Canonical lone pair orbitals

CANONICAL LONE PAIR ORBITALS Figure 1 Canonical molecular orbitals for the water molecule... 

It follows from the above analysis that the rabbit-ears and canonical MO representations of the water s lone pairs are both perfectly correct, as they lead to equivalent wave functions for the ground state of water, as well as for its two ionized states. Both representations account for the two ionization potentials that are observed experimentally. This example illustrates the well-known fact that, while the polyelectronic wave function for a given state is unique, the orbitals from which it is constructed are not unique, and this holds true even in the MO framework within which a standard localization procedure generates the rabbit-ear lone pairs while leaving the total wave function unchanged. Thus, the question what are the true lone-pair orbitals of water is not very meaningful. 

For the r[ -coordination, the crucial orbitals of thiophene are the high-energy occupied sulfur lone pairs the 7t-lone pair 2b] (S(p ), 45%) and the cr-lone pair 6ai (S(3s), 25% S(3p ), 58%) separated by 0.801 eV. In free thiophene, these canonical lone pairs of sulfur come from different symmetries due to the sp hybridization. 

The strategy which we examined more in detail in the following years may be summarized thus. The orthogonal transformation of canonical SCF MO orbitals (p ir) into LO s A,(r) is followed by a deletions of the Tails , i.e. of the components not describing the main portion of A,-. This manipulation may be described as a projection of A, in the basis set subspace spanned by functions belonging to the chemically significant portion of 2, (it may be an atom, in the case of inner shell or lone pair orbitals, and respectively, a couple of adjacent atoms, in the case of a or banana orbitals, Pab respectively, or larger portions of the molecule in the case of... 

In equations (l7)-(20), A is the antisymmetrizer and U is a unitary matrix. Since the wavefiinctions in the different molecular orbital bases differ at most by their signs, the electron density and all molecular properties are invariant under the transformation of equation (18). Pople made use of this relationship to transform the canonical orbitals (the eigenfunctions of the Fock operator) of water to a set of equivalent orbitals , consisting of two equivalent O-H bond orbitals and two equivalent oxygen lone pair orbitals. Pople also noted that if one writes the total closed shell energy as the sum of one- and two-electron terms. 

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. 

In the case of two valence electrons there is hardly any difference between the localized orbital and the canonical valence orbital, except for the fact that the localization has separated the valence shell somewhat from the other shells. — In the case of four valence electrons, the sigma bonding and the sigma antibonding canonical orbitals yield two equivalent localized orbitals which resemble distorted atomic (2s) orbitals on each of the two atoms. They are precursors of what will be seen to be sigma lone pairs and are denoted by oC and ok . The absence of a bond can be ascribed to the nonbonded repulsion between these orbitals. This corresponds to the case of the unstable Be2 molecule. —... 

The determinant (= total molecular wavefunction T) just described will lead to (remainder of Section 5.2) n occupied, and a number of unoccupied, component spatial molecular orbitals i//. These orbitals i// from the straightforward Slater determinant are called canonical (in mathematics the word means in simplest or standard form ) molecular orbitals. Since each occupied spatial ip can be thought of as a region of space which accommodates a pair of electrons, we might expect that when the shapes of these orbitals are displayed ( visualized Section 5.5.6) each one would look like a bond or a lone pair. However, this is often not the case for example, we do not find that one of the canonical MOs of water connects the O with one H, and another canonical MO connects the O with another H. Instead most of these MOs are spread over much of a molecule, i.e. delocalized (lone pairs, unlike conventional bonds, do tend to stand out). However, it is possible to combine the canonical MOs to get localized MOs which look like our conventional bonds and lone pairs. This is done by using the columns (or rows) of the Slater T to create a T with modified columns (or rows) if a column/row of a determinant is multiplied by k and added to another column/row, the determinant remains kD (Section 4.3.3). We see that if this is applied to the Slater determinant with k = 1, we will get a new determinant corresponding to exactly the same total wavefunction, i.e. to the same molecule, but built up from different component occupied MOs i//. The new T and the new i// s are no less or more correct than the previous ones, but by appropriate manipulation of the columns/rows the i// s can be made to correspond to our ideas of bonds and lone pairs. These localized MOs are sometimes useful. 

