Wavepacket electronic

Much of the previous section dealt with two-level systems. Real molecules, however, are not two-level systems for many purposes there are only two electronic states that participate, but each of these electronic states has many states corresponding to different quantum levels for vibration and rotation. A coherent femtosecond pulse has a bandwidth which may span many vibrational levels when the pulse impinges on the molecule it excites a coherent superposition of all tliese vibrational states—a vibrational wavepacket. In this section we deal with excitation by one or two femtosecond optical pulses, as well as continuous wave excitation in section A 1.6.4 we will use the concepts developed here to understand nonlinear molecular electronic spectroscopy. 

Figure Al.6.14. Schematic diagram showing the promotion of the initial wavepacket to the excited electronic state, followed by free evolution. Cross-correlation fiinctions with the excited vibrational states of the ground-state surface (shown in the inset) detennine the resonance Raman amplitude to those final states (adapted from [14].

Figure Al.6.21. Bra and ket wavepacket dynamics which detennine the coherence overlap, (( ) ( ) ). Vertical arrows mark the transitions between electronic states and horizontal arrows indicate free propagation on the potential surface. Full curves are used for the ket wavepacket, while dashed curves indicate the bra wavepacket. (a) Stimulated emission, (b) Excited state (transient) absorption (from [41]).

Figure Al.6.24. Schematic representation of a photon echo in an isolated, multilevel molecule, (a) The initial pulse prepares a superposition of ground- and excited-state amplitude, (b) The subsequent motion on the ground and excited electronic states. The ground-state amplitude is shown as stationary (which in general it will not be for strong pulses), while the excited-state amplitude is non-stationary. (c) The second pulse exchanges ground- and excited-state amplitude, (d) Subsequent evolution of the wavepackets on the ground and excited electronic states. Wlien they overlap, an echo occurs (after [40]).

So far we have exclusively discussed time-resolved absorption spectroscopy with visible femtosecond pulses. It has become recently feasible to perfomi time-resolved spectroscopy with femtosecond IR pulses. Flochstrasser and co-workers [M, 150. 151. 152. 153. 154. 155. 156 and 157] have worked out methods to employ IR pulses to monitor chemical reactions following electronic excitation by visible pump pulses these methods were applied in work on the light-initiated charge-transfer reactions that occur in the photosynthetic reaction centre [156. 157] and on the excited-state isomerization of tlie retinal pigment in bacteriorhodopsin [155]. Walker and co-workers [158] have recently used femtosecond IR spectroscopy to study vibrational dynamics associated with intramolecular charge transfer these studies are complementary to those perfomied by Barbara and co-workers [159. 160], in which ground-state RISRS wavepackets were monitored using a dynamic-absorption technique with visible pulses. 

In light of tire tlieory presented above one can understand tliat tire rate of energy delivery to an acceptor site will be modified tlirough tire influence of nuclear motions on tire mutual orientations and distances between donors and acceptors. One aspect is tire fact tliat ultrafast excitation of tire donor pool can lead to collective motion in tire excited donor wavepacket on tire potential surface of tire excited electronic state. Anotlier type of collective nuclear motion, which can also contribute to such observations, relates to tire low-frequency vibrations of tire matrix stmcture in which tire chromophores are embedded, as for example a protein backbone. In tire latter case tire matrix vibration effectively causes a collective motion of tire chromophores togetlier, witliout direct involvement on tire wavepacket motions of individual cliromophores. For all such reasons, nuclear motions cannot in general be neglected. In tliis connection it is notable tliat observations in protein complexes of low-frequency modes in tlie... 

In an electron scattering or recombination process, the free center of the incoming electron has the functions Wi = ui U u, and the initial state of the free elechon is some function v/ the width of which is chosen on the basis of the electron momentum and the time it takes the electron to aiTive at the target. Such choice is important in order to avoid nonphysical behavior due to the natural spreading of the wavepacket. 

Central to the description of this dynamics is the BO approximation. This separates the nuclear and electionic motion, and allows the system evolution to be described by a function of the nuclei, known as a wavepacket, moving over a PES provided by the (adiabatic) motion of the electrons. 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

In Section III.A, it was shown that the nuclear wavepacket can be represented by a packet associated with each electronic state, %i. Each of these packets can... 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

In the classical picture developed above, the wavepacket is modeled by pseudo-particles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the Uajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of confignrations, X/(, t)tlt,(r, t)... 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Figure 1.3. Real-time femtosecond spectroscopy of molecules can be described in terms of optical transitions excited by ultrafast laser pulses between potential energy curves which indicate how different energy states of a molecule vary with interatomic distances. The example shown here is for the dissociation of iodine bromide (IBr). An initial pump laser excites a vertical transition from the potential curve of the lowest (ground) electronic state Vg to an excited state Vj. The fragmentation of IBr to form I + Br is described by quantum theory in terms of a wavepacket which either oscillates between the extremes of or crosses over onto the steeply repulsive potential V[ leading to dissociation, as indicated by the two arrows. These motions are monitored in the time domain by simultaneous absorption of two probe-pulse photons which, in this case, ionise the dissociating molecule.

Figure 35. Electronic excitation of a LiH wave packet from the outer classical turning point 6ao) of the ground X S" state. The X —> B transition is considered. The initial wavepacket is the shifted ground vibrational state. Taken from Ref. [37].

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