Diatomic Species by Molecular Orbital Theory
Even rather simple molecular orbital (MO) theory can be used to predict which homonuclear diatomic species will exist, explain many properties (why O2 is a paramagnetic diradical), and identify the important frontier molecular orbitals (FMOs).
Atomic orbitals, solutions derived from the Schrödinger wave equation, are wavefunctions, where waves are well understood mathematically. Waves can be added together or subtracted from each other. Consider the addition of two sine waves. The product is a superposition, here:
Likewise, atomic orbitals can be added together to give a superposition called a molecular orbital or MO. Molecular orbitals are bonding when the orbital phase considerations are favourable:
The bonding MO wave function, Ψ (psi), can be squared, |Ψ|2 (psi squared), to represent electron density. A bonding MO shows a build up of electron density between the two positively charged nuclei.
If the atomic orbitals are "out of phase", the AO/AO interaction will exhibit a phase node a regions in space with zero electron density between the positive nuclei. There will be nothing to attract the nuclei together and the MO is (said to be) antibonding:
The diagram below shows the atomic orbitals, AOs, on a pair of adjacent atoms interacting with each other.
Note that by convention we start reading from the bottom of the diagram because this is how MO diagrams are constructed, from the lower energies... up.
The diagram of the bonding and antibonding MOs is shown below:
Electrons are added to the MOs (bonding and antibonding) using the same rules that are used for adding electrons to atomic orbitals:
Simple, except that when electrons are added, sometimes the relative energies of the orbitals can change...
We can now examine diatomic species. We will start with the interaction of two hydrogen atoms, each with a single electron in a 1s AO.
The two atoms come together and the two electrons go into the σ (sigma) 1s MO with is bonding. H2 is known to exist.
For dihydrogen, H2, we can identify the frontier molecular orbitals (FMOs).
The highest occupied molecular orbital (or HOMO) is the σ (sigma) 1s MO. The lowest unoccupied MO (LUMO) is the σ* (sigma star) 1s MO which is antibonding.
Bond order is defined as the number of electrons in bonding MOs (for H2 this is two) minus the number of electrons in antibonding MOs (zero) divided by two.
Thus, hydrogen has a bond order of 1.
Now consider two helium atoms approaching. Two electrons go into the σ 1s bonding MO, and the next two into the σ* antibonding MO.
As antibonding MOs are more antibonding than bonding MOs are bonding, He2 (dihelium), is not expected to exist
And dihelium has a bond order of zero and the molecule is unknown:
Now consider two lithium atoms interacting.
Dilithium, Li2, is known in the gas phase, it has a bond order of one, and it has HOMO + LUMO FMOs:
Now consider two beryllium atoms interacting. Diberyllium, Be2, has a bond order of zero and is unknown:
Quantum mechanics is all about patterns. The problem is that just when we think we are starting to understand the patterns... Mother Nature shows that she has other ideas...
On going Li2, B2, C2, N2, O2, F2, Ne2 the complexity of the molecular orbitals develop in two ways:
First we see how the energy of 3σg MO drops below the 1πu MO of going from N2 to O2. (Note that σ and π bonding MOs are coloured black and the σ• and π• antibonding MOs are brown):
Next the electrons are added according to the aufbau (build-up) principle, the Pauli exclusion principle & Hund's rule.
This analysis gives rise to some very common species (N2, O2 & F2), some rare species (B2 & C2) and some unknown species (Be2 & Ne2). Indeed, existence can be predicted by bond-order in that the bond-order = 0 species are unknown:
Heteronuclear Diatomic Molecules & Molecular Ions
MO theory can be used to describe heteronuclear diatomic molecules & molecular Ions such as:
As the electronegativity differences increases the interacting orbitals will be at different energies. The result is that the covalent bonding energy decreases, but this is counteracted by an increasing electrostatic +/ attraction that is not represented on MO diagrams (see the Klopman equation, elsewhere in this web book.)
The heteronuclear diatomic ions cyanide ion, CN, and nitrosonium ion, NO+, are also electronic with nitrogen, N2, and carbon monoxide. The only difference between the MO diagrams are the relative energies of the orbitals.
Please note that the primary aim of this page has been produced to introduce the idea of molecular orbitals and to show that the stable homonuclear diatomic species possess either LUMO and HOMO frontier molecular orbitals, or are diradical species.
The following is a message from Dr Eugen Schwarz in Germany for which I am most grateful concerns the above analysis:
© Mark R. Leach 1999-
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