Visual tool in quantum chemistry
A molecular orbital diagram, or MO diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms of molecular orbital theory in general and the linear combination of atomic orbitals (LCAO) method in particular. A fundamental principle of these theories is that as atoms bond to form molecules, a certain number of atomic orbitals combine to form the same number of molecular orbitals, although the electrons involved may be redistributed among the orbitals. This tool is very well suited for simple diatomic molecules such as dihydrogen, dioxygen, and carbon monoxide but becomes more complex when discussing even comparatively simple polyatomic molecules, such as methane. MO diagrams can explain why some molecules exist and others do not. They can also predict bond strength, as well as the electronic transitions that can take place.
History.
Qualitative MO theory was introduced in 1928 by Robert S. Mulliken and Friedrich Hund. A mathematical description was provided by contributions from Douglas Hartree in 1928 and Vladimir Fock in 1930.
Basics.
Molecular orbital diagrams are diagrams of molecular orbital (MO) energy levels, shown as short horizontal lines in the center, flanked by constituent atomic orbital (AO) energy levels for comparison, with the energy levels increasing from the bottom to the top. Lines, often dashed diagonal lines, connect MO levels with their constituent AO levels. Degenerate energy levels are commonly shown side by side. Appropriate AO and MO levels are filled with electrons by the Pauli Exclusion Principle, symbolized by small vertical arrows whose directions indicate the electron spins. The AO or MO shapes themselves are often not shown on these diagrams. For a diatomic molecule, an MO diagram effectively shows the energetics of the bond between the two atoms, whose AO unbonded energies are shown on the sides. For simple polyatomic molecules with a "central atom" such as methane (CH4) or carbon dioxide (CO2), a MO diagram may show one of the identical bonds to the central atom. For other polyatomic molecules, an MO diagram may show one or more bonds of interest in the molecules, leaving others out for simplicity. Often even for simple molecules, AO and MO levels of inner orbitals and their electrons may be omitted from a diagram for simplicity.
In MO theory molecular orbitals form by the overlap of atomic orbitals. Because σ bonds feature greater overlap than π bonds, σ bonding and σ* antibonding orbitals feature greater energy splitting (separation) than π and π* orbitals. The atomic orbital energy correlates with electronegativity as more electronegative atoms hold their electrons more tightly, lowering their energies. Sharing of molecular orbitals between atoms is more important when the atomic orbitals have comparable energy; when the energies differ greatly the orbitals tend to be localized on one atom and the mode of bonding becomes ionic. A second condition for overlapping atomic orbitals is that they have the same symmetry.
Two atomic orbitals can overlap in two ways depending on their phase relationship (or relative signs for real orbitals). The phase (or sign) of an orbital is a direct consequence of the wave-like properties of electrons. In graphical representations of orbitals, orbital phase is depicted either by a plus or minus sign (which has no relationship to electric charge) or by shading one lobe. The sign of the phase itself does not have physical meaning except when mixing orbitals to form molecular orbitals.
Two same-sign orbitals have a constructive overlap forming a molecular orbital with the bulk of the electron density located between the two nuclei. This MO is called the bonding orbital and its energy is lower than that of the original atomic orbitals. A bond involving molecular orbitals which are symmetric with respect to any rotation around the bond axis is called a sigma bond (σ-bond). If the phase cycles once while rotating round the axis, the bond is a pi bond (π-bond). Symmetry labels are further defined by whether the orbital maintains its original character after an inversion about its center; if it does, it is defined gerade, "g". If the orbital does not maintain its original character, it is ungerade, "u".
Atomic orbitals can also interact with each other out-of-phase which leads to destructive cancellation and no electron density between the two nuclei at the so-called nodal plane depicted as a perpendicular dashed line. In this anti-bonding MO with energy much higher than the original AO's, any electrons present are located in lobes pointing away from the central internuclear axis. For a corresponding σ-bonding orbital, such an orbital would be symmetrical but differentiated from it by an asterisk as in σ*. For a π-bond, corresponding bonding and antibonding orbitals would not have such symmetry around the bond axis and be designated π and π*, respectively.
The next step in constructing an MO diagram is filling the newly formed molecular orbitals with electrons. Three general rules apply:
The filled MO highest in energy is called the highest occupied molecular orbital (HOMO) and the empty MO just above it is then the lowest unoccupied molecular orbital (LUMO). The electrons in the bonding MO's are called bonding electrons and any electrons in the antibonding orbital would be called antibonding electrons. The reduction in energy of these electrons is the driving force for chemical bond formation. Whenever mixing for an atomic orbital is not possible for reasons of symmetry or energy, a non-bonding MO is created, which is often quite similar to and has energy level equal or close to its constituent AO, thus not contributing to bonding energetics. The resulting electron configuration can be described in terms of bond type, parity and occupancy for example dihydrogen 1σ"g"2. Alternatively it can be written as a molecular term symbol e.g. 1Σg+ for dihydrogen. Sometimes, the letter n is used to designate a non-bonding orbital.
For a stable bond, the bond order defined as
formula_1
must be positive.
