---
title: "What one infrared peak can tell us"
description: "Start with a vibrating bond, distinguish infrared and Raman selection rules, and see why one frequency axis can hide different molecular histories."
canonical_url: "https://sandboxphysics.com/stories/multidimensional-spectroscopy/vibrational-fingerprints/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/multidimensional-spectroscopy/vibrational-fingerprints/"
last_modified: "2026-10-28"
---

[Multidimensional spectroscopy](https://sandboxphysics.com/stories/multidimensional-spectroscopy/)

L1 · Chapter 1

# What one infrared peak can tell us

Sandbox Physics · 2026-10-28

An infrared peak marks frequencies at which a molecule absorbs light readily. Bond stiffness, atomic mass and the surrounding environment all leave traces. Yet different molecular histories can produce nearly the same peak. A second measurement asks where a vibration that started at one frequency appears later.

## Before you begin

You need to read a graph and know force, energy and frequency. Treat light as an oscillating electric field. Molecular structure and normal modes can be studied in greater depth later; this chapter begins with the phenomenon.

- Explain a vibrational frequency using inertia and restoring force.
- Distinguish the molecular properties measured by infrared absorption and Raman scattering.
- Describe the information added by a second frequency axis.

## 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.

$$
\mu=\frac{m_1m_2}{m_1+m_2},\qquad \omega=\sqrt{\frac{k}{\mu}}
$$

The reduced mass controls relative acceleration. The angular frequency is measured in radians per second.

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 $c$.

$$
E_v=\hbar\omega\left(v+\frac12\right),\qquad \omega=2\pi c\widetilde\nu
$$

The level formula is harmonic. Use centimetres per second for the speed of light when wavenumbers are in inverse centimetres.

Anharmonic level spacings shrink: excitation from the first level encounters a lower transition frequency. Diagrammatic levels, not to scale.

## 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

## Try the prediction

### [Revisit Raman selection rules](https://sandboxphysics.com/labs/spontaneous-raman-spectrometer/)

1. Compare two polarization or vibration settings and observe the Raman intensity.
2. Explain why this intensity cannot be read directly as an infrared transition dipole.

### [Find diagonal and cross features](https://sandboxphysics.com/labs/coupled-vibrations-2d-ir/)

1. Keep the default coupling and locate signed pairs close to and away from the diagonal.
2. Predict what removing coupling should change, then change that condition alone.

## Exercises and explanations

1. At fixed force constant, what happens when the reduced mass becomes four times larger?

   Read the explanation

   The frequency halves because it is inversely proportional to the square root of the reduced mass. The restoring force is unchanged, but the inertia increases.

2. Do identical linear absorption spectra imply identical molecular dynamics?

   Read the explanation

   No. The frequency distribution and total linewidth may agree even when environmental exchange rates or connections between vibrations differ. A waiting-time series supplies an additional observation.

[Next: Why a spectral line broadens](https://sandboxphysics.com/stories/multidimensional-spectroscopy/coherence-and-linewidth/)

## Sources and calculation scope

1. [Andrei Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (June 2011, v1)](https://knowledge.uchicago.edu/records/3tv7m-5ha50)

   Author lecture notes, nonlinear-spectroscopy section: density-matrix perturbation theory; printed pp. 23–25, Eqs. (51)–(55), for rephasing and nonrephasing; pp. 29–30 for the vibrational harmonic limit. The course examples are new teaching calculations.

2. [Tokmakoff, Two-dimensional spectroscopy chapter (June 2011)](https://doi.org/10.6082/uchicago.3314)

   In the same deposited notes, the section whose printed page numbers restart at 1: pp. 4–5, Eqs. (1)–(3), connect response, Fourier transformation and heterodyne detection; pp. 10–11 discuss coupled-vibration peak pairs. No source figure is redistributed.