Canonical orbitals representation

While the canonical orbitals of a system are unique, aside from degeneracies due to multidimensional representations, this is not always the case for localized orbitals, and there may be several sets of localized orbitals in a particular molecule. This situation is related to the fact that the localization sum of Eq. (28) may have several relative maxima under suitable conditions. If one of these maxima is considerably higher than the others, then the corresponding set of molecular orbitals would have to be considered as the localized set. In some cases, however, the two maxima are equal in value, so that there exist two sets of localized orbitals with equal degree of localization. 23) In such a case there... 

There exists no uniformity as regards the relations between localized orbitals and molecular symmetry. Consider for example an atomic system consisting of two electrons in an (s) orbital and two electrons in a (2px) orbital, both of which are self-consistent-field orbitals. Since they belong to irreducible representations of the atomic symmetry group, they are in fact the canonical orbitals of this system. Let these two self-consistent-field orbitals be denoted by Cs) and (2p), and let (ft+) and (ft ) denote the two digonal hybrid orbitals defined by... 

The simplest way to illustrate physical meaning of these quantities is to consider the perturbations of orthogonally twisted ethylene for which SAB = yAB = <5ab = yab = 0 holds via (1) return to planarity or (2) substitution at one end of the C=C bond. For (1), localized orbitals interact, yAB 0, but their energies are the same, 5AB = 0. Since delocalized orbitals become eventually HOMO and LUMO of planar ethylene, they do not have the same energy, Sab 0, but they do not interact, yab = 0. For (2), orthogonal-substituted ethylene, the situation is different. In the localized basis SAB 0, but the interaction is not present due to the symmetry yAn = 0. (A and 2 S belong to different irreducible representations.) For the delocalized description the energies of these orbitals are the same 5ab = 0 since the orbitals are equally distributed over both carbon atoms. But yab 0, since a and b are not canonical orbitals. 

Canonical transformations from the tight-binding (atomic orbitals) representation to the eigenstate (molecular orbitals) representation play an important role, and we consider it in detail. Assume, that we find two unitary matrices SR and SR, such that the Hamiltonians of the left part Hi and of the right part Hi can be diagonalized by the canonical transformations... 

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 ... 

As was done for the ground state (Eq. 5.15), here too, Equation 5.17 can be expanded in terms of elementary determinants (see Exercise 5.3), leading to a unique determinant that is nothing else but the wave function for the lowest energy ionized state in the canonical MO representation, with two electrons in n and one electron in the pure p orbital ... 

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. 

The SCF wave functions we have used to calculate V (r) are written in terms of canonical one-electron orbitals Canonical form is not able to give a simple and evident visualization of a single bond or chemical group. A better representation of the wave function for this purpose is in terms of localized orbitals (LO s), which give a chemically more expressive picture of the electron distribution. It is well known that a one-determinant wave function, written in terms of canonical orbitals localized orbitals A. It is merely necessary to perform a suitable unitary transformation on the set

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The distribution of the nonbonding electrons on bicovalent oxygen is extremely diffuse (Kirby, 1983). Electron density can be seen as filling a kidney-shaped space around the oxygen nucleus. All orbital representations of these nonbonding electrons hold some validity the canonical a + n orbital representation (Lowe et al., 1988) the localized sp orbital representation favoured by Deslongchamps (1983), Kirby (1983) and Gorenstein... 

Do the output orbitals belong to the irreducible representations of the symmetry group (see Appendix C available at booksite.elsevier.com/978-0-444-59436-5 on p. el7) of the Hamiltonian Or, if we set the nuclei in the configuration corresponding to symmetry group G, will the canonical orbitals transform according to some irreducible representations of... 

In Section 14.1, we discussed perturbation theory in general terms, without specifying the zero-order Hamiltonian. The success of perturbation theory depends critically on our ability to provide a suitable zero-order operator. In electronic-structure theory, the most commcm zero-order Hamiltonian is the Fock operator, which in the canonical spin-orbital representation may be written in terms of orbital energies as... 

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... 

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

In this case, it can be proved that the canonical SCF orbitals, being solutions of Eq. (26), are symmetry orbitals, i.e. that they belong to irreducible representations of the symmetry group. 12) If the number of molecular orbitals is larger than the dimension of the largest irreducible representation of the symmetry group, it must then be concluded that the set of all N molecular orbitals form a reducible representation of the group which is the direct sum of all the irreducible representations spanned by the CMO s. 

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 ... 

Bonding. —In the valence-bond description of XeF2, Coulson has emphasized the dominance of the canonical forms (F-Xe)+F and F (Xe-F) in the resonance hybrid. This representation accounts well for the polarity FXe F , indicated by nmr, Moss-bauer, ESCA, and thermodynamic data. It is particularly impressive that the enthalpy of sublimina-tion derived for the XeFy case, by Rice and his co-workers in 1963, on the basis of the charge distribution "FXe+F , is 13.3 kcal mol , whereas the experimental value reported in 1968 is 13.2 kcal mol . It should be recognized that the Coulson valence-bond model is not, in the final analysis, significantly different from the Rundle and Pimentel three-center molecular orbital description or the Bilham and Linnett one-electron-bond description, but it does provide for a more straightforward estimation of thermodynamic stabilities of compounds than the other approaches do. 

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