Intermolecular Forces – Advanced University Notes
(Prepared for a MIT‑level Chemistry course – “COMEDK” – “Intermolecular Forces”)
Goal: Build a rigorous, quantitative understanding of all non‑covalent interactions between molecules from first principles, and explore subtle edge cases that surface in modern chemistry and materials science.
1. Overview & Classification
| Category | Typical Interaction | Distance Dependence | Key Feature |
|---|---|---|---|
| Electrostatic | Permanent multipole–multipole | (r^{-n}) with (n=3) for dipole–dipole, (n=4) for dipole–quadrupole, … | Long‑range, orientation‑dependent |
| Induction | Permanent multipole–induced dipole | (r^{-6}) | “Polarization” of the field |
| Dispersion (London) | Instantaneous dipole–induced dipole | (r^{-6}) | Quantum‑mechanical, present in all atoms/molecules |
| Hydrogen Bond | Donor (H) + Acceptor (X) | (r^{-6}) (dominant) | Directional, partially covalent |
| Short‑range | Pauli (exchange) repulsion | (\sim e^{-a r}) | Overlap of electron clouds |
Total Potential Energy
[ V_{\text{tot}}(r, \Omega) = V_{\text{el}}(r,\Omega) + V_{\text{ind}}(r,\Omega) + V_{\text{disp}}(r,\Omega)+ V_{\text{rep}}(r) ] where (\Omega) denotes relative orientation.
2. Electrostatic Interactions – Multipole Expansion from First Principles
2.1. Classical Derivation
Let two charge distributions (\rho_A(\mathbf{r})) and (\rho_B(\mathbf{r})) be separated by a vector (\mathbf{R}) with (|\mathbf{R}| \gg) molecular dimensions. The interaction energy is
[ V_{\text{el}} = \frac{1}{4\pi\varepsilon_0}\int !!\int \frac{\rho_A(\mathbf{r}_A)\rho_B(\mathbf{r}_B)}{|\mathbf{R}+\mathbf{r}_B-\mathbf{r}_A|}, d^3 r_A d^3 r_B . ]
Expanding the denominator in powers of (\mathbf{r}_A,\mathbf{r}_B) (Legendre series), we obtain the multipole expansion:
[ V_{\text{el}} = \frac{1}{4\pi\varepsilon_0}\sum_{l,m}\sum_{l',m'} \frac{(-1)^{l'} T^{(l)}{m}(\mathbf{R}) T^{(l')}{-m'}(\mathbf{R})}{R^{l+l'+1}} \langle Q^{(l)}m Q^{(l')}{-m'} \rangle . ]
- (T^{(l)}_m(\mathbf{R})) are spherical tensor components of (\mathbf{R}).
- (Q^{(l)}_m) are the multipole moments of each molecule.
Key Result: For neutral molecules, the leading term is dipole‑dipole (\sim R^{-3}).
[
V_{\text{dd}} = \frac{1}{4\pi\varepsilon_0}\frac{ \mathbf{\mu}_A !\cdot! \mathbf{\mu}_B - 3(\mathbf{\mu}_A !\cdot! \hat{\mathbf{R}})(\mathbf{\mu}_B !\cdot! \hat{\mathbf{R}})}{R^3}.
]
2.2. Quantum Mechanical Perspective
Using the Born–Oppenheimer Born–Oppenheimer approximation and treating the nuclei as fixed, the electrostatic energy is simply the Coulomb integral between charge densities of the two subsystems. In the point‑dipole approximation:
[ \hat{V}_{\text{el}} = -\frac{1}{4\pi\varepsilon_0}\frac{ \hat{\boldsymbol{\mu}}_A !\cdot! \hat{\boldsymbol{\mu}}_B - 3(\hat{\boldsymbol{\mu}}_A