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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. This method is called the linear combination of atomic orbitals (LCAO ) approximation and is used in computational chemistry. An additional unitary transformation can be applied on the system to accelerate the convergence in some computational schemes. Molecular orbital theory was seen as a competitor to valence bond theory in the 1930s, before it was realized that the two methods are closely related. Because each of the molecular orbitals (ψ) describes how electrons spread out over the entire molecule and belong to the molecule as a whole rather than to any one particular atom, it is in this way separated from valence bond theory.


 * This method is called the linear combination of atomic orbitals (LCAO ) approximation and is used in computational chemistry. An additional unitary transformation can be applied on the system to accelerate the convergence in some computational schemes. Molecular orbital theory was seen as a competitor to valence bond theory in the 1930s, before it was realized that the two methods are closely related. Because each of the molecular orbitals (ψ) describes how electrons spread out over the entire molecule and belong to the molecule as a whole rather than to any one particular atom, it is in this way separated from valence bond theory.*************

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 multi-electron systems such as molecules. 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. 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 only valence electrons are considered in molecular orbital theory, as core electrons are localized on a particular atom and do not contribute to 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.

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 that atom.

Interactions contributing to bond formation
Bonding and anti-bonding interactions of s, p, and d orbitals are shown in the following diagrams. 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.

Creation of a Molecular Orbital Diagram
General steps and the Case for H2

H2 Molecular Orbitals
First we will discuss the placement of atomic orbitals. The atomic orbitals associated with your 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. Where to place your atomic orbitals will depend on the atoms in your molecule. There are general tables and graphs that show the potential energy of some atoms in their corresponding atomic orbitals, which would be useful to refer to when deciding on the energy levels of your atomic orbitals (orbital potential energies). A general guideline that may be employed is that the more electronegative the atom, the lower in energy the atomic orbital will be.

Next you will use the LCAO of your atoms. There are two possible molecular orbitals that can be made from two 1s orbitals. This will show you what the interaction between your 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 your molecule, becomes very important. But here we are using a simple system, so let’s look at how this will affect H2. Ψ+ describes constructive interference between atomic orbitals and is called a bonding molecular orbital, while Ψ- describes destructive interference and is called antibonding. When thinking about how to draw the interaction between the two H atoms we can look at the equations. 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 when we look at 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 your MO diagram you get Figure 3. Since we know that our bonding orbitals are lower in energy than our antibonding orbitals we can then connect the molecular orbitals that interact. Our diagram now looks like Figure 4.

We know that there is a bonding and antibonding interaction, but we also need to know what kind of interaction it is. 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 there is only this σ type interaction. To relate this to our bonding and antibonding orbitals we will have a σ-σ interaction and a σ-σ* interaction, where the asterisk in σ* indicates antibonding. Now the MO diagram can be labeled accordingly. Also, we know that 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 used to fill the molecular orbitals created. 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.

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 being examined. Note that there is error associated with interpreting photoelectron spectra using Koopman's theorem and there can be some ambiguity in peak assignment. In such cases, computational approaches are a complementary technique.

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.

To simplify the model, sigma bonding only is considered, and a set group orbitals representing all ligands is constructed from the set of 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’s d, s, and p orbitals (which have symmetries of eg/t2g, a1g, and t1u, respectively) to form 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.

Effect of π acceptors and donors on Molecular Orbitals
Coordination complexes have a central metal ion surrounded by ligands which are held in close proximity by electrostatics. These ligands can act as π acceptors or donors depending on the type of ligand. All ligands have the capacity to act as sigma donors.

If a ligand has an empty d or π* orbitals, it is a π acceptor. By virtue of these empty orbitals, the ligand can accept electron density from a filled d orbital of the metal. This is referred to as π back bonding. The metal uses the dxy, dyz, and the dxz orbitals to interact with the empty π* orbital of the ligand to form bonding and anti-bonding orbitals. The d orbitals of the metal are lower in energy than the π* orbitals of the ligand. The resulting bonding MOs are lower in energy than the metal d orbitals. The resulting anti-bonding MOs are higher in energy. CO is an example of a π acceptor ligand. If a ligand has filled π or d orbitals, it is a π donor. The filled π or d orbitals interact with the dxy, dyz, and the dxz orbitals of the metal. In this case, the metal d orbitals are raised in energy, and the π orbitals of the ligand are lowered in energy. Halogens are an example of π donor ligands. The π accepting or donating ability of a ligand has a direct effect on its crystal field stabilization energy (Δo). When placed in an octahedral field, the degeneracy of the d orbitals is removed. For π acceptors, the t2g set is lower leading to a higher Δo. The opposite effect is seen in π donors. The spectrochemical series provides a means of classifying the approximate degree of π accepting or donating ability of a ligand based on its Δo.

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.