Localized Hartree-Fock orbitals

In the literature we may find the procedure for creating localized Hartree-Fock orbitals via an energy minimization based on a Cl procedure employing monoexcitations (see for instance Reference [24]). The scheme starts from a set of given (guess) orbitals and solves iteratively the Hartree-Fock equations via the steps ... 

Rubio J, Povill A, Malrieu J-P, Reinhardt PJ (1997) Direct determination of localized Hartree-Fock orbitals as a step toward Nscaling procedures. J Chem Phys 107 10044... 

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),... 

Table I. Selected values of the Raffenetti-Hartree-Fock orbitals Isg and 2shf for Be, of their locally-scaled transformed functions IsJ, and 2sgr and of their differenees di, = Ish, - Isgj,) and = (2siif — 2sgp). [Reproduced with permission from Table I Ludeiia et al. [Ill]]...

A number of different localized MP2 procedures ( LMP2 ) have been developed. The idea is to localize the Hartree-Fock orbitals prior to their use in the MP2 procedure. For sufficiently large molecules, this significantly reduces the number of integrals (ij ab) which need to be calculated and processed and leads to reduction in both computational effort and overall memory and disk requirements. Localized MP3 and MP4 models are not presently available. 

In addition to density functional models, MP2 models provide a good account of activation energies for organic reactions (see discussion in Chapter 9). Unfortunately, computer time and even more importantly, memory and disk requirements, seriously limit their application. One potential savings is to base the MP2 calculation on Hartree-Fock orbitals which have been localized. This has a relatively modest effect on overall cost, but dramatically reduces memory and disk requirements, and allows the range of MP2 models to be extended. 

LMP2 Model. An MP2 Model in which the Hartree-Fock orbitals are first localized. 

Let us emphasize that the Kohn-Sham orbitals of DFT and the Hartree-Fock orbitals of ab initio theory, though normally quite similar numerically, are distinctly different from a theoretical point of view. Hartree-Fock orbitals satisfy a single-particle equation with a mathematically non-local (i.e. orbital-dependent)... 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

Such a transformation can be used for relocalizing a given set of delocalized molecular orbitals in conformity with the chemical formula. For instance, the occupied orbitals of methane can be transformed into orbitals very close to simple two-center MO s constructed from tetrahedral sp3 hybrid orbitals and Is hydrogen orbitals 24,25,26) a. unitary transformation can hardly modify the wave function, except for an immaterial phase factor therefore, it leads to a description which is as valid as that in terms of the canonical delocalized Hartree-Fock orbitals. Of course, the localization obtained in this way is not perfect, but it is usually much better than is often believed. In the case of methane, the best localized orbitals are uniquely determined by symmetry 27> for less symmetric molecules one needs a criterion for best localization 28 29>, a problem on which we shall not insist here. A careful inspection reveals that there are three classes of compounds ... 

England, W., Salmon, L. S., and Ruedenberg, K., Localized MO s A bridge between chemical intuition and molecular quantum mechanics, in Topics in Current Chemistry, Springer-Verlag, 1971 Caldwell, D. and Eyring, H., Localized orbitals in spectroscopy, Adv. Quantum Chem. 11 93 (1978). These two references discuss the invariance of Hartree-Fock orbitals to a unitary transformation, and the physical interpretation of particular... 

Hence, after a decade of false starts, chemists finally learned that the correct basis set should consist of functions that could represent the atomic Hartree-Fock orbitals plus allow for contraction and polarization corrections in the region where they are largest. Similarly it was realized that the Hartree-Fock virtual molecular orbitals were too diffuse for representing the correction to the SCF wavefunction due to electron correlation. Rather, correlation effects are best represented using excitations to nonphysical molecular orbitals that are of the same size as the occupied MOs. Initially this was learned by transforming existing wavefunctions to natural orbital form. Later, MCSCF orbital optimizations were used to obtain these localized correlating orbitals. 

Let us first consider the calculations on the H2 system. In Table 12.2, the MP2 natural orbitals are used in the first iteration in Table 12.3, we use the canonical Hartree-Fock orbitals. Because of the different choices of orbitals, the optimizations proceed rather differently. For the optimization based on the MP2 natural orbitals, the optimization begins in the local region each step corresporxls to a Newton step with no step-length restrictions. Quadratic convergence is therefore observed in all outer iterations - see the reduction in the gradient and step norms in Table 12.2. The ratio parameter r (12.3.21), which probes the quadratic dominance of the energy function, is close to 1 in all iterations. 

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