Tunneling, quantum-mechanical

Department of Chemistry, Brookhaven National Laboratory, Upton, L. I., N. Y. 11973 

However, a moving particle of atomic or electronic mass does not obey Newtonian mechanics. Instead, It behaves as a wave packet with a wavelength given by the de Broglie expression 

When such a particle encounters a barrier, represented by an Increase In potential energy, it does not behave like a macroscopic particle. Instead, there Is a finite probability of leak- 

Is less than the potential energy at the barrier summit conr versely, there is a finite probability of reflection even if the kinetic energy is greater than this. The extent of tunneling, the transmission probability k. Is defined by k = (A /A ), where Aj and Af. are the wave function amplitudes for the incident and transmitted waves (Fig. 1). 

As one would expect, k depends on the mass of the particle, its velocity, and the shape and height of the barrier. IVo convenient parameters are 

---

While field ion microscopy has provided an effective means to visualize surface atoms and adsorbates, field emission is the preferred technique for measurement of the energetic properties of the surface. The effect of an applied field on the rate of electron emission was described by Fowler and Nordheim [65] and is shown schematically in Fig. Vlll 5. In the absence of a field, a barrier corresponding to the thermionic work function, prevents electrons from escaping from the Fermi level. An applied field, reduces this barrier to 4> - F, where the potential V decreases linearly with distance according to V = xF. Quantum-mechanical tunneling is now possible through this finite barrier, and the solufion for an electron in a finite potential box gives... 

The presence of nonlinearity in an Arrhenius plot may indicate the presence of quantum mechanical tunnelling at low temperatures, a compound reaction mechanism (i.e., the reaction is not actually elementary) or the unfreezing of vibrational degrees of freedom at high temperatures, to mention some possible sources. 

Devault, D. Quantum mechanical tunnelling in biological systems. Quart. Rev. Biophys. 13 (1980) 387-564. 

Torsional barriers are referred to as n-fold barriers, where the torsional potential function repeats every 2n/n radians. As in the case of inversion vibrations (Section 6.2.5.4a) quantum mechanical tunnelling through an n-fold torsional barrier may occur, splitting a vibrational level into n components. The splitting into two components near the top of a twofold barrier is shown in Figure 6.45. When the barrier is surmounted free internal rotation takes place, the energy levels then resembling those for rotation rather than vibration. 

In 1979, a viable theory to explain the mechanism of chromium electroplating from chromic acid baths was developed (176). An initial layer of polychromates, mainly HCr3 0 Q, is formed contiguous to the outer boundary of the cathode s Helmholtz double layer. Electrons move across the Helmholtz layer by quantum mechanical tunneling to the end groups of the polychromate oriented in the direction of the double layer. Cr(VI) is reduced to Cr(III) in one-electron steps and a colloidal film of chromic dichromate is produced. Chromous dichromate is formed in the film by the same tunneling mechanism, and the Cr(II) forms a complex with sulfate. Bright chromium deposits are obtained from this complex. 

DeVault, D., 1984, Quantum-Mechanical Tunneling in Biological Systems (Cambridge Univ. Press, Cambridge). 

Because STM measures a quantum-mechanical tunneling current, the tip must be within a few A of a conducting surface. Therefore any surface oxide or other contaminant will complicate operation under ambient conditions. Nevertheless, a great deal of work has been done in air, liquid, or at low temperatures on inert surfaces. Studies of adsorbed molecules on these surfaces (for example, liquid crystals on highly oriented, pyrolytic graphite ) have shown that STM is capable of even atomic resolution on organic materials. 

We have seen that 10" M s is about the fastest second-order rate constant that we might expect to measure this corresponds to a lifetime of about 10 " s at unit reactant concentration. Yet there is evidence, discussed by Grunwald, that certain proton transfers have lifetimes of the order 10 s. These ultrafast reactions are believed to take place via quantum mechanical tunneling through the energy barrier. This phenomenon will only be significant for very small particles, such as protons and electrons. 

Concept of quantum-mechanical tunnelling in proton-uansfer reactions introduced (without experimental evidence) by several authors. 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

The transfer of electrons in proteins by a quantum mechanical tunnelling mechanism is now firmly established. Electron transfer within proteins... 

A large primary kinetic isotopeeffect has been reported by several groups the most comprehensive study having been made by Bell and Onwood in an attempt to assess the role of quantum-mechanical tunnelling. This was found to be unimportant and the rate coefficients at infinite dilution were expressed as... 

A model which takes into account the spin-rotation interaction has been found to satisfactorily explain the 0 rotation band of PHg. The millimetre-wave spectra of HCP and DCP have been compared with those of HCN and DCN. A method of estimating frequencies of bands in this region due to processes such as pseudorotation has been suggested. This new approach involves calculation of the rovibronic energy levels from the effects of quantum-mechanical tunnelling. ... 

DeVault D (1984) Quantum-mechanical tunneling in biological systems, Cambridge Univ Press, Cambridge... 

Quantum Mechanical Tunneling in Organic Reactive Intermediates... 

QUANTUM MECHANICAL TUNNELING IN ORGANIC REACTIVE INTERMEDIATES... 

Subsequently, very thorough work by these investigators demonstrated conclusively that these intramolecular H-abstractions occurred via quantum mechanical tunneling. Multiple criteria were invoked to support this proposal as follows ... 

---