Molecular Orbital Theory for Polyatomic Molecules
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In diatomic species, there is only one important coordinate, defined as the internuclear axis. We discuss the molecular orbitals in terms of the overlap of the atomic orbitals along this axis. In polyatomic species, this description must be extended to allow for the fact that there are many internuclear axes.

Let us consider the water molecule, H2O. A simple description would suggest that the two O-H bonds are formed from the overlap of an O(2p) orbital and an H(1s) orbital. The bonding orbitals would be described by the expression

The valence electrons, 8 in H2O (the O 2s (2) + O 2p (4) + H 1s (1+1)), are contained in the molecular orbitals to create two bonds O-H (2 x 2 = 4 electrons), and the remaining 4 are contained in the non-bonding O(2s) and one remaining O(2p) orbitals.

If this description were accurate, the H-O-H bond angle would be 90o, as the two p orbitals involved in the bonding are perpendicular to each other. However, the actual bond angle is 104.5o. This can be explained by noting that the O-H bonds actually result from the overlap of the H(1s) orbital and a hybrid orbital on O.

Hybridization in the sp-valence system

If we treat the s and p valence orbitals as contributing to bonding in the molecule, they can form four molecular orbitals (there are four basis atomic orbitals and hence four molecular orbitals).

Often, the molecular orbitals are not simply formed from overlap of the basis atomic orbitals and the atomic orbitals an another atom, but are formed from the overlap of hybrid orbitals on the central atom and the other atom.

These hybrid orbitals are so-called because they are formed from mixing together the basis atomic orbitals on the central atom, to give another basis set which better reflects the geometry of the molecule.

In the sp-valence system, the 4 orbitals (1 s and 3 p orbitals) can mix in different ways to give different types of sp-hybrid orbital

sp1-hybrids are formed from the mixing of the s orbital and one of the p orbitals (the pz orbital). The two orbitals formed, h1 and h2, are directional, pointing in opposite directions along an axis through the atom (the z-axis).

sp2-hybrids are formed from the mixing of the s orbital and two of the p orbitals (the pz and py orbitals). The three orbitals formed, h1, h2 and h3, are directional, pointing to the corners of a planar triangle in the yz-plane.

 

sp3-hybrids are formed from the mixing of the s orbital and the three p orbitals. The four orbitals formed, h1, h2, h3 and h4, are directional, pointing to the vertices of a regular tetrahedron centered on the origin. These orbitals are shown as arrows on the diagram below.

In water, there is sp3 hybridization, forming two bonding orbitals, where the H-O-H bonding angle is close to the tetrahedral angle of 109.5o (the reduction to 104.5o can be explained by VSEPR), and two non-bonding orbitals, which contain lone pairs of electrons. Hence, each bonding orbital is described by a wavefunction with the form

This picture of the formation of the molecular orbitals from hybrid atomic orbitals is a good approximation for the first period elements, but lower down the periodic table, where the degree of overlap between the s and p orbitals is smaller, the picture of bonding through the original basis s and p atomic orbitals is a better one.

Multiple bonding

Let us consider the series ethane, ethene, and ethyne. These have single, double and triple bonds between the C atoms respectively. This can be explained in terms of the hybridization of the C(2s) and C(2p) orbitals.

Ethyne, C2H2, forms three σ bonds: they are H(1s)-C1(sp1), C1(sp1)-C2(sp1) and C2(sp1)-H(1s). This means that the C atoms are sp1 hybridized to form the linear sp1 orbitals shown above, and the ethyne molecule is linear. The remaining p orbitals on C overlap to form a set of two π bonding orbitals, which are degenerate, and so overall there are three bonds between the C atoms.

Ethyne, C2H4, forms five σ bonds: they are H(1s)-C1(sp2) (x2), C1(sp2)-C2(sp2) and C2(sp2)-H(1s) (x2). This means that the C atoms are sp2 hybridized to form the trigonal planar sp2 orbitals shown above, and the ethene molecule is therefore planar. The remaining p orbitals on the C atoms overlap to form a π bonding orbital and so overall there are three bonds between the C atoms.

The molecule as a whole is planar as this geometry maximizes the amount of overlap between the C p orbitals, and therefore creates the strongest π bond.

In ethane, C2H6, both C atoms are sp3 hybridized, so all the bonds in the molecule are σ bonds. The three C-H and one C-C bond about each C atom are tetrahedrally arranged, and the two tetrahedra are joined by a shared vertex. The two tetrahedra are free to rotate about the C-C axis.

Molecular Orbital diagrams of polyatomic species

In the diatomic species, in constructing the molecular orbital diagram, we simply show the energies of the atomic orbitals of the two species on either side of the diagram and the energies of the molecular orbitals in the center. Lines are drawn to connect the molecular orbitals with their basis atomic orbitals.

In the polyatomic case, it is not so simple, as we would need more dimensions to show the overlap in this way. Instead, the MO diagram is generally drawn with two sets of basis atomic orbitals on either side, and again with the resultant molecular orbitals in the center. These basis orbitals are hybrid orbitals of the atoms, and reflect the symmetry of the molecule.

A simple MO diagram for water is as shown in the table.

MO diagram for water

Thus we can see in the center of the diagram the 2 bonding orbitals (containing 4 electrons) the two non-bonding orbitals (containing 4 electrons) and the 2 antibonding orbitals (which are vacant) formed from the basis set of six O(2s), O(2p) (x3) and H(1s) (x2) atomic orbitals.

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