Time-dependent tunneling

T.G. Douvropoulos, C.A. Nicolaides, Time-dependent tunnelling via path integrals. Connection to results of the quantum mechanics of decaying states, J. Phys. B (At. Mol. Opt.) 35 (2002) 4453. 

There are many other important topics that could not be treated in this book such as effects of environment, chaos-assisted tunneling or chaotic tunneling, time-dependent tunneling phenomena, and macroscopic tunneling. Those who are interested in these topics should refer to the following books [17,21-24,233,250,251]. Tunneling time, which cannot be defined definitely, presents another interesting subject that we did not discuss in this book. The reader should refer to the books [21,48], for instance. 

Figure 12.45. Illustration of time dependent tunneling through a barrier.

Figure B2.5.23. Mode-specific stereomiitation tunnelling in hydrogen peroxide time-dependent probability...

In the genuine low-temperature chemical conversion, which implies the incoherent tunneling regime, the time dependence of the reactant and product concentrations is detected in one way or another. From these kinetic data the rate constant is inferred. An example of such a case is the important in biology tautomerization of free-base porphyrines (H2P) and phtalocyanins (H2PC), involving transfer of two hydrogen atoms between equivalent positions in the square formed by four N atoms inside a planar 16-member heterocycle (fig. 42). 

Flurbiprofen and indomethacin, which comprise the third class of inhibitors, cause a slow, time-dependent inhibition of COX-1 and COX-2, apparently via formation of a salt bridge between a carboxylate on the drug and Arg , which lies in the tunnel. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

As we will see later, the tunneling barriers, and hence the relaxation times of the tunneling centers, are distributed. This would lead to a time-dependent heat capacity. Ignoring this complication for now, the classical, long-time heat capacity is easy to estimate aheady (assuming it exists). Since our degrees of... 

The STM postulated tunneling matrix element distribution P(A) oc 1 /A implies a weakly (logarithmically) time-dependent heat capacity. This was pointed out early on by Anderson et al. [8], while the first specific estimate appeared soon afterwards [93]. The heat capacity did indeed turn out time dependent however, its experimental measures are indirect, and so a detailed comparison with theory is difficult. Reviews on the subject can be found in Nittke et al. [99] and Pohl [95]. Here we discuss the A distribution dictated by the present theory, in the semiclassical limit, and evaluate the resulting time dependence of the specific heat. While this limit is adequate at long times, quantum effects are important at short times (this concerns the heat condictivity as well). The latter are discussed in Section VA. 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

Even Anderson et al. [39] pointed out that an important consequence of the tunnelling model was the (logarithmic) dependence of the measured specific heat on the time needed for the measurement of c. The latter phenomenon was due to the large energy spread and relaxation time of TLS. In 1978, Black [45], by a critic revision of the tunnelling theory, has been able to explain the time dependence of the low-temperature specific heat. 

However, TDSE permits an analysis of the time-dependent population of the molecular orbitals during the tunneling, which is unavailable with the TISE. 

Fig. 2 Electronic conduction of a benzene ring between two conducting electrodes. These calculations are performed by the time-dependent method presented here (solid line) and by the ESQC method (dashed line). The electrodes are connected either in ortho (left column) or meta (right column) position. Two regimes are investigated tunneling with v = —0.25 ev (upper row), pseudoballistic with v = —2 ev (lower row). The vertical dashed lines represent the energy of the benzene s molecular orbitals...

Bar-Joseph I, Gurvitz SA (1991) Time-dependent approach to resonant tunneling and inelastic scattering. Phys Rev B 44 3332... 

The traditional treatment of molecules relies upon a molecular Hamiltonian that is invariant under inversion of all particle coordinates through the center of mass. For such a molecular Hamiltonian, the energy levels possess a well-defined parity. Time-dependent states conserve their parity in time provided that the parity is well defined initially. Such states cannot be chiral. Nevertheless, chiral states can be defined as time-dependent states that change so slowly, owing to tunneling processes, that they are stationary on the time scale of normal chemical events. [22] The discovery of parity violation in weak nuclear interactions drastically changes this simple picture, [14, 23-28] For a recent review, see Bouchiat and Bouchiat. [29]... 

An improved and direct correlation between the experimental rate constant and [obtained using Eq. (49)] is observed if v = /zd is used instead of v = 1/Tt, the solvent-dependent tunneling factor is utilized, and only AG (het) of Eq. (8) is used in Eq. (49) (see triangles in Fig. 18). Furthermore, the inverse of the longitudinal solvent relaxation time Xi is not necessarily the relevant one to use as the frequency factor v (see empty circles in Fig. 18). Similar conclusions were reached by Barbara and Jerzeba for the electron transfer reaction in homogeneous solutions. Barbara and Jerzeba measured the electron transfer time... 

The most extensively used theoretical method for the understanding of the MIM tunneling junction is the time-dependent perturbation approach developed by Bardeen (1960). It is sufficiently simple for treating many realistic cases, and has been successfully used for describing a wide variety of effects (Duke, 1969 Kirtley, 1982). 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

The polarization, or the van der Waals interaction, can be accounted for by a stationary-state perturbation theory, effectively and accurately. The exchange interaction or tunneling can be treated by time-dependent perturbation theory, following the method of Oppenheimer (1928) and Bardeen (1960). In this regime, the polarization interaction is still in effect. Therefore, to make an accurate description of the tunneling effect, both perturbations must be considered simultaneously. This is the essence of the MBA. 

Furthermore, an approximate relation for the time dependence of D, which has been successfully used for tunneling kinetics in glasses [38], is applied ... 

A typical example of a special state is as follows. Electron transfer reactions at an atom are aided by vibrations that equilibrate the interatomic distances that differ for the two oxidation states. Thus a low-energy, high-amplitude vibration is desirable. The vibration could have the further function that it provided a time-dependent fluctuation of the redox potential. As I and Goldanskii in this volume have pointed out, this allows a precise matching of the redox potential of one redox couple with another leading to tunneling of electrons. 

---