Numerical Hartree-Fock Methods for Molecules

L. Laaksonen, P. Pyykkd, and D. Sundholm, Comput. Phys. Rep., 4, 313 (1986). Fully Numerical Hartree-Fock Methods for Molecules. 

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Basis Sets Correlation Consistent Sets Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory Applications to Transition Metal Problems G2 Theory Integrals of Electron Repulsion Integrals Overlap Linear Scaling Methods for Electronic Structure Calculations Localized MO SCF Methods Mpller-Plesset Perturbation Theory Monte Carlo Quantum Methods for Electronic Structure Numerical Hartree-Fock Methods for Molecules Pseudospectral Methods in Ab Initio Quantum Chemistry Self-consistent Reaction Field Methods Symmetry in Hartree-Fock Theory. 

ACES II Basis Sets Correlation Consistent Sets Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Integrals of Electron Repulsion Numerical Hartree-Fock Methods for Molecules Symmetry in Chemistry TURBOMOLE. 

E.A. McCullough, Numerical Hartree-Fock methods for diatomic molecules A partial wave expansion approach. Comp. Phys. Rep. 4 (1986) 265. 

Since the exact solution of the Hartree-Fock equation for molecules also proved to be impossible, numerical methods approximating the solution of the Schrodinger s equation at the HF limit have been developed. For example, in the Roothan-Hall SCF method, each SCF orbital is expressed in terms of a linear combination of fixed orbitals or basis sets ((Pi). These orbitals are fixed in the sense that they are not allowed to vary as the SCF calculation proceeds. From n basis functions, new SCF orbitals are generated by... 

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-Fock equations 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 functions. Any type of basis function may in principle be used expo ... 

The best possible wavefunction of the form of (10) is called the Hartree-Fock wavefunction. For molecules it is difficult to solve (11) numerically. The most widely used procedure was proposed by Roothaan.28 This involves expressing the molecular orbitals t/> (.x) as a linear combination of basis functions (normally atomic orbitals) and varying the coefficients in this expansion so as to find the best possible solutions to (11) within the limits of a given basis set. This procedure is called the self-consistent field (SCF) method. As the size and flexibility of the basis set is increased the SCF orbitals and energy approach the true Hartree-Fock ones. 

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. 

As a final comment, it is interesting to note that this FS(K) study of the hydrogen molecule offers a new and simple illustration of the behavior of sophisticated Hartree-Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient numerical example of instabilities in the standard Hartree-Fock method. It is important to see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to the correct dissociation limit. However, the UHF and PHF schemes only correct the wave function for large enough interatomic distances and the effect of projection in the PHF scheme even results in a spurious minimum. The EHF scheme is thus the only one which shows a lowering of the energy with respect to RHF for all interatomic distances. 

A decade ago Laaksonen et al. published a paper giving an outline of the finite difference (FD) (or numerical) Hartree-Fock (HF) method for diatomic molecules and several examples of its application to a series of molecules (1). A summary of the FD HF calculations performed until 1987 can be found in (2). The work of Laaksonen et al. can be considered a second attempt to solve numerically the HF equations for diatomic molecules exactly. The earlier attempt was due to McCullough who in the mid 1970s tried to tackle the problem using the partial wave expansion method (3). This approach had been extended to study correlation effects, polarizabilities and hyper-fine constants and was extensively used by McCullough and his coworkers (4-6). Heinemann et al. (7-9) and Sundholm et al. (10,11) have shown that the finite element method could also be used to solve numerically the HF equations for diatomic molecules. 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

It is shown that the LCAO molecular Hartree-Fock equations for a closed-shell configuration can be reduced to a form identical with that of the Hoffmann extended Hiickel approximation if (i) we accept the Mulliken approximation for overlap charge distributions and (ii) we assume a uniform charge distribution in calculating two-electron integrals over molecular orbitals. Numerical comparisons indicate that this approximation leads to results which, while unsuitable for high accuracy calculations, should be reasonably satisfactory for molecules that cannot at present be handled with facility by standard LCAO molecular Hartree-Fock methods. 

Numerical calculation for systems with o or with tt electrons indicate that the approximations considered produce results which, while not appropriate for a requirement of high accuracy, should be reasonably satisfactory for molecules that cannot at the present time be conveniently handled by full molecular Hartree-Fock methods. 

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. 

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