Let us compare the molecular-orbital and valence-bond treatments of the $\mathrm{H}{2}$ ground state.
If $\phi{a}$ symbolizes an atomic orbital centered on nucleus $a$, the spatial factor of the unnormalized LCAO-MO wave function for the $\mathrm{H}_{2}$ ground state is
\(
\begin{equation}
\left[\phi{a}(1)+\phi{b}(1)\right]\left[\phi{a}(2)+\phi{b}(2)\right] \tag{13.106}
\end{equation}
\)
In the simplest treatment, $\phi$ is a $1 s$ AO. The function (13.106) equals
\(
\begin{equation}
\phi{a}(1) \phi{a}(2)+\phi{b}(1) \phi{b}(2)+\phi{a}(1) \phi{b}(2)+\phi{b}(1) \phi{a}(2) \tag{13.107}
\end{equation}
\)
What is the physical significance of the terms? The last two terms have each electron in an atomic orbital centered on a different nucleus. These are covalent terms, corresponding to equal sharing of the electrons between the atoms. The first two terms have both electrons in AOs centered on the same nucleus. These are ionic terms, corresponding to the chemical structures
\(
\mathrm{H}^{-} \mathrm{H}^{+} \text {and } \mathrm{H}^{+} \mathrm{H}^{-}
\)
The covalent and ionic terms occur with equal weight, so this simple MO function gives a $50-50$ chance as to whether the $\mathrm{H}{2}$ ground state dissociates to two neutral hydrogen atoms or to a proton and a hydride ion. Actually, the $\mathrm{H}{2}$ ground state dissociates to two neutral H atoms. Thus the simple MO function gives the wrong limiting value of the energy as $R$ goes to infinity.
How can we remedy this? Since $\mathrm{H}_{2}$ is nonpolar, chemical intuition tells us that ionic terms should contribute substantially less to the wave function than covalent terms. The simplest procedure is to omit the ionic terms of the MO function (13.107). This gives
\(
\begin{equation}
\phi{a}(1) \phi{b}(2)+\phi{b}(1) \phi{a}(2) \tag{13.108}
\end{equation}
\)
We recognize (13.108) as the Heitler-London function (13.100).
Although interelectronic repulsion causes the electrons to avoid each other, there is some probability of finding both electrons near the same nucleus, corresponding to an ionic structure. Therefore, instead of simply dropping the ionic terms from (13.107), we might try
\(
\begin{equation}
\phi{\mathrm{VB}, \mathrm{imp}}=\phi{a}(1) \phi{b}(2)+\phi{b}(1) \phi{a}(2)+\delta\left[\phi{a}(1) \phi{a}(2)+\phi{b}(1) \phi_{b}(2)\right] \tag{13.109}
\end{equation}
\)
where $\delta(R)$ is a variational parameter and the subscript imp indicates an improved VB function. In the language of valence-bond theory, this trial function represents ioniccovalent resonance. Of course, the ground-state wave function of $\mathrm{H}{2}$ does not undergo a time-dependent change back and forth from a covalent function corresponding to the structure $\mathrm{H}-\mathrm{H}$ to ionic functions. Rather (in the approximation we are considering), the wave function is a time-independent mixture of covalent and ionic functions. Since $\mathrm{H}{2}$ dissociates to neutral atoms, we know that $\delta(\infty)=0$. A variational calculation done by Weinbaum in 1933 using $1 s$ AOs with an orbital exponent gave the result that at $R_{e}$ the parameter $\delta$ has the value 0.26 ; the orbital exponent was found to be 1.19 , and the dissociation energy was calculated as 4.03 eV , a modest improvement over the Heitler-London-Wang value of 3.78 eV . With $\delta$ equal to zero in (13.109), we get the VB function (13.108). With $\delta$ equal to 1 , we get the LCAO-MO function (13.107). The optimum value of $\delta$ turns out to be closer to zero than to 1 , and, in fact, the Heitler-London-Wang VB function gives a better dissociation energy than the LCAO-MO function.
