Structural Biochemistry/Molecular Orbitals

Molecular orbital theory uses group theory to describe the bonding patterns of molecular orbitals. This is determined by the symmetry and energies of the orbitals involved in bonding.

Requirements for overlap that result in bonding

1. Symmetry of the orbitals of the overlap must have the same sign of ϕ. 2. The energies between the orbitals must be similar for a bonding to occur. This energy difference is ~14-15eV max for two orbitals to bond. 3. The distance between the atoms must be short enough that they provide good overlap. However, this can't be too short because it would lead to an interference from repulsive forces. 4. When two orbitals bond, their new overall energy orbital must be lower than the two original bonding orbitals.

Molecular Orbitals Combinations

The two orbitals gradually move closer to each other that their electron clouds overlap, leading to a larger electron cloud. The signs of the orbitals are determined by the shading (dark vs. light) between the two orbitals. The bonding molecular orbital is where this larger electron cloud resides that was combined from the two smaller original orbitals. This is lower in energy than both original bonding orbitals due to stabilization. The opposite of this, called the anti-bonding molecular orbitals, involves a region called the node where there's no electron density from the cancellation of two wave functions. This results in a higher energy than the two original bonding orbitals. An asterisk is usually used to designate an anti-bonding orbital. Nonbonding orbitals can also result from the incompatible symmetry between two atomic orbitals.

Molecular orbital combinations are great when trying to understand how molecules interact and bond! As stated above, molecular orbitals try to show how two electron densities overlap together ultimately binding two molecules. There is always either a positive charge or negative charge on the densities, or one might call it an antibonding property and bonding property due to two different spins. In essence, when two electron densities meet, they either repel or attract. The molecular orbital combinations try to explain how this happens.

Molecular orbitals need to be considered in order to understand how electrons move and exist in a bonded structure. An orbital is a quantum mechanical description of wave function, or in other words where electrons reside. Due to the unpredictable nature of electrons, the molecular orbital theory is not absolutely stable, but we use it more as a tool to help us understand rather than as the absolute rule in chemistry. It is important to remember molecular orbitals are used as tools to explain rather than being the explanation itself!

Diatomic Molecular Orbital Energies

Table 1. Calculated MO energies for diatomic molecules in Hartrees
	H2	Li2	B2	C2	N2	O2	F2
1σg	-0.5969	-2.4523	-7.7040	- 11.3598	- 15.6820	- 20.7296	-26.4289
1σu		-2.4520	-7.7032	-11.3575	-15.6783	-20.7286	-26.4286
2σg		-0.1816	-0.7057	-1.0613	-1.4736	-1.6488	-1.7620
2σu			-0.3637	-0.5172	-0.7780	-1.0987	-1.4997
3σg					-0.6350	-0.7358	-0.7504
1πu			-0.3594	-0.4579	-0.6154	-0.7052	-0.8097
1πg						-0.5319	-0.6682
1s (AO)	-0.5	-2.4778	-7.6953	-11.3255	-15.6289	-20.6686	-26.3829
2s (AO)		-0.1963	-0.4947	-0.7056	-0.9452	-1.2443	-1.5726
2p (AO)			-0.3099	-0.4333	-0.5677	-0.6319	-0.7300

Correlation Diagram

Correlation diagrams, also known as Walsh diagrams, show the mixing of orbitals of the same energy. This shows the effect of moving two atoms together and combining two nuclei into one nucleus. Non-crossing rules also result from the correlation diagram. This rule states that orbitals of the same symmetry will never have their energies interact.

HOMO vs. LUMO

HOMO stands for highest occupied molecular orbital and LUMO stands for the lowest unoccupied molecular orbital. These are called frontier orbitals because they are the first of their respective occupied and unoccupied orbitals. HOMO and LUMO are significant in molecular orbital bonding because it has a greater contribution from the lower energy atomic orbital. Their electron density is also concentrated on the atom with the lower energy level.

