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In chemistry, molecular orbital theory (MOT) is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule. MOT uses group theory to describe bonding in molecules with the basis for a molecular orbital (MO) being the atomic orbital associated with it. In this theory, each molecule has a set of molecular orbitals, in which it is assumed that the molecular orbital wave function ψf may be written as a simple weighted sum of the n constituent atomic orbitals χi, according to the following equation:


 * $$ \psi_j = \sum_{i=1}^{n} c_{ij} \chi_i$$

The cij coefficients may be determined numerically by substitution of this equation into the Schrödinger equation and application of the variational principle. Molecular orbital theory is useful for many areas of chemistry including being able to provide insight into the electronic configuration for a molecule or complex ion, give the bond order, predict stability, predict and explain spectroscopic properties, predict reactivity, and explain the magnetism of a molecule.

History
Molecular orbital theory was developed, in the years after valence bond theory had been established (1927), primarily through the efforts of Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones. MO theory was originally called the Hund-Mulliken theory. The word orbital was introduced by Mulliken in 1932. By 1933, the molecular orbital theory had become accepted as a valid and useful theory. According to German physicist and physical chemist Erich Hückel, the first quantitative use of molecular orbital theory was the 1929 paper of Lennard-Jones. The first accurate calculation of a molecular orbital wavefunction was that made by Charles Coulson in 1938 on the hydrogen molecule. By 1950, molecular orbitals were completely defined as eigenfunctions (wave functions) of the self-consistent field Hamiltonian and it was at this point that molecular orbital theory became fully rigorous and consistent. This rigorous approach is known as the Hartree–Fock method for molecules although it had its origins in calculations on atoms. In calculations on molecules, the molecular orbitals are expanded in terms of an atomic orbital basis set, leading to the Roothaan equations. This led to the development of many ab initio quantum chemistry methods. In parallel, molecular orbital theory was applied in a more approximate manner using some empirically derived parameters in methods now known as semi-empirical quantum chemistry methods.

Molecular Orbital Theory Background
The Schrödinger equation for a hydrogen atom can be solved analytically, giving rise to the hydrogenic (atomic) orbitals. However, an analytical solution is not possible for molecules because they are complex multi-electron systems. A simple approach to creating a molecular wavefunction is to build molecular orbitals from a linear combination of atomic orbitals. In this approach, referred to as the LCAO-MO approach, the molecular wavefunction $$\Psi$$ is written as a weighted sum of atomic orbitals $$\phi_n$$:

$$ \Psi=N\sum c_n\phi_n $$

where N is a normalization constant, adjusted such that ʃΨΨ*dτ=1, and $$c_n$$ are adjustable coefficients that indicate the relative contributions of each atomic orbital.

When two or more atomic orbitals are in close proximity, a covalent bond, described by molecular orbitals, will be formed from the atomic orbitals provided two conditions are met. First, the two atomic orbitals must have the same symmetry so that the wavefunctions describing these atomic orbitals are able to overlap and constructively and destructively interfere. In practical terms, this means that the orbitals must have the same shape so that overlap can occur. Second, the atomic orbitals must have similar energy; if their energies are dissimilar, the energy of electrons in the molecular orbital will not be significantly different from the energy of electrons in the atomic orbitals and a bond will not form. Note that usually only valence electrons are explicitly considered, as core electrons do not participate in bonding.

When atomic orbitals of the same symmetry and similar energy interact, bonding and antibonding molecular orbitals are formed. Bonding orbitals arise from constructive interference of the atomic wavefunctions, which leads to increased electron density between the atoms. For each bonding molecular orbital, there is a corresponding antibonding molecular orbital (denoted by an asterisk) in which the atomic wavefunctions interfere destructively, leading to decreased electron density between the atoms. Three types of bonds, distinguished by the number of nodal planes they contain, are possible: sigma bonds, pi bonds, and delta bonds.

Bonding and anti-bonding interactions of s, p, and d orbitals are shown below. Sigma (σ) and pi (π) denote bonding orbitals while σ* and π* interactions represent anti-bonding orbitals. Each of these orbitals represents an energy level in the MO diagram for a given molecule.

If there is no atomic orbital of suitable symmetry and energy to interact with a particular atomic orbital, a nonbonding molecular orbital is formed. In this case, the energy of the new molecular orbital usually roughly the same as that of the original atomic orbital, and any electrons populating this molecular orbital are localized on the atom from which they arose.

After a molecular orbital diagram has been constructed, the molecular orbitals are filled with electrons according to the Aufbau principle, Hund's rule, and the Pauli exclusion principle.

