---
title: "Coupled Vibrations in Two-Dimensional IR Interactive Lab | Sandbox Physics"
description: "How does coupling between two vibrations appear in a two-dimensional infrared spectrum?"
canonical_url: "https://sandboxphysics.com/labs/coupled-vibrations-2d-ir/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/coupled-vibrations-2d-ir/"
---

FR16-B · Two bonds leave a shared fingerprint

# Coupled Vibrations in Two-Dimensional IR

**How does coupling between two vibrations appear in a two-dimensional infrared spectrum?** Coupling mixes the two vibrations, shifts their frequencies and can create off-diagonal peaks. The positive fundamental and negative excited-state absorption peaks carry different parts of the response. Their pattern also depends on anharmonicity and dipole orientation.

Turn one transition dipole and change the coupling between two weakly anharmonic vibrations. Watch the bonds move, then follow the signed peaks reconstructed from the complex pulse response.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This oriented two-mode model retains the ground, one-quantum and two-quantum states and uses broadband impulsive pulses. It includes ground-state bleach, stimulated emission and excited-state absorption in rephasing and nonrephasing channels. A common phenomenological dephasing and signal lifetime are imposed. It omits orientational averaging, frequency disorder, chemical exchange and finite-pulse overlap. Spectra and records are synthetic; bond displacements and optical delays are magnified.

Synthetic complex signal → Fourier reconstruction

## Two frequency axes reveal the shared response

### Absorptive spectrum · red positive, blue negative

### Detection-frequency slice

### Saved rephasing quadratures

Positive peaks combine ground-state bleach and stimulated emission. Excited-state absorption has the opposite sign under this phase convention. Each image uses the full saved signal.

Fourier signs and saved fields

$$
S=\operatorname{Re}\!\sum_{j,k}w_jw_k\Delta t^2\left[R_{jk}e^{-i\Omega_1t_j+i\Omega_3t_k}+N_{jk}e^{i\Omega_1t_j+i\Omega_3t_k}\right]
$$

The record saves both complex grids before transformation, the seed, all model parameters, sample spacing and rotating reference. Trapezoidal weights halve the boundary samples. Peak color uses the largest absolute value of the current image; compare positions and signs across presets.

[Continue through the nonlinear spectroscopy course](https://sandboxphysics.com/stories/multidimensional-spectroscopy/)

Physics tutorial

## Coupled Vibrations in Two-Dimensional IR

**Background** A linear spectrum puts every absorption on one frequency axis. Two-dimensional IR remembers which frequency was excited before measuring which frequency responds.

**Why it matters** Two coupled vibrations can respond together even when a linear spectrum looks like two ordinary bands. The off-diagonal peaks show where exciting one frequency changes the response at another.

Start with the essentials

- **Focus question**: How does coupling between two vibrations appear in a two-dimensional infrared spectrum?
- **One-sentence intuition**: A cross peak follows the full third-order response, including the cancellation between fundamental and excited-state transitions.

### Core mathematical model

#### Two weakly anharmonic vibrations

$$
H/(hc)=\sum_{i=1}^{2}\left[\widetilde\nu_i n_i-\frac{\Delta}{2}n_i(n_i-1)\right]+J(a_1^\dagger a_2+a_2^\dagger a_1)
$$

Wavenumbers and coupling are in inverse centimeters. The local frequencies are 1630 and 1670 inverse centimeters. The six retained states contain at most two total quanta, sufficient for the zero-temperature third-order response.

#### Why the harmonic signal cancels

$$
R_{\mathrm R}(t_3,0,t_1)\propto 2e^{i\omega_{01}t_1}\left[e^{-i\omega_{01}t_3}-e^{-i\omega_{12}t_3}\right]
$$

For an isolated mode, ground-state bleach and stimulated emission give the first term; excited-state absorption gives the second. Equal transition frequencies make the two terms cancel. Anharmonicity separates their positive and negative peaks.

#### Reconstruct the absorptive spectrum

$$
S(\omega_1,\omega_3)=\operatorname{Re}\int_0^\infty\!dt_1\int_0^\infty\!dt_3\,[R_{\mathrm R}e^{-i\omega_1t_1+i\omega_3t_3}+R_{\mathrm{NR}}e^{i\omega_1t_1+i\omega_3t_3}]
$$

The two coherence intervals use opposite signs on the excitation axis for the two channels. The browser applies this transform to the saved finite complex grids. Both frequency axes display absolute wavenumber after restoring the rotating reference.

### Common difficulties

#### A negative peak is a different transition

**Typical misconception** A negative peak means the molecule emits negative energy.

**Better mental model** The sign belongs to the change in absorption under the chosen phase convention. Here excited-state absorption has the opposite sign to ground-state bleach and stimulated emission. The transition energies remain positive.

### Run the experiment

1. 01

   #### Switch the coupling off

   Predict which features will survive, then select Independent vibrations. Compare with Bonds talk to each other.

   **What to observe:** At zero coupling the response is the sum of two independent single-mode responses. Their positive fundamental and negative excited-state absorption bands remain, but mixed-mode contributions cancel.

2. 02

   #### Make the level spacings equal

   Select Harmonic cancellation. Increase coupling while leaving the anharmonic shift at zero.

   **What to observe:** The third-order signal stays zero. Coupled harmonic modes still form a linear system; coupling alone cannot make its response nonlinear.

3. 03

   #### Rotate the second dipole

   Return to the coupled preset and drag the cyan bond, or move Second dipole angle. Keep the two frequency axes fixed.

   **What to observe:** The second local dipole becomes dark at a right angle to the field. Coupling can still mix its motion with the first dipole, so some mixed-mode signal can survive.

Physical conclusion

Read peak positions, signs and dipole weights together before interpreting a cross peak.

## Related experiments and articles

Multidimensional spectroscopy · Reading path

### Vibrations, echoes and frequency memory

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **What one infrared peak can tell us**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/vibrational-fingerprints/)
2. [Lab **Spontaneous Raman Spectrometer** Revisit Raman selection rules](https://sandboxphysics.com/labs/spontaneous-raman-spectrometer/)
3. [Article **How two vibrations meet in a 2D spectrum**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/coupled-vibrations/)
4. [Article **Reconstruct a 2D spectrum from a complex record**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/signal-to-spectrum/)
5. [Lab **Spectral Diffusion and Frequency Memory** Compare a consistently processed waiting-time series](https://sandboxphysics.com/labs/spectral-diffusion-inference/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)