Orbitals localized

A Hartree-Fock wave function can be written as a single Slater determinant, composed of a set of orthonormal MOs (eq. (3.20)). 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

The traditional view of molecular bonds is that they are due to an increased probability of finding electrons between two nuclei, as compared to a sum of the contributions of the pure atomic orbitals. The canonical MOs are delocalized over the whole molecule and do not readily reflect this. There is, furthermore, little similarity between MOs for systems which by chemical measures should be similar, such as a series of alkanes. The canonical MOs therefore do not reflect the concept of functional groups. 

One of the goals of Localized Molecular Orbitals (LMO) is to derive MOs which are approximately constant between structurally similar units in different molecules. A set of LMOs may be defined by optimizing the expectation value of an two-electron operator The expectation value depends on the n, parameters in eq. (9.19), i.e. this is again a function optimization problem (Chapter 14). In practice, however, the localization is normally done by performing a series of 2 x 2 orbital rotations, as described in Chapter 13. 

Since all observable properties depend only on the total electron density, and not the individual MOs, there is no unique choice for H, 

The traditional view of molecular bonds is that they are due to an increased probability of finding electrons between two nuclei, as compared to a sum of the 

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Flead and Silva used occupation numbers obtained from a periodic FIF density matrix for the substrate to define localized orbitals in the chemisorption region, which then defines a cluster subspace on which to carry out FIF calculations [181]. Contributions from the surroundings also only come from the bare slab, as in the Green s matrix approach. Increases in computational power and improvements in minimization teclmiques have made it easier to obtain the electronic properties of adsorbates by supercell slab teclmiques, leading to the Green s fiinction methods becommg less popular [182]. 

Head J D and Silva S J 1996 A localized orbitals based embedded cluster procedure for modeling chemisorption on large finite clusters and infinitely extended surfaces J. Chem. Phys. 104 3244... 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

There is always a transformation between symmetry-adapted and localized orbitals that can be quite complex. A simple example would be for the bonding orbitals of the water molecule. As shown in Figure 14.1, localized orbitals can... 

The individual gauge for localized orbitals (IGLO) and localized orbital... 

ADF uses a STO basis set along with STO fit functions to improve the efficiency of calculating multicenter integrals. It uses a fragment orbital approach. This is, in essence, a set of localized orbitals that have been symmetry-adapted. This approach is designed to make it possible to analyze molecular properties in terms of functional groups. Frozen core calculations can also be performed. 

The ah initio methods available are RHF, UHF, ROHE, GVB, MCSCF along with MP2 and Cl corrections to those wave functions. The MNDO, AMI, and PM3 semiempirical Hamiltonians are also available. Several methods for creating localized orbitals are available. 

LORG (localized orbital-local origin) technique for removing dependence on the coordinate system when computing NMR chemical shifts LSDA (local spin-density approximation) approximation used in more approximate DFT methods for open-shell systems LSER (linear solvent energy relationships) method for computing solvation energy... 

For c/rwo-boranes and for the larger open-cluster boranes it becomes increasingly difficult to write a simple satisfactory localized orbital structure, and a full MO treatment is required. Intermediate cases, such a.s B5H9, require several resonance hybrids in the localized orbital... 

Carbon oxohalides are reactive gases or volatile liquids which feature planar molecules of C2t, symmetry they are isoelectronic with BX3 (p. 196) and the bonding is best described in terms of molecular orbitals spanning all 4 atoms rather than in terms of localized orbitals as... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

Mueller, C. R., and Eyring, H., J. Chan. Phys. 19, 1495, Semi-localized orbitals. I. The hydrogen molecule. Combination of Inui (1941) and Coulson-Fischer (1949). 

Manning, P. P., Proc. Roy. Soc. London) A225, 136, 244, The molecular orbital theory of chemical valency. XVII. Higher approximations. XVIII. Localized orbitals in conjugated molecules."... 

CC and CH bond orbitals but also for the CTL, ami CH3 group orbitals. If the local symmetry elements are preserved in the full molecule, the 7r (or a) local orbitals can combine to give v (or o) molecular orbitals. The reader should, therefore, not be surprised to find, for instance, tt type molecular orbitals in cyclopropane which are delocalized over the CH2 groups. 

Let us now construct in turn the various basic localized orbitals and consider their most significant features. 

The procedure for determining the six localized orbitals of a CII3 group (Fig. 9), is similar to that for determining the four localized orbitals of a CH2 group. This time the basic valence orbitals can be divided into three sets a orbitals with either cylindrical or pseudo-cylindrical symmetry about the x axis a ttv orbitals, which... 

Now that we have established the major types of localized orbitals, we must inquire as to how they combine in a molecule to form delocalized combihations. 

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

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