chemical bonding - molecular orbital theory

Molecular orbital (MO) theory is a method for predicting molecular bonds and structure in which electrons are not assigned to individual bonds between atoms – as in valence shell electron pair repulsion (VESPR) theory – but as interacting with the nuclei in the molecule as a whole. Electrons are allowed to move around atomic nuclei in trajectories explained by mathematical functions called orbitals. As atoms bond to form molecules, their atomic orbitals combine to form molecular orbitals. Atomic orbital functions are described by Schrödinger’s wave equation. The Schrödinger wave equation is a linear equation and thus molecular orbitals can be described by simple addition of the atomic orbitals; the linear combinations of atomic orbitals (LCAO).

With advances that have been made in computational chemistry, the LCAO approximations can be further refined by applying the density functional theory or Hartree–Fock models to the Schrödinger equation.

A molecular orbital is really a mathematical function that describes the wave-like behavior of an electron in a molecule and used to determine the probability of finding an electron in any particular location. Molecular orbitals are delocalized over the entire molecule; they can surround many atoms in a molecule and thus can contain many valence electrons, therefore any electron in a molecule may be found anywhere in that molecule. At the same time, in accordance with Hund’s rule and the Pauli exclusion principle, each MO can contain only 2 electrons each with opposite spin.

When two atoms bond, the electrons occupy a molecular orbital whose wave function is analogous to that of an atomic orbital. For example, in the case of a diatomic molecule, LCAO declares that the molecular wave function can be built as a linear combination of the wavefunctions for each atom. Thus, two molecular orbital wavefunctions are formed

\[\ce{Ψ\,=\,c_{1}ψ_{1}\,+\,c_{2}ψ_{1}}\]

\[\ce{Ψ} *{=\,c_{1}ψ_{1}\, -\,c_{2}ψ_{1}}\]

where \(\ce{ψ_{1}}\) and \(\ce{ψ_{2}}\) are the atomic wavefunctions for atoms 1 and 2; \(\ce{c_{1}}\) and \(\ce{c_{2}}\) are variable coefficients that change depending on the energies of the atomic orbitals. \(\ce{Ψ}\) and \(\ce{Ψ}\)* represent the two independent quantum states for electrons in the diatomic molecule - the molecular wavefunctions for the bonding and antibonding molecular orbitals which will be described in the next section.

Molecular wavefunctions are thought of as combinations of atomic wavefunctions. When two atoms come into close proximity their orbitals overlap producing an area of high electronic probability density and a molecular orbital is formed where the atoms are held together by electrostatic attraction between the positively charged nuclei and negatively charged electrons.

These combinations of wavefunctions can be constructive (in-phase) or destructive (out of phase). When they are constructive, regions of higher probability of electron density are produced and when destructive, there is no chance of finding an electron in that region.

constructive, in-phase

destructive, out-of-phase

The term phase is a consequence of the mathematical description of the wavefunction. Phase is usually discussed in terms of + sign or – sign or in different colors. The sign of the phase doesn’t have meaning except to distinguish constructive or destructive interference when combining atomic orbitals to form molecular orbitals.

Bonding molecular orbitals are formed when atomic orbitals have constructive interaction and have lower energy than the individual atomic orbitals. Bonding MOs are stabilizing since the bound atoms have less energy than unbound atoms. Anti-bonding MOs are formed when atomic orbitals have destructive interaction – there is a nodal plane where the wavefunction of the anti-bonding orbital is zero between the two atoms and there is no probability of finding an electron.

EXAMPLE. Constructive and destructive interaction of \(\ce{H_{2}}\) molecular orbitals.

When atomic orbitals come into proximity and overlap, the overlap in-phase (+ with +) or out-of-phase (- with +). When the s orbitals of \(\ce{H_{2}}\) overlap, their linear combination can be an in-phase \(\ce{σ_{s}}\) bonding molecular orbital or an out-of-phase \(\ce{σ_{s}}\)* anti-bonding molecular orbital.

Anti-bonding MOs have decreased electron density and higher energy than the individual atomic orbitals. Non-bonding MOs are formed when atomic orbitals are have no interaction beause their symmetries are not compatible. Therefore, the original energy of the individual atomic orbitals is retained. As seen in the example above, when two atomic s-orbitals are combined, two types of molecular orbitals are formed. The in-phase, constructive form produces a low energy, bonding, molecular orbital denoted \(\ce{σ_{s}}\) with the electrons attracted to both atomic nuclei and the electron probability density concentrated between the atomic nuclei. This is the single bond seen in a Lewis structure and VSEPR, it corresponds to the sharing of one pair of electrons. The electron probability density concentrated between the two nuclei is the essential component of a σ-bond. On the other hand, the out-of-phase, destructive form produces a high energy, anti-bonding molecular orbital denoted \(\ce{σ_{s}}\)* that forms a node of low-to-zero electronic probability density between the atomic nuclei. With the electrons in \(\ce{σ_{s}}\)* so far away from the center of the atomic nuclei, the attraction between the nuclei and the electrons pulls the nuclei apart. An antibonding molecular orbital weakens the atomic bond because it has higher energy than the two atoms separately. When a nodal plane is formed in an anti-bonding orbital it means that the orbital’s electron density is mostly outside the bonding region and the nuclei are thus pulled away from each other. If the bonding orbitals are filled, then electrons will start occupying anti-bonding orbitals. As anti-bonding orbitals have higher energy than bonding orbitals, bonding is not favored.

