Molecular Orbital Theory

The goal of molecular orbital theory is to describe molecules in a similar way to how we describe atoms, that is, in terms of orbitals, orbital diagrams, and electron configurations. For example, to give you a glimpse at where we are headed, the following are orbital diagrams for O₂ and O.

O₂ O

Each line in the molecular orbital diagram represents a molecular orbital, which is the volume within which a high percentage of the negative charge generated by the electron is found. The molecular orbital volume encompasses the whole molecule. We assume that the electrons would fill the molecular orbitals of molecules like electrons fill atomic orbitals in atoms.

• The molecular orbitals are filled in a way that yields the lowest potential energy for the molecule.

• The maximum number of electrons in each molecular orbital is two. (We follow the Pauli exclusion principle.)

• Orbitals of equal energy are half filled with parallel spin before they begin to pair up. (We follow Hund's Rule.)

Before we continue with a description of a model used to generate molecular orbital diagrams, lets get a review of light and electron waves and how two waves can interact. The wave description of light describes the effect that the light has on the space around it. This effect is to generate an oscillating electric and magnetic fields. These fields can vary in intensity, which is reflected in varying brightness of light.

The wave description of the electron describes the variation in the intensity of negative charge generated by the electron.

Light waves can interact in-phase, which leads to an increase in the intensity of the light (brighter) and out-of-phase, which leads to a decrease in the intensity of the light (less bright). Electron waves can also interact in-phase and out-of-phase. In-phase interaction leads to an increase in the intensity of the negative charge. Out-of-phase interaction leads to a decrease in the intensity of the negative charge.

One common approximation that allows us to generate molecular orbital diagrams for some small diatomic molecules is called the Linear Combination of Atomic Orbitals (LCAO) approach. The following assumptions lie at the core of this model.

• Molecular orbitals are formed from the overlap of atomic orbitals.

• Only atomic orbitals of about the same energy interact to a significant degree.

• When two atomic orbitals overlap, they interact in two extreme ways to form two molecular orbitals, a bonding molecular orbital and an antibonding molecular orbital.

For example, our model assumes that two 1s atomic orbitals can overlap in two extreme ways to form two molecular orbitals. One of the ways the atomic orbitals interact is in-phase, which leads to wave enhancement similar to the enhancement of two in-phase light waves. Where the atomic orbitals overlap, the in-phase interaction leads to an increase in the intensity of the negative charge in the region where they overlap. This creates an increase in negative charge between the nuclei and an increase in the plus-minus attraction between the electron charge and the nuclei for the atoms in the bond. The greater attraction leads to lower potential energy. Because electrons in the molecular orbital are lower potential energy than in separate atomic orbitals, energy would be required to shift the electrons back into the 1s orbitals of separate atoms. This keeps the atoms together in the molecule, so we call this orbital a bonding molecular orbital. The molecular orbital formed is symmetrical about the axis of the bond. Symmetrical molecular orbitals are called sigma, σ, molecular orbitals. The symbol σ_1s is used to describe the bonding molecular orbital formed from two 1s atomic orbitals.

The second way that two atomic orbitals interact is out-of-phase. Where the atomic orbitals overlap, the out-of-phase interaction leads to a decrease in the intensity of the negative charge. This creates a decrease in negative charge between the nuclei and a decrease in the plus-minus attraction between the electron charge and the nuclei for the atoms in the bond. The lesser attraction leads to higher potential energy. The electrons are more stable in the 1s atomic orbitals of separate atoms, so electrons in this type of molecular orbital destabilize the bond between atoms. We call molecular orbitals of this type antibonding molecular orbitals. The molecular orbital formed is symmetrical about the axis of the bond, so it is a sigma molecular orbital with a symbol of σ*_1s. The asterisk indicates an antibonding molecular orbital.

The following diagram shows the bonding and antibonding molecular orbitals formed from the interaction of two 1s atomic orbitals.

When two larger atoms atoms combine to form a diatomic molecule (like O₂, F₂, or Ne₂), more atomic orbitals interact. The LCAO approximation assumes that only the atomic orbitals of about the same energy interact. For O₂, F₂, or Ne₂, the orbital energies are different enough so only orbitals of the sameenergy interact to a significant degree.

Like for hydrogen, the 1s from one atom overlaps the 1s from the other atom to form a σ_1s bonding molecular orbital and a σ*_1s antibonding molecular orbital. The shapes would be similar to those formed from the 1s orbitals for hydrogen. The 2s atomic orbital from one atom overlaps the 2s from the other atom to form a σ_2s bonding molecular orbital and a σ*_2s antibonding molecular orbital. The shapes of these molecular orbitals would be similar to those for the σ_1s and σ*_1s molecular orbitals. Both σ_2s and σ*_2s molecular orbitals are higher energy and larger than the σ_1s and σ*_1s molecular orbitals.

