Deducing Anharmonic Coupling Matrix Elements from Picosecond Time-Resolved Photoelectron Spectra

Katharine Reid

(Julia Davies, Alistair Green)

School of Chemistry, University of Nottingham

Intramolecular vibrational energy redistribution(IVR)

Timescale: tens of picoseconds

1. What is the timescale?

2. What is the mechanism (which dark states are involved)?

3. Can we influence the process? (Bond-selective chemistry, coherent control, mode-specificity)

4. What can we learn about chemical reactivity?

Questions

Time-resolved photoelectron spectroscopy

t = 0 t1 t2

Photoelectron imaging

Laser system

Pulse width 1 ps

Bandwidth ~15 cm-1

Independently tunable (and scannable!) pump and probe

IVR in toluene

6a1/10b116b1

Picosecond absorption spectrum

The 6a1/10b116b1 Fermi resonance in S1 toluene

Characterized by dispersed fluorescence – Hickman, Gascooke and Lawrance JCP (1996)

Excitation of the eigenstate at 462 cm-1

(predominantly 10b116b1)

Excitation of the eigenstate at 457 cm-1

(predominantly 6a1)

This should provide a good test of our technique …

The 6a1/10b116b1 Fermi resonance in S1 toluene

Probing a two-level prepared wavepacket

In photoelectron spectroscopy (PES) we can, depending on our resolution, see “signatures” of harmonic oscillator levels a and b.

If we use a laser pulse of appropriate duration we expect that the photoelectron peak intensities will oscillate at 12.

Different photoelectron peaks may have different sensitivities to the wavepacket dynamics.

Picosecond photoelectron spectrum at t = 0

“one-colour” Hammond and Reid, 2006

“SEVI” Davies et al., 2010

Photoelectron images

0 ps 3 ps 6 ps

Time-resolved photoelectron spectra

One-colour:

Hammond and Reid, 2006

Two-colour:

Davies et al., 2010

Time dependence of the ion origin peak

5.14 cm-1 5.49 cm-1

So … there must be a third state involved

The S1 frequency of mode 16a is given as 228 cm-1, so 16a2 is expected at ~456 cm-1 and is a likely candidate …

Courtesy of Warren Lawrance

“The two spectra show different intensity for transitions terminating in 162

(16a2), consistent with it being involved in the Fermi Resonance”

At this level of excitation there are not many options

Possible explanation

12 23 12 23( ) cos( ) cos( ) cos(( ) )I t A t B t C t D

0 ps

3 ps

(a) (b)

(c)

(d)

But what do the other ion states tell us?

Ion vibrational states: (a) = 00, (b) = 6a1, (c) = 10b116b1, (d) = 16a2

Time dependence of peaks (a), (c) and (d)

Time delay / picoseconds

00

10b116b1

16a2

5.14 cm-1 5.49 cm-1

Fourier transforms for peaks (a) and (c)

What about the 16a2 peak?

On resonance

Red-shifted

Fourier transforms

5.14 cm-1 5.49 cm-1

4.92 cm-1 5.69 cm-1

On resonance

Red shifted

But there is no plausible coupling of zero-order vibrational states, or torsion-vibration coupling that could cause this …

Torsional populations at 10 K

Energy level scheme

12 = 5.14 cm-1 12 = 5.49 cm-1

23 = 5.69 cm-1 23 = 4.92 cm-1

ij ijE

(… or the other way round)

Formalism (thanks to Felker and Zewail)

a cba b b a1 1 1 21 111 6 10 16 16

a cba b b a1 1 1 22 222 6 10 16 16

a cba b b a1 1 1 23 333 6 10 16 16

and similarly for eigenstates |n> in the other torsional ladder.

The eigenstates can be expressed as:

2 2 21 2 3 1

ia ib ic2 2 2 1

Normalization requires

Formalism

( ) cos cos cos

cos cos cos

kt

k t

S t A t A t A t e

A t A t A t e D D

12 12 23 23 13 13

12 12 23 23 13 13

ij i j ia ja i jA Kp p2

2

For a given observed ion state

where

and pn depends on the light intensity at the energy of eigenstate |n>

This enables us to simulate the observed beating patterns for chosen coupling matrix elements, i, and compare with those observed experimentally. The most stringent test is the 16a2 beating pattern.

Simulation of the 16a2 beating patterns

Cosine fit

Simulation

(a) on resonance

Simulation of the 16a2 beating patterns

Cosine fit

Simulation

(b) off resonance

Coupling matrix elements

Or to put it another way …

Summary

Simulations based on the proposed energy level scheme reproduce all observed beating patterns; thus we have determined the anharmonic coupling constants connecting three zero-order states in S1 toluene.

Time-resolved photoelectron spectra can be treated quantitatively in favourable circumstances.

The Fermi resonance originally believed to be a two-level system has been shown to be a three-level system, which is “doubled” as a consequence of small changes in vibrational frequencies in two torsional ladders.

This provides an explanation for the apparently complex IVR behaviour that has been observed for molecular systems containing methyl rotors, even at quite low densities of states.

Acknowledgements

Julia DaviesAlistair Green

Paul Hockett

Warren Lawrance

EPSRC