When a molecule is irradiated by an utrashort laser pulse, it is electronically excited. In molecular spectroscopy it is normally assumed that the nuclear con-figuration is fixed due to the relatively slow velocity of the nuclei in comparison with the electronic. This principle is well described within the Franck-Condon approximation. Excitation of the molecule leads to a vertical transition, as it is demonstrated in Figure 2.2.
If laser pulse is short compared with the vibrational period of the molecule, several vibrational levels become populated. As a result it forms a coherent super-position of eigenstates with a well-defined phase evolution. Such a supersuper-position is called a wavepacket, which is spatially localized and its motion can be temporally deciphered using ultrashort laser pulses. The time evolution of eigenfunctions can be described as
Nuclear Coordinate
Energy
S1
S0
Emission Absorption
Figure 2.2:Schematic energy level diagram. Blue (absorption) and green (emission) arrows show electronic transitions, grey arrows illustrate nonradiative vibrational relaxation from the out of equilibrium position, where the molecule is found immediately after an electronic transition.
Ψ(t) =X
n
cnψnexp[−i(wnt−φn)] (2.1) where ψn andωn are the eigenfunction and transition frequency of the n-th vibrational level, respectively;cn andφn represent its amplitude and phase. Ma-nipulation of the amplitudes and phases of the laser pulses allows the control of the quantum amplitude and phases of the molecular wavepacket.
When the molecular wavepacket is formed, it periodically oscillates back and forth in the harmonic potential. If there is no external perturbation, it continues oscillating without losing energy until it decays from the excited state.
In an anharmonic potential, the wavepacket exhibits broadning [72] during a period of time and changes its shape. This occurs because the slower high energy (’blue’) components of the molecular wavepacket are delayed with respect to the faster low energy (’red’) components (see Figure 2.3 a). Using a tailored laser pulse it is possible to suppress broadning, as shown in Figure 2.3 b. It is also observed that after a certain revival time, the wavepacket reforms with its initial phase [73]. This phenomenon is well described theoretically and experimentally,
t
0t
1Figure 2.3: Explanation for wavepacket dispersion. a) The wavepacket is excited att0by an unchirped laser pulse.
Because the vibrational spacing in the high energy range
∆EH is smaller than the vibrational spacing in the low-energy range∆EL, thereby the oscillation timesTHare longer thanTL. Thus after some oscillations at a timet1
>t0, the lower-energy parts of the wavepacket advance the higher-energy parts. b) The dispersion can be sup-pressed by chipping a laser pulse in the way to start the
’slow’ blue components earlier than the ’fast’ red ones.
Adapted from [9].
and is observed in various systems.
Since most chemical reactions take place in the liquid phase, it is important to study the effect of this environment on the molecular systems. Generally, interac-tions with the environment induce dephasing of the coherently formed molecular wavepacket. It can be expressed in two forms: 1) coupling with internal modes of the solute following Internal Vibrational Redistribution (IVR), 2) interaction with solvent molecules leading to efficient depopulation channels of coherently excited vibrational modes. Solute-solvent interaction can change the potential energy sur-faces on which the wavepacket evolves [74], which can result in fast dephasing.
When we deal with an ensemble of molecules in the liquid phase, their excita-tion leads to the formaexcita-tion of a wavepacket whose frequencies are distributed for different molecules. At first they are all in phase and coherence is preserved, but after a certain time, the interaction with the surrounding solute molecules breaks
the initial coherence and all phases become randomized. The dephasing time of an ensemble of wave packets varies in different environments, and it is faster for solvents that undergo a strong interaction with the solutes and have fast fluctuation time scales. To overcome this, instantaneous excitation is necessary. This means that the duration of the laser pulse should be much shorter than the oscillation pe-riod, so that the evolution of the excited wavefunction during the interaction with the field can be considered negligible.
Example
Here we show an example of how a pump-probe approach allows the observation of the evolution of the molecular wavepacket during its motion along a potential energy surface.
As an example investigated by our group [10,75], fluorescence can be used as an observable for this purpose. Here we focus on the molecular systems consisting of the aromatic amino-acids Trp and Tyr. They are two of the 20 amino-acids that made up proteins and the main contributors in protein fluorescence. They have both absorption bands centred at 270 nm. While the second band which is char-acteristic for Trp originates from π −π∗ transition to the 1Ba, 1Ba states, Trp fluorescence originates from the two transitions1La,1Lain the firstπ,π∗ excited singlet state. Figure 2.4 shows a schematic of this process. First, a short UV pulse excites molecules from theS0 toS1 states forming a coherent superposition of vibrational states, i.e. the molecular wavepacket. This evolves in time and fluores-cence is eventually observed. The evolution of the wavepacket can be probed by an IR pulse, which transfers the population to higher lying ionizing and dissociative states, thereby depleting fluorescence.
Fluorescence depletion for Trp and Tyr already exhibits different time-resolved dynamics under the excitation with FT-limed pulses. One can see from Figure 2.5 that Trp undergoes rapid fluorescence depletion reaching a minimum at 600 fs, which can be attributed to the opening of a Franck-Condon window toward higher lying ionizing states. In contrast, Tyr fluorescence decreases until 600 fs and then continues less abruptly until 7 ps. The sampling of the transient dynamical con-figuration explored by the molecular wavepacket allows one to easily discriminate between the two main amino acids, Trp and Tyr, responsible for protein fluores-cence.
1La, 1Lb
Figure 2.4: Absorption spectra of Trp in PH 7 aqueous solution. In-set: Schematic representation of fluorescence depletion.
Picture is taken from [10].
Figure 2.5: Fluorescence depletion traces for Trp (a) and Tyr (b) un-der excitation with FT-limited pulse represented in black triangles. Picture is taken from [10]
This example demonstrates that probing the motion of molecular wavepackets for the two essential amino-acids Trp and Tyr allows to discriminate them using pump-probe technique. Nevertheless, the question rises how to discriminate when time-resolved fluorescence depletion is the same for both molecules, for example in more complex molecules such as peptides and proteins, described in more detail in Chapters 4 and 5.