Localized MOs

In systems such as [A... A ]+ where an electron (or a hole) hesitates or oscillates between two equivalent positions on subsystems A or A, symme breakings may occur when the effective transfer integral between the two sites is weak. Hiis will be the case when A and A are far apart, when they are bridged by an "insulating" ligand, or when the two localized MOs concerned by the electron transfer have a very we spatial overlap. 

The two solutions may be reached independently since they belong to different symmetries. Then one may define localized MOs, on A and A respectively ... 

As the SIBFA approach relies on the use of distributed multipoles and on approximation derived form localized MOs, it is possible to generalize the philosophy to a direct use of electron density. That way, the Gaussian electrostatic model (GEM) [2, 14-16] relies on ab initio-derived fragment electron densities to compute the components of the total interaction energy. It offers the possibility of a continuous electrostatic model going from distributed multipoles to densities and allows a direct inclusion of short-range quantum effects such as overlap and penetration effects in the molecular mechanics energies. 

Fig. 2. Schematic diagrams of occupied canonical MO s and the corresponding localized MO s in homonuclear diatomic molecules. The number of doubly occupied valence orbitals is denoted by n... 

Figure 1 illustrates for four molecular waveftinctions that split-localized MOs generate Cl expansions whose configurational convergence is markedly faster than that of the Cl expansions generated by natural orbitals. Plotted are the truncation errors due to Cl truncations versus the number of determinants in the truncated Cl expansions. 

Type LL Both localized MOs are lone-pair orbitals on the same atom. 

Type LB One localized MO is a lone-pair orbital, the other is a bond... 

Type VV Any two localized MOs that are separated by only one other localized MO ( vicinal localized MOs). 

Hybrid orbitals may be considered as perfectioned AOs, adopted in the calculation of localized MOs in polyatomic molecules, with the LCAO method (cf. section 1.17.1). In the case of hybrid orbitals sp, the four linear combinations of s and p orbitals (Tbi, Tc2, Te, Te ) that lead to tetrahedral symmetry are... 

HC 0 plane, the rotational strength as a function of HCOH dihedral angle can be detemiined for the chatge flow model. Similar calculations using the localized MO model showed that recoil of the C causes the oxygen lone pairs to rotate, generating the magnetic moment for this model. 

An AO is a region of space in an atom in which an electron may exist. An HO is mathematically fabricated from some number of AO s to explain equivalency of bonds. An MO is a region of space about the entire molecule capable of accommodating electrons. A localized MO is a region of space between a pair of bonded atoms in which the bonding electrons are assumed to be present. 

Ik this chapter we explore how symmetry considerations can be applied to one of the most pervasive concepts in all of chemistry bonding between atoms by the sharing of pairs of electrons. Though the idea of an electron-pair bond was first introduced in 1916 by G. N. Lewis, it was only after the advent of quantum mechanics that it could be given a proper theoretical basis. This came about through the development of two theories valence bond (VB) theory and localized MO theory both of which describe the electron pair in terms of orbitals of the component atoms of the bond. 

Symmetry plays an important role in localized MO theory since the orbitals used in the construction of the MOs y>A and xpB of eqn (11-1.1), must be symmetric about the bond axis (for the present we will limit our discussion to o-bonding). The most natural, though not mandatory, building blocks to use for tpk and tpB are the atomic orbitals (AOs) of the component atoms (A and B). Jn some cases there is available a single AO on A and a single AO on B, both of which are symmetric about the bond axis and therefore meet our requirements. But more often, and particularly when A has to form several bonds, there are not the required number of atomic orbitals with the appropriate symmetry and it is necessary to synthesize xpk (or tpB) from several AOs of A (or B). For example methane CH4 is a tetrahedral molecule with four equivalent C—H bonds pointing to the comers of a tetrahedron and each localized MO is made up of an orbital from the... 

A necessary prelude to determining the combinations of AOs which give a hybrid orbital of correct symmetry is the classification of the AOs of the central atom A in terms of the irreducible representations of the point group to which the molecule belongs. This is discussed in 11-2. In 11-4 we consider 77--bonding systems and in the final section we discuss the relationship between localized and non-localized MO theory. 

In constructing a localized MO for the bond A—B it is necessary to specify an orbital centred on A (tpA) and an orbital centred on B (y ). In principle, provided symmetry about the bond axis is preserved (we are still considering only cr-bonded systems), our choice of tpA and pB is not restricted and we could use any well-defined mathematical function or combination of functions. Common sense, however, dictates that the most sensible functions to use for this purpose are the AOs of the free atoms A and B. There are three reasons why this is a sensible choice one mathematical, one chemical, and one practical. 

In open-shell electronic states, the orbitals are not all doubly occupied, and the preceding procedure is not applicable. However, if the wave function can be written as a single Slater determinant, one can use a modified procedure to obtain energy-localized MOs here also. The procedure is to deal with the a spin-orbitals and the jS spin-orbitals separately, using two different unitary transformation matrices Ba and B in (2.85). 

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