General Steps to Creating a Molecular Orbital Diagram: H2 Molecular Orbitals
An important first step in constructing a MO diagram is the placement of the atomic orbitals. The atomic orbitals associated with the molecule will coincide with the molecular orbitals produced (n atomic orbitals will produce n molecular orbitals). Each H atom in H-H possesses a 1s atomic orbital. These atomic orbitals are the start for the diagram. Because they are the same molecule their 1s orbitals are located at the same energy. The atoms in the molecule will direct the placement of the atomic orbitals. A table showing the orbital potential energy levels of atoms should be employed. A general guideline that is accepted is the more electronegative the atom, the lower in energy the atomic orbital will be.

The linear combination of atomic orbitals (LCAO) of the atoms is also a very important part in constructing the diagram. There are two possible molecular orbitals that can be made from two 1s orbitals. These come from the two different wave functions. This will show what the orbital interaction between the atoms looks like.

$$ \Psi_+=\phi_{1s} (A) + \phi_{1s} (B) $$

$$ \Psi_-=\phi_{1s} (A) + \phi_{1s} (B) $$

In some systems it is often possible to infer what the interaction will look like without performing the calculations behind it, but as the MO diagram increases in difficulty it becomes more of a challenge. As the difficulty increases a good understanding of how to use character tables, which describes the symmetry of the molecule, becomes very important. We are looking at a simple H2 system. Ψ+ describes constructive interference between atomic orbitals and is called a bonding molecular orbital, while Ψ- describes destructive interference and is called antibonding. These equations show us how to draw the interaction between the two H atoms. Where the two atoms are both (+) as in the equation $$ \Psi_+=\phi_{1s} (A) + \phi_{1s} (B) $$ they will both be assigned the same phase, but in the second equation where one is (+) and the other is (–), $$ \Psi_-=\phi_{1s} (A) + \phi_{1s} (B) $$, they will be opposite phases. When they are the same phase they are closely bound and therefore become lower in energy (Figure 2). Being of opposite phases means that there is a nodal plane between the two nuclei, which causes it to be less stable and therefore higher in energy. When you adapt this to the MO diagram you produce Figure 3. Because we know that our bonding orbitals are lower in energy than our antibonding orbitals, we can then connect the molecular orbitals that interact (Figure 4).



There is a bonding and antibonding interaction. In the case of 1s orbitals it is a sigma (σ) interaction. This type of interaction indicates that the MO is symmetrical with respect to rotation about the bond axis and because our H-H molecule is linear with a 1s orbital we know that this is the only σ interaction. To relate this to the bonding and antibonding orbitals we will have a σ-σ interaction and a σ-σ* interaction, where the asterisk in σ* indicates antibonding. Each H atom in this system will contribute 1 electron from its 1s orbital. This electron has been placed in the corresponding atomic orbital representative of each H. All of the electrons contributing to the atomic orbitals are counted and then placed into the molecular orbitals. Therefore the two electrons available will then be put into the σ bonding orbital and the σ* antibonding orbital will remain empty. This is the representative MO diagram of H2 and the bond order of H2 is 1.



Molecular Orbitals of More Complex Molecules
A similar approach to that outlined above may be used to construct molecular orbitals for more complex homonuclear and heteronuclear diatomics. Note that when orbitals of similar, but not equal, energy have the symmetry and are thus allowed to interact, mixing may occur, which changes the ordering of the molecular orbitals.

The general approach for building molecular orbitals for three or more atoms is as follows:


 * Determine the point group of the molecule
 * Consider all pendant atoms (i.e. all atoms other than the central atom) as a single group and build a set of symmetry adapted linear combinations of atomic orbitals (SALCs). These SALCs describe how the orbitals of the pendant atoms interact with the orbitals of the central atom.
 * Knowing the symmetry of the SALCs, find the atomic orbitals of the central atom that are of suitable symmetry to interact with the SALCs
 * Use this information to build a molecular orbital diagram

Experimental Evidence Supporting Molecular Orbital Theory
Molecular orbital theory is experimentally supported by its ablilty to predict a variety of molecular properties, such as bond strength, reactivity, and magnetism. More direct experimental evidence for molecular orbital theory is provided by photoelectron spectroscopy. In this technique, a sample is irradiated with a beam of specified energy and electrons are ejected with some kinetic energy as a result of the photoelectric effect. These data can be interpreted using Koopmans' theorem which states that the ionization energy, which is calculated from the energy of the beam and the kinetic energy of ejected electrons, is equal to the negative of the energy of the orbital from which the electron was ejected. Note that although in general, it works well for interpreting photoelectron spectra, Koopman's theorem is only an approximation and is known to break down under certain conditions. Photoelectron spectroscopy can be used to experimentally verify the relative energy levels of atomic orbitals, particularly for more complex molecules, in which ordering is somewhat ambiguous. Analysis of photoelectron spectra is complemented by performing quantum mechanical calculations on the molecule in question.

