LCAO and the aufbau principle for molecules

The ``aufbau'' concept applied to the MOs produced from LCAO follows the same rules as for atoms. That is, we have to obey the Pauli exclusion principle and apply Hund's rule. Thus, for the molecule H$_2$, we have two electrons: Each H$_2$ donates one in a 1s orbital to the LCAO procedure. We already know that combining two 1s orbitals gives us the $\sigma_{g1s}$ and $\sigma^*_{u1s}$ MOs, and since they are different in energy, we do not need to worry about Hund's rule. We simply place both electrons (in opposite spin configurations) into the $\sigma_{g1s}$ orbital as the correlation diagram below illustrates:

Figure 3: H$_2$ correlation diagram

For the molecule He$_2^+$, we have three electrons. On one of the He atoms, there are two electrons both in a 1s orbital, while on the He$^+$ atom, there will be one electron in a 1s orbital. Note, however, that these 1s orbitals are only hydrogen-like but have a $Z$ value of $Z=2$. That is, the 1s orbital is

$$\psi_{100}(r,\phi,\theta) = \left({8 \over \pi a_0^3}\right)^{1/2}e^{-2r/a_0}$$

Hence, if the He nuclei are at positions ${\bf r}_A$ and ${\bf r}_B$, the guess wave function for the LCAO procedure would take the form

$$\psi_g({\bf r}) = C_A \psi_{1s(Z=2)}({\bf r}-{\bf r}_A) + C_B \psi_{1s(Z=2)}({\bf r}-{\bf r}_B)$$

The procedure would still yield $\psi_+({\bf r})$ and $\psi_-({\bf r})$ orbitals, but these would also use $Z=2$ rather than $Z=1$ in their construction. Now, since only two electrons can occupy the $\psi_+({\bf r})$ or $\sigma_{g1s}$ orbital that LCAO gives, the third electron must be placed in the $\psi_-({\bf r})$ or $\sigma^*_{u1s}$ orbital. This gives a correlation diagram that appears as shown below:

Figure 4: He$_2^+$ correlation diagram

The fact that an electron must be placed in an antibonding orbitals means that there will be some destabilization of the bond, and indeed, we find that He$_2^+$ is more weakly bonded than H$_2$.

For the helium dimer He$_2$, there are four electrons, each He atom contributing 2 electrons in 1s orbitals. When placing these electrons into the $\sigma_{g1s}$ and $\sigma^*_{u1s}$ MOs, we are forced to place two electrons into each, which means two electrons in the bonding orbital and two in the antibonding orbital. LCAO predicts that two electrons in an antibonding orbital generates as much destabilization of the bond as there is stabilization from the two bonding electrons. Thus, LCAO would predict that He$_2$ cannot form a stable bond. In fact, a very weak bond stabilization of roughly 10$^{-8}$ Ry exists between to helium atoms, resulting in a very weak He$_2$ dimer. Of course, LCAO cannot predict this. Nevertheless, LCAO gives us an easy way to estimate the stability of a chemical bond which is qualitatively reasonable (even if it misses exotic objects like the weak He$_2$ dimer). Within LCAO, we can define a bond order

$${\rm Bond\ order} = {1 \over 2} \left[\char93 {\rm electrons... ...} - \char93 {\rm electrons\ in\ antibonding\ orbitals}\right]$$

Applying this formula, we can obtain bond orders for the molecules we have considered thus far:

$${\rm For\ H}_2^+: {\rm Bond\ order} = {1 \over 2}[1-0] = {1 \over 2}$$

$${\rm For\ H}_2: {\rm Bond\ order} = {1 \over 2}[2-0] = 1$$

$${\rm For\ He}_2^+: {\rm Bond\ order} = {1 \over 2}[2-1] = {1 \over 2}$$

$${\rm For\ He}_2: {\rm Bond\ order} = {1 \over 2}[2-2] = 0$$

As expected, the weaker bonds have lower bond orders.

The electronic configurations of H$_2^+$, H$_2$ and He$_2^+$ are written in an manner analogous to the atomic case:

$${\rm For\ H}_2^+: \left(\sigma_{g1s}\right)^1$$

$${\rm For\ H}_2: \left(\sigma_{g1s}\right)^2$$

$${\rm For\ He}_2^+: \left(\sigma_{g1s}\right)^2\left(\sigma^*_{u1s}\right)^1$$

At this point, we state the general rules for constructing MOs from AOs in the LCAO procedure:

1.

Form linear combinations of the minimum number of AOs to generate MOs. The number of MOs must equal the number of AOs from which they are generated.

2.

Arrange the MOs in order from lowest to highest in energy.

3.

Occupy or place electrons in MOs starting from MOs of lowest energy following the Pauli exclusion principle (at most 2 electrons per MO) and Hund's rule for MOs of similar energy.