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# LCAO for heteronuclear diatomic molecules

In a heteronuclear diatomic molecule, there is an electronegativity difference between the atoms, which leads to an asymmetric distribution of the electronic probability density, weighted more heavily toward the element with the greater electronegativity.

Consider constructing a MO from two 2s orbitals for nuclei with different values. Let atom A have an atomic number and atom B have an atomic number . For a single electron interacting with the two nuclei, an LCAO guess wave function could be

Recall the energies and that were defined for LCAO in the homonuclear case:

For the heteronuclear case, . If atom B is the more electronegative, then its will be larger (in the second period), and the energy will be more negative due to the large Coulomb attraction between the electrons and the nuclei. Thus, in this case (remember and are both negative). Now, if we calculate the guess of the ground-state energy

we find the more general result

and then perform the minimization of :

we obtain the conditions

If we make the simplifying approximation that the overlap , then these conditions lead to the following relationship between and :

In order to make the analysis a bit easier, let us expand the square root using the fact that

Then

Now, we know that

First, look at the . If , is more electronegative because its energy is lower (think of the Bohr formula , which shows that as increases, the energy decreases). Generally, a not so easy calculation of shows that . Thus, if , then

and this means that or , which is what we would expect if is more electronegative. Similarly, if , is more electronegative. We would then find

and , and , which is what we would expect if is more electronegative.

Another consequence we can derive from the above result is that if or , then the coefficients will be very different, e.g. or , in which case, the resulting MO is not really an MO at all, but rather more like the AO with the larger coefficient. Thus, it is clear that and cannot be very different if we are to have a reasonable amount of mixing of the two AOs. This supports the idea that only orbitals with similar energies can be combined in the LCAO scheme.

We can no longer use the g'' and u'' designators because the orbitals have no particular symmetry when . That is

Thus, we denote simply as and simply as .

Let us now construct a correlation diagram for the heteronuclear diatomic BO. Boron has an electronic configuration

while oxygen's is

Since we are interested in the chemical bond that forms between them, we only consider the valence electrons explicitly, and these are the electrons in the shell. In BO, oxygen is the more electronegative, so its orbitals are lower in energy than those of boron. This must be indicated on the correlation diagram. Thus, the correlation diagram appears as in the figure below:
Note that the ordering of the MOs follows the pattern we would expect for boron rather than oxygen. Only high-level calculations can predict this, but physically, the simple explanation is that there is only one electron in the orbital, and only one of the two atoms has a large nuclear charge (unlike in O, where they both do). There is, therefore, insufficient Coulomb attraction for this one electron to pull the energy of the orbital below the energy of the bonding orbitals.

The electronic configuration of BO is, therefore

and the bond order is (1/2)(7-2)=5/2. The fraction bond order indicates that the molecule is paramagnetic, as with O.

As another example, consider the molecule NO. NO has 11 valence electrons and has the electronic configuration:

The bond order is also 5/2, and the molecule is paramagnetic as well.

What about the molecule HF? Here, the 1s and 2s orbitals of F are so low in energy compared to the 1s orbital in H that they cannot be combined to form MOs. At the same time, the and orbitals of F have an insignificant spatial overlap with the 1s orbital in H (assuming that the two nuclei lie along the -axis) that they also do not form MOs. Only the orbital of F has significant overlap with the 1s orbital in H and can mix with it energetically. Thus, the LCAO guess wave function takes the form

where atom A is H and atom B is F. If H lies to the left of F, then this orbital has antibonding character, while the orbital

has bonding character because there is significant amplitude in the region between the nuclei. See the figure below:
The orbital is denoted simply as while the orbital is denoted for bonding and antibonding, respectively. The 2s, and orbitals of F do not mix with anything and are, therefore, called nonbonding orbitals. They are denoted and and , respectively. The orbital ordering is (since it is a low-energy 2s orbital), followed by , then the two nonbonding orbitals and finally . Therefore, the correlation diagram for HF is as shown in the figure below:
The bond order of HF is easily seen to be 1.

Next: Frontier molecular orbitals Up: lecture_15 Previous: lecture_15
Mark E. Tuckerman 2011-11-21