The p atomic orbitals of the two atoms can interact in two different ways, parallel or end-on. The molecular orbitals are different for each type of interaction. The end-on interaction between two 2p_x atomic orbitals yields sigma molecular orbitals, which are symmetrical about the axis of the bond.

The two 2p_y atomic orbitals overlap in parallel and form two pi molecular orbitals. Pi molecular orbitals are asymmetrical about the axis of the bond.

The 2p_z-2p_z overlap generates another pair of π_2p and π*_2p molecular orbitals. The 2p_z-2p_z overlap is similar to the The 2p_y-2p_y overlap. To visualize this overlap, picture all of the orbitals in the image above rotated 90 degrees so the axes that run through the atomic and molecular orbitals are perpendicular to the screen (paper). The molecular orbitals formed have the same potential energies as the molecular orbitals formed from the 2p_y-2p_yoverlap.

There is less overlap for the parallel atomic orbitals. When the interaction is in-phase, less overlap leads to less electron charge enhancement between the nuclei. This leads to less electron charge between the nuclei for the pi bonding molecular orbital than for the sigma bonding molecular orbital. Less electron character between the nuclei means less plus-minus attraction, less stabilization, and higher potential energy for the pi bonding molecular orbital compared to the sigma bonding molecular orbital.

When the interaction is out-of-phase, less overlap leads to less shift of electron charge from between the nuclei. This leads to more electron charge between the nuclei for the pi antibonding molecular orbital than for the sigma antibonding molecular orbital. More electron charge between the nuclei means more plus-minus attraction and lower potential energy for the pi antibonding molecular orbital compared to the sigma antibonding molecular orbital.

The expected molecular orbital diagram from the overlap of 1s, 2s and 2p atomic orbitals is as follows. We will use this diagram to describe O₂, F₂, Ne₂, CO, and NO.

We use the following procedure when drawing molecular orbital diagrams.

• Determine the number of electrons in the molecule. We get the number of electrons per atom from their atomic number on the periodic table. (Remember to determine the total number of electrons, not just the valence electrons.)

• 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.

We describe the stability of the molecule with bond order.

bond order = 1/2 (#e- in bonding MO's - #e- in antibonding MO's)

We use bond orders to predict the stability of molecules.

• If the bond order for a molecule is equal to zero, the molecule is unstable.

• A bond order of greater than zero suggests a stable molecule.

• The higher the bond order is, the more stable the bond.

We can use the molecular orbital diagram to predict whether the molecule is paramagnetic or diamagnetic. If all the electrons are paired, the molecule is diamagnetic. If one or more electrons are unpaired, the molecule is paramagnetic.

EXAMPLES:

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

• The bond order is 1. Bond Order = 1/2(2 - 0) = 1

• The bond order above zero suggests that 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.

• The bond order is 0 for He₂. Bond Order = 1/2(2 - 2) = 0

• The zero bond order for He₂ suggests that He₂ is unstable.

• If He₂ did form, it would be diamagnetic.

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

• O₂ has a bond order of 2. Bond Order = 1/2(10 - 6) = 2

• The bond order of two suggests that the oxygen molecule is stable.

• The two unpaired electrons show that O₂ is paramagnetic.

4. The molecular orbital diagram for a diatomic fluorine molecule, F₂, is

• F₂ has a bond order of 1. Bond Order = 1/2(10 - 8) = 1

• The bond order of one suggests that the fluorine molecule is stable.

• Because all of the electrons are paired, F₂ is diamagnetic.

5. The molecular orbital diagram for a diatomic neon molecule, Ne₂, is

• Ne₂ has a bond order of 0. Bond Order = 1/2(10 - 10) = 0

• The zero bond order for Ne₂ suggests that Ne₂ is unstable.

• If Ne₂ did form, it would be diamagnetic.

We can describe diatomic molecules composed of atoms of different elements in a similar way. The bond between the carbon and oxygen in carbon monoxide is very strong despite what looks like a strange and perhaps unstable Lewis Structure.

The plus formal charge on the more electronegative oxygen and the minus formal charge on the less electronegative carbon would suggest instability. The molecular orbital diagram predicts CO to be very stable with a bond order of three.

We predict the nitrogen monoxide molecule to be unstable according to the Lewis approach to bonding.

The unpaired electron and the lack of an octet of electrons around nitrogen would suggest an unstable molecule. NO is actually quite stable. The molecular orbital diagram predicts this by showing the molecule to have a bond order of 2.5.

http://www.mpcfaculty.net/mark_bishop/molecular_orbital_theory.htm

Molecular Orbital Theory

The goal of molecular orbital theory is to describe molecules in a similar way to how we describe atoms, that is, in terms of orbitals, orbital diagrams, and electron configurations. For example, to give you a glimpse at where we are headed, the following are orbital diagrams for O₂ and O.

