAO calculations of crystalUne systems [100] based on the idempotency relations of the density matrix. The results... 

Summary of the steps in a single-point Hartree-Fock (SCF) calculation using the Roothaan-Hall LCAO expansion of the MO s... 

The LCAO methods can treat all electrons and need not make shape approximations to the potential. However, as for Hartree-Fock band calculations, there is a very large number of electron-electron repulsion integrals, and care must be taken in truncating their sums. A number of... 

The first ab initio calculations on interaction dipoles were performed by Matcha and Nesbet They considered the systems HeNe, HeAr and NeAr as super-molecules and did ordinary Hartree-Fock-LCAO-SCF calculations in the range R = 2.0 to 5.5aQ. Because of the Hartree-Fock approximation they did not obtain the dispersion contribution to the dipole moment (cf. sect. 2), but only exchange, penetration and overlap-induction contributions. Their ab initio dipoles could be fitted quite well by a single exponential, which supported the assumption made earlier by Van Kranendonk ... 

Compounds containing B—O Bonds.—Calculated bond lengths [r(H—B) = 2.1913 Bohr(1.1596 A) r(B—0) = 2.2284 Bohr(1.1792 A)] for the recently detected, linear HBO species were obtained from a restricted Hartree-Fock LCAO-MO-SCF study. ... 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

The total electron energy Ee in the Hartree-Fock LCAO method is calculated from the relation... 

The SCF-Xa scattered wave technique has been used by Basiri and Pan for calculation of the MO energies of the octahedral boranes BeHe z = 0, 2, and 4) this technique requires only a small fraction of the computer time of ab initio Hartree—Fock LCAO methods. These calculations confirm the stability of BeHe " by 110.8 kj/mol with respect to the monoanion and by 176.8 kJ/mol with respect to the neutral molecule. 

Ravenek W and Geurts EMM 1986 Hartree-Fock-Slater-LCAO implementation of the moderately large-embedded-cluster approach to chemisorption. Calculations for hydrogen on lithium (100) J. Chem. Phys. 84 1613-23... 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

Since the first formulation of the MO-LCAO finite basis approach to molecular Hartree-Fock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — are calculated and stored on external storage. The second step then consists of the iterative solution of the Roothaan equations, where the integrals from the first step are read once for every iteration. 

There is some small print to the derivation the orbitals must not change during the ionization process. In other words, the orbitals for the cation produced must be the same as the orbitals for the parent molecule. Koopmans (1934) derived the result for an exact HF wavefunction in the numerical Hartree-Fock sense. It turns out that the result is also valid for wavefunctions calculated using the LCAO version of HF theory. 

Even an exact Hartree-Fock calculation cannot be exempt from the correlation error. A practical method of evaluation has been proposed by Hollister and Sinanoglu 16>. An LCAO SCF method has been applied to the calculation of the heat of various simple reactions by Snyder and Basch 17>. They have evaluated the correlation error by the method of Hollister and Sinanoglu 16>. 

In view of the impracticability of the Hartree-Fock calculation for common molecules, the LCAO MO spatial functions may be used in place of Hartree-Fock ones. The MO s a and b are given by... 

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. 

A Hartree-Fock calculation leads to the following LCAOs, each occupied by two electrons ... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

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