In the preceding section, we used the approximate functions (13.57) and (13.58) for the two lowest $\mathrm{H}{2}^{+}$electronic states. Now we construct approximate functions for further excited states so as to build up a supply of $\mathrm{H}{2}^{+}$-like molecular orbitals. We shall then use these MOs to discuss many-electron diatomic molecules qualitatively, just as we used hydrogenlike AOs to discuss many-electron atoms.
To get approximations to higher $\mathrm{H}{2}^{+} \mathrm{MOs}$, we can use the linear-variation-function method. We saw that it was natural to take variation functions for $\mathrm{H}{2}^{+}$as linear combinations of hydrogenlike atomic-orbital functions, giving LCAO-MOs. To get approximate MOs for higher states, we add in more AOs to the linear combination. Thus, to get approximate wave functions for the six lowest $\mathrm{H}_{2}^{+} \sigma$ states, we use a linear combination of the three lowest $m=0$ hydrogenlike functions on each atom:
\(
\phi=c{1} 1 s{a}+c{2} 2 s{a}+c{3}\left(2 p{0}\right){a}+c{4} 1 s{b}+c{5} 2 s{b}+c{6}\left(2 p{0}\right){b}
\)
As found in the preceding section for the function (13.43), the symmetry of the homonuclear diatomic molecule makes the coefficients of the atom- $b$ orbitals equal to $\pm 1$ times the corresponding atom- $a$ orbital coefficients:
\(
\begin{equation}
\phi=\left[c{1} 1 s{a}+c{2} 2 s{a}+c{3}\left(2 p{0}\right){a}\right] \pm\left[c{1} 1 s{b}+c{2} 2 s{b}+c{3}\left(2 p{0}\right){b}\right] \tag{13.69}
\end{equation}
\)
where the upper sign goes with the even $(g)$ states.
Consider the relative magnitudes of the coefficients in (13.69). For the two electronic states that dissociate into a $1 s$ hydrogen atom, we expect that $c{1}$ will be considerably greater than $c{2}$ or $c{3}$, since $c{2}$ and $c{3}$ vanish in the limit of $R$ going to infinity. Thus the Dickinson function (13.68) has the $2 p{0}$ coefficient equal to one-seventh the $1 s$ coefficient at $R{e}$. (This function does not include a $2 s$ term, but if it did, we would find its coefficient to be small compared with the $1 s$ coefficient.) As a first approximation, we therefore set $c{2}$ and $c_{3}$ equal to zero, taking
\(
\begin{equation}
\phi=c{1}\left(1 s{a} \pm 1 s_{b}\right) \tag{13.70}
\end{equation}
\)
as an approximation for the wave functions of these two states (as we already have done). From the viewpoint of perturbation theory, if we take the separated atoms as the unperturbed problem, the functions (13.70) are the correct zeroth-order wave functions.
The same argument for the two states that dissociate to a $2 s$ hydrogen atom gives as approximate wave functions for them
\(
\begin{equation}
\phi=c{2}\left(2 s{a} \pm 2 s_{b}\right) \tag{13.71}
\end{equation}
\)
since $c{1}$ and $c{3}$ will be small for these states. The functions (13.71) are only an approximation to what we would find if we carried out the linear variation treatment. To find rigorous upper bounds to the energies of these two $\mathrm{H}_{2}^{+}$states, we must use the trial function (13.69) and solve the appropriate secular equation (8.58) (or use matrix algebra-Section 8.6).
In general, we have two $\mathrm{H}{2}^{+}$states correlating with each separated-atoms state, and rough approximations to the wave functions of these two states will be the LCAO functions $f{a}+f{b}$ and $f{a}-f{b}$, where $f$ is a hydrogenlike wave function. The functions (13.70) give the $\sigma{g} 1 s$ and $\sigma{u}^{*} 1 s$ states. Similarly, the functions (13.71) give the $\sigma{g} 2 s$ and $\sigma{u}^{*} 2 s$ molecular orbitals. The outer contour lines for these orbitals are like those for the corresponding MOs made from $1 s$ AOs. However, since the $2 s$ AO has a nodal sphere while the $1 s$ AO does not, each of these MOs has one more nodal surface than the corresponding $\sigma{g} 1 s$ or $\sigma_{u}^{*} 1 s \mathrm{MO}$.
