Hartree-Fock theory size-extensivity

In Subsection 2.2.5 we presented our minimal basis H2 model, which has only one occupied molecular orbital and one virtual molecular orbital. With the description of the Is minimal STO-3G basis set given in the last subsection we are now in a position to illustrate ah initio Hartree-Fock calculations on H2. The model is simple but extension to larger basis sets is relatively straightforward and most of the aspects of Hartree-Fock theory that we wish to illustrate here are independent of the actual size of the basis set. Unfortunately, however, the model is too simple to be able to illustrate the iterative nature of the SCF procedure. In the next subsection we describe a minimal basis calculation on HeH", in order to illustrate this aspect of Hartree-Fock theory. 

The purpose of the present chapter is to discuss the structure and construction of restricted Hartree-Fock wave functions. We cover not only the traditional methods of optimization, based on the diagonalization of the Fock matrix, but also second-order methods of optimization, based on an expansion of the Hartree-Fock eneigy in nonredundant orbital rotations, as well as density-based methods, required for the efficient application of Hartree-Fock theory to large molecular systems. In addition, some important aspects of the Hartree-Fock model are analysed, such as the size-extensivity of the energy, symmetry constraints and symmetry-broken solutions, and the interpretation of orbital energies in the canonical representation. 

Density Functional theory [4] (DFT) has been widely recognized as a powerful alternative computational method to traditional ab initio schemes, particularly in studies of transition metal complexes where large size of basis set and an explicit treatment of electron correlation are required. The local spin density approximation [5] (LDA) is the most frequently applied approach within the families of approximate DFT schemes. It has been used extensively in studies on solids and molecules. Most properties obtained by the LDA scheme are in better agreement with experiments [4a] than data estimated by ab initio calculations at the Hartree-Fock level. However, bond energies are usually overestimated by LDA. Thus, gradient or nonlocal corrections [6] have been introduced to rectify the shortcomings in the LDA. The non-... 

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

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