Group Orbitals

Group orbitals are collections of matching orbitals on the outer atoms. The ion FHF- is a good example of group orbitals because of the strong hydrogen bonding. The lowest energy group orbitals come from the 2s orbitals of fluorine. The two 2s fluorine orbitals can have matching signs or they can have opposite signs in term of their wave functions. The 2px and 2py orbitals are the same, except they are separated by hydrogen's 1s orbital. As a result, H will only bond to the 2s orbital and pz orbital of fluorine. As a result, five out of the six p orbitals of fluorine will be shown as nonbonding.

Steps to Drawing MO Diagram

1. The point group of the molecule must first be determined. If the molecule is linear with infinite rotation such D∞h, it's often useful to use a similar point group like D2h to approximate.

2. x, y, z axis are assigned to each of the atoms. In general, the highest rotation axis is known as the z-axis. In most non-linear molecules, the y axis is used to point directly at the central atom.

3. A reducible representation is then formed from the s-orbitals of the central atoms. It is then found for all other outer orbitals if the s-orbital is not the most outer orbital. A value of 1 is returned from the reducible representation when the symmetry operation returns exactly the original position of the orbitals. A value of 0 is returned if the symmetry operation changes the location of the orbitals. A value of -1 is returned if the symmetry operation inverts the sign of the orbitals (eg: a p orbital's phase is changed but location stays the same). These values are additive, so a value of E (same position) on four orbitals, it will return a value of 1 x 4 = 4.

4. The reducible representation is then converted to its irreducible representations to determine the symmetry-adapted linear combinations (SALCS) of the orbitals. The total values for the irreducible representations must add up to the reducible representation.

5. The reducible and irreducible representation is then determined for the central atom.

6. The atomic orbitals of the central atom is then matched with the atomic orbitals of the group orbitals based on their symmetry and overlap.

Example (CO2):

1. Since CO2 has a D∞h, the D2h point group is used as a substitute. 2. The z-axis chosen as the primary rotation axis - C2 3. The reducible representation of the outer atom orbitals are determined. There will be four total, 2s, 2px, 2py, 2pz.

	E	C2 (z)	C2 (y)	C2 (x)	i	σ(xy)	σ(xz)	σ(yz)
Reducible Representation (s)	2	2	0	0	0	0	2	2

	E	C2 (z)	C2 (y)	C2 (x)	i	σ(xy)	σ(xz)	σ(yz)
Reducible Representation (2pz)	2	2	0	0	0	0	2	2
Reducible Representation (2px)	2	-2	0	0	0	0	2	-2
Reducible Representation (2py)	2	-2	0	0	0	0	-2	2

4. From the reducible presentation, the irreducible representation is then determined from the character table of D2hhere

This is the irreducible representation for the outer 2s orbitals

	E	C2 (z)	C2 (y)	C2 (x)	i	σ(xy)	σ(xz)	σ(yz)
Ag	1	1	1	1	1	1	1	1
B1u	1	1	-1	-1	-1	-1	1	1

Ag and B1u are the unique sets in the character table. It should be known that the irreducible representation adds up to the reducible representation.

What these symbols mean: A - singly degenerate (only 1 orbital transform): Symmetric with respect to the primary rotation axis B - singly degenerate (only 1 orbital transform)): Anti-symmetric with respect to the primary rotation axis E - doubly degenerate (two orbitals transform together) T - triply degenerate (three orbitals transform together) Subscript 1 - symmetric with respect to perpendicular C2 Subscript 2 - anti-symmetric with respect to perpendicular C2 ' - symmetric with respect to σh '' - anti-symmetric with respect to σh

5. The symmetry of the outer orbitals is matched to the symmetry of the central atom. The central atom has: 2s - Ag 2px - B3u 2py - B2u 2pz - B1u

6. Finally, the molecular orbitals are formed. In the group 1 and 2 orbitals, the oxygen 2s have Ag and B1u symmetry, which can be matched with the Ag symmetry of the 2s carbon orbital. In the group 3 and 4 orbitals, the oxygen 2pz orbital has Ag and B1u symmetry. This allows group 3 to bond with 2s of carbon and group 4 with the 2pz of carbon. This is then done for the rest of the orbitals, matched and drawn in the MO diagram. Like all bonding orbitals, their energies decrease when they bond due to stabilization.

Reference

Miessler, Gary. Inorganic Chemistry. 4th Edition.