The molecular orbital Model

We have seen that the localized electron model is of great value in interpreting the structure and bonding of molecules. However, there are some problems with this model. For example, it incorrectly assumes that electrons are localized, and so the concept of resonance must be added. Also, the model does not deal effectively with molecules containing unpaired electrons. And finally, the model gives no direct information about bond energies. 

Another model often used to describe bonding is the molecular orbital model. To introduce the assumptions, methods, and results of this model, we will consider the simplest of all molecules, H2, which consists of two protons and two electrons. A very stable molecule, H2 is lower in energy than the separated hydrogen atoms by 432 kJ/mol. 

Since the hydrogen molecule consists of protons and electrons, the same components found in separated hydrogen atoms, it seems reasonable to use a theory similar to the atomic theory discussed in Chapter 7, which assumes that the electrons in an atom exist in orbitals of a given energy. Can we apply this same type of model to the hydrogen molecule Yes. In fact, describing the H2 molecule in terms of quantum mechanics is quite straightforward. 

However, even though it is formulated rather easily, this problem cannot be solved exactly. The difficulty is the same as that in dealing with polyelectronic atoms—the electron correlation problem. Since we do not know the details of the electron movements, we cannot deal with the electron-electron interactions in a specific way. We need to make approximations that allow a solution of the problem but do not destroy the model s physical integrity. The success of these approximations can only be measured by comparing predictions based on theory with experimental observations. In this case we will see that the simplified model works very well. 

Just as atomic orbitals are solutions to the quantum mechanical treatment of atoms, molecular orbitals (MOs) are solutions to the molecular problem. Molecular orbitals have many of the same characteristics as atomic orbitals. Two of the most important are that they can hold two electrons with opposite spins and that the square of the molecular orbital wave function indicates electron probability. 

AE is equal to E -Eo (given in eV). A large value of f corresponds to a strong absorption band and a short lifetime of the excited state. The maximum value is f= 1. 

c is the velocity of light, m and e are the mass and charge of an electron, respectively, and N is Avogadro s number. The factor F, which reflects solvent effects and depends on the refractive index of the absorbing medium, is close to unity. max the extinction coefficient at the maximum of an absorption band, is a measure of the intensity (magnitude) of the band and an indicator of the al-lowedness of the corresponding electronic transition. 

The simple MO model is based on several assumptions. For instance, a and n orbitals are assumed not to interact. Moreover, molecules are described by localized orbitals each covering two nuclei only. Delocalized orbitals involving more than two nuclei are thought to exist only in the case of Tr-bonding in conjugated systems. 

When a molecule in its ground state absorbs a photon, an electron occupying a cr, 7T or n orbital is promoted to a higher-energy cr or tz orbital. In principle, the following transitions are possible cr cr, tt — tt, n tt, and n cr. As 

Electron transition Absorption region (nm) Extinction coefficient (L mol cm ) 

IBLG See questions from The Molecular Orbital Model  

Molecular orbital theory parallels the atomic theory discussed In Chapter 7. 

In the models of chemical bonding we have discussed up to now, we have assumed that the electrons that interpose themselves between adjacent nuclei (the bonding electrons ) are in orbitals associated with one or the other of the parent atoms. In the simple Lewis and VSEPR models, these were just the ordinary s, p, and d orbitals. The more sophisticated hybridization model recognized that these orbitals will be modified by the interaction with other atoms, and the concept of mixed (hybrid) orbitals was introduced. 

These models in which the bonding electrons are regarded as occupying atomic-type 

The main difficulty with the valence-bond approach is that the atomic orbitals, whether hybridized or not, result from the interaction of electrons with a single central force field— that of the atomic nucleus. An electron that spends most of its time between two nuclei will find itself in a very different, two-center force field, and this will give rise to new types of orbitals that are better characterized as molecular, rather than as atomic orbitals. 

OnPage 12 it was pointed out that an electron that occupies the region of space between two nuclei exerts a mutual attraction on the two positive centers, leading to a net binding effect. Conversely, if the electron is off to one side, in an anti-binding region, it actually adds to the repulsion between the two nuclei. 

