Localized molecular orbitals limitations

Two relevant topics have been ignored completely in this short chapter the treatment of electron correlation with more sophisticated methods than DFT (that remains unsatisfactory from many points of view) and the related subject of excited states. Wave function-based methods for the calculation of electron correlation, like the perturbative Moller-Plesset (MP) expansion or the coupled cluster approximation, have registered an impressive advancement in the molecular context. The computational cost increases with the molecular size (as the fifth power in the most favorable cases), especially for molecules with low symmetry. That increase was the main disadvantage of these electron correlation methods, and it limited their application to tiny molecules. This scaling problem has been improved dramatically by modern reformulation of the theory by localized molecular orbitals, and now a much more favorable scaling is possible with the appropriate approximations. Linear scaling with such low prefactors has been achieved with MP schemes that the... 

Linear scaling can be achieved when localized molecular orbitals (LMOs) are used to create sparsity within the Slater matrix. The density of a LMO is confined to limited region of space around its centroid, so only a few LMOs need to be evaluated for each electron. The second consideration on the way to linear scaling is to accelerate the transformation from basis function to LMO s. Linear scaling QMC methods have been a popular field of research and several algorithms have already been published. 

Using time-dependent density functional cubic response theory, a scheme has been designed to analyze the static and dynamic second hyperpolarizabilities in terms of y densities as well as in terms of contributions from natural bond orbitals (NBOs) and natural localized molecular orbitals (NLMOs). This approach, which has been implemented for both hybrid and nonhybrid TDDFT schemes and which is based on Slater-type basis functions, constitutes an extension of previously proposed schemes limited to the static responses. 

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

At the other limit of 8 much larger than /3, the delocalized functions (10) are not suitable as a basis because the delocalization is destroyed by the presence of the traps. It is convenient to begin with the localized excitation functions ,. The secular equation is formally the same as that for the molecular-orbital secular problem based on atomic orbitals, as given in Eq. (15). 

Analogously to all similar UHF wave functions describing dissociation of a closed shell molecule Into two odd-electron fragments, the UHF/1 wave fimctions is characterized by a pair of molecular orbitals of opposite spins, which are more and more localized on the boron and on the hydrogen, respectively, as the intemuclear distance R increases. (In the limit R- >, one of these orbitals becomes a pure boron, another a pure hydrogen atomic orbital). 

---