Direct Hartree-Fock theory

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. 

An alternative to Hartree-Fock theory is density-functional (DF) theory in which certain elements in the Hamiltonian are evaluated at fixed points directly from the electron densities at those points. This can circumvent the need for calculating the enormous numbers of electron-electron repulsion integrals encountered in Hartree-Fock calculations. We here briefly describe density-functional theory based upon reviews by von Barth (1986) and Srivastava and Weaire (1987). 

A fundamental characteristic of the FPA is the dual extrapolation to the one-and n-particle electronic-structure limits. The process leading to these limits can be described as follows (a) use families of basis sets, such as the correlation-consistent (aug-)cc-p(wC)VnZ sets [51,52], which systematically approach completeness through an increase in the cardinal number n (b) apply lower levels of theory with extended [53] basis sets (typically direct Hartree-Fock (HF) [54] and second-order Moller-Plesset (MP2) [55] computations) (c) use higher-order valence correlation treatments [CCSD(T), CCSDTQ(P), even FCI] [5,56] with the largest possible basis sets and (d) lay out a two-dimensional extrapolation grid based on the assumed additivity of correlation increments followed by suitable extrapolations. FPA assumes that the higher-order correlation increments show diminishing basis set dependence. Focal-point [2,49,50,57-62] and numerous other theoretical studies have shown that even in systems without particularly heavy elements, account must also be taken for core correlation and relativistic phenomena, as well as for (partial) breakdown of the BO approximation, i.e., inclusion of the DBOC correction [28-33]. 

The development of MOLFDIR came to an end in 2001 and some of the developers of this program joined forces with a new Scandinavian program, Dirac, that emerged in the mid 1990s [518]. Dirac contains an elegant implementation of Dirac-Hartree-Fock theory as a direct SCF method [317] in terms of quaternion algebra [318,319]. For the treatment of electron correla-... 

In Section 3.2.1, we discussed the two main sets of approximations available to us when considering the interaction terms present in the Hamiltonian operator. The first is ab initio theory, which has as its basis Hartree-Fock theory the second is density functional theory, which recasts the basic equations in terms of the electron density rather than the wavefunction directly. 

These considerations call our attention that individual molecular orbitals have no direct physical significance in a many-electron system. It is merely the occupied subspace which has physical meaning in the Hartree-Fock theory. Any transformation among occupied MOs is permitted without affecting the validity of the Brillouin condition, the value of the ground-state energy, or that of any physical observable obtained as an expectation value by the Hartree-Fock wave function. 

With ab initio methods there are two distinct approaches to calculating time dependent properties. One is the analytic derivative approach, where explicit expressions are obtained for the derivatives of the dipole moment or of the pseudoenergy. The other approach, known as response theory calculates the response function directly by time dependent perturbation theory. In general finite field methods are not applicable for dynamit properties except that, e.g., j8(ru 0, ru) could be calculated by the finite difference of a ay,a)) calculated at two different field strengths. Consider first methods based on Hartree-Fock theory, known collectively as TDHF. 

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