Skip to main content
Sandbox Physics
Multidimensional spectroscopy

L3 · Chapter 5

How two vibrations meet in a 2D spectrum

Sandbox Physics ·

When two vibrations couple, an excitation no longer belongs entirely to one bond. A two-dimensional spectrum connects excitation on one transition with detection on another. Signed peak pairs also reveal that a molecule starting in an excited state encounters different energy gaps from one starting in its ground state.

Coupling mixes local excitations

Specify local fundamentals ν~1,ν~2\widetilde\nu_1,\widetilde\nu_2, downward anharmonic shifts Δ1,Δ2\Delta_1,\Delta_2 and coupling JJ, all in inverse centimetres. The number operator njn_j counts excitation quanta; creation and annihilation operators add or remove one.

Divide energy by hchc so that eigenvalues are wavenumbers. The coupling transfers one quantum between modes. We neglect off-resonant terms that create or remove two quanta together.

Hhc=∑j=12[ν~jnj−Δj2nj(nj−1)]+J(a1†a2+a2†a1)\frac{H}{hc}=\sum_{j=1}^{2}\left[\widetilde\nu_jn_j-\frac{\Delta_j}{2}n_j(n_j-1)\right]+J(a_1^\dagger a_2+a_2^\dagger a_1)
A low-excitation, weak-anharmonicity model with rotating-wave coupling. Ground, one-quantum and two-quantum manifolds are all needed for excited-state absorption.

Reading: Tokmakoff

Worked example for the one-quantum levels

The one-quantum block has two basis states: excitation on the first or second mode. Choose local frequencies 1680, 1720 cm−11680,\ 1720\,\mathrm{cm}^{-1} and J=15 cm−1J=15\,\mathrm{cm}^{-1}. Setting the two-by-two determinant to zero gives 1675, 1725 cm−11675,\ 1725\,\mathrm{cm}^{-1}. The separation grows from 40 to 50 inverse centimetres.

Each new state mixes the local excitations. The anharmonic terms vanish in this block because no mode holds two quanta. They matter in the two-quantum block, where they shift overtone and combination states and hence the negative features.

H1hc=(168015151720)\frac{H_1}{hc}=\begin{pmatrix}1680&15\\15&1720\end{pmatrix}
Every entry is in inverse centimetres.
(1680−λ)(1720−λ)−152=0⟹(λ−1700)2=625(1680-\lambda)(1720-\lambda)-15^2=0\quad\Longrightarrow\quad(\lambda-1700)^2=625
The two roots give the coupled frequencies.
ν~±=ν~1+ν~22±(ν~1−ν~22)2+J2\widetilde\nu_\pm=\frac{\widetilde\nu_1+\widetilde\nu_2}{2}\pm\sqrt{\left(\frac{\widetilde\nu_1-\widetilde\nu_2}{2}\right)^2+J^2}
These are eigenfrequencies of the two-state block. The numbers are a teaching example, not a fit to a named molecule.

Why the map has opposite signs

Pulses change the ground- and excited-state populations. Removing ground-state population reduces absorption: ground-state bleach. Stimulated emission also reduces net absorption. Excited-state absorption adds a different process, promoting an excited molecule into the two-quantum manifold.

Our sign convention shows bleach and stimulated emission as positive, and excited-state absorption as negative. Some papers reverse the overall sign, so inspect the colour scale first. Anharmonicity separates their detection frequencies into signed pairs. Taking an absolute value discards that information.

Reading: Golonzka et al. · Tokmakoff

Two limits test the entire spectrum

Set coupling to zero and assume completely independent modes with an additive dipole. Exciting one mode cannot change the response of the other. Summed pathways must cancel the true coherent cross response. Overlapping line tails can still colour pixels away from the diagonal; those pixels alone do not establish a cross peak.

Next set every anharmonic shift to zero. A harmonic system with a dipole linear in displacement and consistent damping is linear. Bleach and stimulated-emission weights are cancelled by excited-state absorption because μ12=2μ01\mu_{12}=\sqrt2\mu_{01} and the transition frequencies coincide. A two-level truncation would fail this test.

2μ014−μ012μ122=0whenμ12=2μ012\mu_{01}^{4}-\mu_{01}^{2}\mu_{12}^{2}=0\quad\text{when}\quad\mu_{12}=\sqrt2\mu_{01}
Harmonic cancellation requires coincident frequencies, a linear dipole and consistent pathway damping. Electrical anharmonicity or level-dependent damping can break it.

Reading: Andrei Tokmakoff

Dipole directions set polarization weights

Each interaction projects a transition dipole onto a field polarization. Four such projections multiply in a third-order signal. Changing the angle between local dipoles changes effective coupled-state dipoles and the relative strengths of pathways.

An oriented sample and an isotropic solution require different averages. The experiment states its chosen orientation and polarization model. Inferring a molecular angle from intensity ratios also needs transition moments, spectral coverage and all relevant pathways. Golonzka and colleagues demonstrated this structural sensitivity in 2001; our few-level model exposes its mechanism.

Reading: Golonzka et al. · Andrei Tokmakoff

Try the prediction

Test cross features in two limits

  1. Save the default spectrum, then reduce coupling to zero. Distinguish a cross response from overlapping tails.
  2. Restore coupling and reduce anharmonicity to zero. Explain why positive and negative contributions approach and cancel.
  3. Restore the default and change dipole angle. Read the orientation and polarization assumption before interpreting peak heights.

Exercises and explanations

  1. Why do resolved one-quantum frequencies not determine all negative-peak locations?

    Read the explanation

    Negative peaks involve transitions from one-quantum to two-quantum states. Anharmonicity and combination-state mixing shift those final energies, which are not fixed by the one-quantum levels alone.

  2. What is wrong with normalizing a near-zero harmonic signal into a bright map?

    Read the explanation

    That would magnify roundoff into an apparently finite physical response. Display the actual near-zero amplitude and explain that the selected linear-dipole model has no third-order harmonic response.

Sources and calculation scope

  1. Andrei Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (June 2011, v1)

    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)

    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.

  3. Golonzka et al., Vibrational Anharmonicities Revealed by Coherent Two-Dimensional Infrared Spectroscopy (2001)

    Physical Review Letters 86, 2154–2157, published 5 March 2001. Publisher abstract checked for paired features and dipole-angle sensitivity. The full paper was not used for a figure or numerical-data reproduction; the explicit model follows the author lecture notes.

FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy