What one infrared peak can tell us

Stretch a bond a little

Two atoms repel when too close and attract when pulled apart. Near equilibrium, their restoring force is approximately proportional to the displacement. The reduced mass combines the inertia of both atoms in their relative motion.

Let $q$ be the bond extension in metres, $k$ the force constant in newtons per metre, and $m_1,m_2$ the atomic masses. A stiffer bond oscillates faster; increasing the reduced mass slows the motion.

Reading: Andrei Tokmakoff

Energy arrives in steps

A quantum vibration has discrete energies, indexed by $v=0,1,2,\ldots$. The reduced Planck constant is $\hbar=h/(2\pi)$. An ideal harmonic spring has equal gaps of $\hbar\omega$. Real bonds generally soften at larger extensions, so their level spacings can shrink. This departure from the harmonic model is called anharmonicity. Later, it will appear as a measurable separation between spectral features.

Infrared spectroscopy commonly reports wavenumber $\widetilde\nu$ in inverse centimetres. Wavenumber is distinct from angular frequency. The conversion includes a full turn as well as the speed of light $c$.

How light drives the motion

For infrared absorption, a vibration must change the molecular dipole. A field polarized along that change couples effectively to the motion. Transition strength depends on the projection of the transition dipole onto the field polarization.

Raman scattering probes the change in polarizability: how easily the light induces a dipole as the atoms move. A Raman-active vibration need not be infrared-active. For a molecule with inversion symmetry, the usual electric-dipole infrared and ordinary Raman selection rules are mutually exclusive. This distinction matters when we later use infrared intensities to discuss dipole directions.

Reading: Andrei Tokmakoff

Worked example from wavenumber to time

Take a vibration at $1700\,\mathrm{cm}^{-1}$. With $c=0.0299792458\,\mathrm{cm\,ps}^{-1}$, its ordinary frequency is about $50.96\,\mathrm{ps}^{-1}$ and its period is $0.0196\,\mathrm{ps}$, or 19.6 femtoseconds. This is the optical oscillation period. A 2D IR pulse can span several carrier cycles; envelope timing and phase-sensitive detection resolve the slower environmental evolution.

Now let two environments shift the same vibration to $1685\,\mathrm{cm}^{-1}$ and $1715\,\mathrm{cm}^{-1}$. If each population has a broad line, their sum may look like one broad peak. That spectrum records the amount of absorption at each frequency. To determine whether the populations are connected, we need an additional observation.

A second axis records a connection

A two-dimensional infrared spectrum pairs an initial excitation frequency with a later detection frequency. Equal frequencies lie near the diagonal. A connection between different frequencies can create a cross peak. Coupling, energy transfer and chemical exchange can all create connections; waiting time and polarization help distinguish them.

Think of the spectrum as a map of signed connections. First compare two equally broad lines and see how an echo reveals different broadening mechanisms.

Reading: Tokmakoff