Molecular Orbital Theor

1. Introduction

Modern chemistry has depended upon the use of models of increasing comlexity. Atoms can be represented as spheres connected by cyclinders or sticks. In order to understand the mechanism of many reactions, Lewis Theory, developed by Robinson and Ingold, can provide a succesful answer.
Lewis Theory uses curly arrows to denote electron migration during a chemical reaction and has led to a greater understanding of the factors controlling chemical reactions.
Pauling with others, developed Resonance Theory, which provided the rationale to an all-embracing orbital theory. The use of "canonical forms" and "resonance hybrids", alonng with extensive use of curvy arrows has provided the fundamental background to modern organic theory, but for eg. Diels-Alder and pericyclic reactions, the curly arrow format is not very clear and in some instances the reactions are described as no-machanism reactions. Woodward and Hoffmann showed that by examining the interaction of the frontier molecular orbitals (ie. the Highest Occupied, HOMO and Lowest Unoccupied, LUMO) both the regio- and stereospecificity could be accountred for.
Woodward and Hoffmann work was assimilated into general organic reaction theory.

2. Atomic and Molecular Orbitals

By sharing electron, molecules can form bonds, and it is possible to regard the sharing of two electrons by two atoms as constituting a chemical bond. Atoms can share one, two or three electrons (forming single, double and triple bonds).
A hydrogen atom consists of a nucleus (a proton) and an electron. It is not possible to accurately determine the position of the electron, but it is possible to calculate the probability of findng the electron at any point around the nucleus. With a hydrogen atom the probability distribution is spherical around the nucleus and it is possible to draw a spherical boundary surface, inside which there is a 95% possibility of finding the electron. The electron has a fixed energy and a fixed spatial distribution called an orbital. In the helium atom there are two electrons associated with the helium nucleus. The electrons have the same spatial distribution and energy (ie. they occupy the same orbital), but they differ in their spin (Pauli exlusion principle). In general: electrons in atomic nuclei occupy orbitals of fixed energy and spatial distribution, and each orbital only contains a maximum of two electrons with anti-parallel spins.
In physics, periodic phenomena are associated with a "wave equation", and in atomic theory the relevant equation is called the "Schrödinger Equation". The wave equation predicts discrete solutions in one dimension for a particle confined to a box with infinite walls, The solutions can be shown as in the figure below:

y₁ - y₄ represent solutions of increasing energy. In three dimensions, the equation determines the energy and defines the spatial distribution of each electron. Solutions of the wave equations in three-dimensions allows calculation of the "shape" of each orbital. The first five solutions of the wave equation for an electron associated with a proton can be shown in the figure below:

In the hydrogen atom, the 1s atomic orbital has the lowest energy, while the remainder (2s, 2p_x, 2p_y and 2p_z) are of equal energy (ie.degenerate), but for all other atoms, the 2s atomic orbital is of lower enegry than the 2p_x, 2p_y and 2p_z orbitals, which are degenerate.
In atoms, electrons occupy atomic orbitals, but in molecules they occupy similar molecular orbitals which surround the molecule. The simplest molecule is hydrogen, which can be considered to be made up of two seperate protons and electrons. There are two molecular orbitals for hydrogen, the lower energy orbital has its greater electron density between the two nuclei. This is the bonding molecular orbital - and is of lower energy than the two 1s atomic orbitals of hydrogen atoms making this orbital more stable than two seperated atomic hydrogen orbitals. The upper molecular orbital has a node in the electronic wave function and the electron density is low between the two positively charged nuclei. The energy of the upper orbital is greater than that of the 1s atomic orbital, and such an orbital is called an antibonding molecular orbital.
Normally, the two electrons in hydrogen occupy the bonding molecular orbital, with anti-parallel spins. If molecular hydrogen is irradiated by ultra-violet (UV) light, the molecule may absorb the energy, and promote one electron into its antibonding orbital (s^*), and the atoms will seperate. The energy levels in a hydrogen molecule can be represented in a diagram - showing how the two 1s atomic orbitals combine to form two molecular orbitals, one bonding (s) and one antibonding (s^*). This is shown below - by clicking upon either the s or s^* molecular orbital in the diagram - it will show graphically in a window to the right:

Methane:
The valence molecular orbitals of methane are delocalized over the entire nuclear skeleton - that is, it is not easy to assign any one orbital to a particular C-H bond. It is possible to see how complex the orbital structure becomes with the increase in energy. Methane has four valence molecular orbitals (bonding), consisting of one orbital with one nodal plane (lowest occupied) and three degenerate (equal energy) orbitals that do have a nodal plane.
For the energy diagram and pictorial view of the orbitals - please see below:

Ethane:
The ethane molecule has fourteen valence electrons occupying seven bonding molecular orbitals. As can be seen from the energy diagram - four of the molecular orbitals occur as degenerate pairs. Like in methane - the molecular orbitals of ethane show increasing nodal structure with increasing orbital energy.
For the energy diagram and pictorial view of the orbitals - please see below:

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Molecules with double bonds

In molecules where the number of bonding electron pairs exceeds the number of unions between atoms, the extra electrons occupy higher energy molecular orbitals than the orbitals found in molecules where the number of bonding electron pairs equals the number of unions between atoms. These are double bonds, and the orbitals have a nodal plane containig the atoms sharing these p-type orbitals.

