LON-CAPA MOTheory

Molecular Orbital Theory

Introduction

In our discussion of valence bond theory, we saw that chemical bonds are formed when two nuclei share a pair of electrons between them. The sharing lowers the potential energy of both electrons by exposing them to increased nuclear attractions. The decrease in potential energy is greatest when the two electrons are confined to a region between the two nuclei. This type of bond is described as a localized bond. For example, in methane, CH₄, each pair of electrons is considered to be confined to the region between the carbon nucleus and a hydrogen nucleus. One of the limitations of valence bond theory is that it assumes all bonds are localized bonds. As we will see, this is not a valid assumption.

Another limitation of valence bond theory is that is sometimes makes incorrect predictions. The case of dioxygen provides a good example. Consider the two Lewis structures for dioxygen shown in Figure 1.

Figure 1

Two Lewis Structures for O₂

Exercise 1 Which of the alternatives in Figure 1 best represents the structure of dioxygen according to valence bond theory? 1 2

Since structure 2 implies that all of the electrons are paired, you should expect dioxygen to be diamagnetic. It is not. Several lines of experimental evidence indicate that dioxygen is paramagnetic, i.e. it contains at least one unpaired electron.

Whenever a theory makes incorrect predictions or is shown to be inconsistent with experimental fact, chemists have two choices:

Modify the theory to accomodate the experimental facts.
Create an alternative theory.

We will see how chemists have modified valence bond theory to deal the limitations of localized bonds when we discuss resonance theory. But first we will consider molecular orbital (MO) theory as an alternative to valence bond theory.

Molecular Orbital Theory

Chemists view molecules as combinations of atoms. They consider molecular orbitals as combinations of atomic orbitals, specifically as linear combinations of atomic orbitals. Don't let this term put you off. It simply means that molecular orbitals are formed by adding and subtracting atomic orbitals. Remember, electrons behave like waves, and their wavelike behavior may be described by mathematical functions similar to the sine or cosine function. These mathematical functions are what we call orbitals. The addition of atomic wave functions is analogous to constructive interference that occurs with sound waves. The subtraction corresponds to destructive interference.

Consider the simplest molecule, dihydrogen. In our discussion of Lewis structures, we imagined a process in which two hydrogen atoms came together to form a molecule of dihydrogen. That process represents a linear combination of the two hydrogen atoms' 1s atomic wave functions. Mathematically molecular orbital theorists describe the process by the equation y = 1s_A + 1s_B, where y (psi) stands for the molecular orbital, while 1s_A and 1s_B represent the atomic orbitals for H_A and H_B, respectively. y is the molecular orbital equivalent of a s bond in valence bond theory.

A fundamental rule of molecular orbital theory is that the number of molecular orbitals must be equal to the number of atomic orbitals. For two hydrogen atoms, there are two atomic orbitals, which means that there must be two molecular orbitals for dihydrogen. The second molecular orbital is described by the equation y^* = 1s_A - 1s_B. There is no valence bond equivalent of y^*.

Figure 2 illustrates the energy changes that accompany these linear combinations of atomic orbitals. The molecular orbital y corresponds to the minimum of the potential energy diagram we considered during our introductory discussion of valence bond theory.

Figure 2

The Molecular Orbital Diagram for Dihydrogen

Figure 3 offers an alternative description of the information shown in Figure 2. The colored spheres and elipses represent regions of electron density about the nuclei, which are shown as dots at the centers of the two 1s atomic orbitals.

Figure 3

An Orbital Picture of the Formation of Dihydrogen

There are several features of Figure 3 that deserve comment:

The electron distribution about the separated hydrogen atoms is spherically symmetrical. In contrast, the electron distribution in the MO y is not. Rather, the electron density is concentrated in the inter-nuclear region where both electrons can experience the Coulombic attraction of both nuclei. (This is the result of constructive interference.)
In the ground state there are no electrons in y^*. However, if an electron is excited from y to y^*, the probability of the resulting electron density being in the inter-nuclear region of y^*is very low. (This is the result of destructive interference.)
As we've seen previously, the inter-nuclear distance in y is less than the sum of the atomic radii of the two hydrogen atoms. The inter-nuclear distance in y^*is greater than the sum of the atomic radii of the two hydrogen atoms.

Before we turn our attention to the MO diagram of dioxygen, there is one additional aspect of Figures 1 and 2 that you should know. The MOs y and y^*are called the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO), respectively. These terms will be useful when we discuss chemical reactivity and spectroscopy. Chemical reactions involve the transfer of electron density from the HOMO of one reactant to the LUMO of another. Spectroscopy is the interaction of light with matter, an interaction which alters the populations of different energy states.

Exercise 2 According to the Bohr model of the atom, what value of n corresponds to the LUMO of dihydrogen?

Our discussion of the MO description of dihydrogen was a necessary interlude so that we could understand how MO theory accounts for the paramagnetic behavior of dioxygen. Figure 4 shows the MO diagram for dioxygen. Only the valence shell atomic orbitals are shown.

Figure 4

An MO Diagram for Dioxygen

Don't worry about the details of this MO diagram. The important feature of the figure is that there are two HOMOs that have the same energy. Each one contains a single electron. According to Hund' Rule the energy of the system will be lower if the spins of these two electrons are unpaired than if they are paired. In other words, the most stable form of dioxygen should be paramagnetic, not diagmagnetic.

Exercise 3 Which of the structures in Figure 1 is more consistent with the MO diagram shown in Figure 4? 1 2

As mentioned earlier, we will consider MO theory again when we discuss chemical reactivity and spectroscopy. For the moment the message is that there is more than one way to think about molecules. Each approach has advantages and disadvantages. MO theory is superior to VB theory in several respects. Despite its shortcomings however, the convenience and simplicity of valence bond theory make it the method of choice for most organic chemists.

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