We now begin the study of the electronic energies of molecules. We shall use the Born-Oppenheimer approximation, keeping the nuclei fixed while we solve, as best we can, the Schrödinger equation for the motion of the electrons. We shall usually be considering an isolated molecule, ignoring intermolecular interactions. Our results will be most applicable to molecules in the gas phase at low pressure. For inclusion of solvent effects, see Sections 15.17 and 17.6.
We start with diatomic molecules, the simplest of which is $\mathrm{H}{2}^{+}$, the hydrogen molecule ion, consisting of two protons and one electron. Just as the one-electron H atom serves as a starting point in the discussion of many-electron atoms, the one-electron $\mathrm{H}{2}^{+}$ ion furnishes many ideas useful for discussing many-electron diatomic molecules. The electronic Schrödinger equation for $\mathrm{H}_{2}^{+}$is separable, and we can get exact solutions for the eigenfunctions and eigenvalues.
Figure 13.3 shows $\mathrm{H}{2}^{+}$. The nuclei are at $a$ and $b ; R$ is the internuclear distance; $r{a}$ and $r_{b}$ are the distances from the electron to nuclei $a$ and $b$. Since the nuclei are fixed, we have a one-particle problem whose purely electronic Hamiltonian is [Eq. (13.5)]
\(
\begin{equation}
\hat{H}{\mathrm{el}}=-\frac{\hbar^{2}}{2 m{e}} \nabla^{2}-\frac{e^{2}}{4 \pi \varepsilon{0} r{a}}-\frac{e^{2}}{4 \pi \varepsilon{0} r{b}} \tag{13.31}
\end{equation}
\)
FIGURE 13.3 Interparticle distances in $\mathrm{H}_{2}^{+}$.
The first term is the electronic kinetic-energy operator; the second and third terms are the attractions between the electron and the nuclei. In atomic units the purely electronic Hamiltonian for $\mathrm{H}_{2}^{+}$is
\(
\begin{equation}
\hat{H}{\mathrm{el}}=-\frac{1}{2} \nabla^{2}-\frac{1}{r{a}}-\frac{1}{r_{b}} \tag{13.32}
\end{equation}
\)
In Fig. 13.3 the coordinate origin is on the internuclear axis, midway between the nuclei, with the $z$ axis lying along the internuclear axis. The $\mathrm{H}_{2}^{+}$electronic Schrödinger equation is not separable in spherical coordinates. However, separation of variables is possible using confocal elliptic coordinates $\xi$, $\eta$, and $\phi$. The coordinate $\phi$ is the angle of rotation of the electron about the internuclear $(z)$ axis, the same as in spherical coordinates. The coordinates $\xi$ (xi) and $\eta$ (eta) are defined by
\(
\begin{equation}
\xi \equiv \frac{r{a}+r{b}}{R}, \quad \eta \equiv \frac{r{a}-r{b}}{R} \tag{13.33}
\end{equation}
\)
The ranges of these coordinates are
\(
\begin{equation}
0 \leq \phi \leq 2 \pi, \quad 1 \leq \xi \leq \infty, \quad-1 \leq \eta \leq 1 \tag{13.34}
\end{equation}
\)
We must put the Hamiltonian (13.32) into these coordinates. We have
\(
\begin{equation}
r{a}=\frac{1}{2} R(\xi+\eta), \quad r{b}=\frac{1}{2} R(\xi-\eta) \tag{13.35}
\end{equation}
\)
We also need the expression for the Laplacian in confocal elliptic coordinates. One way to find this is to express $\xi, \eta$, and $\phi$ in terms of $x, y$, and $z$, the Cartesian coordinates of the electron, and then use the chain rule to find $\partial / \partial x, \partial / \partial y$, and $\partial / \partial z$ in terms of $\partial / \partial \xi, \partial / \partial \eta$, and $\partial / \partial \phi$. We then form $\nabla^{2} \equiv \partial^{2} / \partial x^{2}+\partial^{2} / \partial y^{2}+\partial^{2} / \partial z^{2}$. The derivation of $\nabla^{2}$ is omitted. (For a discussion, see Margenau and Murphy, Chapter 5.) Substitution of $\nabla^{2}$ and (13.35) into (13.32) gives $\hat{H}{\text {el }}$ of $\mathrm{H}{2}^{+}$in confocal elliptic coordinates. The result is omitted.
