Molecular Orbital Theory

The molecular orbital theory is a technique for modeling the chemical bonding and geometry of molecules and polyatomic ions.

Molecular orbital theory helps explain why some compounds are colored, why an unpaired electron is stable in certain species, and why some molecules have resonance structures.

The molecular orbital theory builds off of valence bond theory and valence shell electron pair repulsion theory to better describe the interactions of electrons within a given molecule and the effects that has on the molecule's physical and chemical properties.

Atomic orbitals are wavefunctions describing the probability distribution of an electron orbiting an atom. While it is impossible to know the exact location of an electron at a given time, the orbital can be used to determine the energy of the electron. Electron energy is important to understanding the behavior and properties of atoms, for example, predicting which electrons will transfer from one atom to another during chemical reactions.

The frequency of the wave function corresponds to the energy of the bond, with a larger frequency signifying higher energy. Since covalent-bonds in molecules require the sharing of electrons between two atoms, the spatial distribution of electrons is important to forming bonds. Each atomic orbital is comprised of a unique, valid set of quantum numbers.

The Schrödinger equation can be used to derive the energies and orbitals of electrons around a single atom.

Whereas an atomic orbital is localized around a single atom, a molecular orbital is delocalized, extending over all the atoms in a molecule. Theoretically, the Schrödinger equation could be used to solve molecular orbitals, but in practice the equation becomes impossible to solve, even for the simplest molecules, without making approximations. Molecular orbital theory approximates the solution to the Schrödinger equation for a molecule.

Energy calculations are used to test the validity of the proposed molecular orbital. Nature minimizes the energy of each orbital, so the best possible orbital is the one with the lowest energy. The simplest guesses that work well in molecular orbital theory are linear combinations of atomic orbitals (LCAOs), which are similar to a weighted average of the atomic orbitals. The number of molecular orbitals in a given molecule is always equal to the sum of all the atomic orbitals in the atoms making up the molecule.

Animation showing the relative energies of atomic and molecular orbitals. Hydrogen's 1s orbitals are on the left and right. The antibonding orbital is on top (highest energy), and the bonding orbital is on the bottom (lowest energy).

Bonding molecular orbitals are always lower in energy than the parent atomic orbitals, where antibonding molecular orbitals are always higher in energy than the parent atomic orbitals. Electrons seek the lowest possible energy state in molecular orbitals, just as they do in atomic orbitals. Therefore, bonding molecular orbitals are stabilizing to the molecule and antibonding molecular orbitals are destabilizing.

The molecular orbitals (MOs) of \(\ce{H2}\) are a simple example system. Each hydrogen atom has a single 1s atomic orbital. Since \(\ce{H2}\) is comprised of two atoms, it has two molecular orbitals. The bonding orbital is an equally weighted sum of the two 1s atomic orbitals and is lower in energy than those orbitals. This is a \( σ\) orbital. The waves of the atomic orbitals show constructive interference, meaning there is an increased probability of finding electrons in the center of the \( σ\) orbital. The antibonding orbital, \( σ∗\), is shaped by destructive interference between the two atomic orbitals. The result is a node in the middle of the orbital, where there is zero probability of finding an electron.

Atomic orbitals other than \(s\) orbitals can interact to form molecular orbitals. Since \(p, d,\) and \(f\) orbitals are not spherically symmetrical, the orbitals must be plotted on a coordinate system to show how they are oriented in space. The shapes of the orbitals gets increasingly complex as the number of electrons increases.

Bonding and anti-bonding in s,p, and d orbitals [3]

Energy level diagrams are used to determine whether or not a molecule will be stable. The atomic orbitals of the two atoms are drawn on the sides of the diagram, with the molecular orbitals in the middle. This diagram can be used to calculate the bond order of the diatomic molecule. To find the bond order, subtract the number of electrons in antibonding molecular orbitals. If the bond order is positive, meaning more atoms are in bonding orbitals than antibonding orbitals, the molecule is stable in nature. A zero or negative bond order indicates an unstable molecule that probably does not exist. In general, a larger bond order correlates to a shorter bond length and a higher bond energy. In other words, the larger the bond order, the closer the atoms are to each other and the harder it will be to separate them.

Energy level diagram for \(\ce{H2}\) [2]

What is the bond order for \(\ce{H2}\)?

Number of electrons in bonding MO's: 2

Number of electrons in antibonding MO's: 0

\((2-0)/2=1\)

We therefore expect \(\ce{H2}\) to be stable, and it is, in fact, a naturally occurring gas.

[1] Image from https://commons.wikimedia.org/wiki/File:H2OrbitalsAnimation.gif under Creative Commons licensing for reuse and modification.

[2] Image from https://commons.wikimedia.org/wiki/File:H2str.png under Creative Commons licensing for reuse and modification.

[3] Image from https://commons.wikimedia.org/wiki/File:BondingandAnti-bondinginteractionsofs,p,andd_orbitals.png under Creative Commons licensing for reused and modification.