Time-Dependent Energy Levels

We will now study the more useful case when the nuclei are moving by classical mechanics with a Bom-Oppenheimer potential energy surface (PES). The points passed by the nuclei are called the trajectory and denoted Q(t). Q(t) is the spatial coordinates for the heavy particles (nuclei) in the system  

We have to solve the wave equation for the electrons as the nuclei move along this trajectory. A general solution, given by the Russian physicist E. E. Nikitin, is sketched below. 

The wave equation for an electronic Born-Oppenheimer state n is written  

H(Q) is the electronic Hamiltonian in Q along the trajectory. We try to satisfy Equation 7.13 by expressing the time dependence in the following way  

In analogy with Equation 7.4, the time-dependent wave function may be expanded as 

---

Figure 4.4 (a) Time-dependent energy levels for the one-dimensional model of thioacetyla-cetone for a few-cycle driving field similarto Fig. 4.3. In (b) we show the resulting rates...

Fig. 6. The time-dependent energies of the first three vibrational levels of the coupling oscillator on the two diabatic surfaces according to the MTDM (see text). The numbering in the two panels refers to the initial conditions of Fig. 1(c). As explained in the text, the plotted energies are obtained considering that the motion along the tuning oscillator is a classical trajectory on the initial diabatic surface. As in Fig. 2, the points are placed at the crossing which are relevant for the selected initial condition, while the arrows indicate possible paths.

Note that the diagonal elements of the matrix, ap and hp, correspond to the populations in the energy levels, a and b, and contain no time dependence, while the off-diagonal elements, called the coherences, contain all the time dependence. 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

We refer to this equation as to the time-dependent Bom-Oppenheimer (BO) model of adiabatic motion. Notice that Assumption (A) does not exclude energy level crossings along the limit solution q o- Using a density matrix formulation of QCMD and the technique of weak convergence one can prove the following theorem about the connection between the QCMD and the BO model ... 

Hagedorn, G. A. Electron energy level crossing in the time-dependent Born-Oppenheimer approximation. Theor. Chim. Acta 67 (1990) 163-190... 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

This apparent time dependent cell disruption is caused because of the statistically random distribution of the orientation of the cells within a flow field and the random changes in that distribution as a function of time, the latter is caused as the cells spin in the flow field in response to the forces that act on them. In the present discussion this is referred to as apparent time dependency in order to distinguish it from true time-dependent disruption arising from anelastic behaviour of the cell walls. Anelastic behaviour, or time-dependent elasticity, is thought to arise from a restructuring of the fabric of the cell wall material at a molecular level. Anelasticity is stress induced and requires energy which is dissipated as heat, and if it is excessive it can weaken the structure and cause its breakage. 

Hyun et al. [345] prepared PbS Q-dots in a suspension and tethered them to Ti02 nanoparticles with a bifunctional thiol-carboxyl linker molecule. Strong size dependence due to quantum confinement was inferred from cyclic voltammetry measurements, for the electron affinity and ionization potential of the attached Q-dots. On the basis of the measured energy levels, the authors claimed that pho-toexcited electrons should transfer efficiently from PbS into T1O2 only for dot diameters below 4.3 nm. Continuous-wave fluorescence spectra and fluorescence transients of the PbS/Ti02 assembly were consistent with electron transfer from small Q-dots. The measured charge transfer time was surprisingly slow ( 100 ns). Implications of this fact for future photovoltaics were discussed, while initial results from as-fabricated sensitized solar cells were presented. 

The time-dependent Schrddinger equation (2.30) for the particle in a box has an infinite set of solutions tpn(x) given by equation (2.40). The first four wave functions tpn(x) for = 1, 2, 3, and 4 and their corresponding probability densities ip (x) are shown in Figure 2.2. The wave function ipiix) corresponding to the lowest energy level Ei is called the ground state. The other wave functions are called excited states. 

---