For the hydrogen atom, whose Hamiltonian has spherical symmetry, the electronic angular-momentum operators $\hat{L}^{2}$ and $\hat{L}{z}$ both commute with $\hat{H}$. The $\mathrm{H}{2}^{+}$ion does not have spherical symmetry, and one finds that $\left[\hat{L}^{2}, \hat{H}{\mathrm{el}}\right] \neq 0$ for $\mathrm{H}{2}^{+}$. However, $\mathrm{H}{2}^{+}$does have axial symmetry, and one can show that $\hat{L}{z}$ commutes with $\hat{H}{\mathrm{el}}$ of $\mathrm{H}{2}^{+}$. Therefore, the electronic wave functions can be chosen to be eigenfunctions of $\hat{L}{z}$. The eigenfunctions of $\hat{L}{z}$ are [Eq. (5.81)]
\(
\begin{equation}
\text { constant } \cdot(2 \pi)^{-1 / 2} e^{i m \phi}, \quad \text { where } m=0, \pm 1, \pm 2, \pm 3, \ldots \tag{13.36}
\end{equation}
\)
The $z$ component of electronic orbital angular momentum in $\mathrm{H}{2}^{+}$is $m \hbar$ or $m$ in atomic units. The total electronic orbital angular momentum is not a constant for $\mathrm{H}{2}^{+}$.
The "constant" in (13.36) is a constant only as far as $\partial / \partial \phi$ is concerned, so the $\mathrm{H}{2}^{+}$ wave functions have the form $\psi{\mathrm{el}}=F(\xi, \eta)(2 \pi)^{-1 / 2} e^{\text {im } \phi}$. One now tries a separation of variables:
\(
\begin{equation}
\psi_{\mathrm{el}}=L(\xi) M(\eta)(2 \pi)^{-1 / 2} e^{i m \phi} \tag{13.37}
\end{equation}
\)
Substitution of (13.37) into $\hat{H}{\mathrm{el}} \psi{\mathrm{el}}=E{\mathrm{el}} \psi{\mathrm{el}}$ gives an equation in which the variables are separable. One gets two ordinary differential equations, one for $L(\xi)$ and one for $M(\eta)$. Solving these equations, one finds that the condition that $\psi{\mathrm{el}}$ be well-behaved requires that, for each fixed value of $R$, only certain values of $E{\text {el }}$ are allowed. This gives a set of different electronic states. There is no algebraic formula for $E{\mathrm{el}}$; it must be calculated numerically for each desired value of $R$ for each state. In addition to the quantum number $m$, the $\mathrm{H}{2}^{+}$ electronic wave functions are characterized by the quantum numbers $n{\xi}$ and $n{\eta}$, which give the number of nodes in the $L(\xi)$ and $M(\eta)$ factors in $\psi_{\mathrm{el}}$.
For the ground electronic state, the quantum number $m$ is zero. At $R=\infty$, the $\mathrm{H}{2}^{+}$ground state is dissociated into a proton and a ground-state hydrogen atom; hence $E{\mathrm{el}}(\infty)=-\frac{1}{2}$ hartree. At $R=0$, the two protons have come together to form the $\mathrm{He}^{+}$ ion with ground-state energy: $-\frac{1}{2}(2)^{2}$ hartrees $=-2$ hartrees. Addition of the internuclear repulsion $1 / R$ (in atomic units) to $E{\mathrm{el}}(R)$ gives the $U(R)$ potential-energy curve for nuclear motion. Plots of the ground-state $E{\mathrm{el}}(R)$ and $U(R)$, as found from solution of the electronic Schrödinger equation, are shown in Fig. 13.4. At $R=\infty$ the internuclear repulsion is 0 , and $U$ is $-\frac{1}{2}$ hartree.
