MO Theory Intro

While atomic orbitals are localized around or near atoms, molecular orbitals can span the entire molecule.  Molecular Orbital (MO) theory is derived from quantum mechanic's.  There are both quantitative (requires math) and qualitative (pictorial) approaches to MO Theory.

In quantum mechanics to solve the Shrodinger equation, we guess at the structure of the molecular orbitals and see if they make sense.  Our guess is simply a Linear Combination of Atomic Orbitals (LCAO).  Let's examine what this means and build up the molecular orbitals for molecular hydrogen (H-H) both semi-mathematically and pictorially.  In doing so we will describe the molecular orbitals that form the σ bond in molecular hydrogen.

The wave function is simply a mathematical function that describes the probability of finding an electron in a given region.  For a hydrogen 1s electron it would have the following form;

φ = ae^-br

We can write this same equation for hydrogen atoms 1 and 2, for example;

φ₁ = a₁e^-b₁r₁  and  φ₂ = a₂e^-b₂r₂

There is two possible linear combinations which is simply adding them together or subtracting one from the other.  The additive MO, σ, is called the bonding MO, while the subtractive MO, σ*, is called the anti-bonding MO.  Think back to our discussion of electrons behaving as waves and constructive/destructive interference.  The additive MO is constructive while the subtractive is destructive interference.

σ  = φ₁ + φ₂

σ* = φ₁ - φ₂

These are shown pictorially below.  Note the bonding MO, σ, does not have a node between it, while σ* anti-bonding has a node between the two orbitals.  A node is a region of zero electron density.  The number of nodes always increases as you go up the MO diagram.  The 3D Jmol applet shows the molecular orbitals as calculated using quantum mechanics.

The bond order (i.e. # of bonds) is related to the number of electrons in the bonding MO and the antibonding MO as follows.

Thus for the diatomic hydrogen atom the bond order is

Bond Order = (2-0)/2 = 1.  This means that two hydrogen atoms can form a single bond.

The σ* antibondng MO (top) and the σ-bonding MO (bottom) for H₂.

When a system has electron in an anti-bonding orbital, the bond order diminishes. Thus for the diatomic Helium atom the bond order is  (2-2)/2 = 0.  This means that two helium atoms can not form a single bond between them.  Interestingly the He₂²⁺ ion does have a single σ bond, since there are two less electrons.

We can examine π bonds with Qualitative MO theory.  The following example is the MO diagram for the two p orbitals that interact to make up the π bond in an alkene such as ethene.  What is the bond order?