Next we have the combinations
\(
\begin{equation}
\left(2 p{0}\right){a} \pm\left(2 p{0}\right){b}=\left(2 p{z}\right){a} \pm\left(2 p{z}\right){b} \tag{13.72}
\end{equation}
\)
giving the $\sigma{g} 2 p$ and $\sigma{u}^{*} 2 p$ MOs (Fig. 13.11). These are $\sigma$ MOs even though they correlate with $2 p$ separated AOs, since they have $m=0$.
The preceding discussion is oversimplified. For the hydrogen atom, the $2 s$ and $2 p$ AOs are degenerate, and so we can expect the correct zeroth-order functions for the $\sigma{g} 2 s, \sigma{u}^{} 2 s, \sigma{g} 2 p$, and $\sigma{u}^{} 2 p$ MOs of $\mathrm{H}{2}^{+}$to each be mixtures of $2 s$ and $2 p$ AOs rather than containing only $2 s$ or $2 p$ character. [In the $R \rightarrow \infty$ limit, $\mathrm{H}{2}^{+}$consists of an H atom perturbed by the essentially uniform electric field of a far-distant proton. Problem 9.23
FIGURE 13.11 Formation of $\sigma{g} 2 p$ and $\sigma{u}^{*} 2 p \mathrm{MOs}$ from $2 p_{z}$ AOs. The dashed lines indicate nodal surfaces. The signs on the contours give the sign of the wave function. The contours are symmetric about the $z$ axis. (Because of substantial $2 s-2 p$ hybridization, these contours are not accurate representations of true MO shapes. For accurate contours, see the reference for Fig. 13.20.)
showed that the correct zeroth-order functions for the $n=2$ levels of an H atom in a uniform electric field in the $z$ direction are $2^{-1 / 2}\left(2 s+2 p{0}\right), 2^{-1 / 2}\left(2 s-2 p{0}\right), 2 p{1}$, and $2 p{-1}$. Thus, for $\mathrm{H}{2}^{+}, 2 s$ and $2 p{0}$ in Eqs. (13.71) and (13.72) should be replaced by $2 s+2 p{0}$ and $2 s-2 p{0}$.] For molecules that dissociate into many-electron atoms, the separated-atoms $2 s$ and $2 p$ AOs are not degenerate but do lie close together in energy. Hence the first-order corrections to the wave functions will mix substantial $2 s$ character into the $\sigma 2 p$ MOs and substantial $2 p$ character into the $\sigma 2 s$ MOs. Thus the designation of an MO as $\sigma 2 s$ or $\sigma 2 p$ should not be taken too literally. For $\mathrm{H}{2}^{+}$and $\mathrm{H}{2}$, the united-atom designations of the MOs are preferable to the separated-atoms designations, but we shall use mostly the latter.
