Computational quantum mechanics Hartree-Fock equations

Fundamental to almost all applications of quantum mechanics to molecules is the use of a finite basis set. Such an approach leads to computational problems which are well suited to vectoris-ation. For example, by using a basis set the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients - a set of matrix equations. The absolute accuracy of molecular electronic structure calculations is ultimately determined by the quality of the basis set employed. No amount of configuration interaction will compensate for a poor choice of basis set. 

In most fields of physical chemistry, the use of digital computers is considered indispensable. Many things are done today that would be impossible without modem computers. These include Hartree-Fock ab initio quantum mechanical calculations, least-squares refinement of x-ray crystal stmctures with hundreds of adjustable parameters and mar r thousands of observational equations, and Monte Carlo calculations of statistical mechanics, to mention only a few. Moreover computers are now commonly used to control commercial instalments such as Fourier transform infrared (FTIR) and nuclear magnetic resonance (FT-NMR) spectrometers, mass spectrometers, and x-ray single-crystal diffractometers, as well as to control specialized devices that are part of an independently designed experimental apparatus. In this role a computer may give all necessary instaic-tions to the apparatus and record and process the experimental data produced, with relatively little human intervention. 

Computational quantum mechanical methods, such as the Hartree-Fock method (Hehre et al., 1986 Szabo and Ostlund, 1989 Levine, 2000), were developed to convert the many-body Schrodinger equation into a singleelectron equation, which can then be solved tractably with modern computational power. The single-electron equation is an approach by which the state (or wavefunction) of each electron is computed within the field... 

Another branch of computational quantum mechanics, quantum Monte Carlo, is described in Chapter 3 by Professor James B. Anderson. Quantum Monte Carlo techniques, such as variational, diffusion, and Green s function, are explained, along with applications to atoms, molecules, clusters, liquids, and solids. Quantum Monte Carlo is not as widely used as other approaches to solving the Schrodinger equation for the electronic structure of a system, and the programs for running these calculations are not as user friendly as those based on the Hartree-Fock approach. This chapter sheds much needed light on the topic. 

The Hartree-Fock approach derives from the application of a series of well defined approaches to the time independent Schrodinger equation (equation 3), which derives from the postulates of quantum mechanics [27]. The result of these approaches is the iterative resolution of equation 2, presented in the previous subsection, which in this case is solved in an exact way, without the approximations of semiempirical methods. Although this involves a significant increase in computational cost, it has the advantage of not requiring any additional parametrization, and because of this the FIF method can be directly applied to transition metal systems. The lack of electron correlation associated to this method, and its importance in transition metal systems, limits however the validity of the numerical results. 

An alternative to the static lattice approach is to use quantum mechanical methods in the calculation of defect energies. These methods essentially rely on solution of the Schrodinger equation using the Hartree-Fock approximation. The two main techniques involve calculation on an embedded defect cluster, i.e., the defect and surrounding lattice. Alternatively, calculations may be performed on a defect supercell in which the defect is periodically repeated on a superlattice. Ab initio Haitree-Fock calculations are computationally intensive, and a number of approximations have been made in order to simplify these calculations to obtain reasonable computerprocessing times. Foramore detailed description of ab initio calculations readers are referred to the recent review by Pyper. ... 

Solutions to the eigenvalue equation (16) can be obtained by any of the standard quantum chemical methods, such as Hartree-Fock SCF, multiconfiguration SCF (MCSCF), Mpller-Plesset perturbation, coupled cluster, or density functional theories. The matrix elements of Hr, a one-electron operator, are readily computed, thus formally the inclusion of solvent effects in the quantum mechanical description of the solute molecule appears quite simple. Moreover, gradients of the eigenvalue E are readily computed. 

---