Sigma Bonds

Overlap of Atomic Orbitals

When two atoms are brought close together, an electron can simultaneously experience a strong attraction to both nuclei. The electron exists in a molecular orbital that is distributed between the two atoms, producing the chemical bond. To a rough approximation, the molecular orbital itself can be modeled as a linear combination of the wave functions for two (or more) atomic orbitals.

In this context one can envision a molecular orbital as arising from the overlap of two atomic orbitals. The overlap produces one of two results, depending upon the relative signs of the respective atomic wave functions. If the two wave functions have the same sign in the region of overlap, a constructive interaction between the two wave functions occurs and a bonding orbital is produced. If the two wave functions have opposite signs in the region of overlap, a destructive interaction between the two wave functions occurs and an anti-bonding orbital is produced.

Suppose the atomic wave functions for atoms A and B are ψA and ψB, respectively. The two molecular orbital wave functions, φ1 and φ2, are (to a rough approximation) given by

φ1 = N1 ( ψA + ψB )

φ2 = N2 ( ψA - ψB )

One of the wave functions, φ1 or φ2, describes the bonding orbital and and the other describes the anti-bonding orbital. The terms N1 and N2 are normalization constant whose purpose is to ensure that the integrals of φ11 and φ22 over all space are unity. This normalization procedure is necessary, because the electron must exist somewhere.

In the H2 molecule, the sigma bond was formed by the overlap of two 1s orbitals. Other orbital combinations can also give rise to a sigma bond, provided the atoms are oriented along the appropriate axis. The most common sigma bonds arise from s-s, s-p, and p-p overlap. The formation of sigma bonding and anti-bonding orbitals through 2pz-2pz overlap in the F2 molecule is shown below.



Orbital Overlap

The overlap of two 2pz orbitals is shown below. The two nuclei both lie on the z axis. Use the slider to move the two atoms close together. Notice that as the atoms approach each other, the two 2pz orbitals overlap with the region of overlap occurring along the line connecting the two nuclei. The overlap region itself occurs directly between the two nuclei.

This type of overlap is designated sigma (σ). The energy diagram for the interaction is shown at the lower left. The energies of the isolated 2pz orbitals are shown on the far left and right of the energy diagram. When the orbitals are brought close together, two molecular orbitals are formed and these molecular orbitals are shown in the center of the diagram.

Molecular Orbital Diagram


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Sigma Bonding Orbital

Of the two molecular orbitals, the lower energy orbital is the bonding orbital. The sigma bonding orbital formed by overlap of the 2pz orbitals in the F2 molecule are shown below.

Observe that electron density is concentrated in the region between the two nuclei. This is characteristic of a bonding orbital.

Observe that the electron density is concentrated directly between the two nuclei; the electron density is centered on the line passing through the two nuclei. This is characteristic of a sigma-type interaction.

The strength of a bond is directly related to the match in energy levels for the two atomic orbitals and in the quality of the overlap. For diatomic molecules like H2 and F2, the energy match is obviously excellent.

In the case of the H2 molecule, there was excellent overlap between the two 1s orbitals, with the resulting bond being relatively strong (DH-H = 436 kJ/mole).

In the case of the F2 molecule, the overlap between the two 2pz orbitals is very poor. This fact is most evident in the electron density plot shown at the lower left. The two fluorine nuclei cannot approach sufficiently close to produce good overlap, and the electron density at the point halfway between the two nuclei is not particularly great. The result is a very weak bond (DF-F = 158 kJ/mole). (See the molecular orbital diagram for F2.)




Sigma Anti-Bonding Orbital

The higher energy molecular orbital is the antibonding orbital. The sigma bonding orbital formed by overlap of the 2pz orbitals in the F2 molecule are shown below.

Observe that the region between the two nuclei is void of electron density. In fact, there is a nodal surface separating the two nuclei. This is characteristic of a anti-bonding orbital.

Observe that the electron density is centered on the line passing through the two nuclei. This is characteristic of a sigma-type interaction.

Bonding orbitals are called "bonding" because electrons are more stable in such orbitals than in the atomic orbitals that overlapped to form the bonding orbital. The system would therefore prefer to keep the electrons in the bonding orbital (because of the lower energy) and this is possible only if the atoms remain close to each other. This is the essence of chemical bonding.

Antibonding orbitals are called "antibonding" because electrons are more stable in the atomic orbitals that overlapped to form the antibonding orbital than in the antibonding orbital itself. The system would therefore prefer to keep the electrons in atomic orbitals (because of the lower energy) and this is possible only if the atoms separate so that no overlap of the atomic orbitals occur. This effect opposes bonding.

For the simple overlap of two atomic orbitals, the increase in energy for the antibonding orbital is always greater (frequently much greater) than the decrease in energy for the bonding orbital. Thus if both bonding and antibonding orbitals are filled, the net effect is antibonding and no chemical bond will exist.



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