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Molecular Orbital Theory

A fundamental knowledge of the electronic states in molecules or complexes is the foundation of a thorough understanding of photochemical processes that these substances can undergo. The molecular orbital theory provides such an understanding.

The molecular orbital theory assumes, that bonds between atoms are a linear combination of the individual atom orbitals, forming the molecular orbitals. The electrons in a molecule are localized in these molecular orbitals. The theoretical foundation of this concept is the Schrödinger equation.

The first assumption is, that core electrons of an atom or better the electrons in the inner shells do not interact with the electrons of other atoms. These electrons are the so called non binding electrons. Responsible for the interaction with other atoms are only the valence electrons, or more precisely, the electrons in the outermost electronic shell of the atom. The Schrödinger equation describes the electrons as a wave. Waves can interfere in two way with each other. Firstly, constructive interference, that reduces the energy level of an orbital and secondly destructive interference, that increases the energy of an orbital. The electrons in the low energy orbitals are called bonding electrons, the electrons in the high energy orbitals are called antibonding electrons. An easy to imagine picture of the difference between a bonding and an anti-bonding orbital is, that the electron density distribution is concentrated between the atom cores for bonding orbitals. Despite this, the electron density of anti-bonding orbitals is not between the two nuclei. A non-bonding molecular orbital (NBMO) neither increases nor decreases the bond order between the involved atoms (see equation 1). Non-bonding orbitals are often referred to by the letter n. The energy level of a non-bonding orbital is typically in between the lower energy of a bonding orbital and the higher energy of a corresponding anti-bonding orbital.[1]

The nature of the bond depends on the 3 dimensional probability density of the orbitals. It is possible to distinguish between σ-orbitals, π-orbitals and δ-orbitals. σ-orbitals are orbitals where the linear combination of the atomic orbitals (LCAO) lead to a cylindrical geometry. The electron density is highest directly on the axis between the nuclei. This can be found for bonds consisting of two s-orbitals or one s-orbital and one py-orbital (see figure 1). π-orbitals are orbitals where the two atom orbitals are orientated parallel and where two lobes of the atom orbitals overlap. The p-orbital stands perpendicular to the σ-orbital. This can occur for two py or two pz orbitals (see figure 1). δ-orbitals only occur by the parallel orientation of two d-orbitals, where all four lobes of the orbital overlap (e.g. two dxy-orbitals).[1] Examples of all bonding-types are shown in figure 2.


Figure 1: Different atomic orbitals.

sigma-bond pi-bond   deltabond

                             (a)                                                         (b)                                                         (c)

Figure 2: Electron density distributions of different bonding molecular orbitals:  (a) σ-orbital[2], (b) π-orbital[3] and (c) δ-orbital[4].


Figure 3: MO schemes of (a) O2 and (b) CNmolecule.[1]

Based on the Grotrian-diagram, that shows the energetic level of the separate atomic orbitals, it is possible to create a MO-scheme with the newly created orbitals by the LCAO. The MOscheme of O2 and CNare shown in figure 3 as examples. The higher-energy MO or antibonding is illustrated with an , thus σ and π are the bonding orbitals, σand πthe anti-bonding orbitals. It is possible to calculate a bond order b with the following equation:                   



with n being the bonding electrons and nthe anti-bonding electrons. As obvious in the MOscheme, a higher b means a more stable molecule due to the lower energy level.

The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecule orbital (LUMO) are of special interests. For the example of CN, the HOMO is the 4σ-orbital and the LUMO is the 5π-orbital as shown in the MO scheme in figure 3 (b). HOMO-LUMO transitions are typical in photochemistry.

For more detailed information on the molecular orbital theory, the reader should refer to textbooks of general chemistry.


  1. ATKINS, P. W.; DE PAULA, J.: Physikalische Chemie, 2013, Wiley VCH Verlag GmbH, ISBN 3527332472.
  2. BENJAH-BMM27: Dihydrogen-HOMO-phase-3D-balls, https://commons. wikimedia.org/wiki/File:Dihydrogen-HOMO-phase-3D-balls.png.
  3. BEN MILLS: Ethylene-HOMO-Spartan-3D-balls, https://upload.wikimedia. org/wikipedia/commons/e/e6/Ethylene-HOMO-Spartan-3D-balls. png.
  4. BEN MILLS: Dimolybdenum-Mo2-delta-bond-Spartan-HF-3-21G-3D-side, https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Dimolybdenum-Mo2-delta-bond-Spartan-HF-3-21G-3D-side.png/ 200px-Dimolybdenum-Mo2-delta-bond-Spartan-HF-3-21G-3D-side. png.


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