The first quantum-mechanical treatment of the hydrogen molecule was by Heitler and London in 1927. Their ideas have been extended to give a general theory of chemical bonding, known as the valence-bond (VB) theory. The valence-bond method is more closely related to the chemist's idea of molecules as consisting of atoms held together by localized bonds than is the molecular-orbital method. The VB method views molecules as composed of atomic cores (nuclei plus inner-shell electrons) and bonding valence electrons. For $\mathrm{H}_{2}$, both electrons are valence electrons.
The first step in the Heitler-London treatment of the $\mathrm{H}_{2}$ ground state is to approximate the molecule as two ground-state hydrogen atoms. The wave function for two such noninteracting atoms is
\(
f{1}=1 s{a}(1) 1 s_{b}(2)
\)
where $a$ and $b$ refer to the nuclei and 1 and 2 refer to the electrons. Of course, the function
\(
f{2}=1 s{a}(2) 1 s_{b}(1)
\)
is also a valid wave function. This then suggests the trial variation function
\(
\begin{equation}
c{1} f{1}+c{2} f{2}=c{1} 1 s{a}(1) 1 s{b}(2)+c{2} 1 s{a}(2) 1 s{b}(1) \tag{13.96}
\end{equation}
\)
This linear variation function leads to the determinantal secular equation $\operatorname{det}\left(H{i j}-S{i j} W\right)=0$ [Eq. (8.57)], where $H{11}=\left\langle f{1}\right| \hat{H}\left|f{1}\right\rangle, S{11}=\left\langle f{1} \mid f{1}\right\rangle, \ldots$
We can also consider the problem using perturbation theory (as Heitler and London did). A ground-state hydrogen molecule dissociates to two neutral ground-state hydrogen atoms. We therefore take as the unperturbed problem two ground-state hydrogen atoms at infinite separation. One possible zeroth-order (unperturbed) wave function is $1 s{a}(1) 1 s{b}(2)$. However, electron 2 could just as well be bound to nucleus $a$, giving the unperturbed wave function $1 s{a}(2) 1 s{b}(1)$. These two unperturbed wave functions belong to a doubly degenerate energy level (exchange degeneracy). Under the perturbation of molecule formation, the
doubly degenerate level is split into two levels, and the correct zeroth-order wave functions are linear combinations of the two unperturbed wave functions:
\(
c{1} 1 s{a}(1) 1 s{b}(2)+c{2} 1 s{a}(2) 1 s{b}(1)
\)
This leads to a $2 \times 2$ secular determinant that is the same as (8.56), except that $W$ is replaced by $E^{(0)}+E^{(1)}$; see Prob. 9.20.
We now solve the secular equation. The Hamiltonian is Hermitian, all functions are real, and $f{1}$ and $f{2}$ are normalized. Therefore
\(
H{12}=H{21}, \quad S{12}=S{21}, \quad S{11}=S{22}=1
\)
Consider $H{11}$ and $H{22}$ :
\(
\begin{aligned}
H{11} & =\left\langle 1 s{a}(1) 1 s{b}(2)\right| \hat{H}\left|1 s{a}(1) 1 s{b}(2)\right\rangle \
H{22} & =\left\langle 1 s{a}(2) 1 s{b}(1)\right| \hat{H}\left|1 s{a}(2) 1 s{b}(1)\right\rangle
\end{aligned}
\)
Interchange of the coordinate labels 1 and 2 in $H{22}$ converts $H{22}$ to $H{11}$, since this relabeling leaves $\hat{H}$ unchanged. Hence $H{11}=H{22}$. The secular equation $\operatorname{det}\left(H{i j}-S_{i j} W\right)=0$ becomes
\(
\left|\begin{array}{cc}
H{11}-W & H{12}-W S{12} \tag{13.97}\
H{12}-W S{12} & H{11}-W
\end{array}\right|=0
\)
This equation has the same form as Eq. (13.49), and by analogy to Eqs. (13.51), (13.57), and (13.58) the approximate energies and wave functions are
\(
\begin{array}{cl}
W{1}=\frac{H{11}+H{12}}{1+S{12}}, & W{2}=\frac{H{11}-H{12}}{1-S{12}} \
\phi{1}=\frac{f{1}+f{2}}{\sqrt{2}\left(1+S{12}\right)^{1 / 2}}, & \phi{2}=\frac{f{1}-f{2}}{\sqrt{2}\left(1-S{12}\right)^{1 / 2}} \tag{13.99}
\end{array}
\)
The numerators of (13.99) are
\(
f{1} \pm f{2}=1 s{a}(1) 1 s{b}(2) \pm 1 s{a}(2) 1 s{b}(1)
\)
From our previous discussion, we know that the ground state of $\mathrm{H}{2}$ is a ${ }^{1} \Sigma$ state with the antisymmetric spin factor (11.60) and a symmetric spatial factor. Hence $\phi{1}$ must be the ground state. The Heitler-London ground-state wave function is
\(
\begin{equation}
\frac{1 s{a}(1) 1 s{b}(2)+1 s{a}(2) 1 s{b}(1)}{\sqrt{2}\left(1+S_{12}\right)^{1 / 2}} \frac{1}{\sqrt{2}}[\alpha(1) \beta(2)-\alpha(2) \beta(1)] \tag{13.100}
\end{equation}
\)
The Heitler-London wave functions for the three states of the lowest ${ }^{3} \Sigma$ term are
\(
\frac{1 s{a}(1) 1 s{b}(2)-1 s{a}(2) 1 s{b}(1)}{\sqrt{2}\left(1-S_{12}\right)^{1 / 2}}\left{\begin{array}{l}
\alpha(1) \alpha(2) \tag{13.101}\
2^{-1 / 2}[\alpha(1) \beta(2)+\beta(1) \alpha(2)] \
\beta(1) \beta(2)
\end{array}\right.
