Tunneling is a supreme quantum effect. Every introductory text [1] on quantum mechanics gives the paradigm example of a particle tunneling through a one-dimensional potential barrier despite having a total energy less than the barrier height. Indeed, the reader typically works through a number of exercises, all involving one-dimensional potential barriers of one form or another modeling several key physical phenomena ranging from atom transfer reactions to the decay of α-particles [2]. However, one seldom encounters coupled multidimenisonal tunneling in such texts since an analytical solution of the Schro¨dinger equation in such cases is not possible. Interestingly, the richness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimensional systems [3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as one approaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedom to the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result in strong mode-specificity and fluctuations in the reaction rates [4]. In fact, a proper description of tunneling of electrons and hydrogen atoms is absolutely essential [5,6] even in molecular systems as large as enzymes and proteins. Although one usually assumes tunneling effects to be significant in molecules involving light atom transfers, it is worth pointing out that neglecting the tunneling of even a heavy atom like carbon is the difference between a reaction occurring or not occurring. In particular, one can underestimate rates by nearly hundred orders of magnitude [7]. Interestingly, and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barrier tunneling have been obtained from a phase-space perspective [8,9]. The contributions by Creagh, Shudo and Ikeda, and Takahashi in the present volume provide a detailed account of the latest advances in the phasespace-based understanding of multidimensional barrier tunneling.
Tunneling is a supreme quantum effect. Every introductory text [1] on quantum mechanics gives the paradigm example of a particle tunneling through a one-dimensional potential barrier despite having a total energy less than the barrier height. Indeed, the reader typically works through a number of exercises, all involving one-dimensional potential barriers of one form or another modeling several key physical phenomena ranging from atom transfer reactions to the decay of α-particles [2]. However, one seldom encounters coupled multidimenisonal tunneling in such texts since an analytical solution of the Schro¨dinger equation in such cases is not possible. Interestingly, the richness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimensional systems [3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as one approaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedom to the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result in strong mode-specificity and fluctuations in the reaction rates [4]. In fact, a proper description of tunneling of electrons and hydrogen atoms is absolutely essential [5,6] even in molecular systems as large as enzymes and proteins. Although one usually assumes tunneling effects to be significant in molecules involving light atom transfers, it is worth pointing out that neglecting the tunneling of even a heavy atom like carbon is the difference between a reaction occurring or not occurring. In particular, one can underestimate rates by nearly hundred orders of magnitude [7]. Interestingly, and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barrier tunneling have been obtained from a phase-space perspective [8,9]. The contributions by Creagh, Shudo and Ikeda, and Takahashi in the present volume provide a detailed account of the latest advances in the phasespace-based understanding of multidimensional barrier tunneling.
It is rather obvious that one-dimensional theory is not good enough to comprehend quantum mechanical tunneling phenomena. For instance, the tunneling splitting estimated using the one-dimensional potential along the intrinsic reaction coordinate gives a few orders of magnitude smaller value [56]. There is a more crucial example that demonstrates the conceptual deficiency of one-dimensional treatment and shows interesting features peculiar to multidimensional tunneling phenomena. This is explained in Chapter 4. In this sense, multidimensional treatment is definitely needed. Herewe describe the two-dimensionalWKB theory developed by Takada and Nakamura [30].
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].
Everyday macroscopic objects follow the laws of classical mechanics. For microscopic objects, such as atoms or nuclei, the wave character of their dynamics has to be taken into account within the framework of quantum mechanics. While wave-like behaviour can show up in various ways, one of the most striking is the tunnelling effect: some events that are classically forbidden for energetic reasons happen because the quantum particles can tunnel under a potential barrier. Similarly, events that are prohibited due to some other dynamical constraints can occur via a process known as dynamical tunnelling. Indeed, tunnelling plays a major role in a variety of physical phenomena, ranging from α-particle radioactivity to the current–voltage characteristics of transistors.
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.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.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.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.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.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.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.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 ar
In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.