PHY.K02UF Molecular and Solid State Physics

## Valence Bond Theory

Valence bond theory is similar to Molecular orbital theory but is mathematically simpler. The starting point is the many-particle Hamltonian,

$$$\label{eq:htotal} H_{\text{mp}}= -\sum\limits_i \frac{\hbar^2}{2m_e}\nabla^2_i -\sum\limits_a \frac{\hbar^2}{2m_a}\nabla^2_a -\sum\limits_{a,i} \frac{Z_ae^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_a|}+\sum\limits_{i< j} \frac{e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_j|}+\sum\limits_{a< b} \frac{Z_aZ_be^2}{4\pi\epsilon_0 |\vec{r}_a-\vec{r}_b|} .$$$

As in molecule orbital theory, the Born-Oppenheimer approximation is used. Since the nuclei are much heavier than the electrons, the electrons will move much faster than the nuclei. We may therefore fix the positions of the nuclei while solving for the electron states. The electron states are described by the electronic Hamiltonain $H_{\text{elec}}$.

$$$\label{eq:helec} H_{\text{elec}}= -\sum\limits_i \frac{\hbar^2}{2m_e}\nabla^2_i -\sum\limits_{a,i} \frac{Z_ae^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_a|}+\sum\limits_{i< j} \frac{e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_j|} +\sum\limits_{a< b} \frac{Z_aZ_be^2}{4\pi\epsilon_0 |\vec{r}_a-\vec{r}_b|}.$$$

The kinetic energy of the nuclei do not appear in $H_{\text{elec}}$ since the nuclei have been fixed. The last term in $H_{\text{elec}}$ does not depend on the positions of the electrons so it just adds a constant to the energy. Since a constant can always be added to or subtracted from the energy, this term will be neglected as we solve for the motion of the electrons. The electronic Hamiltonian cannot be solved analytically; it can only be solved numerically. To make further progress with an analytical solution in molecular orbital theory, we negleted the electron-electron interaction terms at this point. In valence bond theory we associate an each electron to an atom and neglect the electron-electron interactions and the interactions between an electron and the nuclei of other atoms that it is not associated with. The resulting Hamiltonian is a sum of atomic Hamiltonians.

$$$\label{eq:helecred} H_{\text{vb}}= -\sum\limits_i \frac{\hbar^2}{2m_e}\nabla^2_i -\sum\limits_{i} \frac{Z_{ai}e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_{ai}|}= \sum\limits_i H_{\text{atom}_i}.$$$

Here $Z_{ai}$ is the nuclear charge of the atom associated with electron $i$ and $\vec{r}_{ai}$ is the position of the atom associated with electron $i$. This Hamiltonian can be solved by the separation of variables. Since the solutions to the atomic Hamiltonians are the atomic orbitals, the solution to $H_{\text{vb}}$ is an antisymmetrized product of atomic orbitals. This is an exact solution to $H_{\text{vb}}$ and an approximate solution to $H_{\text{elec}}$. It can be used to find the approximate energy of the many electron system using the equation,

$$$E= \frac{\langle \Psi |H_{\text{elec}}|\Psi\rangle}{\langle \Psi|\Psi\rangle}.$$$

The shape of a molecule in terms of bond lengths and bond angles is determined by finding the arrangement of the atoms that minimizes this energy.

Heitler-London theory
The most famous application of valence bond theory is Heitler and London's description of the hydrogen molecule. This appeared in 1927, just two years after Schrödinger proposed his wave equation. For H2, all terms in the many-particle Hamiltonian are neglected except,

$$$H_{\text{vb}}= - \frac{\hbar^2}{2m_e}\nabla^2_1 -\frac{e^2}{4\pi\epsilon_0 |\vec{r}_1-\vec{r}_a|}- \frac{\hbar^2}{2m_e}\nabla^2_2 -\frac{e^2}{4\pi\epsilon_0 |\vec{r}_2-\vec{r}_b|}.$$$

Here $\vec{r}_{1,2}$ are the positions of the two electrons and $\vec{r}_{a,b}$ are the positions of the two protons. Notice that the interaction between electron 1 and proton b, the interaction between electron 2 and proton a, and the electron-electron interaction have been neglected. This Hamiltonian separates into two Hamiltonians for atomic hydrogen. The lowest energy solutions of the atomic Hamiltonians are $\phi^H_{1\text{s}}(\vec{r}-\vec{r}_a)$ and $\phi^H_{1\text{s}}(\vec{r}-\vec{r}_b)$. An exact solution to $H_{\text{vb}}$ is an antisymmetrized product of these atomic orbitals. There are four possibilities for the spins: $\uparrow\uparrow$, $\downarrow\downarrow$, $\uparrow\downarrow$, and $\downarrow\uparrow$. The corresponding Slater determinants are,

