Summary of Molecular Spectroscopy: Fundamentals and Applications
Molecular Spectroscopy: Fundamentals and Applications Guide
Introduction
Vibrational spectroscopy probes how atoms in a molecule move relative to each other. When bonds stretch, compress or bend, the system has quantized vibrational energy levels. Measuring transitions between these levels (most commonly with infrared light) reveals information about bond strength, masses, molecular structure and chemical environment.
Definition: Vibrational spectroscopy is the set of experimental methods that measure transitions between quantized vibrational energy levels of molecules, typically using infrared (IR) or Raman techniques.
1. Types of Vibrational Motion
- Stretching: change in bond length. Can be symmetric or asymmetric in polyatomic molecules.
- Bending: change in bond angle (scissoring, rocking, wagging, twisting).
- Internal modes: combinations of stretches and bends in larger molecules.
Definition: A normal mode is an independent collective vibration of atoms in a molecule that can be excited without coupling to other modes (within the harmonic approximation).
1.1 Examples
- Diatomic molecule: only one vibrational mode — the bond stretch.
- CO2: has symmetric stretch (IR inactive), asymmetric stretch (IR active), and two degenerate bending modes (IR active depending on orientation).
2. Harmonic Approximation
Near the equilibrium bond length $R_e$ we expand the potential energy $V(x)$ where $x=R-R_e$ using a Taylor series and keep the quadratic term.
$$V(x) \approx \tfrac{1}{2}k_f x^2$$
- $k_f$ is the force constant: $k_f = \left.\dfrac{d^2V}{dx^2}\right|_{x=0}$.
- Stiff bonds: large $k_f$, higher vibrational frequency and steep potential. Soft bonds: small $k_f$, lower frequency and shallow potential.
Quantum solution (harmonic oscillator):
$$E_\nu = \left(\nu + \tfrac{1}{2}\right)\hbar\omega,\quad \nu = 0,1,2,\dots$$
$$\omega = \sqrt{\dfrac{k_f}{m_{\text{eff}}}}$$
- Effective mass for a diatomic: $m_{\text{eff}}=\dfrac{m_1 m_2}{m_1 + m_2}$.
- Convert energies to spectroscopically convenient wavenumbers (in cm$^{-1}$):
$$\tilde{\nu} = \dfrac{\omega}{2\pi c}$$
Example (brief)
Given $k_f$ for HCl is $516\ \mathrm{N,m^{-1}}$ and $m_{\text{eff}}\approx1.628\times10^{-27}\ \mathrm{kg}$, one computes
$$\omega = \sqrt{\dfrac{516}{1.628\times10^{-27}}}=5.631\times10^{14}\ \mathrm{s^{-1}}$$
$$\tilde{\nu}=\dfrac{\omega}{2\pi c}=2906\ \mathrm{cm^{-1}}$$
Definition: Wavenumber $\tilde{\nu}$ is frequency divided by the speed of light, with units of cm$^{-1}$; spectroscopists commonly report vibrational energies this way.
3. Selection Rules for Infrared (IR) Spectroscopy
- Gross selection rule: A vibration is IR-active only if it produces a change in the electric dipole moment of the molecule.
- Specific harmonic selection rule: $\Delta\nu = \pm 1$ (absorption: $+1$, emission: $-1$).
- At room temperature most molecules are in $\nu=0$, so the fundamental $0\rightarrow1$ transition dominates.
Table: IR activity examples
| Vibration type | Change in dipole? | IR activity |
|---|---|---|
| HCl stretch | Yes | Active |
| N$_2$ stretch | No | Inactive |
| CO$_2$ symmetric stretch | No | Inactive |
| CO$_2$ asymmetric stretch | Yes | Active |
4. Anharmonicity and the Morse Potential
The harmonic oscillator cannot describe bond dissociation and predicts equally spaced levels. Real molecular potentials are anharmonic and levels get closer as energy increases.
A commonly used anharmonic potential is the Morse potential:
$$V(x)=hc,D_e\left(1-e^{-ax}\right)^2$$
Parameters and consequences:
- $D_e$ is the dissociation energy (in energy units divided by $hc$ when using $hcD_e$
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Vibrational Spectroscopy Basics
Klíčové pojmy: Vibrational energy levels are quantized: $E_\nu=\left(\nu+\tfrac{1}{2}\right)\hbar\omega$, Harmonic approximation: $V(x)\approx\tfrac{1}{2}k_f x^2$ with $\omega=\sqrt{k_f/m_{\text{eff}}}$, Effective mass for diatomics: $m_{\text{eff}}=\dfrac{m_1 m_2}{m_1+m_2}$, IR activity requires a change in molecular dipole moment during vibration, Harmonic selection rule: $\Delta\nu=\pm1$; anharmonicity allows overtones ($\Delta\nu=2,3$), Morse potential models anharmonicity: $V(x)=hcD_e(1-e^{-ax})^2$, Number of normal modes: nonlinear $3N-6$, linear $3N-5$, Wavenumber $\tilde{\nu}=\dfrac{\omega}{2\pi c}$ is used in spectroscopy, Anharmonicity reduces level spacing: $\tilde{\nu}_{\nu\rightarrow\nu+1}=\tilde{\nu}-2(\nu+1)x_e\tilde{\nu}$, CO$_2$ example: symmetric stretch IR inactive, asymmetric stretch IR active