O₂ O

Each line in the molecular orbital diagram represents a molecular orbital, which is the volume within which a high percentage of the negative charge generated by the electron is found. The molecular orbital volume encompasses the whole molecule. We assume that the electrons would fill the molecular orbitals of molecules like electrons fill atomic orbitals in atoms.

• The molecular orbitals are filled in a way that yields the lowest potential energy for the molecule.

• The maximum number of electrons in each molecular orbital is two. (We follow the Pauli exclusion principle.)

• Orbitals of equal energy are half filled with parallel spin before they begin to pair up. (We follow Hund's Rule.)

Before we continue with a description of a model used to generate molecular orbital diagrams, lets get a review of light and electron waves and how two waves can interact. The wave description of light describes the effect that the light has on the space around it. This effect is to generate an oscillating electric and magnetic fields. These fields can vary in intensity, which is reflected in varying brightness of light.

The wave description of the electron describes the variation in the intensity of negative charge generated by the electron.

Light waves can interact in-phase, which leads to an increase in the intensity of the light (brighter) and out-of-phase, which leads to a decrease in the intensity of the light (less bright). Electron waves can also interact in-phase and out-of-phase. In-phase interaction leads to an increase in the intensity of the negative charge. Out-of-phase interaction leads to a decrease in the intensity of the negative charge.

One common approximation that allows us to generate molecular orbital diagrams for some small diatomic molecules is called the Linear Combination of Atomic Orbitals (LCAO) approach. The following assumptions lie at the core of this model.

• Molecular orbitals are formed from the overlap of atomic orbitals.

• Only atomic orbitals of about the same energy interact to a significant degree.

• When two atomic orbitals overlap, they interact in two extreme ways to form two molecular orbitals, a bonding molecular orbital and an antibonding molecular orbital.

For example, our model assumes that two 1s atomic orbitals can overlap in two extreme ways to form two molecular orbitals. One of the ways the atomic orbitals interact is in-phase, which leads to wave enhancement similar to the enhancement of two in-phase light waves. Where the atomic orbitals overlap, the in-phase interaction leads to an increase in the intensity of the negative charge in the region where they overlap. This creates an increase in negative charge between the nuclei and an increase in the plus-minus attraction between the electron charge and the nuclei for the atoms in the bond. The greater attraction leads to lower potential energy. Because electrons in the molecular orbital are lower potential energy than in separate atomic orbitals, energy would be required to shift the electrons back into the 1s orbitals of separate atoms. This keeps the atoms together in the molecule, so we call this orbital a bonding molecular orbital. The molecular orbital formed is symmetrical about the axis of the bond. Symmetrical molecular orbitals are called sigma, σ, molecular orbitals. The symbol σ_1s is used to describe the bonding molecular orbital formed from two 1s atomic orbitals.

The second way that two atomic orbitals interact is out-of-phase. Where the atomic orbitals overlap, the out-of-phase interaction leads to a decrease in the intensity of the negative charge. This creates a decrease in negative charge between the nuclei and a decrease in the plus-minus attraction between the electron charge and the nuclei for the atoms in the bond. The lesser attraction leads to higher potential energy. The electrons are more stable in the 1s atomic orbitals of separate atoms, so electrons in this type of molecular orbital destabilize the bond between atoms. We call molecular orbitals of this type antibonding molecular orbitals. The molecular orbital formed is symmetrical about the axis of the bond, so it is a sigma molecular orbital with a symbol of σ*_1s. The asterisk indicates an antibonding molecular orbital.

The following diagram shows the bonding and antibonding molecular orbitals formed from the interaction of two 1s atomic orbitals.

When two larger atoms atoms combine to form a diatomic molecule (like O₂, F₂, or Ne₂), more atomic orbitals interact. The LCAO approximation assumes that only the atomic orbitals of about the same energy interact. For O₂, F₂, or Ne₂, the orbital energies are different enough so only orbitals of the sameenergy interact to a significant degree.

Like for hydrogen, the 1s from one atom overlaps the 1s from the other atom to form a σ_1s bonding molecular orbital and a σ*_1s antibonding molecular orbital. The shapes would be similar to those formed from the 1s orbitals for hydrogen. The 2s atomic orbital from one atom overlaps the 2s from the other atom to form a σ_2s bonding molecular orbital and a σ*_2s antibonding molecular orbital. The shapes of these molecular orbitals would be similar to those for the σ_1s and σ*_1s molecular orbitals. Both σ_2s and σ*_2s molecular orbitals are higher energy and larger than the σ_1s and σ*_1s molecular orbitals.