$$\begin{matrix} \Psi_{\uparrow\uparrow}=\frac{1}{\sqrt{2}}\left|\begin{matrix} \phi_{1s}^{\text{H}}\uparrow(\vec{r}_1-\vec{r}_a) & \phi_{1s}^{\text{H}}\uparrow(\vec{r}_1-\vec{r}_b) \\ \phi_{1s}^{\text{H}}\uparrow(\vec{r}_2-\vec{r}_a) & \phi_{1s}^{\text{H}}\uparrow(\vec{r}_2-\vec{r}_b) \end{matrix}\right| = \frac{1}{\sqrt{2}}\left(\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_a)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_b)-\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_b)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_a)\right)\uparrow(\vec{r}_1)\uparrow(\vec{r}_2), \\ \\ \Psi_{\downarrow\downarrow}=\frac{1}{\sqrt{2}}\left|\begin{matrix} \phi_{1s}^{\text{H}}\downarrow(\vec{r}_1-\vec{r}_a) & \phi_{1s}^{\text{H}}\downarrow(\vec{r}_1-\vec{r}_b) \\ \phi_{1s}^{\text{H}}\downarrow(\vec{r}_2-\vec{r}_a) & \phi_{1s}^{\text{H}}\downarrow(\vec{r}_2-\vec{r}_b) \end{matrix}\right| = \frac{1}{\sqrt{2}}\left(\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_a)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_b)-\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_b)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_a)\right)\downarrow(\vec{r}_1)\downarrow(\vec{r}_2), \\ \\ \Psi_{\uparrow\downarrow}=\frac{1}{\sqrt{2}}\left|\begin{matrix} \phi_{1s}^{\text{H}}\uparrow(\vec{r}_1-\vec{r}_a) & \phi_{1s}^{\text{H}}\downarrow(\vec{r}_1-\vec{r}_b) \\ \phi_{1s}^{\text{H}}\uparrow(\vec{r}_2-\vec{r}_a) & \phi_{1s}^{\text{H}}\downarrow(\vec{r}_2-\vec{r}_b) \end{matrix}\right| = \frac{1}{\sqrt{2}}\left(\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_a)\uparrow(\vec{r}_1)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_b)\downarrow(\vec{r}_2)-\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_b)\downarrow(\vec{r}_1)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_a)\uparrow(\vec{r}_2)\right), \\ \\ \Psi_{\downarrow\uparrow}=\frac{1}{\sqrt{2}}\left|\begin{matrix} \phi_{1s}^{\text{H}}\downarrow(\vec{r}_1-\vec{r}_a) & \phi_{1s}^{\text{H}}\uparrow(\vec{r}_1-\vec{r}_b) \\ \phi_{1s}^{\text{H}}\downarrow(\vec{r}_2-\vec{r}_a) & \phi_{1s}^{\text{H}}\uparrow(\vec{r}_2-\vec{r}_b) \end{matrix}\right| = \frac{1}{\sqrt{2}}\left(\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_a)\downarrow(\vec{r}_1)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_b)\uparrow(\vec{r}_2)-\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_b)\uparrow(\vec{r}_1)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_a)\downarrow(\vec{r}_2)\right). \\ \\ \end{matrix}$$

By constructing the linear combinations of the last two wave functions $\Psi_{\uparrow\downarrow+\downarrow\uparrow} = \Psi_{\uparrow\downarrow}+\Psi_{\downarrow\uparrow}$ and $\Psi_{\uparrow\downarrow-\downarrow\uparrow} = \Psi_{\uparrow\downarrow}-\Psi_{\downarrow\uparrow}$, the wave functions factor into an orbital part and a spin part,

$$$\Psi_{\uparrow\downarrow+\downarrow\uparrow}(\vec{r}_1,\vec{r}_2)= \frac{1}{\sqrt{2}}\left(\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_a)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_b)-\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_a)\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_b)\right)\left(\uparrow(\vec{r}_1)\downarrow(\vec{r}_2)+\downarrow(\vec{r}_1)\uparrow(\vec{r}_2)\right), \nonumber$$$ $$$\Psi_{\uparrow\downarrow-\downarrow\uparrow}(\vec{r}_1,\vec{r}_2)= \frac{1}{\sqrt{2}}\left(\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_a)\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_b)+\phi_{1s}^{\text{H}}(\vec{r}_2-\vec{r}_a)\phi_{1s}^{\text{H}}(\vec{r}_1-\vec{r}_b)\right)\left(\uparrow(\vec{r}_1)\downarrow(\vec{r}_2)-\downarrow(\vec{r}_1)\uparrow(\vec{r}_2)\right). \nonumber$$$

The three wave functions with an antisymmetric orbital part, $\Psi_{\uparrow\uparrow}$, $\Psi_{\downarrow\downarrow}$, and $\Psi_{\uparrow\downarrow+\downarrow\uparrow}$ all have the same energy when evaluated with $H_{\text{elec}}$. This is the triplet state. The singlet state, $\Psi_{\uparrow\downarrow-\downarrow\uparrow}$ has a different energy when evaluated with $H_{\text{elec}}$. For H2, the singlet state has a lower energy than the triplet state so the singlet state is the molecular ground state. The bond potential for H2 can be approximated by evaluating,

$$$E= \frac{\langle \Psi_{\uparrow\downarrow-\downarrow\uparrow} |H_{\text{elec}}|\Psi_{\uparrow\downarrow-\downarrow\uparrow}\rangle}{\langle \Psi_{\uparrow\downarrow-\downarrow\uparrow}|\Psi_{\uparrow\downarrow-\downarrow\uparrow}\rangle},$$$

as a function of the distance between the atoms.

It is interesting to compare the ground state wave function found by Heitler and London to that found by molecular orbital theory:

$$\Psi(\vec{r}_1,\vec{r}_2) = \frac{1}{2\sqrt{2}}\left(\phi_{\text{1s}}^H(\vec{r}_1-\vec{r}_a) + \phi_{\text{1s}}^H(\vec{r}_1-\vec{r}_b)\right)\left(\phi_{\text{1s}}^H(\vec{r}_2-\vec{r}_a) + \phi_{\text{1s}}^H(\vec{r}_2-\vec{r}_b)\right)\left(\uparrow(\vec{r}_1)\downarrow(\vec{r}_2)-\downarrow(\vec{r}_1)\uparrow(\vec{r}_2)\right).$$

The ground state found by molecular orbital theory has two additional terms compared to the valence bond wave function.