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# Molecular Orbital Theory

The Molecular Orbital Theory, initially developed by Robert S. Mullikan, incorporates the wave like characteristics of electrons in describing bonding behavior. In Molecular Orbital Theory, the bonding between atoms is described as a combination of their atomic orbitals. While the Valence Bond Theory and Lewis Structures sufficiently explain simple models, the Molecular Orbital Theory provides answers to more complex questions. In the Molecular Orbital Theory, the electrons are delocalized. Electrons are considered delocalized when they are not assigned to a particular atom or bond (as in the case with Lewis Structures). Instead, the electrons are “smeared out” across the molecule. The Molecular Orbital Theory allows one to predict the distribution of electrons in a molecule which in turn can help predict molecular properties such as shape, magnetism, and Bond Order.

### Introduction

Atoms form bonds by sharing electrons. Atoms can share two, four, or six electrons, forming single, double, and triple bonds respectively. Although it is impossible to determine the exact position of an electron, it is possible to calculate the probability that one will find the electron at any point around the nucleus using the Schrödinger Equation. This equation can help predict and determine the energy and spatial distribution of the electron, as well as the shape of each orbital. The figure below shows the first five solutions to the equation in a three dimensional space. The colors show the phase of the function. In this diagram, blue stands for negative and red stands for positive. Note, however, that the 2s orbital has 2 phases, one of which is not visible because it is inside the other.

### Principles of Molecular Orbital Theory

In molecules, atomic orbitals combine to form molecular orbitals which surround the molecule. Similar to atomic orbitals, molecular orbitals are wave functions giving the probability of finding an electron in certain regions of a molecule. Each molecular orbital can only have 2 electrons, each with an opposite spin. Take a hydrogen molecule (H2) for example. It has two molecular orbitals, an antibonding orbital and a bonding orbital. Compared to the original atomic orbitals, a bonding molecular orbital has lower energy and is therefore more stable. Where the atomic orbitals overlap, there is an increase in electron density and therefore an increase in the intensity of the negative charge. This increase in negative charge causes the nuclei to be drawn closer together. Due to the lower potential energy in molecular bonds than in separate atomic orbitals, it is more energy efficient for the electrons to stay in a molecular bond rather than be pushed back into the 1s orbitals of separate atoms. This is what keeps bonds from breaking apart. A bonding orbital can only be formed if the orbitals of the constituent atoms have the same phase (here represented by colors). The wave functions of electrons of the same phase interfere constructively which leads to bonding.

Principle Details/Examples

Total number of molecular orbits is equal to the total number of atomic orbitals from combining atoms

The molecule H2 is composed of two H atoms. Both H atoms have a 1s orbital, so when bonded together, there are therefore two molecular orbitals.

Bonding molecular orbitals have less energy than the constituent atomic orbitals (before bonding).

Antibonding molecular orbitals have greater energy than the constituent atomic orbitals (before bonding).

Bonding molecular orbitals help stabilize a system of atoms since less energy is associated with bonded atoms as opposed to a system of unbound atoms.

Likewise, antibonding molecular orbitals cause a system to be unstabilized since more energy is associated with bonded atoms than that of a system of unbound atoms.

Following both the Pauli exclusion principle and Hund's rule, electrons fill in orbitals of increasing energy.

Electrons fill orbitals with the lowest energy first. No more than 2 electrons can occupy 1 molecular orbital at a time. Furthermore, all orbitals at an energy level must be filled with one electron before they can be paired.  (see second diagram below)

Molecular orbitals are best formed when composed of Atomic orbitals of like energies.

Molecular Orbital Configuration of Li2:

(s1s)2(s*1s)2(s2s)2

The bonding (s1s)2 and antibonding (s*1s)2 cancel each other out, leaving (s2s)2 as the valence electrons involved in the atoms' bonding.

If the atomic orbitals have the different phases, they interfere destructively and an antibonding molecular orbital is formed (see the top part of the figure below). Antibonding molecular orbitals have a higher energy than the atomic orbitals of their constituent atoms. When antibonds are formed, the interaction creates a decrease in the intensity of the negative charge, which causes a decrease in the plus minus attraction in the molecular bond. This smaller attraction leads to the higher potential energy. This type of bond destabilizes the attraction between atoms, so the number of antibonding orbitals in a molecule must be less than the number of bonding orbitals.

### $$\sigma$$ Bonds

Molecular orbitals that are symmetrical about the axis of the bond are called sigma molecular orbitals, often abbreviated by the Greek letter $$\sigma$$. The diagram to the left shows the 1s orbitals of 2 Hydrogen atoms forming a sigma orbital. There are two types of sigma orbitals formed, antibonding sigma orbitals (abbreviated $$\sigma^*$$), and bonding sigma orbitals (abbreviated $$\sigma$$). In sigma bonding orbitals, the in phase atomic orbitals overlap end to end causing an increase in electron density along the bond axis. Where the atomic orbitals overlap, there is an increase in electron density and therefore an increase in the intensity of the negative charge. This increase in negative charge causes the nuclei to be drawn closer together. In sigma antibonding orbitals ($$\sigma^*$$), the out of phase 1s orbitals interfere destructively which results in a low electron density between the nuclei as seen on the top of the diagram.

