Perturbation theory tunneling

The coefficient of the 8-function reflects the pile-up of the two-level systems that would have had a value of e < S were it not for quantum effects. These fast two-level systems will contribute to the short-time value of the heat capacity in glasses. The precise distribution in Eq. (69) was only derived within perturbation theory and so is expected to provide only a crude description of the interplay of clasical and quantum effects in forming low-barrier TLS. Quantitative discrepancies from the simple perturbative distribution may be expected owing to the finite size of a tunneling mosaic cell, as mentioned earlier. 

Wong K-Y, Gao J (2008) Systematic approach for computing zero-point energy, quantum partition function, and tunneling effect based on Kleinert s variational perturbation theory. J Chem Theory Comput 4(9) 1409-1422... 

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. 

Fig. 1.20. The Bardeen approach to tunneling theory. Instead of solving the Schrddinger equation for the coupled system, a, Bardeen (1960) makes clever use of perturbation theory. Starting with two free subsystems, b and c, the tunneling current is calculated through the overlap of the wavefunctions of free systems using the Fermi golden rule.

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. 

As shown in Fig. 7.2, at a shorter proton-proton separation (/ <16 a.u. or <8 A), the electron in the Is state in the vicinity of one proton has an appreciable probability of tunneling to the Is state in the vicinity of another proton. The tunneling matrix element can be evaluated using the perturbation theory we presented in Chapter 2. A schematic of this problem is shown in Fig. 7.3. By defining a pair of one-center potentials, Ul and Ur, we define the right-hand-.side states and the left-hand-side states. Because the potential Ul is different from the potential of a free proton, Uro, the wavefunction i ii, and the energy level Eo are different from the Is state of a free hydrogen atom. (The same is true for Ur and We will come back to the effect of such a distortion later in this section. 

For R>7 bohr, the difference between the exact solution and the leading terms (Equations (7.33) and (7.34)) is less than 1 meV. Therefore, the interpretation of the resonance energy in terms of tunneling is verified quantitatively in the case of the hydrogen molecular ion. Furthermore, the comparison with the soluble case of the hydrogen molecular ion is also a verification of the accuracy of the perturbation theory presented in Chapter 2. 

Fig. 6.17 Tunnelling and saddle point ionization in Li. (a) Experimental map of the energy levels of Li m = 1 states in a static field. The horizontal peaks arise from ions collected after laser excitation. Energy is measured relative to the one-electron ionization limit. Disappearance of a level with increasing field indicates that the ionization rates exceed 3 x 105 s 1. The dotted line is the classical ionization limit given by Eqs. (6.35) and (6.36). One state has been emphasized by shading, (b) Energy levels for H (n = 18-20, m = 1) according to fourth order perturbation theory. Levels from nearby terms are omitted for clarity. Symbols used to denote the ionization rate are defined in the key. The tick mark indicates the field where the ionization rate equals the spontaneous radiative rate, (c) Experimental map as in (a) except that the collection method is sensitive only to states whose ionization rate exceeds 3 x 105 s-1. At high fields, the levels broaden into the continuum in agreement with tunnelling theory for H (from ref. 32).

Through 1960, Per-Olov had authored and coauthored some fifty papers. He added nearly two hundred in the following thirty-five years. These covered topics in quantum genetics, proton tunneling, and science in society in addition to further pursuits in quantum theory. The series Studies in Perturbation Theory I-XIV demonstrated a search for economy and elegance in presentation, which was important to Per-Olov and became one of his trademarks as a scientist. Per-Olov frequently referred to the economy of thinking" as shorthand for elegance... 

When the coupling to the leads is weak, electron-electron interaction results in Coulomb blockade, the sequential tunneling is described by the master equation method [169-176] and small cotunneling current in the blockaded regime can be calculated by the next-order perturbation theory [177-179], This theory was used successfully to describe electron tunneling via discrete... 

Keywords. Electron transport low-conductance transport non-equilibrium transport perturbation theory density functional theory scanning tunneling microscopy. 

When a tunneling calculation is undertaken, many simplifications render the task easier than a complete transport calculation such as the one of [32]. Let us take the formulation by Caroli et al. [16] using the change induced by the vibration in the spectral function of the lead. In this description, the current and thus the conductance are proportional to the density of states (spectral function) of the leads (here tip and substrate). This is tantamount to using some perturbational scheme on the electron transmission amplitude between tip and substrate. This is what Bardeen s transfer Hamiltonian achieves. The main advantage of this approximation is that one can use the electronic structure calculated by some standard way, for example plane-wave codes, and use perturbation theory to account for the inelastic effect. In [33], a careful description of the Bardeen approximation in the context of inelastic tunneling is given, and how the equivalent of Tersoff and Hamann theory [34,35] of the STM is obtained in the inelastic case. 

This equation says that there is an increase of conductance due to the modulation of the wave function by the vibration. The spatial resolution of the wave function carries the information of the exponential decay in vacuum of the tunneling probability. Hence, during the vibration this tunneling probability will be modulated, in a way given by the change of the wave function. The change of wave function is calculated in perturbation theory ... 

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