We now begin the study of molecular quantum mechanics. If we assume the nuclei and electrons to be point masses and neglect spin-orbit and other relativistic interactions (Sections 11.6 and 11.7), then the molecular Hamiltonian operator is
\(
\begin{equation}
\hat{H}=-\frac{\hbar^{2}}{2} \sum{\alpha} \frac{1}{m{\alpha}} \nabla{\alpha}^{2}-\frac{\hbar^{2}}{2 m{e}} \sum{i} \nabla{i}^{2}+\sum{\alpha} \sum{\beta>\alpha} \frac{Z{\alpha} Z{\beta} e^{2}}{4 \pi \varepsilon{0} r{\alpha \beta}}-\sum{\alpha} \sum{i} \frac{Z{\alpha} e^{2}}{4 \pi \varepsilon{0} r{i \alpha}}+\sum{j} \sum{i>j} \frac{e^{2}}{4 \pi \varepsilon{0} r_{i j}} \tag{13.1}
\end{equation}
\)
where $\alpha$ and $\beta$ refer to nuclei and $i$ and $j$ refer to electrons. The first term in (13.1) is the operator for the kinetic energy of the nuclei. The second term is the operator for the kinetic energy of the electrons. The third term is the potential energy of the repulsions between the nuclei, $r{\alpha \beta}$ being the distance between nuclei $\alpha$ and $\beta$ with atomic numbers $Z{\alpha}$ and $Z{\beta}$. The fourth term is the potential energy of the attractions between the electrons and the nuclei, $r{i \alpha}$ being the distance between electron $i$ and nucleus $\alpha$. The last term is the potential energy of the repulsions between the electrons, $r_{i j}$ being the distance between electrons $i$ and $j$. The zero level of potential energy for (13.1) corresponds to having all the charges (electrons and nuclei) infinitely far from one another.
As an example, consider $\mathrm{H}{2}$. Let $\alpha$ and $\beta$ be the two protons, 1 and 2 be the two electrons, and $m{p}$ be the proton mass. The $\mathrm{H}_{2}$ molecular Hamiltonian operator is
\(
\begin{align}
\hat{H}= & -\frac{\hbar^{2}}{2 m{p}} \nabla{\alpha}^{2}-\frac{\hbar^{2}}{2 m{p}} \nabla{\beta}^{2}-\frac{\hbar^{2}}{2 m{e}} \nabla{1}^{2}-\frac{\hbar^{2}}{2 m{e}} \nabla{2}^{2} \
& +\frac{e^{2}}{4 \pi \varepsilon{0} r{\alpha \beta}}-\frac{e^{2}}{4 \pi \varepsilon{0} r{1 \alpha}}-\frac{e^{2}}{4 \pi \varepsilon{0} r{1 \beta}}-\frac{e^{2}}{4 \pi \varepsilon{0} r{2 \alpha}}-\frac{e^{2}}{4 \pi \varepsilon{0} r{2 \beta}}+\frac{e^{2}}{4 \pi \varepsilon{0} r{12}} \tag{13.2}
\end{align}
\)
The wave functions and energies of a molecule are found from the Schrödinger equation:
\(
\begin{equation}
\hat{H} \psi\left(q{i}, q{\alpha}\right)=E \psi\left(q{i}, q{\alpha}\right) \tag{13.3}
\end{equation}
\)
where $q{i}$ and $q{\alpha}$ symbolize the electronic and nuclear coordinates, respectively. The molecular Hamiltonian (13.1) is formidable enough to terrify any quantum chemist. Fortunately, a very accurate, simplifying approximation exists. Since nuclei are much heavier than electrons ( $m{\alpha} \gg m{e}$ ), the electrons move much faster than the nuclei. Hence, to a good
approximation as far as the electrons are concerned, we can regard the nuclei as fixed while the electrons carry out their motions. Speaking classically, during the time of a cycle of electronic motion, the change in nuclear configuration is negligible. Thus, considering the nuclei as fixed, we omit the nuclear kinetic-energy terms from (13.1) to obtain the Schrödinger equation for electronic motion:
\(
\begin{equation}
\left(\hat{H}{\mathrm{el}}+V{N N}\right) \psi{\mathrm{el}}=U \psi{\mathrm{el}} \tag{13.4}
\end{equation}
\)
where the purely electronic Hamiltonian $\hat{H}_{\text {el }}$ is
\(
\begin{equation}
\hat{H}{\mathrm{el}}=-\frac{\hbar^{2}}{2 m{e}} \sum{i} \nabla{i}^{2}-\sum{\alpha} \sum{i} \frac{Z{\alpha} e^{2}}{4 \pi \varepsilon{0} r{i \alpha}}+\sum{j} \sum{i>j} \frac{e^{2}}{4 \pi \varepsilon{0} r_{i j}} \tag{13.5}
\end{equation}
\)
The electronic Hamiltonian including nuclear repulsion is $\hat{H}{\text {el }}+V{N N}$. The nuclear-repulsion term $V_{N N}$ is
\(
\begin{equation}
V{N N}=\sum{\alpha} \sum{\beta>\alpha} \frac{Z{\alpha} Z{\beta} e^{2}}{4 \pi \varepsilon{0} r_{\alpha \beta}} \tag{13.6}
\end{equation}
\)
The energy $U$ in (13.4) is the electronic energy including internuclear repulsion. The internuclear distances $r_{\alpha \beta}$ in (13.4) are not variables, but are each fixed at some constant value. Of course, there are an infinite number of possible nuclear configurations, and for each of these we may solve the electronic Schrödinger equation (13.4) to get a set of electronic wave functions and corresponding electronic energies. Each member of the set corresponds to a different molecular electronic state. The electronic wave functions and energies thus depend parametrically on the nuclear coordinates:
\(
\psi{\mathrm{el}}=\psi{\mathrm{el}, n}\left(q{i} ; q{\alpha}\right) \quad \text { and } \quad U=U{n}\left(q{\alpha}\right)
\)
where $n$ symbolizes the electronic quantum numbers.
