Tunneling times and excited state interactions between chromophores

R. Stomphorst

Research output: Thesisinternal PhD, WU

Abstract

<p>This Thesis consists of two related parts. The first part (Chapters 2 and 3) presents the results of a fundamental theoretical study of the interpretation of quantum mechanics, and particularly of tunneling processes. Part two (Chapters 4, 5, and 6) describes the experimental results of spectroscopic investigations of quantum mechanical interactions between chromophores, which may play a role in electron transport by means of tunneling.</p><p>The formulation of the quantum theory at the beginning of the twentieth century resulted in a scientific revolution. The impact of this theory on our physical world picture is still visible today. The quantum theory is impressively successful in its predictions of the outcomes of experiments, but the interpretation of the theory is still shaky. The predictions of quantum mechanics about the outcome of experiments are detailed and testable, but the meaning of the theory for quantum systems, in which no measurement has taken place is not clear. Various interpretations exist. Two of these are compared in Chapter 2. The first is the orthodox or Copenhagen interpretation, which was advocated by the Copenhagen-based Niels Bohr. His followers claimed that, in the orthodox interpretation, nothing can be said about the properties of systems if no suitable measurement has been carried out. The alternative is the causal interpretation of quantum mechanics, in which a particle, e.g. an electron, has a well-defined position and velocity at each instant of time independent of a measurement act. The causal theory does not assign a special role to the observer. In the causal interpretation the location as a function of time amounts to trajectories. The two interpretations are empirically equivalent, i.e. they predict the same experimental outcomes.</p><p>In Chapter 2 the meaning of the two interpretations is clarified by the notion of tunneling time. Tunneling is the quantum mechanical phenomenon that a particle can cross a barrier even if its energy is less than the barrier energy. The time a particle takes to tunnel is the tunneling time. Time is a confusing concept: quantum theory appears not to know how to handle this topic. Time appears in the quantum mechanical formulae as a parameter, and not as an operator that corresponds to an observable quantity. Since an unambiguous time operator is missing, it is interesting to see what different interpretations have to say about tunneling times. It is strictly speaking impossible to define tunneling times in the orthodox interpretation because of the lack of a clear time operator, of which the expectation value can be observed by experiment. In the causal interpretation, every trajectory corresponds to a possible path of an electron. The density functions determine the probability that particles move on particular trajectories. This is how the stochastic character of quantum mechanics is accounted for in this interpretation. The transmission time is defined as the time spent inside the barrier by the electrons that eventually cross the barrier. The average transmission time can be defined by means of the trajectories, both in the case of stationary wave functions and in that of time-dependent, Gaussian wave packets. The Gaussian wave packets travel from minus infinity to plus infinity. On the way they meet potential barriers, which cause them to be reflected or transmitted. The process by which an electron moves from one chromophore to another can be described as tunneling because of the high potential barrier formed by the medium between the chromophores. In this Thesis the experimental media used are polymer films and apolar solutions. The model described in Chapter 2 is generally used in the literature but appears not to be applicable for a system of two chromophores and a tunneling electron, because electrons belonging to chromophores neither start in infinity nor go to infinity. A better model, a double potential well, is used in Chapter 3. The conclusion of Chapter 2 is that the transmission time can be defined in the causal interpretation when time-dependent wave packets are used.</p><p>Chapter 3 illustrates the preceding by tunneling of a particle from one potential well to another passing a potential barrier. This model is the so-called double potential well. Examples are primary charge separation in the photosynthetic reaction center and in artificial model systems of chromophores in solutions and polymer films. The latter two cases are investigated spectroscopically in this Thesis. In these molecular systems, effective electron transport between chromophores after absorption of light takes place, yielding charge separation, i.e. the formation of chromophore cations. During their transport between natural or artificial chromophores, the released electrons often have to pass weak conducting media, corresponding to high potential barriers. If the energy of the electron exceeds the energy of the potential barrier, electron transport can be explained by classical theories. However, if the energy of the electron falls below the energy of the potential barrier, this is no longer possible. The transport mechanism is then understood as quantum mechanical tunneling. Experimental proof of tunneling has been found at low temperatures in photosynthesis.</p><p>In Chapter 3 the chromophores and the tunneling electrons are described by double potential wells. Each well represents a chromphore and the space between the molecules a potential barrier. The formalism of the causal interpretation is applied to define a transmission time. The transmission time can be found using trajectories. An alternative method provides the average transmission time without calculating the trajectories. Arrival time distribution functions are defined and these are used to determine the average arrival time of the tunneling electrons at the entrance and at the exit of the barrier. Because arrival time distribution functions and hence both average arrival times and transmission times are not determined in an experimental context, we conclude that these definitions of transmission times are meaningless in the orthodox interpretation. The question posed at the end of Chapter 3 is whether, and if so how, these transmission times are experimentally accessible.