For the other two $2 p$ atomic orbitals, we can use either the $2 p{+1}$ and $2 p{-1}$ complex functions or the $2 p{x}$ and $2 p{y}$ real functions. If we want MOs that are eigenfunctions of $\hat{L}_{z}$, we will choose the complex $p$ orbitals, giving the MOs
\(
\begin{array}{r}
\left(2 p{+1}\right){a}+\left(2 p{+1}\right){b} \
\left(2 p{+1}\right){a}-\left(2 p{+1}\right){b} \
\left(2 p{-1}\right){a}+\left(2 p{-1}\right){b} \
\left(2 p{-1}\right){a}-\left(2 p{-1}\right){b} \tag{13.76}
\end{array}
\)
From Eq. (6.114) we have, since $\phi{a}=\phi{b}=\phi$,
\(
\begin{equation}
\left(2 p{+1}\right){a}+\left(2 p{+1}\right){b}=\frac{1}{8} \pi^{-1 / 2}\left(r{a} e^{-r{a} / 2} \sin \theta{a}+r{b} e^{-r{b} / 2} \sin \theta{b}\right) e^{i \phi} \tag{13.77}
\end{equation}
\)
Since $\lambda=|m|=1$, this is a $\pi$ orbital. The inversion operation amounts to the coordinate transformation (Fig. 13.12)
\(
\begin{equation}
r{a} \rightarrow r{b}, \quad r{b} \rightarrow r{a}, \quad \phi \rightarrow \phi+\pi \tag{13.78}
\end{equation}
\)
We have $e^{i(\phi+\pi)}=(\cos \pi+i \sin \pi) e^{i \phi}=-e^{i \phi}$. From Fig. 13.12 we see that inversion converts $\theta{a}$ to $\theta{b}$ and vice versa. Thus inversion converts (13.77) to its negative, meaning it is a $u$ orbital. Reflection in the plane perpendicular to the axis and midway between the nuclei causes the following transformations (Prob. 13.24):
\(
\begin{equation}
r{a} \rightarrow r{b}, \quad r{b} \rightarrow r{a}, \quad \phi \rightarrow \phi, \quad \theta{a} \rightarrow \theta{b}, \quad \theta{b} \rightarrow \theta{a} \tag{13.79}
\end{equation}
\)
This leaves (13.77) unchanged, so we have an unstarred (bonding) orbital. The designation of (13.77) is then $\pi{u} 2 p{+1}$.
FIGURE 13.12 The effect of inversion of the electron's coordinates in $\mathrm{H}{2}^{+}$. We have $r{\mathrm{a}}^{\prime}=r{b}, r{b}^{\prime}=r_{a}$, and $\phi^{\prime}=\phi+\pi$.
FIGURE 13.13 Cross section of the $\pi{u} 2 p{+1}$ (or $\pi{u} 2 p{-1}$ ) molecular orbital. To obtain the three-dimensional contour surface, rotate the figure about the $z$ axis. The $z$ axis is a nodal line for this MO (as it is for the $2 p_{+1}$ AO.)
The function (13.77) is complex. Taking its absolute value, we can plot the orbital contours of constant probability density (Section 6.7). Since $\left|e^{i \phi}\right|=1$, the probability density is independent of $\phi$, giving a density that is symmetric about the $z$ (internuclear) axis. Figure 13.13 shows a cross section of this orbital in a plane containing the nuclei. The three-dimensional shape is found by rotating this figure about the $z$ axis, creating a sort of fat doughnut.
The MO (13.75) differs from (13.77) only in having $e^{i \phi}$ replaced by $e^{-i \phi}$ and is designated $\pi{u} 2 p{-1}$. The coordinate $\phi$ enters the $\mathrm{H}{2}^{+}$Hamiltonian as $\partial^{2} / \partial \phi^{2}$. Since $\partial^{2} e^{i \phi} / \partial \phi^{2}=\partial^{2} e^{-i \phi} / \partial \phi^{2}$, the states (13.73) and (13.75) have the same energy. Recall (Section 13.4) that the $\lambda=1$ energy levels are doubly degenerate, corresponding to $m= \pm 1$. Since $\left|e^{i \phi}\right|=\left|e^{-i \phi}\right|$, the $\pi{u} 2 p{+1}$ and $\pi{u} 2 p{-1}$ MOs have the same shapes, just as the $2 p{+1}$ and $2 p_{-1}$ AOs have the same shapes.
The functions (13.74) and (13.76) give the $\pi{g}^{*} 2 p{+1}$ and $\pi{g}^{*} 2 p{-1}$ MOs. These functions do not give charge buildup between the nuclei; see Fig. 13.14.