The $U(R)$ curve is found to have a minimum at $R{e}=1.9972$ bohrs $=1.057 \AA$, indicating that the $\mathrm{H}{2}^{+}$ground electronic state is a stable bound state. The calculated value of $E{\mathrm{el}}$ at 1.9972 bohrs is -1.1033 hartrees. Addition of the internuclear repulsion $1 / R$ gives $U\left(R{e}\right)=-0.6026$ hartree, compared with -0.5000 hartree at $R=\infty$. The ground-state binding energy is thus $D_{e}=0.1026$ hartree $=2.79 \mathrm{eV}$. This corresponds to $64.4 \mathrm{kcal} / \mathrm{mol}=269 \mathrm{~kJ} / \mathrm{mol}$. The binding energy is only $17 \%$ of the total energy at the equilibrium internuclear distance. Thus a small error in the total energy can correspond to a large error in the binding energy. For heavier molecules the situation is even worse, since chemical binding energies are of the same order of magnitude for most diatomic molecules, but the total electronic energy increases markedly for heavier molecules.
FIGURE 13.4 Electronic energy with $(U)$ and without ( $E{\mathrm{e}}$ ) internuclear repulsion for the $\mathrm{H}{2}^{+}$ground electronic state.
FIGURE $13.5 U(R)$ curves for several $\mathrm{H}_{2}^{+}$electronic states. Dashed lines are used to help distinguish closely spaced states. Not visible on the scale of this diagram is a slight minimum in the curve of the first excited electronic state at 12.5 bohrs with a well depth of 0.00006 hartrees.
Note that the single electron in $\mathrm{H}{2}^{+}$is sufficient to give a stable bound state.
Figure 13.5 shows the $U(R)$ curves for the first several electronic energy levels of $\mathrm{H}{2}^{+}$, as found by solving the electronic Schrödinger equation.
The angle $\phi$ occurs in $\hat{H}{\text {el }}$ of $\mathrm{H}{2}^{+}$only as $\partial^{2} / \partial \phi^{2}$. When $\psi{\text {el }}$ of (13.37) is substituted into $\hat{H}{\mathrm{el}} \psi{\mathrm{el}}=E{\mathrm{el}} \psi{\mathrm{el}}$, the $e^{i m \phi}$ factor cancels, and we are led to differential equations for $L(\xi)$ and $M(\eta)$ in which the $m$ quantum number occurs only as $m^{2}$. Since $E{\text {el }}$ is found from the $L(\xi)$ and $M(\eta)$ differential equations, $E_{\text {el }}$ depends on $m^{2}$, and each electronic level with $m \neq 0$ is doubly degenerate, corresponding to states with quantum numbers $+|m|$ and $-|m|$. In the standard notation for diatomic molecules [F. A. Jenkins, J. Opt. Soc. Am., 43, 425 (1953)], the absolute value of $m$ is called $\lambda$ :
\(
\lambda \equiv|m|
\)
(Some texts define $\lambda$ as identical to $m$.) Similar to the $s, p, d, f, g$ notation for hydrogen-atom states, a letter code is used to specify $\lambda$, the absolute value (in atomic units) of the component along the molecular axis of the electron's orbital angular momentum:
| $\lambda$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| letter | $\sigma$ | $\pi$ | $\delta$ | $\phi$ | $\gamma$ |
Thus the lowest $\mathrm{H}{2}^{+}$electronic state is a $\sigma$ state.