The p atomic orbitals of the two atoms can interact in two different ways, parallel or end-on. The molecular orbitals are different for each type of interaction. The end-on interaction between two 2p_x atomic orbitals yields sigma molecular orbitals, which are symmetrical about the axis of the bond.

The two 2p_y atomic orbitals overlap in parallel and form two pi molecular orbitals. Pi molecular orbitals are asymmetrical about the axis of the bond.

The 2p_z-2p_z overlap generates another pair of π_2p and π*_2p molecular orbitals. The 2p_z-2p_z overlap is similar to the The 2p_y-2p_y overlap. To visualize this overlap, picture all of the orbitals in the image above rotated 90 degrees so the axes that run through the atomic and molecular orbitals are perpendicular to the screen (paper). The molecular orbitals formed have the same potential energies as the molecular orbitals formed from the 2p_y-2p_yoverlap.

There is less overlap for the parallel atomic orbitals. When the interaction is in-phase, less overlap leads to less electron charge enhancement between the nuclei. This leads to less electron charge between the nuclei for the pi bonding molecular orbital than for the sigma bonding molecular orbital. Less electron character between the nuclei means less plus-minus attraction, less stabilization, and higher potential energy for the pi bonding molecular orbital compared to the sigma bonding molecular orbital.

When the interaction is out-of-phase, less overlap leads to less shift of electron charge from between the nuclei. This leads to more electron charge between the nuclei for the pi antibonding molecular orbital than for the sigma antibonding molecular orbital. More electron charge between the nuclei means more plus-minus attraction and lower potential energy for the pi antibonding molecular orbital compared to the sigma antibonding molecular orbital.

The expected molecular orbital diagram from the overlap of 1s, 2s and 2p atomic orbitals is as follows. We will use this diagram to describe O₂, F₂, Ne₂, CO, and NO.

We use the following procedure when drawing molecular orbital diagrams.

• Determine the number of electrons in the molecule. We get the number of electrons per atom from their atomic number on the periodic table. (Remember to determine the total number of electrons, not just the valence electrons.)

• 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.

We describe the stability of the molecule with bond order.

bond order = 1/2 (#e- in bonding MO's - #e- in antibonding MO's)

We use bond orders to predict the stability of molecules.

• If the bond order for a molecule is equal to zero, the molecule is unstable.

• A bond order of greater than zero suggests a stable molecule.

• The higher the bond order is, the more stable the bond.

We can use the molecular orbital diagram to predict whether the molecule is paramagnetic or diamagnetic. If all the electrons are paired, the molecule is diamagnetic. If one or more electrons are unpaired, the molecule is paramagnetic.

EXAMPLES:

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

• The bond order is 1. Bond Order = 1/2(2 - 0) = 1

• The bond order above zero suggests that 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.

• The bond order is 0 for He₂. Bond Order = 1/2(2 - 2) = 0

• The zero bond order for He₂ suggests that He₂ is unstable.

• If He₂ did form, it would be diamagnetic.

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

• O₂ has a bond order of 2. Bond Order = 1/2(10 - 6) = 2

• The bond order of two suggests that the oxygen molecule is stable.

• The two unpaired electrons show that O₂ is paramagnetic.

4. The molecular orbital diagram for a diatomic fluorine molecule, F₂, is

• F₂ has a bond order of 1. Bond Order = 1/2(10 - 8) = 1

• The bond order of one suggests that the fluorine molecule is stable.

• Because all of the electrons are paired, F₂ is diamagnetic.

5. The molecular orbital diagram for a diatomic neon molecule, Ne₂, is

• Ne₂ has a bond order of 0. Bond Order = 1/2(10 - 10) = 0

• The zero bond order for Ne₂ suggests that Ne₂ is unstable.

• If Ne₂ did form, it would be diamagnetic.

We can describe diatomic molecules composed of atoms of different elements in a similar way. The bond between the carbon and oxygen in carbon monoxide is very strong despite what looks like a strange and perhaps unstable Lewis Structure.

The plus formal charge on the more electronegative oxygen and the minus formal charge on the less electronegative carbon would suggest instability. The molecular orbital diagram predicts CO to be very stable with a bond order of three.

We predict the nitrogen monoxide molecule to be unstable according to the Lewis approach to bonding.

The unpaired electron and the lack of an octet of electrons around nitrogen would suggest an unstable molecule. NO is actually quite stable. The molecular orbital diagram predicts this by showing the molecule to have a bond order of 2.5.

http://www.mpcfaculty.net/mark_bishop/molecular_orbital_theory.htm