Hartree-Fock theory introduction

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. 

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. 

MOs and the configuration expansion. To be successful, we must choose the parametrization of the MCSCF wave function with care and apply an algorithm for the optimization that is robust as well as efficient. The first attempts at developing MCSCF optimization schemes, which borrowed heavily from the standard first-order methods of single-configuration Hartree-Fock theory, were not successful. With the introduction of second-order methods and the exponential parametrization of the orbital space, the calculation of MCSCF wave functions became routine. Still, even with the application of second-order methods, the optimization of MCSCF wave functions can be difficult - more difficult than for the other wave functions treated in this book. A large part of the present chapter is therefore devoted to the discussion of MCSCF optimization techniques. 

As we said in the introduction, the only consistent framework for a relativistic many-electron system is QED. By means of the Hartree-Fock limit of this theory, after renormalization, and using gradient techniques, Engel and Dreizler [22] found a complete energy functional where both terms of the two previous sections appear naturally. 

A detailed, very advanced introduction to basic Hartree-Fock, Cl and MP theory. Well-known as a rigorous introduction to the mathematical fundamentals. 

Application of the Hartree-Fock Method. - Since numerical Hartree-Fock programs dealing with complex numbers are available in many research groups, it seemed natural to apply this scheme also to the scaled Bom-Oppenheimer Hamiltonian (4.15). As a consequence, some numerical results were obtained before the theory was developed, and - as we have emphasized in the Introduction - some features seemed rather astonishing. 

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. 

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. 

It has been mentioned in the introduction that many authors [4] believed that the model-Hamiltonian Hg = OH would give a better basis for the semi-empirical quantum theory than the derivations starting from e.g. the Hartree-Fock Hamiltonian. One has previously had the dilemma that the parameters in the semi-empirical approach determined fi om selected experiments were usually rather different from those calculated by means of the ab-initio methods. This applied e.g. to Slater s F- and G-integrals in the theory of atomic spectra, to Hiickel s parameters a and P in the theory of conjugated systems, or to the y parameter in the Pariser-Parr-Pople scheme. Careful studies by Karl Freed and his group [9] in Chicago have shown that the discrepancy between the two sets of parameters disappears, if one bases the semi-... 

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