Chemical Bonding & Molecular Structure
MIT‑level, first‑principles treatment for the KVPY “Historic” chemistry course
Goal of these notes – to give you a rigorous, quantum‑mechanical understanding of how atoms come together to form molecules, and how the resulting electronic structure dictates geometry, spectroscopy and reactivity.
All derivations start from the time‑independent Schrödinger equation and proceed through the Born–Oppenheimer approximation, LCAO and Crystal‑Field Theory before touching on DFT and post‑Hartree–Fock corrections.
1. The Quantum Mechanical Foundation
1.1 Schrödinger Equation (SE) for an N‑electron atom
[ \hat H\Psi = E\Psi,\qquad \hat H = \sum_{i=1}^N\left(-\frac{\hbar^2}{2m_e}\nabla_i^2 - \frac{Z e^2}{4\pi\varepsilon_0 r_i}\right)
- \sum_{i<j}\frac{e^2}{4\pi\varepsilon_0 r_{ij}} ]
- Electron–nucleus attraction (first sum)
- Electron–electron repulsion (second sum)
Because the SE has no analytic solution for (N>2), we use approximate solutions:
- Hartree–Fock (HF) – mean‑field, antisymmetrized wavefunction (Slater determinant).
- Post‑HF – configuration interaction (CI), coupled cluster (CC), Møller–Plesset (MP2/MP3), multireference methods.
- DFT – Kohn–Sham orbitals, exchange–correlation functional.
1.2 Born–Oppenheimer Approximation (BOA)
Assumes nuclei move on a fixed electronic potential energy surface:
[ \Psi(\mathbf{r},\mathbf{R}) = \Phi_{\text{el}}(\mathbf{r};\mathbf{R})\chi_{\text{nuc}}(\mathbf{R}) ]
- Electronic Hamiltonian solved with nuclei fixed at (\mathbf{R}).
- Gives potential energy surfaces (E_{\text{el}}(\mathbf{R})).
- Vibrational, rotational, and translational motions are then treated as nuclear dynamics on this surface.
2. Atomic Electronic Structure
| Quantum Number | Interpretation | Allowed values | Energy ordering in a hydrogenic atom |
|---|---|---|---|
| (n) | Principal | (\ge 1) | (E_n \propto -1/n^2) |
| (l) | Azimuthal | (0,\dots,n-1) | (E) independent of (l) (hydrogenic) |
| (m_l) | Magnetic | (-l,\dots,+l) | Degenerate |
| (s) | Spin | (\pm \tfrac12) | Degenerate |
Aufbau principle: Fill orbitals in order of increasing energy (1s, 2s, 2p, 3s, 3p,\dots).
Hund’s rule: Maximize total spin in a degenerate set.
Pauli Exclusion Principle: No two electrons share all four quantum numbers.
3. Bond Formation
3.1 Ionic Bonding
Energy change:
[ \Delta E_{\text{ionic}} = \Delta E_{\text{ionization}} + \Delta E_{\text{electron affinity}} + \Delta E_{\text{Coulomb}} ]
- For metal–nonmetal pairs, ionization energy of the metal is usually low; electron affinity of the nonmetal is high.
- Resulting lattice energy dominates; solids form.
3.2 Covalent Bonding – Classical Lewis View
- Electron pair model: Two atoms share an electron pair to achieve octets.
- Formal charge:
[ \mathrm{FC} = N_{\text{valence}} - (N_{\text{non‑bonding electrons}} + \tfrac12 N_{\text{bonding electrons}}) ]
- Resonance: Delocalized electrons across multiple bonds (e.g., benzene).
3.3 Molecular Orbital (MO) Theory
3.3.1 LCAO–SCF
[ \psi_k = \sum_i c_{ik} \phi_i ]
- (\phi_i) are atomic orbitals (AOs).
- Coefficients (c_{ik}) determined by solving the secular determinant.
3.3.2 Bonding & Antibonding
- (\sigma) (over