\)
where $S_{12}$ is given in Prob. 13.33.
Now consider the ground-state energy expression. We write the molecular electronic Hamiltonian as the sum of two H -atom Hamiltonians plus perturbing terms:
\(
\begin{equation}
\hat{H}=\hat{H}{a}(1)+\hat{H}{b}(2)+\hat{H}^{\prime} \tag{13.102}
\end{equation}
\)
\(
\hat{H}{a}(1)=-\frac{1}{2} \nabla{1}^{2}-\frac{1}{r{a 1}}, \quad \hat{H}{b}(1)=-\frac{1}{2} \nabla{2}^{2}-\frac{1}{r{b 2}}, \quad \hat{H}^{\prime}=-\frac{1}{r{b 1}}-\frac{1}{r{a 2}}+\frac{1}{r_{12}}
\)
The Heitler-London calculation does not introduce an effective nuclear charge into the $1 s$ function. Hence $1 s{a}(1)$ is an eigenfunction of $\hat{H}{a}(1)$ with eigenvalue $-\frac{1}{2}$ hartree, the hydrogen-atom ground-state energy. Using this result, one finds the following expressions for the VB energies (Prob. 13.33):
\(
\begin{equation}
W{1}=-1+\frac{Q+A}{1+S{a b}^{2}}, \quad W{2}=-1+\frac{Q-A}{1-S{a b}^{2}} \tag{13.103}
\end{equation}
\)
where the Coulomb integral $Q$ and the exchange integral $A$ are defined by:
\(
\begin{align}
Q & \equiv\left\langle 1 s{a}(1) 1 s{b}(2)\right| \hat{H}^{\prime}\left|1 s{a}(1) 1 s{b}(2)\right\rangle \tag{13.104}\
A & \equiv\left\langle 1 s{a}(2) 1 s{b}(1)\right| \hat{H}^{\prime}\left|1 s{a}(1) 1 s{b}(2)\right\rangle \tag{13.105}
\end{align}
\)
and the overlap integral $S_{a b}$ is defined by (13.48). The quantity -1 hartree in these expressions is the energy of two ground-state hydrogen atoms. To obtain the $U(R)$ potentialenergy curves, we add the internuclear repulsion $1 / R$ to these expressions.
Many of the integrals needed to evaluate $W{1}$ and $W{2}$ have been evaluated in the treatment of $\mathrm{H}{2}^{+}$in Section 13.5. The only new integrals are those involving $1 / r{12}$. The hardest one is the two-center, two-electron exchange integral:
\(
\iint 1 s{a}(1) 1 s{b}(2) \frac{1}{r{12}} 1 s{a}(2) 1 s{b}(1) d v{1} d v_{2}
\)
Two-center means that the integrand contains functions centered on two different nuclei, $a$ and $b$; two-electron means that the coordinates of two electrons occur in the integrand. This can be evaluated using an expansion for $1 / r{12}$ in confocal elliptic coordinates, similar to the expansion in Prob. 9.14 in spherical coordinates. Details of the integral evaluations are given in Slater, Quantum Theory of Molecules and Solids, Volume 1, Appendix 6. The results of the Heitler-London treatment are $D{e}=3.15 \mathrm{eV}, R{e}=0.87 \AA$. The agreement with the experimental values $D{e}=4.75 \mathrm{eV}$, $R_{e}=0.741 \AA$ is only fair. In this treatment, most of the binding energy is provided by the exchange integral $A$.
Consider some improvements on the Heitler-London function (13.100). One obvious step is the introduction of an orbital exponent $\zeta$ in the $1 s$ function. This was done by Wang in 1928. The optimum value of $\zeta$ is 1.166 at $R{e}$, and $D{e}$ and $R{e}$ are improved to 3.78 eV and $0.744 \AA$. Recall that Dickinson in 1933 improved the Finkelstein-Horowitz $\mathrm{H}{2}^{+}$trial function by mixing in some $2 p_{z}$ character into the atomic orbitals (hybridization). In 1931 Rosen used this idea to improve the Heitler-London-Wang function. He took the trial function
\(
\phi=\phi{a}(1) \phi{b}(2)+\phi{a}(2) \phi{b}(1)
\)
where the atomic orbital $\phi{a}$ is given by $\phi{a}=e^{-\zeta r{a}}\left(1+c z{a}\right)$, with a similar expression for $\phi_{b}$. This allows for the polarization of the AOs on molecule formation. The result is a binding energy of 4.04 eV . Another improvement, the use of ionic structures, will be considered in the next section.