Now consider the more familiar alternative of using the $2 p{x}$ and $2 p{y}$ AOs to make the MOs. The linear combination
\(
\begin{equation}
\left(2 p{x}\right){a}+\left(2 p{x}\right){b} \tag{13.80}
\end{equation}
\)
gives the $\pi{u} 2 p{x} \mathrm{MO}$ (Fig. 13.15). This MO is not symmetrical about the internuclear axis but builds up probability density in two lobes, one above and one below the $y z$ plane, which is a nodal plane for this function. The wave function has opposite signs on each side of this plane. The linear combination
\(
\begin{equation}
\left(2 p{x}\right){a}-\left(2 p{x}\right){b} \tag{13.81}
\end{equation}
\)
gives the $\pi{g}^{*} 2 p{x}$ MO (Fig. 13.15). Since the $2 p{y}$ functions differ from the $2 p{x}$ functions solely by a rotation of $90^{\circ}$ about the internuclear axis, they give MOs differing from those of Fig. 13.15 by a $90^{\circ}$ rotation about the $z$ axis. The linear combinations
\(
\begin{align}
& \left(2 p{y}\right){a}+\left(2 p{y}\right){b} \tag{13.82}\
& \left(2 p{y}\right){a}-\left(2 p{y}\right){b} \tag{13.83}
\end{align}
\)
give the $\pi{u} 2 p{y}$ and $\pi{g}^{*} 2 p{y}$ molecular orbitals. The MOs (13.80) and (13.82) have the same energy. The MOs (13.81) and (13.83) have the same energy. (Note that the $g \pi 2 p$ MOs are antibonding, while the $u \pi 2 p$ MOs are bonding.)
FIGURE 13.14 Cross section of the $\pi{g}^{*} 2 p{+1}$ (or $\pi{g}^{*} 2 p{-1}$ ) MO. To obtain the threedimensional contour surface, rotate the figure about the $z$ axis. The $z$ axis and the xy plane are nodes.
Just as the $2 p{x}$ and $2 p{y}$ AOs are linear combinations of the $2 p{+1}$ and $2 p{-1}$ AOs [Eqs. (6.118) and (6.120)], the $\pi{u} 2 p{x}$ and $\pi{u} 2 p{y}$ MOs are linear combinations of the $\pi{u} 2 p{+1}$ and $\pi{u} 2 p{-1}$ MOs. We can use any linear combination of the eigenfunctions of a degenerate energy level and still have an energy eigenfunction. Just as the $2 p{+1}$ and $2 p{-1}$ AOs are eigenfunctions of $\hat{L}{z}$ and the $2 p{x}$ and $2 p{y}$ AOs are not, the $\pi{u} 2 p{+1}$ and $\pi{u} 2 p{-1}$ MOs are eigenfunctions of $\hat{L}{z}$ and the $\pi{u} 2 p{x}$ and $\pi{u} 2 p{y}$ MOs are not. For the $\mathrm{H}{2}^{+} \pi{u} 2 p$ energy level, we can use the pair of real MOs (13.80) and (13.82), or the pair of complex MOs (13.73) and (13.75), or any two linearly independent linear combinations of these functions.
We have shown the correlation of the $\mathrm{H}{2}^{+}$MOs with the separated-atoms AOs. We can also show how they correlate with the united-atom AOs. As $R$ goes to zero, the $\sigma{u}^{} 1 s$ MO (Fig. 13.9) increasingly resembles the $2 p{z}$ AO, with which it correlates. Similarly, the $\pi{u} 2 p$ MOs correlate with $p$ united-atom states, while the $\pi_{g}^{} 2 p$ MOs correlate with $d$ united-atom states.
An online simulation of $\mathrm{H}_{2}^{+} \mathrm{MOs}$ is at www.falstad.com/qmmo; you can vary the internuclear distance.