Bonding and Antibonding Molecular Orbitals
In MO theory, molecular orbitals form by the overlap of atomic orbitals. The atomic orbital energy correlates with electronegativity, as electronegative atoms hold electrons more tightly, lowering their energies. MO modelling is only valid when the atomic orbitals have comparable energy; when the energies differ greatly, the mode of bonding becomes ionic. A second condition for overlapping atomic orbitals is that they have the same symmetry.
Two atomic orbitals can overlap in two ways, depending on their phase relationship. The phase of an orbital is a direct consequence of the wave-like properties of electrons. In graphical representations of orbitals, the orbital phase is depicted either by a plus or minus sign (which has no relationship to electric charge) or by shading one lobe. The sign of the phase itself does not have physical meaning except when mixing orbitals to form molecular orbitals. Two same-sign orbitals have a constructive overlap, forming a molecular orbital with the bulk of the electron density located between the two nuclei. This MO is called the bonding orbital and its energy is lower than that of the original atomic orbitals. A bond involving molecular orbitals that are symmetric with respect to rotation around the bond axis is called a sigma bond (σ-bond). If the phase changes, the bond becomes a pi bond (π-bond). Symmetry labels are further defined by whether the orbital maintains its original character after an inversion about its center: if it does, it is defined gerade, g; if the orbital does not maintain its original character, it is ungerade, u.
This concept is illustrated in Figure 2, in which a stable molecular orbital is produced by overlap.
Atomic orbitals can also interact with each other out-of-phase. This leads to destructive cancellation and no electron density between the two nuclei at the so-called nodal plane depicted as a perpendicular dashed line. In this anti-bonding MO, with energy much higher than the original AO's, any electrons present are located in lobes pointing away from the central internuclear axis. For a corresponding σ-bonding orbital, such an orbital would be symmetrical, but differentiated from it by an asterisk, as in σ*. For a π-bond, corresponding bonding and antibonding orbitals would not have such symmetry around the bond axis, and are designated π and π*, respectively.The next step in constructing an MO diagram is filling the newly formed molecular orbitals with electrons. Three general rules apply:
- The Aufbau principle states that orbitals are filled starting with the lowest energy
- The Pauli exclusion principle states that the maximum number of electrons occupying an orbital is two, with opposite spins
- Hund's rule states that when there are several MOs with equal energy, the electrons occupy the MOs one at a time before two electrons occupy the same MO.
The filled MO highest in energy is called the Highest Occupied Molecular Orbital, or HOMO. The empty MO just above it is then the Lowest Unoccupied Molecular Orbital, or LUMO. The electrons in the bonding MOs are called bonding electrons and any electrons in the antibonding orbital are called antibonding electrons. The reduction in energy of these electrons is the driving force for chemical bond formation. Whenever mixing for an atomic orbital is not possible for reasons of symmetry or energy, a non-bonding MO is created, which is often quite similar to and has energy levels equal or close to its constituent AO, thus not contributing to bonding energetics. The resulting electron configuration can be described in terms of bond type, parity, and occupancy for example dihydrogen 1σg2. Alternatively, it can be written as a molecular term symbol, e.g.. 1Σg+ for dihydrogen. Sometimes, the letter n is used to designate a non-bonding orbital.
Several of these concepts are illustrated in Figure 1, which depicts the hypothetical dimer of helium. The presence of a filled antibonding orbital, after fulfilling the conditions above, indicates that the bond in this case simply does not exist.