</p><p>The theoretical models developed in Chapters 2 and 3 are too simple to be extrapolated to real systems, such as electron transport between an electron donor and an acceptor, one of which is excited by light. Intermolecular interactions between excited electron donor molecules and ground state surrounding molecules cannot be neglected in chromophore films and solutions with relatively high concentrations of chromophores. These excitonic interactions should be included in adequate descriptions of electron transfer. Chapters 4, 5, and 6 describe these interactions using absorption and emission spectroscopy. In Chapter 4 a theoretical description of the absorption characteristics of chromophores in disordered films is given. At low concentrations, chromophores in films are mainly monomers. On statistical grounds, however, we conclude that some molecules are close enough together to form dimer-like structures, concentration pairs with characteristic absorption spectra. The presence of these concentration pairs influences the emission characteristics of chromophore films as well. When monomer electrons are excited by the absorption of photons, the electrons might return to the ground state, giving off their energy as fluorescence. Light energy can also be transferred to neighboring concentration pairs, in which case the fluorescence is quenched.</p><p>The aim of Chapter 4 is to estimate the influence of potential quenchers in the vicinity of excited monomers and to predict the influence of these quenchers on the absorption spectra. The transfer of energy from monomers to concentration pairs, the so-called Förster transfer, takes place over quite large distances (1-5 nm). Statistics can be used to calculate the number of quenchers in particular concentrations of chromophores. In the literature it is assumed that two monomers that form a concentration pair are randomly oriented with respect to each other. If these pairs consisted of randomly oriented monomers and all fluctuations are taken into account, it appears on theoretical grounds that there would not be enough quenchers to describe the experimental changes. Because most chromophores are elongated or pancake-like, they are not randomly oriented but exhibit preferential orientations. These preferential orientations increase the efficiency of fluorescence quenching and theory and experiment are thereby reconciled. An analytical expression of the spectral changes compared to monomeric spectra of randomly oriented dimers is presented. To describe the interaction between chromophores that have one dominant transition dipole moment, a point dipole exciton model is used. To enhance the description of the spectral changes, homogeneous and inhomogeneous line broadening and monomer-to- monomer distance fluctuations should be taken into account. Inclusion of these factors in the model gives rise to the conclusion that the dimer spectra are broadened compared to the monomeric spectra and that the broadening is stronger at the blue end than at the red end of the spectra. However, to detect the blue-shifted spectral profile of randomly oriented dime-like samples, the concentration needs to be so high that isolated dimers cannot be observed. Then, simulation of the dimers becomes impossible, and all mutual interactions must be taken into account. Modeling of these interactions shows that the blue shift is not visible, and that only a broadening of the absorption spectra with respect to the monomeric spectra remains.</p><p>Chapter 5 describes spectral changes that result from changes in the concentration changes of elongated Erythrosin-B (Ery B) chromophores in polyvinylalcohol (PVA) films. At low concentrations, monomers are numerous,giving monomeric absorption spectra. At higher concentrations, measurements of fluorescence quenching show formations of pairs, which are not visible in the absorption spectra. At higher concentrations still, a broadening of the absorption spectra is observed, as predicted by the theory developed in Chapter 4. Further increase of the concentrations induces order of the chromophores. Preferential orientations for dimers, trimers, and tetramers are determined by use of molecular mechanics. Structures that are energetically favorable according to these calculations are selected. The mutual orientations of these oligomers are used to simulate absorption spectra using an exciton model. All structures show dominant blue-shifted spectra, like the experimental spectra. At very high concentrations, the experimental peak shifts to the red end of spectra. Red shifts can be explained theoretically, by simulation of very densely packed molecules.</p><p>Chapter 6 describes absorption spectra of four porfyrin dimers in toluene solutions. Porphyrines are pancake-like molecules containing two perpendicular transition dipole moments. The experimental absorption spectra are explained with the aid of simple exciton models. To account for the two perpendicular transition dipole moments, the exciton model presented in Chapter 5 is extended. The orientations and rotational freedom of the monomers comprising a dimer explain the spectral changes in the absorption spectra.</p><p>This Thesis uses the quantum theory at various levels. Both tunneling and the excitonic interactions between molecules involved in tunneling processes can be described only by the quantum theory. This theory is well equipped to explain experimental data but at the same time appears to yield more questions than answers. This Thesis reports on the fascinating interplay of question and answer between the quantum mechanics of molecular processes and our knowledge of observable processes</p>
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Wageningen University
Supervisors/Advisors
  • Schaafsma, T.J., Promotor
  • van der Zwan, G., Promotor, External person
  • Uffink, J.B.M., Promotor, External person
Award date27 Apr 2001
Place of PublicationS.l.
Print ISBNs9789058083951
Publication statusPublished - 2001

Keywords

  • erythrosine
  • porphyrins
  • quantum mechanics
  • excited state

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