EXAMPLE. LCAO and MO for hydrogen and helium.

Hydrogen \(\ce{H_{2}}\)

H has one proton, no neutrons, and one electron. When hydrogen atoms are in close proximity atomic orbitals (1s) combine to become molecular orbitals resulting in \(\ce{H_{2}}\) with one bonding molecular orbital of lower energy (1σ) and one anti-bonding molecular orbital of higher energy (1σ*). Because the bonding molecular orbital is more stable and lower energy than the individual atomic orbitals, \(\ce{H_{2}}\) is more stable than two isolated H atoms. An antibonding orbital with two electrons in it would be less stable and does not form.

MO diagram for \(\ce{H_{2}}\).

Each H atom has one electron (blue up and down arrows for spin up and down) in the 1s-orbital. Remember that the Pauli exclusion principle states that two electrons can only share an orbital if their spins are opposite. Since \(\ce{H_{2}}\) has only two electrons, both electrons will enter the lower energy σ-bonding orbital upon \(\ce{H_{2}}\) formation. There are no anti-bonding electrons so \(\ce{H_{2}}\) is a stable molecule.

Helium \(\ce{He_{2}}\)

Helium’s atomic number is 2 so it has two electrons in each atom, both in the 1s-orbital. To form a molecule, the four electrons must be placed in bonding and anti-bonding orbitals as in this MO diagram:

MO diagram for \(\ce{He_{2}}\).

Since both bonding σ and anti-bonding σ* orbitals would be filled upon formation of \(\ce{He_{2}}\) and anti-bonding orbitals are higher in energy than bonding orbitals, \(\ce{He_{2}}\) will not form as two separate He atoms are more stable.

What’s interesting is that \(\ce{He_{2}^{+}}\) does form! This is because \(\ce{He_{2}^{+}}\) is short one electron meaning only one electron will be in the anti-bonding σ* orbital. There is less energy when two electrons are in the bonding orbital and one in the anti-bonding orbital than when there are two separate helium atoms and so formation of \(\ce{He_{2}^{+}}\) is favoured.

Shape of molecular orbitals.

The wave function for p-orbitals is not spherical as for s-orbitals, but with two lobes that are opposite in-phase. When lobes of the same phase overlap, constructive interference occurs and there is increased electron density. When regions of opposite phase overlap, destructive interference occurs decreasing electron density and nodes are formed. When p-orbitals overlap end to end, σ and σ* orbitals are formed. Two overlapping \(\ce{p_{x}}\) orbitals form the low-energy, bonding \(\ce{σ_{px}}\). Electrons in the bonding orbital interact with both nuclei and help hold the two atoms together. The overlapping \(\ce{p_{x}}\) orbitals can also form high-energy, anti-bonding \(\ce{σ_{px}}\)* . The x subscript denotes the x-axis in a cartesian coordinate system and is used for visualization purposes. In the same way p-orbitals can form on y- and z-axis.

If the p-orbitals overlap side-by-side, π-orbital and π-orbital are formed. This is a double bond between two atoms sharing of two pairs of electrons, as seen in the Lewis approach. One pair of electrons in a σ-bond and the other pair in a π-bond. It follows that a triple bond made up of three shared electron pairs, forming one σ-bond and two π-bonds. The d-orbitals and *f-orbitals combine to form bonding and anti-bonding orbitals in the shape depicted in the figure above.

Combining p-orbitals.

In molecular orbital theory, an electron stabilizes bonding interactions if it is in a bonding orbital and destabilizes bonding effects if it is in an antibonding orbital. The bond strength of a molecule (due to electrons being found in bonding or anti-bonding orbitals) can be found by calculating the bond order that results from filling the molecular orbitals:

\[bond\, order\, = \frac{1}{2} (bonding\, electrons\, - anti\, bonding\, electrons)\]

EXAMPLES. Calculating bond orders.

What is the bond order for hydrogen gas \(\ce{H_{2(g)}}\)?

Answer: Hydrogen atoms each have one electron and this electron is in the s-orbital. Each orbital can hold up to two electrons. When two hydrogen atoms bond together to form \(\ce{H_{2(g)}}\), each atom contributes to complete the others’ s-orbital and thus two bonding orbitals are formed. Since no electrons are left unpaired or forced to move to a higher energy level orbital, no antibonding orbitals are formed. The bonding order can then be found to be

\[bond\, order\, = \frac{1}{2} (2 - 0) = 1\]

What is the bond order for acetylene \(\ce{C_{2}H_{2}}\)?

Answer: This molecule is a bit more involved than \(\ce{H_{2(g)}}\) so lets draw the Lewis structure:

There are two types of bonds in this molecule, the H-C and the triple bond between the carbon atoms. The bond order for H-C = 1 for the same reason that the bond order between H-H was 1. To find the bond order for the triple bond we see that

\[bond\, order\, for\, the\, triple\, bond\, = \frac{1}{2} (6 - 0) = 3\]

You can always draw MO diagrams if that method of visualization helps determine how many bonding and anti-bonding electrons there are.

The Lewis approach to chemical bonding defines bond order as the number of chemical bonds in a molecule. So, in Lewis structures, a single bond has bond order = 1, a double bond has bond order = 2, and a triple bond has bond order = 3. Molecular orbital theory is more accurate in its description of electron distribution but the resulting bond order is usually the same as both methods describe the same phenomenon.