Part VI: Spectroscopy & Molecular Structure
Spectroscopy reveals the internal structure of molecules by probing how they absorb, emit, and scatter electromagnetic radiation. This part develops the theory of rotational, vibrational, and electronic spectroscopy from first principles — connecting quantum-mechanical energy levels to observable spectra and molecular geometry.
3 chapters | Rotational, Vibrational & Electronic Spectroscopy
Key Equations
Rotational & Vibrational
Rigid Rotor Energies: $E_J = B J(J+1), \quad B = \frac{\hbar^2}{2I}$
Rotational Selection Rule: $\Delta J = \pm 1$
Harmonic Oscillator: $E_v = \hbar\omega\!\left(v + \tfrac{1}{2}\right)$
Anharmonic Correction: $E_v = \hbar\omega\!\left(v + \tfrac{1}{2}\right) - \hbar\omega x_e\!\left(v + \tfrac{1}{2}\right)^2$
Vibrational Selection Rule: $\Delta v = \pm 1 \;\text{(harmonic)}$
Electronic & General
Beer-Lambert Law: $A = \varepsilon c l$
Transmittance: $T = \frac{I}{I_0} = 10^{-A}$
Franck-Condon Principle: $I_{v' \leftarrow v''} \propto \left|\langle \chi_{v'} | \chi_{v''} \rangle\right|^2$
Transition Dipole: $\mu_{fi} = \langle \psi_f | \hat{\mu} | \psi_i \rangle$
Einstein Coefficients: $\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$
Chapters
1. Rotational Spectroscopy
The rigid rotor model, classification of molecular rotors (linear, symmetric top, spherical top, asymmetric top), microwave spectroscopy, centrifugal distortion, and determination of bond lengths from rotational constants.
Rigid rotor: $E_J = BJ(J+1)$ with degeneracy $g_J = 2J+1$
Centrifugal distortion: $E_J = BJ(J+1) - DJ^2(J+1)^2$
Transition frequencies: $\tilde{\nu} = 2B(J+1)$
2. Vibrational Spectroscopy
The quantum harmonic oscillator applied to molecular vibrations, infrared absorption, Raman scattering, normal modes of polyatomic molecules, anharmonicity, overtones, and combination bands.
Harmonic oscillator: $E_v = \hbar\omega(v + \tfrac{1}{2})$, $\Delta v = \pm 1$
IR activity: requires $\frac{\partial \mu}{\partial Q} \neq 0$
Raman activity: requires $\frac{\partial \alpha}{\partial Q} \neq 0$ (polarizability change)
3. Electronic Spectroscopy
UV-visible absorption, the Franck-Condon principle, vibronic transitions, term symbols and selection rules for electronic transitions, fluorescence, phosphorescence, and the Beer-Lambert law for quantitative analysis.
Franck-Condon factor: $\text{FC} = \left|\langle \chi_{v'} | \chi_{v''} \rangle\right|^2$
Electronic selection rules: $\Delta S = 0, \; \Delta \Lambda = 0, \pm 1$
Beer-Lambert: $A = \varepsilon c l = -\log_{10}(T)$
What You Will Learn
- ● How rotational spectra reveal bond lengths and molecular geometry
- ● The quantum-mechanical origin of IR and Raman selection rules
- ● Why vibrational spectra show overtones and combination bands through anharmonicity
- ● How Franck-Condon factors govern the intensity pattern in electronic spectra
- ● The physical basis of fluorescence, phosphorescence, and non-radiative decay
- ● Quantitative analysis using the Beer-Lambert law and molar absorptivity
- ● How spectroscopic data connect to molecular structure and bonding