Hartree-Fock calculation introduction

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

The difficulty of determining the Half-Projected Hartree-Fock function has somewhat hampered its utilization [3-10]. Some calculations, however, exist in literature. At present time, because of the increasing computing facilities, as well as the introduction of more powerful convergence techniques, the HPHF model is expected to play a more important role, especially in the field of medium size molecules, in which the use of more sophisticated procedure are not yet possible [9-10]. 

The second chapter introduces the student to orbitals proper and offers a simplified rationalization for why orbital interaction theory may be expected to work. It does so by means of a qualitative discussion of Hartree-Fock theory. A detailed derivation of Hartree-Fock theory making only the simplifying concession that all wave functions are real is provided in Appendix A. Some connection is made to the results of ab initio quantum chemical calculations. Postgraduate students can benefit from carrying out a project based on such calculations on a system related to their own research interests. A few exercises are provided to direct the student. For the purpose of undergraduate instruction, this chapter and Appendix A may be skipped, and the essential arguments and conclusions are provided to the students in a single lecture as the introduction to Chapter 3. 

The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian hF—h+G(R)z becomes h+A + G R), and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople et al.73b gives a good introduction to the method, including a discussion of the computational errors likely to be involved. 

The static polarizabilities, a, of various xanthone analogues including seleno- and telluroxanthen-9-one Id and le and seleno- and telluroxanthen-9-thione 2d and 2e have been estimated by ab initio molecular orbital calculations using the coupled perturbed Hartree-Fock (CPHF) method <1996CPL125>. The results indicate that the introduction of heavy elements in 1 and 2 increases all components of a with a greater effect observed in the case of the thione derivatives 2. 

Ab initio LCAO-MO Methods.—The so called ab initio MO method is defined by the discussion so far. The two approximations made are the MO form of the wave-function and the necessary truncation of the basis set. It is important to realize that all the ab initio calculations on large molecules are far from the Hartree-Fock limit and that even this limit may be inadequate to describe a particular feature of the molecule. An excellent introduction to ab initio calculations is the recent book by... 

In the section that follows this introduction, the fundamentals of the quantum mechanics of molecules are presented first that is, the localized side of Fig. 1.1 is examined, basing the discussion on that of Levine (1983), a standard quantum-chemistry text. Details of the calculation of molecular wave functions using the standard Hartree-Fock methods are then discussed, drawing upon Schaefer (1972), Szabo and Ostlund (1989), and Hehre et al. (1986), particularly in the discussion of the agreement between calculated versus experimental properties as a function of the size of the expansion basis set. Improvements on the Hartree-Fock wave function using configuration-interaction (Cl) or many-body perturbation theory (MBPT), evaluation of properties from Hartree-Fock wave functions, and approximate Hartree-Fock methods are then discussed. 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

In many approximate methods, the error of calculated Hellmann-Feynman forces is significant. Following the introduction of the force method of direct, analytic differentiation of Hartree-Fock and related approximate energies by Pulay and by Pulay and Meyer," "the Hellmann-Feynman theorem is rarely used in computational applications. Note, however, that the Hellmann-Feynman theorem still plays a prominent role in studying various special problems." " ... 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

Outline This review concentrates on work which mainly treats ILs from theoretical considerations and not from an experimental point of view. If calculations play only a supportive role in them, articles may have been neglected on principle. We also refrain from an introduction to methodological aspects and rather refer the reader to good textbooks on the subjects. The review is organized as follows Static QC calculations are discussed in detail in the next section including Hartree-Fock, density functional theory (Sect. 2.2) and correlated (i.e., more sophisticated) methods (Sect. 2.4) as well as semiempirical methods (Sect. 2.1). We start with these kinds of small system calculations because they can be considered as a basis for the other calculations, i.e., an insight into the intermolecular forces is obtained. 

The original ab initio approach to calculating electronic properties of molecules was the Hartree-Fock method [31,32,33,34]. Its appeal is that it preserves the concept of atomic orbitals, one-electron functions, describing the movement of the electron in the mean field of all other electrons. Although there are some inherent deficiencies in the method, especially those referred to the absence of correlation effects. Improvements have included the introduction of many-body perturbation theory by Mollet and Plesset (MP) [35] (MP2 to second-order MP4 to fourth order). The computer power required for Hartree-Fock methods makes their use prohibitive for molecules containing more than very few atoms. 

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