Molecular Orbital Theory (Incomplete)

Molecular Orbital Theory or MO Theory utilizes concepts of atomic orbitals to rationalize general behaviour of chemicals. MO Theory is based on the “mixing” or combining of orbitals. As two atoms form a successful covalent bond, their valence electrons become shared. In the case of atomic orbitals, only major interactions between the nucleus and electrons are considered; no electron-electron interactions are considered. However, when electrons on shared, their behaviour changes drastically.

The simple wavefunctions that defined electron motion with respect to one nucleus is no longer applicable. The complications that arise from the interactions of a second nucleus are very complex, and difficult to accurately model. The resulting mathematical equations are very complex. MO Theory takes a more general approach. Instead of combining the wavefunctions and adding various correction factors, only the visible representations are combined. The resulting linear combinations of the atomic orbitals give rise to a variety of molecular orbitals.

In order for orbitals to mix, they need to be of comparable energy. If the energy levels are two high or low, they have no net interaction with one another. Each suitable pairing has a bonding orbital and an anti-bonding combination. The bonding and antibonding orbitals between two s-orbitals is denoted denoted σ and σ* respectively. Bonding and antibonding between a s-orbital and p-orbital or two p orbitals is denoted by π and π* respectively. This notation will be left off the MO diagrams produced due to space restrictions.

When drawing up the angular portions of the orbitals, different colours were used to denote the various phases. While there exist no real differences between the two phases, there are theoretical considerations. When two phases of the same type align, they generate a mutual or constructive field that leads to bonding. When two phases of the opposite type align, they generate a “destructive” field that leads to anti-bonding.

There will be two examples presented: Hydrogen Fluoride and Carbon Monoxide.

Hydrogen Fluoride (HF)

Hydrogen Fluoride (HF) is an excellent example to introduce the primary concepts of MO Theory.

Hydrogen has only one single valence electron in an s-orbital. It has an electron configuration of 1s1. The 1s refers to the valence shell. The regular “1” refers to its general energy level (1st row element), and the s denotes its shape type (a spherical shape). The superscript “1” means that it has one electron in this shell.

Fluorine has seven valence electrons available for bonding. It has an electron configuration of [He]2s22p5. The [He] symbol means that it has base electron configuration that is the same as helium (1s2). The square bracket notation is used for both convenience and clarity. Any electrons that are included in base element (in this case He) are considered non-reactive. The two electrons in the 1s shell do not participate in bonding. The 2s22p5 means that there are seven electrons available for bonding: two reside in a 2s orbital, and five are in a 2p orbital. This following is a picture of their relative energies.

In total, the combination of the single electron of hydrogen and seven of fluorine means that there are eight electrons available for bonding.  We begin by drawing the orbitals that are available for bonding, and their relative energies (top left). The first thing to note is that the 2s orbital of the F atom is too low to participate in bonding with the 1s of the H (top center). This means that we draw a line of the same energy level as the 2s orbital in the middle.  While it does not participate in bonding it still carries electrons. Also, there are two orbitals that have unfavorable geometry. When the axis of the angular elements is perpendicular to the H atom, successful overlap cannot occur (top right).

Of all the bonding possibilties, the 1s of the H can mix with a p orbital that lies on the same axis. This generates one bonding (low energy) and  one anti-bonding pair high energy (bottom left). A pair of dashed lines are used to draw together the mixing orbitals (bottom center). Now will fill up the diagram with the either electrons available, starting from the bottom up (bottom right). .This completes the diagram for Hydrogen Flouride.

Carbon Monoxide (CO)

Carbon Monoxide (CO) will build upon the general principles that the hydrogen fluoride example introduced, but will be slightly more involved.

Carbon has an electron configuration of [He]2s22p2, so it has four valence electrons available for bonding. Oxygen has six valence electrons from its electronic configuration of [He]2s22p4. This is a total of 10 electrons.

Unlike in the case of hydrogen fluoride, both atoms have s and p orbitals that are available for bonding. As a result, there are more possible interactions. Start with the relative energy levels of all the orbitals (top left).  The s-Orbital of oxygen is far to low to participate in bonding, and is drawn into the middle.  If the s-orbital of carbon, and one of the p-orbitals from carbon align (such that the angular element of the p-orbital is on an axis that runs through the s-orbital), there can be some successful overlap (top right). Another pair with bonding character lie on the axis that runs through both atoms. Imagine a pair of p-orbitals from both atoms that have the same axis of symmetry. One lobe of the p from each of the atoms can mix together and overlap. This creates a moderate bonding orbital (top right).

In the case of the last four orbitals, orbital overlap is a possibility, but it requires that the two atoms be very close together, and the overlap is very minimal. The axis of these orbitals are parallel, and overlap is due mainly to the width of the lobes of the orbitals.  As such, the energy reduction from the orbital mixing is minimal. However, the resulting anti-bonding combination is only slightly higher in energy. (bottom right). The bottom right diagram shows filling of the ten electrons.

Introduction to Quantum Chemistry
Page 1
Page 2
Schrodinger Equation
Spherical Harmonics
Periodic Table
MO Theory