The Time Evolution of an Excited State

We begin by considering an isolated molecule that is initially in a given discrete electronic state Es be the energy of this state. The state interacts with a 

1) Strictly speakmg, truly isolated systems are not known in physics. They are extremely useful idealization. 

To make the discussion simple, we use Dirac notation and accordingly denote the 

Our central problem is now the study of the dynamical behavior of the system, as determined by the exact Hamiltonian 

The function G( ) that appears inside the integrand is the Green s function 

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In this book we shall write the Hamiltonian as an (algebraic) operator using the appropriate Lie algebra. We intend to illustrate by many applications what we mean by this cryptic statement. It is important to emphasize that one way to represent such a Hamiltonian is as a matrix. In this connection we draw attention to one important area of spectroscopy, that of electronically excited states of larger molecules,4 which is traditionally discussed in terms of matrix Hamiltonians, the simplest of which is the so-called picket fence model (Bixon and Jortner, 1968). A central issue in this area of spectroscopy is the time evolution of an initially prepared nonstationary state. We defer a detailed discussion of such topics to a subsequent volume, which deals with the algebraic approach to dynamics. 

Having all the eigenstates of the system calculated, we are now in a position to study the time evolution of an initial polariton excitation, which we choose in the form of a wavepacket k°) built out of the low-energy polariton states ) of the perfect system ... 

Let us consider the time evolution of an initial wavepacket (0) = (t = 0) created by optical excitation out of an initial state With the transition operator T as in Eqs. (31) and (32) we have... 

Fig. 4.11. Real-time evolution of the ion signals for Nan=4...io (taken from [263])(a) and energy level schemes to describe the temporal evolution of the NaJ-signal (X = ground state) for At > 0 Epump = 2.96 eV and F probe = 1.48eV (b) and for At < 0 Epump = 1.48 eV and pump = 2.96eV (c). 1/tq and 1/ts indicate the fragmentation probabilities of an excited state Na for different excitation (Epump) and ionization energies (Eprohe)...

Fig. 8.14 Snapshots of the time evolution of an electronic wavepacket prepared close to Coln-1 for fixed nuclear geometry. The underlying electronic wavepacket is created as the normaUzed superposition of the CASSCF-wavefunctions of ground and first excited state. The depicted motion of the electron density takes place in 1.6 ps...

The time evolution of an initially prepared state by the crossed excitation beams is... 

A major technological innovation that opens up the possibility of novel experiments is the availability of reliable solid state (e.g., TiSapphire) lasers which provide ultra short pulses over much of the spectral range which is of chemical interest. [6] This brings about the practical possibility of exciting molecules in a time interval which is short compared to a vibrational period. The result is the creation of an electronically excited molecule where the nuclei are confined to the, typically quite localized, Franck-Condon region. Such a state is non-stationary and will evolve in time. This is unlike the more familiar continuous-wave (cw) excitation, which creates a stationary but delocalized state. The time evolution of a state prepared by ultra fast excitation can be experimentally demonstrated, [5,7,16] and Fig. 12.2 shows the prin-... 

It is clear that a core-hole represents a very interesting example of an unstable state in the continuum. It is, however, also rather complicated [150]. A simpler system with similar characteristics is a doubly excited state in few-body systems, as helium. Here, it is possible [151-153] to simulate the whole sequence of events that take place when the interaction with a short light pulse first creates a wave packet in the continuum, including doubly excited states, and the metastable components subsequently decay on a timescale that is comparable to the characteristic time evolution of the electronic wave packet itself. On the experimental side, techniques for such studies are emerging. Mauritsson et al. [154] studied recently the time evolution of a bound wave packet in He, created by an ultra-short (350 as) pulse and monitored by an IR probe pulse, and Gilbertson et al. [155] demonstrated that they could monitor and control helium autoionization. Below, we describe how a simulation of a possible pump-probe experiment, targeting resonance states in helium, can be made. 

The extension of density functional theory (DFT) to the dynamical description of atomic and molecular systems offers an efficient theoretical and computational tool for chemistry and molecular spectroscopy, namely, time-dependent DFT (TDDFT) [7-11]. This tool allows us to simulate the time evolution of electronic systems, so that changes in molecular structure and bonding over time due to applied time-dependent fields can be investigated. Its response variant TDDF(R)T is used to calculate frequency-dependent molecular response properties, such as polarizabilities and hyperpolarizabilities [12-17]. Furthermore, TDDFRT overcomes the well-known difficulties in applying DFT to excited states [18], in the sense that the most important characteristics of excited states, the excitation energies and oscillator strengths, are calculated with TDDFRT [17, 19-26]. 

An important aspect of this kind of experiment is the time evolution of the reactive excited complex. It can be expected that for a reaction near the threshold, fine tuning of the optical excitation should yield to drastic changes in the reactive decay time as the spectroscopy already shows for the Ca-FICl system (Keller 1991, Soep et al. 1991, 1992). Real time evolution of binary reactive collisions can be studied through van der Waals complexes, since time t = 0 is defined by the excitation laser, as well as the starting internuclear distance between reactants which is fixed by the ground state geometry. This approach has been used for... 

In the previous section, we examined the time evolution that follows excitation of a ring current in the donor (Fig. la). The ring state Md) is an eigenstate of the donor moiety with the eigenvalue E(KMt ) = d + 2/icos(KUlJa) and with time evolution... 

The first three steps represent the evolution of the solute excited state. Step 1 and Step 2 are described following the time evolution of 9K (0 in Eq. (7-45) where the electronic excitation occurs at t = 0, whereas Step 3 is described by a geometry optimization of the excited state solute in the presence of an equilibrated solvent, which is equivalent to consider dielectric relaxation to be faster than the solute geometry relaxation. Such as assumption has to be verified for the system of interest, and, in all cases where it is not valid, Steps 2 and 3 need to be inverted. 

Further low-pressure studies have shown, as discussed previously, that a purely kinetic treatment of the "reversible" intersystem crossing is inadequate for describing the time evolution of the excited state, since biacetyl is an "intermediate-case" molecule. Nevertheless, a kinetic treatment is useful for tracing the energy flow which leads to photodecomposition in a molecule. 

This section will present two selected examples of electronic spectroscopy on mass-selected metal clusters in the gas phase. In the first example, time-resolved photoelectron spectroscopy is employed to monitor the real time evolution of an electronic excitation leading to the thermal desorption of an adsorbate molecule from a small gold cluster. In the second example, optical absorption-depletion spectroscopy in conjunction with first principles calculations provide insight into the excited state structure of mass-selected metal clusters. 

Figure 2.8 shows the experimental set-up. An above-band-gap, short laser pulse is used to optically excite electron-hole pairs with excess energy. The time evolution of the electron distribution is followed by the use of a second short laser pulse with sufficient photon energy to place the excited electrons above the electron affinity of the semiconductor. The optical generation of the carriers again is not selectively at the surface region of interest—but the probed states are. 

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