Hartree-Fock theory canonical orbitals

Section 3.2 constitutes a derivation of the results of the previous section. The order of presentation of these two sections is such that the derivations of Section 3.2 can be skipped if necessary. For a fuller appreciation of Hartree-Fock theory, however, it is recommended that the derivations be followed. We first present the elements of functional variation and then use this technique to minimize the energy of a single Slater determinant. A unitary transformation of the spin orbitals then leads to the canonical Hartree-Fock equations. 

Similar habits are reinforced by Hartree-Fock theory, where Koopmans s theorem [1] enables one to use canonical orbital energies as estimates of ionization energies and electron affinities. Here, orbitals that are variationally optimized for an N-electron state are used to describe final states with N 1 electrons. Energetic consequences of orbital relaxation in the final states are ignored, as is electron correlation. 

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. 

Although we may keep the redundant parameters fixed (equal to zero) during the optimization of the Hartree-Fock state, we are also free to vary them so as to satisfy additional requirements on the solution - that is, requirements that do not follow from the variational conditions. In canonical Hartree-Fock theory (discussed in Section 10.3), the redundant rotations are used to generate a set of orbitals (the canonical orbitals) that diagonalize an effective one-electron Hamiltonian (the Fock operator). This use of the redundant parameters does not in any way affect the final electronic state but leads to a set of MOs with special properties. 

Contrarily to conventional MP2 theory, the original formulation of MP2-R12 theory (3,4) did not provide the same results when canonical or localized molecular orbitals were used. Indeed, for calculations on extended molecular systems, unphysical results were obtained when the canonical Hartree-Fock orbitals were rather delocalized (5). In order to circumvent this problem, an orbital-invariant MP2-R12 formulation was introduced in 1991, which is the preferred method since then (6),... 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

The many-body perturbation theory [39] [40] [41] was used to model the electronic structure of the atomic systems studied in this work. The theory developed with respect to a Hartree-Fock reference function constructed from canonical orbitals is employed. This formulation is numerically equivalent to the M ler-Plesset theory[42] [43]. 

Mpller-Plesset perturbation theory (MPPT) uses the orbitals and orbital energies obtained from a closed-shell Hartree-Fock-Roothaan (HFR) calculation. The HFR (or canonical) orbitals correspond to the eigenvectors of the inactive Fock matrix... 

This partitioning, when applied in conjunction with the set of canonical Hartree-Fock orbitals (in which is diagonal), corresponds to the Moller-Plesset variant of many-body perturbation theory. A Hartree-Fock determinant, which is an eigenfunction of Pjq, is therefore the natural choice for the zeroth-order wavefunctiond... 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

Density functional theory also offers an attractive computational scheme, the Kohn-Sham (KS) theory [2], similar to the Hartree-Fock (HF) approach, which in principle takes into account both the electron exchange and correlation effects. The canonical KS orbitals thus offer certain interpretative advantages over the widely used HF orbitals, especially for describing the bond dissociation and the open system characteristics, when the electrons are added or removed from the system [3,82,126-130]. For this reason, a determined effort has been made to calculate the reactivity indices from the KS DFT calculations [3,82,83,112,118,119,121, 131-136]. 

The correlated methods discussed up to this point provide a delocalized description of the electronic system. The delocalized nature of these methods arises from their use of canonical orbitals (i.e., the eigenvectors of the Hartree-Fock equations) of Eq. (33). To treat large systems, it is better to express the theory in terms of orbitals that are localized in space, extending over only a few atoms. The virtual excitations then occur predominantly locally in the molecule (among localized occupied and virtual orbitals). As a result, the number of excitation amplitudes increases only linearly with system size. 

An analogy to molecular orbital (MO) theory may help to clarify further what is needed. Chemists prefer to discuss chemical problems in terms of localized MOs rather than in terms of (canonical) delocalized MOs resulting from Hartree-Fock (HF) based quantum chemical calculations. The localized MOs are obtained from the delocalized ones by a transformation ("localization"), which in most cases yields MOs directly related to the bonds of a molecule. The same should be true with regard to localized modes associated with a particular internal coordinate q. The question is only How can we transform from delocalized normal modes to localized internal modes To answer this question we will first summarize the basic theory of vibrational spectroscopy. 

The most direct experimental tests that pertain to these models of electronic structure are measurements of electron binding energies. Photoelectron spectra, for example, provide ionization energies that may be compared with canonical, Hartree-Fock orbital energies. Discrepancies between theory and experiment are generally redressed by improved total energy calculations that consider final-state orbital relaxation and electron correlation in initial and final states. Often these corrections are necessary for correct assignment of the spectra. 

The structure of /,p ( ) may be determined using many-body perturbation theory. If one chooses self-consistence field-restricted Hartree-Fock orbitals, all diagrams contributing to canonical Hartree-Fock are omitted. The second-order expression for (w ) is... 

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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