Step-by-Step Guidance: Molecular Bonds and Molecular Spectra (H₂ Molecule)

Q1. The H₂ Molecule: Semi-Classical Analysis of Bonding and Energy

Background

Topic: Quantum Mechanics and Molecular Physics

This set of questions explores the quantum and classical physics underlying the covalent bond in the H₂ molecule. You'll analyze the arrangement of protons and electrons, potential and kinetic energy, and the quantum mechanical treatment of vibrational and rotational energy levels.

Key Terms and Formulas

• Electrostatic Potential Energy: $U = +\frac{k e^2}{r}$ (for two protons a distance $r$ apart)

• Heisenberg Uncertainty Principle: $\Delta x \Delta p \gtrsim \frac{\hbar}{2}$

• Kinetic Energy Estimate (from Uncertainty Principle): $K_{min} = \frac{\hbar^2}{8 m_e r^2}$

• Total Energy (as a function of $r$): $E_{tot}(r) = -\frac{3 k e^2}{r} + \frac{\hbar^2}{8 m_e r^2}$

• Spring Potential Energy (approximation): $U = \frac{1}{2} k x^2$

• Vibrational Energy Levels: $E_{n, vib} = \left(n + \frac{1}{2}\right) \hbar \sqrt{\frac{k}{\mu}}$

• Reduced Mass: $\mu = \frac{m_1 m_2}{m_1 + m_2}$

• Rotational Energy Levels: $E_{\ell, rot} = \ell(\ell+1) \frac{\hbar^2}{2I}$

• Moment of Inertia: $I = \mu r^2$

Step-by-Step Guidance

1. Part (a): Sketching and Interpreting the Electrostatic Potential Energy - Start by considering two protons separated by a distance $r$. The electrostatic potential energy between them is $U = +\frac{k e^2}{r}$. - Sketch $U$ as a function of $r$. Notice that as $r$ decreases, $U$ increases (becomes more positive), indicating repulsion. - The gradient (slope) of this curve with respect to $r$ gives the force: $F = -\frac{dU}{dr}$. - Think about how this relates to the repulsive force between the protons.

2. Part (b): Adding an Electron and Analyzing Potential Energy - Place one electron in the field of the two protons. The electron is attracted to both protons, so its potential energy is negative. - Compare the potential energy when the electron is midway between the protons (distance $r/2$ from each) versus off to one side. - Use the fact that the electrostatic force decreases with the square of the distance to reason where the electron 'prefers' to be. - Consider the net force on the electron and on each proton in this configuration.

3. Part (c): Calculating Total Electrostatic Potential Energy - When the electron is midway between the protons, calculate the potential energy contributions: proton-proton, proton-electron (for both protons). - Add these contributions to find the total potential energy. You should arrive at $U_{PE} = -\frac{3 k e^2}{r}$.

4. Part (d): Estimating Minimum Kinetic Energy (Quantum) - Model the electron as a particle in a 1D box of width $r$. Use the uncertainty principle to estimate the minimum momentum: $\Delta p \sim \frac{\hbar}{r}$. - The minimum kinetic energy is then $K_{min} = \frac{(\Delta p)^2}{2m_e}$. - Substitute the expression for $\Delta p$ to get $K_{min} = \frac{\hbar^2}{8 m_e r^2}$.

5. Part (e): Sketching Total Energy vs. $r$ - Combine the potential and kinetic energy terms to write $E_{tot}(r) = -\frac{3 k e^2}{r} + \frac{\hbar^2}{8 m_e r^2}$. - Sketch the $-1/r$ and $+1/r^2$ terms separately, then their sum. The total energy curve should have a minimum at some $r^*$.

6. Part (f): Finding the Minimum Energy and Equilibrium Separation - To find the value of $r$ where $E_{tot}(r)$ is minimized, set $\frac{dE_{tot}}{dr} = 0$ and solve for $r^*$. - Substitute $r^*$ back into $E_{tot}(r)$ to find the minimum energy $E_{min}$ (do not compute the final values yet).

Try solving on your own before revealing the answer!

Final Answers and Explanations

Part (f): $r^* = \frac{1}{12} \frac{\hbar^2}{k e^2 m_e}$ and $E_{min} = -\frac{1}{18} \frac{(k e^2)^2 m_e}{\hbar^2}$

These results come from minimizing the total energy with respect to $r$ and substituting back into the energy expression.