Picture of molecular orbitals

Remember that the square of the wave function, or any of the reduced density matrices, are independent of a unitary transformation of the orbitals. Hence, any pair of orbitals is as good as the other. However, the chemical picture of molecular orbitals is easily understood for most of the chemists. In this case, it is easier looking... 

Weakness Pictures of molecular orbitals can be very complex. 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Another broad approach to the description of molecular structure that is of importance in organic chemistry is molecular orbital theory. Molecular orbital (MO) theory pictures electrons as being distributed among a set of molecular orbitals of discrete... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2, but in somewhat different ways. Both assume that electron waves behave like more familiar waves, such as sound and light waves. One important property of waves is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other (in phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2.2). Recall from Section 1.1 that electron waves in atoms are characterized by then- wave function, which is the same as an orbital. For an electron in the most stable state of a hydrogen atom, for example, this state is defined by the I5 wave function and is often called the I5 orbital. The valence bond model bases the connection between two atoms on the overlap between half-filled orbitals of the two atoms. The molecular orbital model assembles a set of molecular- orbitals by combining the atomic orbitals of all of the atoms in the molecule. 

Viewed from the standpoint of molecular orbital theory, as it has developed during the last decade or so3, the above simple pictures of the sulfur bonding in a dialkyl sulfide are somewhat naive but they serve to introduce the subject and act as a basis for discussing the bonding in sulfoxides and sulfones. It will be convenient to use the second of the two pictures as the basis for further discussion, i.e. that involving the use of 3sp3 hybridized orbitals on sulfur. 

We shall shortly draw on both of these symmetry and energy aspects of Fig. 6-1 in the construction of molecular orbitals for the octahedron. First, however, let us extend the picture to molecules with more than two atoms. 

If molecules or atoms form a chemical bond with the surface upon adsorption, we call this chemisorption. To describe the chemisorption bond we need to briefly review a simplified form of molecular orbital theory. This is also necessary to appreciate, at least qualitatively, how a catalyst works. As described in Qiapter 1, the essence of catalytic action is often that it assists in breaking strong intramolecular bonds at low temperatures. We aim to explain how this happens in a simplified, qualitative electronic picture. 

A satisfactory description of the bonding in hypervalent molecules can also be given in terms of molecular orbitals but this does not directly correspond to the very useful picture of five or more localized bonds (see, for example, Mingos, 1998, p. 250). 

The theory of band structures belongs to the world of solid state physicists, who like to think in terms of collective properties, band dispersions, Brillouin zones and reciprocal space [9,10]. This is not the favorite language of a chemist, who prefers to think in terms of molecular orbitals and bonds. Hoffmann gives an excellent and highly instructive comparison of the physical and chemical pictures of bonding [6], In this appendix we try to use as much as possible the chemical language of molecular orbitals. Before talking about metals we recall a few concepts from molecular orbital theory. 

Practitioners of quantum chemistry employed both the visual imagery of nineteenth-century theoretical chemists like Kekule and Crum Brown and the abstract symbolism of twentieth-century mathematical physicists like Dirac and Schrodinger. Pauling s Nature of the Chemical Bond abounded in pictures of hexagons, tetrahedrons, spheres, and dumbbells. Mulliken s 1948 memoir on the theory of molecular orbitals included a list of 120 entries for symbols and words having exact definitions and usages in the new mathematical language of quantum chemistry. 

Problem 10.5 How is the structure of benzene explained by (a) resonance, (b) the orbital picture, (c) molecular orbital theory <... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

For a simple MO picture of molecular electronic structure, the same procedures can be followed to classify the symmetries spanned by bond and lone-pair orbitals. For instance, we can envisage the electronic structure of BF3 as involving sp2 hybridization of the atoms. We would then have B and three F Is core orbitals, two sp2 and one jr symmetry lone pair on each F atom, and three sp2 bond orbitals for the three BF bonds. Our analysis above shows that the four core orbitals comprise 2 a and a doubly degenerate e orbital the in-plane lone-pairs transform as a, a 2, and 2 e orbitals the out-of-plane lone pairs transform as a"2 and e" and the three BF bonds as a and e. ... 

Aromatic systems play a central role in organic chemistry, and a great deal of this has been fruitfully interpreted in terms of molecular orbital theory that is, in terms of electrons moving more-or-less independently of one another in delocalized orbitals. The spin-coupled model provides a clear and simple picture of the motion of correlated electrons in such systems. The spin-coupled and classical VB descriptions of benzene are very similar, except for the small but crucial distortions of the orbitals. The localized character of the orbitals allows the electrons to avoid one another. Nonetheless, the electrons are still able to influence one another directly because of the non-orthogonality of the orbitals. 

Despite the quantitative victory of molecular orbital (MO) theory, much of our qualitative understanding of electronic structure is still couched in terms of local bonds and lone pairs, that are key conceptual elements of the valence bond (VB) picture. VB theory is essentially the quantum chemical formulation of the Lewis concept of the chemical bond [1,2]. Thus, a chemical bond involves spin-pairing of electrons which occupy valence atomic orbitals or hybrids of adjacent atoms that are bonded in the Lewis structure. In this manner, each term of a VB wave function corresponds to a specific chemical structure, and the isomorphism of the theoretical elements with the chemical elements creates an intimate relationship between the abstract theory and the nature of the... 

Let s consider the shape of the MO first. The simplest picture considers molecular orbitals as resulting from the overlap of atomic orbitals. When atoms are separated by their usual bonding distance, their AOs overlap. Where this overlap occurs, either the electron waves reinforce and the electron density increases, or the electron waves cancel and the electron density decreases. The left-hand side of Figure 3.3 shows the overlap of the Is atomic orbitals on two different hydrogens (Hu and H/ ) when these hydrogens are separated by their normal bonding distance. The two atomic orbitals interact to produce two molecular orbitals. The MOs result from a linear combination of the AOs (called the LCAO approximation). Simply, this means that the AOs are either added (lsa + lsfc) or subtracted (1 sa — lst) to get the MOs. 

One-electron picture of molecular electronic structure provides electronic wavefunction, electronic levels, and ionization potentials. The one-electron model gives a concept of chemical bonding and stimulates experimental tests and predictions. In this picture, orbital energies are equal to ionization potentials and electron affinities. The most systematic approach to calculate these quantities is based on the Hartree-Fock molecular orbital theory that includes many of necessary criteria but very often fails in qualitative and quantitative descriptions of experimental observations. 

Within an orbital picture of molecular electronic structure, formation of ion states can be described by removal of electrons from occupied orbitals in a molecule (Figure 1). Koopmans put this model on a quantitative footing by relating the ionization energy of a molecule to the negative of the self-consistent field (SCF) orbital energy, -Bj (equation 5) ... 

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