The easiest way of visualizing a molecular orbital is to start by picturing two isolated atoms and the electron orbitals that each would have separately. These are just the orbitals of the separate atoms, by themselves, which we already understand. We will then try to predict the manner in which these atomic orbitals interact as we gradually move the two atoms closer together. Finally, we will reach some point where the inter-nuclear distance corresponds to that of the molecule we are studying. The corresponding orbitals will then be the molecular orbitals of our new molecule. 

Note that the lone pairs are placed in the plane where they are 120 degrees apart. Accommodating five pairs at the vertices of a trigonal bipyramid can be explained if the xenon atom adopts a set of five dsp orbitals. Each fluorine atom has four electron pairs and can be assumed to be sp hybridized. The Xep2 molecule has a linear arrangement of atoms. 

The combination of hydrogen Is atomic orbitais to form MOs. The phases of the orbitais are shown by signs inside the boundary surfaces. When the orbitais are added, the matching phases produce constructive interference, which gives enhanced eiectron probabiiity between the nuciei. This resuits in a bonding moiecuiar orbitai. When one orbitai is subtracted from the other, destructive interference occurs between the opposite phases, ieading to a node between the nuciei. This is an antibonding MO. 

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We 11 begin our discussion of hydrocarbons by introducing two additional theories of covalent bonding the valence bond model and the molecular orbital model... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m 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 m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Generally speaking the three models offer complementary information Organic chemists use all three emphasizing whichever one best suits a particular feature of struc ture or reactivity Until recently the Lewis and orbital hybridization models were used far more than the molecular orbital model But that is changing... 

Dannenberg J. J. The Molecular Orbital Modeling of Free Radical and Diels-Alder Reactions Adv. Mol. Model. 1990 2 1-63... 

The molecular orbital model developed in this section is more elaborate than the localized bonds described earlier in this chapter. Is this more complicated model necessary to give a thorough picture of chemical bonding Experimental evidence for molecular oxygen suggests that the answer is yes. 

Our treatment of O2 shows that the extra complexity of the molecular orbital approach explains features that a simpler description of bonding cannot explain. The Lewis structure of O2 does not reveal its two unpaired electrons, but an MO approach does. The simple (t-tt description of the double bond in O2 does not predict that the bond in 2 is stronger than that in O2, but an MO approach does. As we show in the following sections, the molecular orbital model has even greater advantages in explaining bonding when Lewis structures show the presence of resonance. 

The molecular orbital model can also be applied to complexes of the d-block elements. In octahedral complexes the d-orbitals of the metal are not degenerate, as they are in the free metal, because of the interaction between the ligand and metal orbitals. The five d-orbitals are split into three t2g (nonbonding) and two e (antibonding) MOs that is ... 

The electronic spectra of niobium(IV) and -(V) and zirconium(IV) complexes 126,127) have been reported but not interpreted. The spectrum of Nb(ethyl-dtp)4 is of particular interest since the compound is probably 8-co-ordinate. Discussion of the spectrum of binuclear molybdenum complexes 130,131) employed the molecular orbital model of Blake, Cotton and Wood for MO2O3LX complexes s). 

In the pentadienyl radical, predict the distribution of the unpaired electron (a) from the resonance model, and (b) from the molecular orbital model. 

The valence bond model of covalent bonding is easy to visualize and leads to a satisfactory description for most molecules. It does, however, have some problems. Perhaps the most serious flaw in the valence bond model is that it sometimes leads to an incorrect electronic description. For this reason, another bonding description called molecular orbital (MO) theory is often used. The molecular orbital model is more complex than the valence bond model, particularly for larger molecules, but sometimes gives a more satisfactory accounting of chemical and physical properties. 

Before leaving this brief introduction to molecular orbital theory, it is worth stressing one point. This model constructs a series of new molecular orbitals by the combination of metal and ligand orbitals, and it is fundamental to the scheme that the ligand energy levels and bonding are, and must be, altered upon co-ordination. Whilst the crystal field model probably over-emphasises the ionic contribution to the metal-ligand interaction, the molecular orbital models probably over-emphasise the covalent nature. 

Concluding the molecular orbital treatment of BeH2, we can see that the two (filled) bonding molecular orbitals as and a- have different shapes and different energies. This is contrary to our intuition for BeTh we expect the two bonds in BeH2 to be identical (in shape as well as in stability) to each other. In any event, this is the picture provided by the molecular orbital model. 

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