Hartree-Fock method Gaussian orbitals

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. 

Table 5.3 Contributions of -orbitals to the total electron density at the iron nucleus (in a.u. ) as a function of oxidation state and configuration. Calculations were done with the spin-averaged Hartree-Fock method and a large uncontracted Gaussian basis set. (17 1 Ip 5d If)... 

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. 

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-FbciTequations may be solved by mapping the orbitals on a set of grid points. These are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known -type of basis function may in principle be used exponential, Gaussian,... 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

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. 

Computational Details. Restricted Hartree-Fock (RHF) calculations were carried out using Gaussian 94 (45) and ACES II (46) on an IBM RISC/6000 computer. The gauge independent atomic orbitals (GIAO) method was used for the shielding calculations (47). All second-order many-body perturbation theory (MBPT2, also referred to as MP2) calculations were performed with ACES II (46). 

---