FR16-C · Molecules forget their frequency

Spectral Diffusion and Frequency Memory

How long does a molecule remember its vibration frequency? A molecule remembers its frequency until its surroundings change enough to shift the vibration. A two-dimensional peak loses its diagonal tilt as that memory fades. Measurements at several waiting times reveal the decay and let a fitted model predict later spectra.

Watch a molecular environment change, then reconstruct a waiting-time series from saved complex signals. Fit the early measurements and predict the later ones with two competing correlation models.

Selected waiting time $0.64\,\mathrm{ps}$

Inferred frequency memory $0.7$

Fitted fast decay time $1.40\,\mathrm{ps}$

Later-time prediction RMS $0.003$

Follow the changing environment

How quickly does memory fade?

Fast correlation time

\tau_c/\mathrm{ps}

1.4

Slow or static fraction

b

0.18

Waiting-time frame

n

2

Drag the sample to pause the waiting-time replay. The first six measurements train the fits; the last three test their predictions.

Calibration and noise

Quadrature noise

\sigma_n

0.002

Spectral acceptance width

B/\mathrm{cm}^{-1}

70

Maximum waiting time

T_{\max}/\mathrm{ps}

8

Random seed

What the model assumes

This is a synthetic isolated positive band. Gaussian frequency pairs remain frozen during the coherence intervals. Excited-state absorption, exchange and overlapping bands are outside the selected band model.

The known linewidth, spectral acceptance and homogeneous decay calibrate the time-domain covariance estimate. The displayed center line also includes finite-window and apodization effects.

Each saved slice uses unit zero-delay amplitude. A fitted log-amplitude intercept allows an unknown common signal scale. Noisy point estimates may exceed correlation bounds; no confidence intervals are calculated. Rejecting weak or negative noisy samples introduces selection and logarithm bias.

R_{\mathrm{R,NR}}=\exp[-v(t_1^2+t_3^2)/2\pm qt_1t_3-\gamma(t_1+t_3)]

Model assumptions

Synthetic signals describe an isolated positive band with Gaussian frequency fluctuations. Frequencies are frozen during each coherence period. A calibrated Gaussian spectral acceptance and known homogeneous decay modify the signals; additive quadrature noise mimics detection. The selected band excludes excited-state absorption. Molecular motion is illustrative; no structural trajectory is reconstructed. Fits assume known linewidth, bandwidth and phase calibration. Exchange, non-Gaussian fluctuations, overlapping bands and pulse overlap require richer models.

Synthetic complex signals → spectra → prediction

Does the fitted memory survive a later measurement?

Absorptive band from the saved complex record

Frequency correlation and two competing predictions

The selected band is reconstructed by Fourier transformation. Filled points train both fits; open points test them.

Fit results and reconstruction details

Reconstructed center-line slope $0.7$

\rho(T)=0.18+0.82e^{-T/(1.4\,\mathrm{ps})}

\rho(T)=0.8e^{-T/(1.4\,\mathrm{ps})}+0.2e^{-T/(10\,\mathrm{ps})}

Early / held-out RMS, decay plus offset $0.003\,/\,0.005$

Early / held-out RMS, two decay times $0.003\,/\,0.005$

Signals contain forty-eight samples per coherence axis, spaced by 0.08 picoseconds. The rotating-frame carrier is 1650 inverse centimeters. A Gaussian apodization of four inverse centimeters smooths the finite-window Fourier reconstruction. Every waiting-time signal and both quadratures are included in the download.

Fenn and Fayer (2011), Eq. 12 and the limits of overlapping bands

Study the full course on multidimensional spectroscopy

Start with the essentials

Focus question: How long does a molecule remember its vibration frequency?
One-sentence intuition: Frequency memory links the initial and final frequencies, tilting a two-dimensional peak along its diagonal.

Core mathematical model

A simple correlation model

\rho(T)=\frac{C(T)}{C(0)}=b+(1-b)e^{-T/\tau_c}

Waiting time and correlation time are in picoseconds. The static fraction stays correlated throughout this measurement. A second model replaces that static part with another exponential decay.

The Gaussian peak and its center line

p(y\mid x)\propto\exp\!\left[-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right],\qquad y_{\max}=\rho x

Here the two frequency shifts are divided by their common standard deviation. The ideal Gaussian peak has a center-line slope equal to its correlation. Finite observation windows, homogeneous decay and spectral acceptance change the reconstructed slope; the fit uses calibrated time signals.

Common difficulties

A round peak does not identify a molecular mechanism

Typical misconceptionThe fitted decay time uniquely measures one solvent rearrangement.

Better mental modelThe model describes frequency correlation. Multiple microscopic processes, overlapping bands and exchange can change similar spectra. Additional polarization, temperature or independent structural measurements are needed to identify the mechanism.

Run the experiment

01
Follow one peak through time
Choose A lasting memory. Drag the sample horizontally, or move Waiting-time frame from the first to the last frame.
What to observe: The peak loses its tilt as the frequency correlation decreases. The colored molecules illustrate correlated frequency pairs, and the heatmap comes from the saved complex signals.
02
Ask the later measurements
Choose Two hidden clocks and compare the solid fitted curves at the three open held-out points.
What to observe: A curve that matches the early points can miss later points. Prediction makes the consequences of the assumed correlation model visible.
03
Shorten the experiment
Choose Stop measuring early. Compare both fits and their fitted decay times, then extend the maximum waiting time under Calibration and noise.
What to observe: A short window makes a slow decay resemble a constant offset. Longer measurements help distinguish the models when the signal precision is sufficient.

Spectral Diffusion and Frequency Memory

Does the fitted memory survive a later measurement?

Watching a frequency lose its memory

Core mathematical model

A simple correlation model

The Gaussian peak and its center line

Common difficulties

A round peak does not identify a molecular mechanism

Run the experiment

Follow one peak through time

Ask the later measurements

Shorten the experiment

Related experiments and articles

Vibrations, echoes and frequency memory