The variables in the electronic Schrödinger equation (13.4) are the electronic coordinates. The quantity $V{N N}$ is independent of these coordinates and is a constant for a given nuclear configuration. Now it is easily proved (Prob. 4.52) that the omission of a constant term $C$ from the Hamiltonian does not affect the wave functions and simply decreases each energy eigenvalue by $C$. Hence, if $V{N N}$ is omitted from (13.4), we get
\(
\begin{equation}
\hat{H}{\mathrm{el}} \psi{\mathrm{el}}=E{\mathrm{el}} \psi{\mathrm{el}} \tag{13.7}
\end{equation}
\)
where the purely electronic energy $E{\mathrm{el}}\left(q{\alpha}\right)$ (which depends parametrically on the nuclear coordinates $q_{\alpha}$ ) is related to the electronic energy including internuclear repulsion by
\(
\begin{equation}
U=E{\mathrm{el}}+V{N N} \tag{13.8}
\end{equation}
\)
We can therefore omit the internuclear repulsion from the electronic Schrödinger equation. After finding $E{\text {el }}$ for a particular configuration of the nuclei by solving (13.7), we calculate $U$ using (13.8), where the constant $V{N N}$ is easily calculated from (13.6) using the assumed nuclear locations.
For $\mathrm{H}{2}$, with the two protons at a fixed distance $r{\alpha \beta}=R$, the purely electronic Hamiltonian is given by (13.2) with the first, second, and fifth terms omitted. The nuclear repulsion $V{N N}$ equals $e^{2} / 4 \pi \varepsilon{0} R$. The purely electronic Hamiltonian involves the six electronic coordinates $x{1}, y{1}, z{1}, x{2}, y{2}, z{2}$ as variables and involves the nuclear coordinates as parameters.
The electronic Schrödinger equation (13.4) can be dealt with by approximate methods to be discussed later. If we plot the electronic energy including nuclear repulsion for a bound state of a diatomic molecule against the internuclear distance $R$, we find a
FIGURE 13.1 Electronic energy including internuclear repulsion as a function of the internuclear distance $R$ for a diatomic-molecule bound electronic state.
curve like the one shown in Fig. 13.1. At $R=0$, the internuclear repulsion causes $U$ to go to infinity. The internuclear separation at the minimum in this curve is called the equilibrium internuclear distance $R{e}$. The difference between the limiting value of $U$ at infinite internuclear separation and its value at $R{e}$ is called the equilibrium dissociation energy (or the dissociation energy from the potential-energy minimum) $D_{e}$ :
\(
\begin{equation}
D{e} \equiv U(\infty)-U\left(R{e}\right) \tag{13.9}
\end{equation}
\)
When nuclear motion is considered (Section 13.2), one finds that the equilibrium dissociation energy $D{e}$ differs from the molecular ground-vibrational-state dissociation energy $D{0}$. The lowest state of nuclear motion has zero rotational energy [as shown by Eq. (6.47)] but has a nonzero vibrational energy-the zero-point energy. If we use the harmonic-oscillator approximation for the vibration of a diatomic molecule (Section 4.3), then this zero-point energy is $\frac{1}{2} h \nu$. This zero-point energy raises the energy for the ground state of nuclear motion $\frac{1}{2} h \nu$ above the minimum in the $U(R)$ curve, so $D{0}$ is less than $D{e}$ and $D{0} \approx D{e}-\frac{1}{2} h \nu$. Different electronic states of the same molecule have different $U(R)$ curves (Figs. 13.5 and 13.19) and different values of $R{e}, D{e}, D_{0}$, and $\nu$.