Let us compare the improved valence-bond trial function (13.109) with the simple LCAO-MO function improved by configuration interaction. The LCAO-MO CI trial function (13.95) has the (unnormalized) form
$\phi{\mathrm{MO}, \text { imp }}=\left[\phi{a}(1)+\phi{b}(1)\right]\left[\phi{a}(2)+\phi{b}(2)\right]+\gamma\left[\phi{a}(1)-\phi{b}(1)\right]\left[\phi{a}(2)-\phi_{b}(2)\right]$
Since we have not yet normalized this function, there is no harm in multiplying it by the constant $1 /(1-\gamma)$. Doing so and rearranging terms, we get
\(
\phi{\mathrm{MO}, \mathrm{imp}}=\phi{a}(1) \phi{b}(2)+\phi{b}(1) \phi{a}(2)+\frac{1+\gamma}{1-\gamma}\left[\phi{a}(1) \phi{a}(2)+\phi{b}(1) \phi_{b}(2)\right]
\)
There is also no harm done if we define a new constant $\delta$ as $\delta=(1+\gamma) /(1-\gamma)$. We see then that this improved MO function and the improved VB function (13.109) are identical. Weinbaum viewed his $\mathrm{H}_{2}$ calculation as a valence-bond calculation with inclusion of ionic terms. We have shown that we can just as well view the Weinbaum calculation as an MO calculation with configuration interaction. (This was the viewpoint adopted in Section 13.9.)
The MO function (13.107) underestimates electron correlation, in that it says that structures with both electrons on the same atom are just as likely as structures with each electron on a different atom. The VB function (13.108) overestimates electron correlation, in that it has no contribution from structures with both electrons on the same atom. In MO theory, electron correlation can be introduced by configuration interaction. In VB theory, electron correlation is reduced by ionic-covalent resonance. The simple VB method is more reliable at large $R$ than the simple MO method, since the latter predicts the wrong dissociation products.
To further fix the differences between the MO and VB approaches, consider how each method divides the $\mathrm{H}_{2}$ electronic Hamiltonian into unperturbed and perturbation Hamiltonians. For the MO method, we write
\(
\hat{H}=\left[\left(-\frac{1}{2} \nabla{1}^{2}-\frac{1}{r{a 1}}-\frac{1}{r{b 1}}\right)+\left(-\frac{1}{2} \nabla{2}^{2}-\frac{1}{r{a 2}}-\frac{1}{r{b 2}}\right)\right]+\frac{1}{r_{12}}
\)
where the unperturbed Hamiltonian consists of the bracketed terms. In MO theory the unperturbed Hamiltonian for $\mathrm{H}{2}$ is the sum of two $\mathrm{H}{2}^{+}$Hamiltonians, one for each electron. Accordingly, the zeroth-order MO wave function is a product of two $\mathrm{H}{2}^{+}$-like wave functions, one for each electron. Since the $\mathrm{H}{2}^{+}$functions are complicated, we approximate the $\mathrm{H}{2}^{+}$-like MOs as LCAOs. The effect of the $1 / r{12}$ perturbation is taken into account in an average way through use of self-consistent-field molecular orbitals. To take instantaneous electron correlation into account, we can use configuration interaction.
For the valence-bond method, the terms in the Hamiltonian are grouped in either of two ways:
\(
\begin{aligned}
& \hat{H}=\left[\left(-\frac{1}{2} \nabla{1}^{2}-\frac{1}{r{a 1}}\right)+\left(-\frac{1}{2} \nabla{2}^{2}-\frac{1}{r{b 2}}\right)\right]-\frac{1}{r{a 2}}-\frac{1}{r{b 1}}+\frac{1}{r{12}} \
& \hat{H}=\left[\left(-\frac{1}{2} \nabla{1}^{2}-\frac{1}{r{b 1}}\right)+\left(-\frac{1}{2} \nabla{2}^{2}-\frac{1}{r{a 2}}\right)\right]-\frac{1}{r{a 1}}-\frac{1}{r{b 2}}+\frac{1}{r{12}}
\end{aligned}
\)
The unperturbed system is two hydrogen atoms. We have two zeroth-order functions consisting of products of hydrogen-atom wave functions, and these belong to a degenerate level. The correct ground-state zeroth-order function is the linear combination (13.100).
The MO method is used far more often than the VB method, because it is computationally much simpler than the VB method. The MO method was developed by Hund, Mulliken, and Lennard-Jones in the late 1920s. Originally, it was used largely for qualitative descriptions of molecules, but the electronic digital computer has made possible the calculation of accurate MO functions (Section 13.14). For a discussion of the relative merits of the MO and VB methods, see R. Hoffman et al., Acc. Chem. Res., 36, 750 (2003).