Other molecular orbitals

As noted above, $L_z$ commutes with the total Hamiltonian for H$_2^+$,i.e.,

$$[L_z,H]=0$$

Thus, the eigenvalue $m$ can be used to characterize molecular orbitals. The following nomenclature has been adopted for designating molecular orbitals from the $m$ value:

$\vert m\vert=0$ $\longrightarrow$ $\sigma$ orbital.

$\vert m\vert=1$ $\longrightarrow$ $\pi$ orbital.

$\vert m\vert=2$ $\longrightarrow$ $\delta$ orbital.

which are clearly analogous to $s$, $p$ and $d$ atomic orbitals.

In addition, molcular orbitals are given another designation based on how they transform under a parity transformation. The parity transformation operator, $\Pi$ or $P$, produces a spatial reflection ${\bf r}\longrightarrow -{\bf r}$. Possible eigenvalues of $\Pi$ are 1 or -1, and an orbital with a parity eigenvalue of 1 is said to be a state of even parity, while an orbital with a parity eigenvalue of -1 is said to be a state of odd parity. Even and odd parity states are designated by a $g$ or a $u$, which derive from the German words gerade and ungerade for ``even'' and ``odd'', respectively. Thus, a $\sigma$ orbital of even parity would be denoted as $\sigma_g$.

To see how to designate the molecular orbitals $\psi_-({\bf r})$ and $\psi_+({\bf r})$, let us look at how they transform under the parity operation:

$$\Pi \psi_-({\bf r}) = \psi_-(-{\bf r}) = C_-\left[e^{-\left\vert-{\bf r} + {R \over 2}\hat{{\bf z}}\right\vert/a_0}+e^{-mpt/a_0}\right]$$

$$= C_-\left[e^{-pt/a_0} + e^{-\left\vert{\bf r} + {R \over 2}\hat{{\bf z}}\right\vert/a_0}\right] = \psi_-({\bf r})$$

$$\Pi \psi_+({\bf r}) = \psi_+(-{\bf r}) = C_+\left[e^{-\left\vert-{\bf r} + {R \over 2}\hat{{\bf z}}\right\vert/a_0}-e^{-mpt/a_0}\right]$$

$$= C_+\left[e^{-pt/a_0} - e^{-\left\vert{\bf r} + {R \over 2}\hat{{\bf z}}\right\vert/a_0}\right] = -\psi_+({\bf r})$$

Thus, we see that $\psi_-({\bf r})$ is an even parity state while $\psi_+({\bf r})$ is an odd parity state. Thus, these molecular orbitals should be designated as

$$\vert\psi_-\rangle \equiv \vert\sigma_g(1s)\rangle$$

$$\vert\psi_+\rangle \equiv \vert\sigma_u(1s)\rangle \nonumber$$

or simply $\sigma_g(1s)$ and $\sigma_u(1s)$, respectively. The $1s$ is also shown explicitly to indicate that these are constructed from $1s$ orbitals of hydrogen. Finally, in order to distinguish the bonding from the anti-bonding state, an asterisk ($^*$) is added to the anti-bonding state, so that its full designation is $\sigma_u^*(1s)$. Note that both orbitals are eigenfunctions of $L_z$ although they are not exact eigenfunctions of $H$, as expected.

In a similar manner, molecular orbitals can be constructed from $2s$ orbitals of hydrogen. These will correspond qualitatively to excited states of H$_2^+$ although their accuracy will be relatively low compared to the exact excited state wave functions. Nevertheless, they can be useful in understanding qualitatively what the electronic distribution will be in such an excited state. It is clear that there will be analogous $\vert\psi_-\rangle$ and $\vert\psi_+\rangle$ states which will correspond to bonding and anti-bonding molecular orbitals, $\sigma_g(2s)$ and $\sigma_u^*(2s)$, respectively. Again, they will be eigenfunctions of $L_z$ but not of $H$.

Orbitals can also be constructed from $2p_z$ atomic orbitals. These will clearly have $m=0$ and hence be $\sigma$ orbitals. The even and odd parity combinations will therefore be

$$\vert\sigma_g(2p_z)\rangle \propto \vert\psi_{2p_z}^{(1)}\rangle + \vert\psi_{2p_z}^{(2)}\rangle$$

$$\vert\sigma_u^*(2p_z)\rangle \propto \vert\psi_{2p_z}^{(1)}\rangle - \vert\psi_{2p_z}^{(2)}\rangle$$

These are both eigenfunctions of $L_z$ with $m=0$ as required for a $\sigma$ orbital. The contours of these orbitals are shown in the figure below (again, the top plot shows the individual contours of the two $2p_z$ orbitals and the middle and bottom show the $g$ and $u$ combinations):

図 5:

From the figure it is clear that $\sigma_g(2p_z)$ must be a bonding state while $\sigma_u^*(2p_z)$ must be anti-bonding.

Finally, we can construct molecular orbitals from $2p_x$ or $2p_y$ orbitals. Since these will be similar in shape and have the same energy, we only need to consider one case. Let us look at the $2p_x$ orbitals. Since the $2p_x$ orbitals have $m=1$, the molecular orbitals will $\pi$ orbitals. In this case, the even and odd parity states will be:

$$\vert\pi_u(2p_x)\rangle \propto \vert\psi_{2p_x}^{(1)}\rangle + \psi_{2p_x}^{(2)}\rangle$$

$$\vert\pi_g^*(2p_x)\rangle \propto \vert\psi_{2p_x}^{(1)}\rangle - \psi_{2p_x}^{(2)}\rangle$$

as can be easily checked. Again, these are both eigenfunctions of $L_z$ with $m=1$ as required for a $\pi$ orbital. The contours of these molecular orbitals are shown in the figure below:

図 6:

In the bonding state, the electron becomes highly delocalized over the entire molecule in a banana shaped orbital, while in the anti-bonding state, the lobes are more well localized.

The energies of these various molecular orbitals are ordered according to the figure below:

図 7:

The figure shows that the bonding orbitals all have lower energies than the anti-bonding orbital as expected.