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.

Normal-mode separation $48.8\,\mathrm{cm}^{-1}$

Local anharmonic shift $22.0\,\mathrm{cm}^{-1}$

Coherence sampling window $7.56\,\mathrm{ps}$

Saved complex samples $8192$

Broadband pulses · oriented dipoles

Find the cross peaks

Vibrational coupling

J/(\mathrm{cm}^{-1})

14

Anharmonic shift

\Delta/(\mathrm{cm}^{-1})

22

Second dipole angle

\theta/{}^\circ

35

Waiting time

T/\mathrm{ps}

0.2

Drag the cyan bond to turn its transition dipole. Click the spectrum below to inspect a frequency slice.

Dephasing, noise and sampling

Coherence lifetime

T_2/\mathrm{ps}

1.8

Quadrature noise standard deviation

0

Random seed

Each channel saves a square grid of sixty-four coherence delays, spaced by 0.12 picoseconds. Independent Gaussian noise is added to the real and imaginary samples before reconstruction.

$\delta\widetilde\nu=4.34\,\mathrm{cm}^{-1},\quad |\widetilde\nu-1650\,\mathrm{cm}^{-1}|<139\,\mathrm{cm}^{-1}$ gives the frequency spacing and Nyquist range. The displayed region is cropped; no peak sharpening is applied.

Model and optical geometry

The cell contains identical oriented two-mode molecules. All pulse polarizations and detection are parallel to the first local dipole. Rotating the second dipole changes its projection; the sample is not isotropically averaged.

Collinear pulse phases select the rephasing and nonrephasing channels. A phase-stable local oscillator mixes with the emitted signal before detection. Pulses are treated as instantaneous and broadband; pulse overlap is outside the model.

The motion shows a magnified, slowed normal-mode amplitude with the optical carrier removed. Waiting-time evolution includes coherent beating and a common six-picosecond signal envelope. It does not simulate chemical exchange or population transfer through a solvent.

Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (2011)

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

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 misconceptionA negative peak means the molecule emits negative energy.

Better mental modelThe 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

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.
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.
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.

Coupled Vibrations in Two-Dimensional IR

Two frequency axes reveal the shared response

Absorptive spectrum · red positive, blue negative

Detection-frequency slice

Saved rephasing quadratures