Molecular Orbital Theory and Coordination Chemistry
A view of the bonding in an octahedral transition metal complex may be created using molecular orbital theory. This approach to understanding bonding in transition metal complexes is known as ligand field theory, and arises from a combination of crystal field theory and molecular orbital theory.

Creation of a Molecular Orbital Diagram for an Octahedral Complex
The most basic approach considers sigma bonding only, since all ligands can act as sigma donors. A set of group orbitals (SALCs) representing the six ligands is constructed from the 6 ligand orbitals. Using a group theoretical approach, the symmetries of the ligand group orbitals can be shown to be a1g, t1u, and eg. These group orbitals interact with the metal d, s, and p orbitals (which have symmetries of eg/t2g, a1g, and t1u, respectively) to form molecular orbitals. The metal dxy, dxz, and dyz orbitals are of t2g symmetry. Because there is no ligand group orbital of suitable symmetry to interact with these orbitals, they form nonbonding molecular orbitals.

The molecular orbitals may then be filled with electrons, with two electrons coming from each of the ligands for a total of 12 electrons. If the metal has any d-electrons, they will populate the t2g and eg* orbitals, as predicted by crystal field theory. The difference between the t2g and eg* orbitals gives Δo, the crystal field splitting parameter.

Effect of π Acceptors and Donors on Molecular Orbitals
In addition to acting as sigma donors, some ligands can act as π donors and acceptors. The group orbitals for π interactions have symmetries of t1g, t2g, t1u, and t2u. The ligand t2g orbital can interact with the metal t2g orbitals, giving new bonding and antibonding molecular orbitals.

A ligand with empty π* orbitals can accept electron density from a metal d-orbital. This type of bonding is referred to as π-backbonding, and a ligand that undergoes π-backbonding is referred to as a π acceptor. The t2g π* ligand orbitals form bonding and antibonding orbitals with the metal t2g orbitals, which were nonbonding orbitals in the sigma-bonding only molecular orbital diagram created above. The result is a decrease in the energy of the t2g orbitals, and a corresponding increase in Δo, as shown below. CO is an example of a π acceptor ligand.

A ligand with filled π orbitals acts as a π donor. The ligand t2g orbitals interact with the metal t2g orbitals, forming bonding and antibonding orbitals. In this case, because the t2g molecular orbital is filled with electrons from the ligand, Δo arises from the difference between the t2g* and eg* orbitals as shown below. Halogens are an example of π donor ligands.

Molecular orbital theory therefore provides an explanation of ordering in the spectrochemical series. π donors decrease Δo relative to σ-bonding only ligands by raising the energy of the t2g set. π acceptors increase Δo relative to σ-bonding only ligands by lowering the energy of the t2g set.

MO Theory and Reactivity
The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are very important in electronic transitions in spectroscopy as well as electron transfer reactions. These orbitals are the least bound energetically, which means they are the most available for interaction with other molecules. Frontier molecular orbital theory states that electron density will be shifted from the HOMO to the LUMO in any given reaction. For a facile reaction to occur between two reactants, Ralph Pearson formulated a set of rules that must be followed.


 * As reactants approach each other, electron density must flow from the HOMO of the donor to the LUMO of the acceptor.
 * The HOMO of the donor and the LUMO of the acceptor must approach each other as to have a net positive overlap.
 * The HOMO of the donor and LUMO of the acceptor must be relatively close (~6eV) in energy.
 * The net effect of the HOMO to LUMO electron transfer must correspond to the bonds to be made and the bonds to be broken during the course of the reaction.

If the reaction follows these four rules, the reaction is symmetry-allowed and has low activation energy. If the rules are not followed, the reaction is termed spin-forbidden and has high activation energy. The energy gap between the HOMO and LUMO can act as a predictor of a compounds ability to undergo a reaction. A large HOMO-LUMO energy gap is indicative of a very stable compound. Conversely, a small HOMO-LUMO energy gap reveals a more reactive compound. Hard and soft acid base theory HSAB theory can also be used to gauge the reactivity of compounds. Using this theory, a compound have a large HOMO-LUMO energy gap could be termed a hard acid, while a compound with a small HUMO-LUMO energy gap could be considered a soft base.