Ethene:
The simplest alkene is ethene. Its chemistry is dominated by two "frontier orbitals", that is the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO). For the ethene orbital energy diagram these are shown as p_CC for the HOMO, and p^*_CC for the LUMO.
An important property of the ethene molecule, and alkenes in general is the existence of a high barrier to rotation about the C=C which tends to hold the molecule flat.
For the energy diagram and pictorial view of the orbitals - please see below:

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Molecules with triple bonds

Ethyne:

For the energy diagram and pictorial view of the orbitals - please see below:

Molecules with electron lone pairs

Hydrogen Fluoride:

A simple diatomic molecule is Hydrogen fluoride. There are eight valence electrons which occupy four molecular orbitals. The two highest energy MO's are degenerate, are p-type and have no electron density associated with the hydrogen atom, ie. they are Non-Bonding Orbitals (NBO) and in Lewis Theory are represented as two "Lone Pairs". Another important difference between Hydrogen Fluoride and previous molecules is that the electron density is not equally distributed about the molecule. There is a much greater electron density around the fluorine atom. This is because fluorine is an exremely electronegative element, and in each bonding molecular orbital, fluorine will take a greater share of the electron density.
For the energy diagram and pictorial view of the orbitals - please see below:

Water:
In the water molecule the highest occupied orbital, (1b₁) is non-bonding and highly localized on the oxygen atom, similar to the non-bonding orbitals of hydrogen fluoride. The next lowest orbital (2a₁) can be thought of as a non-bonding orbital, as it has a lobe pointing away from the two hydrogens. From the lower energy bonding orbitals, it is possible to see that oxygen also takes more than its "fair share" of the total electron density.

Ammonia:
Ammonia has two pairs of degenerate orbitals, one bonding and one antibonding, and like hydrogen fluoride and water has a non-bonding orbital (2a₁). This highest occupied orbital has a lobe pointing away from the three hydrogens, and corresponds to a lone pair orbital localized upon the nitrogen, whereas the three lowest energy MO's lead to the description of the three N-H bonds of the Lewis structure. The lone pair is relatively high in energy, and is responsible for the well known Lewis base properties of ammonia.

The next molecule in the series HF, H₂O and H₃N, is H₄C (methane) - which was discussed earlier - and unlike the other three molecules has no non-bonding orbitals.

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Conjugated and aromatic molecules

p bonds in close proximity will often interact. Some of the delocalized molecular orbitals that result will be stabilized, while others will be destabilized. The individual combinations may be polarized, providing an increase in wave function amplitude on some centers at the expense of a decrease in amplitude on others. This gives rise to the possibility of more varied reactivity patterns than are observed for simple alkenes.

Aromatic molecules exhibit a wide range of reactivity patterns toward both electron rich and electron deficient species. These mainly depend on the structures and energies of the frontier p-type molecular orbitals, the HOMO and LUMO. Except for non-bonded lone pairs, the s framework plays little role in the overall reactivity.
trans-1,3-Butadiene:
The energies of the p-molecular orbitals of conjugated molecules like butadiene, (see below) - occur in pairs, with their energies equal to (a±xb), where a and b are constants. For each bonding orbital of and energy a-xb there is a corresponding antibonding orbital of energy a+xb. The p-molecular orbitals are extended over the whole molecule.
For butadiene, the p manifold contains four electrons, leading to an electronic configuration of p₁²p₂².
For the energy diagram and pictorial view of the p-molecular orbitals - please see below:

Allyl radical
The radicals allyl:

and pentadienyl:

have the same arrangement of p-orbitals, (ie. the occur in pairs of energy a±xb), but because there is an odd number of carbon atoms in the conjugate chain, there must be a non-bonding orbital with energy x=0. Also, because of the pairing properties of the p-molecular orbitals of conjugated chains, there will be a node at every alternate carbon atom in the non-bonding orbital. This is important for the unpaired electron of allyl, which will occupy this non-bonding orbital. If an electron is added to the allyl radical to form the anion, the negative charge will appear at the terminal carbon atoms. If the unpaired electron is removed forming the cation, the resulting positive charge is also spread over the termial carbon atoms.
There are three p-molecular orbitals for allyl, the p₁ is bonding, the p₂ orbital is non-bonding and the p₃ is anti-bonding. In the neutral allyl species - there are a total of seventeen valence electrons - of which three fill the p-orbital manifold. A pictorial representation of the energy diagram for the neutral, cationic and anionic allyl species are shown below - (orbitals are shown only for the cationic species):

In the pentadienyl anion, the negative charge is centred on the carbon atoms in the 1,3 and 5 position - similarly with the positive charge for the cation.
These ions are represented in resonance theory as two or three canonical forms:

The delocalisation of p-electrons is associated with a lowering of the orbital energy. Therefore the total energy of the occupied p-orbitals of butadiene is lower in energy then two isolated ethene-type double bonds. Further delocalisation of p-electrons occurs in aromatic hydrocarbons. A figure showing the comparative energy levels of the p-orbitals of the cyclic molecules C_nH_n, for n=3-6, is shown below:

