Nuclear motion, model systems

Semimetals bismuth (Bi) and antimony (Sb) have been model systems for coherent phonon studies. They both have an A7 crystalline structure and sustain two Raman active optical phonon modes of A g and Eg symmetries (Fig. 2.4). Their pump-induced reflectivity change, shown in Fig. 2.7, consists of oscillatory (ARosc) and non-oscillatory (ARnonosc) components. ARosc is dominated by the coherent nuclear motion of the A g and Eg symmetries, while Af nonosc is attributed to the modification in the electronic and the lattice temperatures. 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

Model quantum-mechanical calculations were carried out on the basis of formulas (11.43) and (11.47), and functions V,(R) and T(7 ) were determined by comparison with the distribution shown in Fig. 15.51 As can be seen from Fig. 15, the measured distribution extends considerably into the region of associative Pgl characterized by condition (II.4a). In this region formula (11.47) was used, and in the Pgl region, characterized by (11.4b), formula (11.43) was used, except for those special values of e and / where, for a certain collision energy, the final state of nuclear motion corresponds to a temporarily bound—or quasibound—HeH+ system. 

The Holstein model was preceded by the Pekar model [34] and in chemistry by the Marcus model [6]. In chemistry donor-acceptor systems are more frequent objects of study than conducting wires but the coupling between electronic and nuclear motion of similar nature. For example if the coupling is large a small nuclear displacement is sufficient to change the wave function much and in a way which corresponds to ET or EET. We use the effective, many-electron Hamiltonian H of eq.(4) and assume that it is solved for donor and acceptor, giving the energies Haa and Hdd, respectively. We use the new nuclear coordinates ... 

The approach has been tested by controlling nuclear wavepacket motion in a two-dimensional model system [23], The relative simplicity of the system makes it possible to compare the semiclassical results with exact quantum ones. Numerical applications to the control of HCN-CNH isomerization [24] demonstrates that the new semiclassical formulation of optimal control theory provides an effective and powerful tool for controlling molecular dynamics with many degrees of freedom. 

Molecular dynamics (MD) has been traditionally linked to purely classical modeling, e.g. based on empirical force fields, or with classical trajectory calculations, based on predetermined potential energy surfaces. Unfortunately, such potentials are extremely expensive to evaluate for chemically interesting systems. Nevertheless, attempts to conduct ab initio molecular dynamics, in which the classical description of nuclear motion is combined with quantum-mechanical determination of the forces, dates back several decades. 

We have focused on the charge transport at interfaces and nonadiabatic interactions between injected electrons and nuclear motions. Our purpose is to establish practical models, which enable us to perform ah initio calculations. We adopted the NEGF formalism, and developed theoretical models combined with a practical ah initio scheme by means of DFT. We chose two systems as examples, the E-M-E junction and photoreaction on metal surfaces. 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

Application of 2D IR spectroscopy to PCET models of Section 17.3.2 is a logical starting point for this type of investigation. 2D methods can unravel the correlated nuclear motion in a PCET reaction and in principle decipher how vibrational coupling in the Dp/Ap interface couples to the ET event between the Ae/De sites. These data can identify the structural dynamics within the interface that promote PCET reactions in much the same way that local hydrogen bonding structure and dynamics mediate excited state PT reactions [239, 240]. In these experiments, the PCET reaction can be triggered by an ultrafast resonant visible laser pulse (as in a standard TA experiment) and a sequence of IR pulses may be employed to build a transient 2D IR spectrum. These experiments demand that systems be chosen so that the ET and PT events occur on an ultrafast timescale. 

---