Orbitals equivalent

To arrive at the electronic configuration of an atom the appropriate number of electrons are placed in the orbitals in order of energy, the orbitals of lower energy being filled first (Aufbau principle ), subject to the proviso that for a set of equivalent orbitals - say the three p orbitals in a set - the electrons are placed one... 

To treat the equivalent-orbital ease one forms a box diagram as in the atom ease ... 

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. 

Because each carbon m acetylene is bonded to two other atoms the orbital hybridization model requires each carbon to have two equivalent orbitals available for CT bonds as outlined m Figure 2 19 According to this model the carbon 2s orbital and one of Its 2p orbitals combine to generate two sp hybrid orbitals each of which has 50% s character and 50% p character These two sp orbitals share a common axis but their major lobes are oriented at an angle of 180° to each other Two of the original 2p orbitals remain unhybridized... 

The total electron density contributed by all the electrons in any molecule is a property that can be visualized and it is possible to imagine an experiment in which it could be observed. It is when we try to break down this electron density into a contribution from each electron that problems arise. The methods employing hybrid orbitals or equivalent orbitals are useful in certain circumsfances such as in rationalizing properties of a localized part of fhe molecule. Flowever, fhe promotion of an electron from one orbifal fo anofher, in an electronic transition, or the complete removal of it, in an ionization process, both obey symmetry selection mles. For this reason the orbitals used to describe the difference befween eifher fwo electronic states of the molecule or an electronic state of the molecule and an electronic state of the positive ion must be MOs which belong to symmetry species of the point group to which the molecule belongs. Such orbitals are called symmetry orbitals and are the only type we shall consider here. 

The importance of the off-diagonal QA/i terms is most easily demonstrated in the case of a shell of N equivalent orbitals. Then Ax may be reduced to... 

Again it is a useful approximation to consider just the set of equivalent orbitals of the outermost subshell. Then... 

At the start, the relative energies of the four orbitals are represented in Fig. 33. The correct procedure requires that we first interact the degenerate orbitals nj and n2, and then mix the resulting combinations with orcc and significantly different energies. This is true even though the overlap between n and n2 is extremely small, because n, and n2 are symmetry-equivalent orbitals. The... 

Now consider the alkynes, hydrocarbons with carbon-carbon triple bonds. The Lewis structure of the linear molecule ethyne (acetylene) is H—O C- H. To describe the bonding in a linear molecule, we need a hybridization scheme that produces two equivalent orbitals at 180° from each other this is sp hybridization. Each C atom has one electron in each of its two sp hybrid orbitals and one electron in each of its two perpendicular unhybridized 2p-orbitals (43). The electrons in the sp hybrid orbitals on the two carbon atoms pair and form a carbon—carbon tr-bond. The electrons in the remaining sp hybrid orbitals pair with hydrogen Ls-elec-trons to form two carbon—hydrogen o-bonds. The electrons in the two perpendicular sets of 2/z-orbitals pair with a side-by-side overlap, forming two ir-honds at 90° to each other. As in the N2 molecule, the electron density in the o-bonds forms a cylinder about the C—C bond axis. The resulting bonding pattern is shown in Fig. 3.23. 

McClure, V. E. (1970) Localized and Equivalent Orbitals, Ph.D. Dissertation, University of California, San Diego. 

A fuller and interesting account of molecular ( and of equivalent ) orbitals will be found in Murrell, Kettle, Tedder, Valency Theory, 2nd ed., Wiley, New York, 1969, p 190. 

There are four equivalent orbitals, each called sp, which point to the comers of a regular tetrahedron (Fig. 1.4). The bond angles of methane (CH4) would thus be expected to be 109° 28, which is the angle for a regular tetrahedron. 

The double bond can also be pictured as consisting of two equivalent orbitals, where the centers of electron density point away from the C—C axis. This is the bent-bond or banana-bond picture. Support for this view is found in Pauling, L. Theoretical Organic Chemistry, The Kekule Symposium Butterworth London, 1959, p. 2 Palke, W.E. J, Am. Chem. Soc., 1986,108, 6543. However, most of the literature of organic chemistry is written in terms of the a-7t picture, and in this book we will use it. 

Innorta, Distefano et al. 53, 72) have reported first ionization potentials for a series of M(CO)jL complexes (L = various phosphines and RNC) and find a linear correlation between these values and the ligand ionization potentials and they report calculations of these ionization potentials using an equivalent orbital method. 

A molecule with a steric number of 6 requires six hybrid orbitals arranged in octahedral geometry. In Chapter 9, sulfur hexafluoride appears as the primary example of a molecule with a steric number of 6 (Figure ). Six equivalent orbitals for sulfur can be constmcted for the inner sulfur atom by combining the 3. S, the three 3 p,... 

E. Honegger and E. Heilbronner, The Equivalent Orbital Model and the Interpretation of PE Spectra, in Theoretical Models of Chemical Bonding, Part 3 (Ed. Z. B. Maksic), Springer-Verlag, Berlin, 1991, p. 99. 

In some atoms, the p and s orbitals are mixed together to form several equivalent, hybridized orbitals. The most common example is carbon, where there are four orbitals that are formed by mixing one s orbital with three p orbitals to give four equivalent orbitals designated as sp3 orbitals. 

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

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