First, let us express the polyelectronic wave functions, limited to the lone pairs, for the two apparently different representations. In the canonical MO representation, the polyelectronic wave function, P mo, is made of the doubly occupied n and p orbitals ... 

In the valence bond or hybridization model for CO2, we have two resonance (or canonical) structures, as shown in Fig. 3.4.7. In both structures, the two a bonds are formed by the sp hybrids on carbon with the 2pz orbitals on the oxygens. In the left resonance structure, the jv bonds are formed by the 2px orbitals on C and Oa and the 2p-y orbitals on C and O. In the other structure, the jv bonds are formed by the 2px orbitals on C and Ob and the 2py orbitals on C and Oa. The real structure is a resonance hybrid of these two extremes. In effect, once again, we get two a bonds, two jv bonds, and four lone pairs on the two oxygens. This description is in total agreement with the molecular orbital picture. The only difference is that electron delocalization in CO2 is... 

A molecule whose bonding can be readily rationalized by the sp2 hybridization scheme is BF3. As indicated by the Lewis formulas shown below, there are three canonical strructures, each with three a bonds, one n bond, and eight lone pairs on the F atoms. The three a bonds are formed by the overlap of the sp2 hybrids (formed by 2s, 2px, and 2py) on B with the 2pz orbital on each F atom, while the n bond is formed by the 2pz orbital on B and the 2p orbital on one of the F atoms. This description is in accord with the experimental FBF bond angles of 120°. Also, it may be concluded that the bond order for the B-F bonds in BF3 is IV3. 

Chemically speaking there is little to say. Canonical Hartree-Fock molecular orbitals leave no place for classical chemical concepts such as bonds between atoms or groups, lone pairs, resonance hybrids, etc. However, chemists still utilize these concepts because they are extremely useful in correlating and understanding chemical facts. Even when one manages to localize the canonical molecular orbitals (which is not always straightforward) in regions such that they could be associated with lone pairs or individual chemical bonds, it is important to bear in mind that the orbitals represent localized one-electron states, and not a two-electron chemical bond between atoms or a lone pair of electrons, as will be discussed further. 

The last equation has been used to analyze occupied-unoccupied localized molecular orbital pair contributions for excitations in chiral metal complexes and metallahelicenes [260, 261], as well as in chiral organic acids derived from amino acids by substitution of the amino group with —OH and —F [170]. The analyses in terms of canonical MOs and LMOs may be considered complementary tools, with the canonical MO analysis generally leading to fewer contributions since the canonical MOs are well adapted to describe electronic excitations. The analysis in terms of LMOs allows one to focus on selected chemist s orbitals of interest, such as contributions to excitations from a given lone pair or localized n orbital, or from metal-centered orbitals, which can also be very useful. 

Among the infinite number of ways of transforming the set of canonical m.o.s without changing the eigenfunction some would conform better to the classical structural formula of the molecule, namely those unitary transformations that lead to Xt functions which are practically localized in each formal bond (or in individual atoms, as electron lone pairs). For example,, by replacing the canonical oi and ctj molecular orbitals of H2O by... 

The Natural Bond Orbital analysis of Weinhold [Foster and Weinhold, 1980 Reed, Weinstock etal., 1985 Reed, Curtiss etal., 1988] generates, departing from canonical MOs, a set of localized one center (core, lone pairs) and two center (jt and a bonds) strongly occupied orbitals, and a set of one center (Rydberg) and two center (a, Jt ) weakly occupied orbitals the NBOs. The Natural Bond Orbitals (NBOs) are obtained by a sequence of transformations from the input basis to give, first, the Natural Atomic Orbitals (NAOs), then the Natural Hybrid Orbitals (NHOs), and finally the Natural Bond Orbitals (NBOs). For NAOs, atomic charges can be calculated as a summation of contributions given by orbitals localized on each atom moreover, from NBOs, bond order can be also calculated. 

It is also interesting to use molecular orbitals localized in core, lone pairs, and bond regions, rather than fully delocalized canonical orbitals. A good choice of the remaining n(n - l)/2 Lagrangian multipliers should lead to those localized orbitals, but such an a priori choice is hardly feasible. Most often, localized orbitals are determined by applying an adequate unitary transformation on previously obtained canonical orbitals ... 

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