The diagram below is a representation of the energy levels of the bonding and antibonding orbitals formed in the hydrogen molecule.  Two molecular orbitals were formed: one antibonding ($$\sigma^*$$) and one bonding ($$\sigma$$).The two electrons in the hydrogen molecule have antiparallel spins. Notice that the $$\sigma^*$$ orbital is empty and has a higher energy than the $$\sigma$$ orbital.

Sigma bonding orbitals and antibonding orbitals can also be formed between p orbitals (shown below). Notice that the orbitals have to be in phase in order to form bonding orbitals. Sigma molecular orbitals formed by p orbitals are often differentiated from other types of sigma orbitals by adding the subscript p below it. So the antibonding orbital shown in the diagram below would be σ*p.

### $$\pi$$ Bonds

The $$\pi$$ bonding bonds as a side to side overlap, which then causes there to be no electron density along the axis, but there is density above and belong the axis. The diagram below shows a $$\pi$$ antibonding molecular orbital and a $$\pi$$ bonding molecular orbital.

2py Orbitals

The two 2py atomic orbitals overlap in parallel to form two $$\pi$$ molecular orbitals which are asymmetrical about the axis of the bond.

2pz orbitals

The two 2pz orbitals overlap to create another pair of pi 2p and pi *2p molecular orbitals. The 2pz-2pz overlap is similar to the 2py-2py overlap because it is just the orbitals of the 2pz rotated 90 degrees about the axis. The new molecular orbitals have the same potential energies as those from the 2py-2py overlap.

Drawing Molecular Orbital Diagrams

• Determine the number of electrons in the molecule.
• Fill the molecular orbitals from bottom to top until all the electrons are added. Describe the electrons with arrows. Put two arrows in each molecular orbital, with the first arrow pointing up and the second pointing down.
• Orbitals of equal energy are half filled with parallel spin before they begin to pair up.

### Bond Orders and Stability of Molecules

Bond Order indicates the strength of the bond with the greater the bond order, the stronger the bond.

$\text{Bond Order}= \dfrac{1}{2} \left(a-b\right)$

where

• $$a$$ is the number of electrons in bonding molecular orbitals and
• $$b$$ is the number of electrons in antibondng molecular orbitals.

If the bond order is zero, then no bonds are produced and the molecule is not stable (for example $$He_2$$). If the Bond Order is 1, then it is a single covalent bond. The higher the Bond Order, the more stable the molecule is. An advantage of Molecular Orbital Theory when it comes to Bond Order is that it can more accurately describe partial bonds (for example in H2+, where the Bond Order=1/2), than Lewis Structures.

### References

1. Petrucci, RH et al. (2007). General Chemistry: Principles and Modern Applications. New Jersey: Pearson Prentice Hall.
2. Dingrando, Laurel, Kathleen Tallman, Nicholas Hainen, and Cheryl Wistrom. Chemistry. Glencoe/McGraw-Hill School Pub Co, 2004.
3. Kotz, John C., Paul M. Treichel, and Gabriela C. Weaver. "Bonding and Molecular Structure:Orbital Hybridization and Molecular Orbitals." Chemistry & Chemical Reactivity. Belmont, CA: Thomson Brooks/Cole, 2006. 457-66. Print.

### Problems

1. What is the molecular orbital diagram for for the diatomic hydrogen molecule, H2? How stable is the molecule? Is it diamagnetic or paramagnetic?
2. What is the molecular orbital diagram for the diatomic helium molecule, He2? How stable is the molecule? Diamagnetic or paramagnetic?
3. What is the molecular orbital diagram for the diatomic oxygen molecule, O2? How stable is the molecule? Diamagnetic or paramagnetic?
4. What is the molecular orbital diagram for the diatomic neon molecule, Ne2? How stable is the molecule? Diamagnetic or paramagnetic?
5. What is the molecular orbital diagram for the diatomic fluorine molecule, F2? How stable is the molecule? Diamagnetic or paramagnetic?

### Solutions

1. The molecular orbital diagram for a diatomic hydrogen molecule, H2, is

• Bond Order  =  1/2(2 - 0)  =  1
• The bond order above zero, so H2 is stable.
• Because there are no unpaired electrons, H2 is diamagnetic.

2. The molecular orbital diagram for a diatomic helium molecule, He2, shows the following.

• Bond Order  =  1/2(2 - 2)  =  0
• bond order is zero so molecule is unstable.
• would be diamagnetic.

3. The molecular orbital diagram for a diatomic oxygen molecule, O2, is

• Bond Order  =  1/2(10 - 6)  =  2
• The bond order is two so the molecule is stable.
• There are two unpaired electrons, so molecule is paramagnetic.

4.The molecular orbital diagram for a diatomic Neon molecule, Ne2, is

• Bond Order  =  1/2(10 - 10)  =  0
• bond order is zero, so Ne2 is unstable.
• diamagnetic

5. The molecular orbital diagram for the diatomic fluorine molecule, F2 is

• B.O. =  1/2(10 - 8)  =  1
• B.O is one so the fluorine molecule is stable.
• Because all of the electrons are paired, F2 is diamagnetic.

22:39, 29 Jan 2015

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