Canonical bonding orbitals

CANONICAL BONDING ORBITALS Figure 3 Canonical molecular orbitals for the water dimer... 

Li, Liu and Lu investigated the electronic structures and the possible aromaticity of some 10 r-electron systems, including the dication, at the HF/6-31G level [118]. The optimised S-S bond length of is 210 pm. Based on the analysis of the bonding characteristics in terms of the canonical molecular orbital and the Foster-Boys localized molecular orbital, they concluded that is of weak aromaticity. This is due to the occupation of the weak antibonding MOs. As a consequence, the bond strengths of the 10 r-electron systems decrease with respect to their 6 r-electron counterparts. 

Step 1. Norbornadiene C7H8 of symmetry C2V contains 8 CH single, 8 CC single and 2 jr-bonds, occupied by 36 electrons. (We disregard the inner carbon ls-orbitals). Accordingly, a SCF treatment yields 18 bonding canonical molecular orbitals (CMOs) irreducible representation Ai, 2 to A2, 4 to B and 5 to 82- We collect these 18 CMOs in a column vector... 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

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 standard MO wave function involves canonical MOs (CMOs), which are permitted to delocalize over the entire molecule. However, it is well known (8,9) that an MO wave function based on CMOs can be transformed to another MO wave function that is based on localized MOs (LMOs), known also as localized bond orbitals (LBOs) (10). This transformation is called unitary transformation, and as such, it changes the representation of the orbitals without affecting the total energy or the total MO wave function. This equivalence is expressed in Equation 3.64 ... 

FIGURE 3.4 Transformation of the valence orbitals of BeH2, from canonical MOs (left-hand side) to localized bond orbitals (right-hand side). This transformation leaves the polyelectronic Hartree-Fock function unchanged. 

LBO Localized bond orbital. For a given molecule, the LBOs can be obtained from the canonical MOs (CMOs) by a unitary transformation that does not change either the total energy or the total wave function. (See Chapter 3.)... 

All these statements, although correct in principle, are not precise from the technical point of view. For example, the zero approximate wave function in the PCILO method is a one-electron approximate function constructed from the bond wave functions determined by an a posteriori localization procedure from an HFR function. Thus the bond orbitals appear after a unitary transformation of the canonical MOs, which correspond to some more or less arbitrary localization criteria [123-125]. 

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. 

Certainly one of the first conceptual problems which arises if one describes the ground state of a molecule in terms of Q-bonds is how to obtain a compatible description of the excited states and ion states which may have B, IT, or A symmetries. A discussion of how a compatible description is obtained has been given recently 11). However, since it is important for later discussions in this work, especially with respect to the connection between valence bond theory, localized molecular orbitals (LMOs) and canonical molecular orbitals (CMOs), a brief account is provided here. 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

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