Besides classifying the states of $\mathrm{H}{2}^{+}$according to $\lambda$, we can also classify them according to their parity (Section 7.5). From Fig. 13.12, inversion of the electron's coordinates
through the origin $O$ changes $\phi$ to $\phi+\pi, r{a}$ to $r{b}$, and $r{b}$ to $r{a}$. This leaves the potentialenergy part of the electronic Hamiltonian (13.31) unchanged. We previously showed the kinetic-energy operator to be invariant under inversion. Hence the parity operator commutes with the Hamiltonian (13.31), and the $\mathrm{H}_{2}^{+}$electronic wave functions can be classified as either even or odd. For even electronic wave functions, we use the subscript $g$ (from the German word gerade, meaning even); for odd wave functions, we use $u$ (from ungerade).
The lowest $\sigma{g}$ energy level in Fig. 13.5 is labeled $1 \sigma{g}$, the next-lowest $\sigma{g}$ level at small $R$ is labeled $2 \sigma{g}$, and so on. The lowest $\sigma{u}$ level is labeled $1 \sigma{u}$, and so on. The alternative notation $\sigma{g} 1 s$ indicates that this level dissociates to a $1 s$ hydrogen atom. The meaning of the star in $\sigma{u}^{*} 1 s$ will be explained later.
For completeness, we must take spin into account by multiplying each spatial $\mathrm{H}_{2}^{+}$ electronic wave function by $\alpha$ or $\beta$, depending on whether the component of electron spin along the internuclear axis is $+\frac{1}{2}$ or $-\frac{1}{2}$ (in atomic units). Inclusion of spin doubles the degeneracy of all levels.
The $\mathrm{H}{2}^{+}$ground electronic state has $R{e}=2.00$ bohrs $=2.00\left(4 \pi \varepsilon{0} \hbar^{2} / m{e} e^{2}\right)$. The negative muon (symbol $\mu^{-}$) is a short-lived (half-life $2 \times 10^{-6}$ s) elementary particle whose charge is the same as that of an electron but whose mass $m{\mu}$ is 207 times $m{e}$. When a beam of negative muons (produced when ions accelerated to high speed collide with ordinary matter) enters $\mathrm{H}{2}$ gas, muomolecular ions that consist of two protons and one muon are formed. This species, symbolized by $(\mathrm{p} \mu \mathrm{p})^{+}$, is an $\mathrm{H}{2}^{+}$ion in which the electron has been replaced by a muon. Its $R{e}$ is found by replacing $m{e}$ with $m{\mu}$ in $R{e}$ :
\(
2.00\left(4 \pi \varepsilon{0} \hbar^{2} / m{\mu} e^{2}\right)=2.00\left(4 \pi \varepsilon{0} \hbar^{2} / 207 m{e} e^{2}\right)=(2.00 / 207) \text { bohr }=0.0051 \AA
\)
The two nuclei in this muoion are 207 times closer than in $\mathrm{H}{2}^{+}$. The magnitude of the vibrational-wave-function factor $S{v}\left(R-R{e}\right)$ in (13.28) is small but not entirely negligible for $R-R{e}=-0.0051 \AA$, so there is some probability for the nuclei in $(\mathrm{p} \mu \mathrm{p})^{+}$to come in contact, and nuclear fusion might occur. The isotopic nuclei ${ }^{2} \mathrm{H}$ (deuterium, D) and ${ }^{3} \mathrm{H}$ (tritium, T ) undergo fusion much more readily than protons, so instead of $\mathrm{H}{2}$ gas, one uses a mixture of $\mathrm{D}{2}$ and $\mathrm{T}_{2}$ gases. After fusion occurs, the muon is released and can then be recaptured to catalyze another fusion. Under the right conditions, one muon can catalyze 150 fusions on average before it decays. Unfortunately, at present, more energy is needed to produce the muon beam than is released by the fusion. (See en.wikipedia.org/wiki/ Muon-catalyzed_fusion.)
In the rest of this chapter, the subscript el will be dropped from the electronic wave function, Hamiltonian, and energy. It will be understood in Chapters 13 to 17 that $\psi$ means $\psi_{\mathrm{el}}$.