The relative order in MO energies and occupancy corresponds with electronic transitions found in photoelectron spectroscopy (PES). In this way it is possible to experimentally verify MO theory. In general, sharp PES transitions indicate nonbonding electrons and broad bands are indicative of bonding and antibonding delocalized electrons. Bands can resolve into fine structure with spacings corresponding to vibrational modes of the molecular cation (see Franck–Condon principle). PES energies are different from ionisation energies which relates to the energy required to strip off the nth electron after the first n − 1 electrons have been removed. MO diagrams with energy values can be obtained mathematically using the Hartree–Fock method. The starting point for any MO diagram is a predefined molecular geometry for the molecule in question. An exact relationship between geometry and orbital energies is given in Walsh diagrams.
s-p mixing.
The phenomenon of s-p mixing occurs when molecular orbitals of the same symmetry formed from the combination of 2s and 2p atomic orbitals are close enough in energy to further interact, which can lead to a change in the expected order of orbital energies.
When molecular orbitals are formed, they are mathematically obtained from linear combinations of the starting atomic orbitals. Generally, in order to predict their relative energies, it is sufficient to consider only one atomic orbital from each atom to form a pair of molecular orbitals, as the contributions from the others are negligible. For instance, in dioxygen the 3σg MO can be roughly considered to be formed from interaction of oxygen 2pz AOs only. It is found to be lower in energy than the 1πu MO, both experimentally and from more sophisticated computational models, so that the expected order of filling is the 3σg before the 1πu. Hence the approximation to ignore the effects of further interactions is valid. 
However, experimental and computational results for homonuclear diatomics from Li2 to N2 and certain heteronuclear combinations such as CO and NO show that the 3σg MO is higher in energy than (and therefore filled after) the 1πu MO. This can be rationalised as the first-approximation 3σg has a suitable symmetry to interact with the 2σg bonding MO formed from the 2s AOs. As a result, the 2σg is lowered in energy, whilst the 3σg is raised. For the aforementioned molecules this results in the 3σg being higher in energy than the 1πu MO, which is where s-p mixing is most evident. Likewise, interaction between the 2σu* and 3σu* MOs leads to a lowering in energy of the former and a raising in energy of the latter. However this is of less significance than the interaction of the bonding MOs.
Diatomic MO diagrams.
A diatomic molecular orbital diagram is used to understand the bonding of a diatomic molecule. MO diagrams can be used to deduce magnetic properties of a molecule and how they change with ionization. They also give insight to the bond order of the molecule, how many bonds are shared between the two atoms.
The energies of the electrons are further understood by applying the Schrödinger equation to a molecule. Quantum Mechanics is able to describe the energies exactly for single electron systems but can be approximated precisely for multiple electron systems using the Born-Oppenheimer Approximation, such that the nuclei are assumed stationary. The LCAO-MO method is used in conjunction to further describe the state of the molecule.
Diatomic molecules consist of a bond between only two atoms. They can be broken into two categories: homonuclear and heteronuclear. A homonuclear diatomic molecule is one composed of two atoms of the same element. Examples are H2, O2, and N2. A heteronuclear diatomic molecule is composed of two atoms of two different elements. Examples include CO, HCl, and NO.
Dihydrogen.
The smallest molecule, hydrogen gas exists as dihydrogen (H-H) with a single covalent bond between two hydrogen atoms. As each hydrogen atom has a single 1s atomic orbital for its electron, the bond forms by overlap of these two atomic orbitals. In the figure the two atomic orbitals are depicted on the left and on the right. The vertical axis always represents the orbital energies. Each atomic orbital is singly occupied with an up or down arrow representing an electron.
Application of MO theory for dihydrogen results in having both electrons in the bonding MO with electron configuration 1σ"g"2. The bond order for dihydrogen is (2-0)/2 = 1. The photoelectron spectrum of dihydrogen shows a single set of multiplets between 16 and 18 eV (electron volts).
The dihydrogen MO diagram helps explain how a bond breaks. When applying energy to dihydrogen, a molecular electronic transition takes place when one electron in the bonding MO is promoted to the antibonding MO. The result is that there is no longer a net gain in energy.
The superposition of the two 1s atomic orbitals leads to the formation of the σ and σ* molecular orbitals. Two atomic orbitals in phase create a larger electron density, which leads to the σ orbital. If the two 1s orbitals are not in phase, a node between them causes a jump in energy, the σ* orbital. From the diagram you can deduce the bond order, how many bonds are formed between the two atoms. For this molecule it is equal to one. Bond order can also give insight to how close or stretched a bond has become if a molecule is ionized.
Dihelium and diberyllium.
Dihelium (He-He) is a hypothetical molecule and MO theory helps to explain why dihelium does not exist in nature. The MO diagram for dihelium looks very similar to that of dihydrogen, but each helium has two electrons in its 1s atomic orbital rather than one for hydrogen, so there are now four electrons to place in the newly formed molecular orbitals. 
The only way to accomplish this is by occupying both the bonding and antibonding orbitals with two electrons, which reduces the bond order ((2−2)/2) to zero and cancels the net energy stabilization. However, by removing one electron from dihelium, the stable gas-phase species He2+ ion is formed with bond order 1/2.
Another molecule that is precluded based on this principle is diberyllium. Beryllium has an electron configuration 1s22s2, so there are again two electrons in the valence level. However, the 2s can mix with the 2p orbitals in diberyllium, whereas there are no p orbitals in the valence level of hydrogen or helium. This mixing makes the antibonding 1σu orbital slightly less antibonding than the bonding 1σg orbital is bonding, with a net effect that the whole configuration has a slight bonding nature. This explains the fact that the diberyllium