In all cyclic polyenes (C_nH_n), the p-molecular orbitals occur in degenerate pairs, except for the lowest p-orbital, and for the cyclic polyenes with even numbers of carbon atoms, the highest p-orbital (see above).
Cyclobutadiene:
From the cyclic polyene diagram - the square molecule cyclobutadiene (C₄H₄) has four p-orbitals, a bonding orbital (p₁), two degenerate non-bonding orbitals (p₂ and p₃) and an anti-bonding orbital (p₄). Four electrons are placed into these four orbitals; twon into the bonding orbital, and one each with parallel spins into the degenerate non-bonding orbitals (Hund's rule) - see below:

There is no reason to expect cyclobutadiene to be square, theoretically calulations show an oblong with two double bonds structre has lower energy. Experimentally it is shown that cyclobutadiene acts a very strained cyclo-olefin, rather than as a bi-radical species.
Cyclopentadiene:
This molecule is a relatively acidic hydrocarbon, and the anion is formed by the treatment of cyclopentadiene with a strong base. From the cyclic polyene diagram it can be seen that cyclopentadiene has three p bonding orbitals which are delocalised over the five carbon atoms. The uppermost p bonding orbitals are a degenerate pair, and are the highest occupied molecular orbitals (HOMO's).
These orbitals are much higher in energy than those in neutral aromatic species such as benzene, indicating that this anion is far more susceptible t attack by electrophiles. The anion is far more capable of coordinating to transition metals with available empty d-orbitals. The anion has six p-electrons, making the system aromatic. The six electrons are arranged as in the diagram below:

Benzene:
Benzene is the archetypal aromatic compound. It has a symmetrical p system and so is not over reactive on any one site. From the cyclic polyene diagram it can be seen that benzene has six p-molecular orbitals, (which contain the six p-electrons), three bonding and three anti-bonding. The upper bonding degenerate pair of orbitals are the HOMO's of benzene. The p orbital manifold, is shown below - but also of interst is that the pattern of the p orbitals is repeated within the s system. The s functions - like the p orbitals are delocalized throughout the carbon skeleton.
The six p electrons are arranged as in the diagram below:

Bonding in benzene
From the above diagram it can be seen that the lowest lying orbital, p₁, the orbital coefficients are such that the bonding charachter between each pair of adjacent carbon atoms is equal. In p₂ bonding only occurs between atoms C₂ and C₃ and between C₅ and C₆ since the coefficients on C₁ and C₄ are zero. In p₃, C₁, which is bonded to C₂ and C₆ and C₄ is bonded to C₃ and C₅, there are anti-bonding interactions between C₂ and C₃ and between C₅ and C₆. Therefore if we consider the pair of orbitals p₂ and p₃ the contribution to the C-C p bonding is equal for each bond. Since there are three occupied bonding orbitals and six CC linkages - the p bond order is ¹/₂. This description is in accord with the two resonating mesomeric forms (or Kekulé structures in a) below in which single and double bond characters alternate around the ring. Conventionally, the diagram in b) is used to show that the six electrons are delocalized around the ring:

Other interesting molecules

A₂ Molecules

Nitrogen:
This molecule has ten electrons. The atomic orbitals combine to produce the following molecular orbital diagram:

Here the 2p_g orbital is occupied by two electrons to give a total bond order of three. This corresponds well with the Lewis structure (), although the orbital approach tells us that there is one s and two p.
Oxygen:
This molecule has twelve electrons, two more than nitrogen - and these extra two are placed in a pair of degenerate p_g orbitals. The atomic orbitals combine to produce the following molecular orbital diagram:

Comparison of the above energy level diagram wit hthat for nitrogen - you can see that the 2s_g level lies lower than p_u. Here, we are starting to fill the anti-bonding orbitals originating from the p orbital interactions and so the bond order decreases from three to two.
The lowest energy arrangment (Hund's rule) - has a single electron, each with parallel spins, in each of the p_g^x and p_g^y orbitals. This produces a paramagnetic molecule, with a double bond and has two unpaired electrons.

Crystal Field Theory

The Crystal Field Theory in now much more wiedly accepted than the valance bond theory. It assumes that

1. Attraction between the central metal and the ligands in a complex is purely electrostatic
2. The transition metal which forms the central atom in the complex is regarded as a positive ion of charge equal to the oxidation state.
3. Central metal atom is surrounded by negative ligands or neutral molecules which have a lone pair of electrons.
4. The electrons on the central metal aer under repulsive forces from those on the ligans , thus the electrons occupy the d orbital furthest away from the direction of approach of ligands.
5. There is no interaction between metal ornitals and ligand orbitals.
6. The d orbitals on the metal all have same energy in the free atom. However, when a complex is formed the ligands destroy the degeneracy of these orbitals, i.e the orbitals now have different energies. 
7. The approach of ligands split the d orbitals in two different energy state

                              a.       eg set of orbitals
                              b.       t2g set of orbitals