Consider an ideal gas composed of diatomic molecules AB. In the limit of absolute zero temperature, all the AB molecules are in their ground states of electronic and nuclear motion, so $D{0} N{\mathrm{A}}$ (where $N{\mathrm{A}}$ is the Avogadro constant and $D{0}$ is for the ground electronic state of AB ) is the change in the thermodynamic internal energy $U$ and enthalpy $H$ for dissociation of 1 mole of ideal-gas diatomic molecules: $N{\mathrm{A}} D{0}=\Delta U{0}^{\circ}=\Delta H{0}^{\circ}$ for $\mathrm{AB}(\mathrm{g}) \rightarrow \mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g})$.
For some diatomic-molecule electronic states, solution of the electronic Schrödinger equation gives a $U(R)$ curve with no minimum. Such states are not bound and the molecule will dissociate. Examples include some of the states in Fig. 13.5.
Assuming that we have solved the electronic Schrödinger equation, we next consider nuclear motions. According to our picture, the electrons move much faster than the nuclei. When the nuclei change their configuration slightly, say from $q{\alpha}^{\prime}$ to $q{\alpha}^{\prime \prime}$, the electrons immediately adjust to the change, with the electronic wave function changing from $\psi{\mathrm{el}}\left(q{i} ; q{\alpha}^{\prime}\right)$ to $\psi{\mathrm{el}}\left(q{i} ; q{\alpha}^{\prime \prime}\right)$ and the electronic energy changing from $U\left(q{\alpha}^{\prime}\right)$ to $U\left(q{\alpha}^{\prime \prime}\right)$. Thus, as the nuclei move, the electronic energy varies smoothly as a function of the parameters defining the nuclear configuration, and $U\left(q_{\alpha}\right)$ becomes, in effect, the potential energy for the nuclear motion. The electrons act like a spring connecting the nuclei. As the internuclear distance
changes, the energy stored in the spring changes. Hence the Schrödinger equation for nuclear motion is
\(
\begin{gather}
\hat{H}{N} \psi{N}=E \psi{N} \tag{13.10}\
\hat{H}{N}=-\frac{\hbar^{2}}{2} \sum{\alpha} \frac{1}{m{\alpha}} \nabla{\alpha}^{2}+U\left(q{\alpha}\right) \tag{13.11}
\end{gather}
\)
The variables in the nuclear Schrödinger equation are the nuclear coordinates, symbolized by $q_{\alpha}$. The energy eigenvalue $E$ in (13.10) is the total energy of the molecule, since the Hamiltonian (13.11) includes operators for both nuclear energy and electronic energy. $E$ is simply a number and does not depend on any coordinates. Note that for each electronic state of a molecule we must solve a different nuclear Schrödinger equation, since $U$ differs from state to state. In this chapter we shall concentrate on the electronic Schrödinger equation (13.4).
In Section 13.2, we shall show that the total energy $E$ for an electronic state of a diatomic molecule is approximately the sum of electronic, vibrational, rotational, and translational energies, $E \approx E{\text {elec }}+E{\text {vib }}+E{\text {rot }}+E{\text {tr }}$, where the constant $E{\text {elec }}$ [not to be confused with $E{\text {el }}$ in (13.7)] is given by $E{\text {elec }}=U\left(R{e}\right)$.
The approximation of separating electronic and nuclear motions is called the BornOppenheimer approximation and is basic to quantum chemistry. [The American physicist J. Robert Oppenheimer (1904-1967) was a graduate student of Born in 1927. During World War II, Oppenheimer directed the Los Alamos laboratory that developed the atomic bomb.] Born and Oppenheimer's mathematical treatment indicated that the true molecular wave function is adequately approximated as
\(
\begin{equation}
\psi\left(q{i}, q{\alpha}\right)=\psi{\mathrm{el}}\left(q{i} ; q{\alpha}\right) \psi{N}\left(q_{\alpha}\right) \tag{13.12}
\end{equation}
\)
if $\left(m{e} / m{\alpha}\right)^{1 / 4} \ll 1$. The Born-Oppenheimer approximation introduces little error for the ground electronic states of diatomic molecules. Corrections for excited electronic states are larger than for the ground state, but still are usually small as compared with the errors introduced by the approximations used to solve the electronic Schrödinger equation of a many-electron molecule. Hence we shall not worry about corrections to the Born-Oppenheimer approximation. For further discussion of the Born-Oppenheimer approximation, see J. Goodisman, Diatomic Interaction Potential Theory, Academic Press, 1973, Volume 1, Chapter 1.
Born and Oppenheimer's 1927 paper justifying the Born-Oppenheimer approximation is seriously lacking in rigor. Subsequent work has better justified the Born-Oppenheimer approximation, but significant questions still remain; "the problem of the coupling of nuclear and electronic motions is, at the moment, without a sensible solution and ... is an area where much future work can and must be done" [B. T. Sutcliffe, J. Chem. Soc. Faraday Trans., 89, 2321 (1993); see also B. T. Sutcliffe and R. G. Woolley, Phys. Chem. Chem. Phys., 7, 3664 (2005), and Sutcliffe and Woolley, arxiv.org/abs/1206.4239].