Local orbitals general

How Localized Orbitals Interact to Create Delocalized Combinations. General Rules for the Interaction between Orbitals of Different Energy... 

In several previous Sections (1.15, 1.19) we considered the interaction of lone-pair orbitals with orbitals localized in other regions of the molecules. In some cases an atom (ether oxygen or thioether sulfur, halogens) may carry several lone pairs. These lone pairs are generally described as localized orbitals pointing in tetrahedral directions on the atom to which they belong as in Fig. 47. The elee-... 

Localized molecular orbital/generalized valence bond (LMO/GVB) method, direct molecular dynamics, ab initio multiple spawning (AIMS), 413-414 Longuet-Higgins phase-change rule conical intersections ... 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

The reader will recall that in Chapter 2 we gave examples of H2 calculations in which the orbitals were restricted to one or the other of the atomic centers and in Chapter 3 the examples used orbitals that range over more than one nuclear center. The genealogies of these two general sorts of wave functions can be traced back to the original Heitler-London approach and the Coulson-Fisher[15] approach, respectively. For the purposes of discussion in this chapter we will say the former approach uses local orbitals and the latter, nonlocal orbitals. One of the principal differences between these approaches revolves around the occurrence of the so-called ionic structures in the local orbital approach. We will describe the two methods in some detail and then return to the question of ionic stmctures in Chapter 8. 

In this chapter we describe four rather different three-electron systems the it system ofthe allyl radical, the HeJ ionic molecule, the valence orbitals ofthe BeHmolecule, and the Li atom. In line with the intent of Chapter 4, these treatments are included to introduce the reader to systems that are more complicated than those of Chapters 2 and 3, but simple enough to give detailed illustrations of the methods of Chapter 5. In each case we will examine MCVB results as an example of localized orbital treatments and SCVB results as an example of delocalized treatments. Of course, for Li this distinction is obscured because there is only a single nucleus, but there are, nevertheless, noteworthy points to be made for that system. The reader should refer back to Chapter 4 for a specific discussion of the three-electron spin problem, but we will nevertheless use the general notation developed in Chapter 5 to describe the results because it is more efficient. 

In a structural formula the bonds are of course localized between pairs of atoms, so that the corresponding bonding and antibonding orbitals are by implication localized in the same way. This picture of localized orbitals is adequate in general for the a orbitals of single bonds, and also for the tt orbitals of isolated double bonds such as the one in formaldehyde. The important difference between diatomic and polyatomic molecules, which was alluded to above, arises when the molecule contains alternating single and... 

An early example implementing the general approach to take into account first the intrabond correlation, is presented by the PCILO - perturbational configuration interaction of localized orbitals method [121,122], As one of its authors, J.-P. Malrieu mentions in [122], the PCILO method opposes the majority of the QC methods in all the fundamental concepts. In contrast to the majority of the methods based on the variational principle, the PCILO method is based on estimating the energy by perturbation theory. Also, the majority of the QC methods use one-electron HFR approximation, at least as an intermediate construct, whereas the PCILO is claimed to addresses directly the V-electron wave function and takes into account all surviving matrix elements of the electron-electron interactions. In contrast with other QC... 

Fig. 2. CASSCF, orthogonal localized, non-orthogonal localized, and generalized valence bond molecular orbitals for the hydrogen molecule.

The matrix defined in Eq.(47) represents a general, non-relativistic spin-independent TV-electron Hamiltonian given by Eq.(2) in a specifically defined model space. The one-electron orbital space is spanned by N orthonormal localized orbitals 4>j, j = 1,2,..., TV and the TV-electron orbital space is one-dimensional with the basis function... 

The recent developments in generalized Valence Bond (GVB) theory have been reviewed by Goddard and co-workers,13 and also the use of natural orbitals in theoretical chemistry,14 15 and the accuracy of computed one-electron properties.18 The Xa method has been reviewed by Johnson,17 and Hurley has discussed high-accuracy calculations on small molecules.18 Several other reviews of interest have appeared in Advances in Quantum Chemistry.17 Localized orbital theory has been reviewed by England, Salmon, and Ruedenberg,19 and the bonding in transition-metal complexes discussed by Brown et a/.20 Finally, the recent developments in computational quantum chemistry have been reviewed by Hall.21... 

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