Junction rule

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero,... 

Here we conventionally assume that the flux entering the junction is positive, whereas the terms that describe the leaving part are taken negative. The advantage of the junction rule is that it allows to avoid the detailed consideration of system s dynamics inside the junction (inside the body of the octopus ) and provides the results of tunneling through the tentacles. 

To show how the junction rule works, consider the above example of tunneling in the double-well potential. In this case we have two nodes connected by just one tunneling path. Let the starting position of the system be in the left well with the ground-state wave function P1 = C il> (r) (Q) are assumed to be normalized, and C is the amplitude in the left well, so that I C I2 is the probability to find the system in this well. The corresponding tail of the WKB ground-state wave function under the barrier should decrease with Q exponentially,... 

Substituting equations (23) and (25) into the corresponding junction-rule equation, /) + j2 = 0, and reducing the equation by common factors, we come to Cf — C = 0. On the other hand, except the factors C and C2, both WKB functions in the wells are normalized. Therefore, the total probability to find the system in both wells is C + C2 = 1. This system of two equations has two real solutions ... 

The junction rule (20) applied to each of the four vertexes results in the following four equations for the amplitudes of the ground-state wave functions in the wells ... 

The other two components of the vibronic triplet term, transforming as y and z, can be obtained by applying symmetry operations of the group Oh to the x component in equation (31). Thus, as in the example of Section 4, the junction rule gives the same result as the symmetry projection techniques. The corresponding eigenvalues are... 

Fig. 6. Rule 3 of thermocouples, where (a) represents a four-junction and (b) a two-junction thermocouple. A and B, the two legs of the thermocouple, are wires of dissimilar materials. Junctions are at temperatures T, and Ty Ey E, < ...

Troisi A, Ratner MA (2006) Molecular transport junctions propensity rules for inelastic electron tunneling spectra. Nano Lett 6(8) 1784-1788... 

Paulsson M, Frederiksen T, Ueba H, Lorente N, Brandbyge M (2008) Unified description of inelastic propensity rules for electron transport through nanoscale junctions. Phys Rev Lett 100(22) 226604... 

The take-home message here is that conductivity measurements in single-molecule junctions are difficult to analyze without the support of quantum mechanical calculations that include the metal electrodes. This is very much the domain of specialists, and the simple rules discussed for analyzing elastic tunneling spectra in other junction types generally do not apply for metal-single-molecule-metal junctions. 

Fig. 21 The variation of the balancing tunneling current of the four branches four electrodes monomolecular Wheatstone bridge connected as presented in (a). In (b), the dashed line is for the current intensity 7W (in absolute value) measured by the ammeter A and deduced from the standard Kirchoff laws calculating each molecular wire tunneling junction resistance of the bridge one after the other from the EHMO-ESQC technique. In (b), Hie full line is the same tunnel current intensity but obtained with the new intramolecular circuit rules discussed in Sect. 2. (c) The resistance of the branch used to balance the bridge as a function of its rotation angle. The minimum accessible resistance by rotation is 78 MQ for the short tolane molecular wire used here...

It has been shown that certain unsaturated cyclic diketones readily decompose via RDA reaction if they are cis at the central ring junction, but not if trans (Scheme 6.49a) It was pointed out that this behavior would be expected if the mass spectral RDA reaction is concerted and follows symmetry rules analogous to those established for thermal reactions. [Ill]... 

The RDA reaction is often observed from steroid molecular ions, and it can be very indicative of steroidal stmcture. [107,110,113,114] The extent of the RDA reaction depends on whether the central ring junction is cis or trans. The mass spectra of A -steroidal olefins, for example, showed a marked dependence upon the stereochemistry of the A/B ring juncture, in accordance with orbital symmetry rules for a thermal concerted process. In the trans isomer the RDA is much reduced as compared to the cis isomer. The effect was shown to increase at 12 eV, and as typical for a rearrangement, the RDA reaction became more pronounced, whereas simple cleavages almost vanished. This represented the first example of such apparent symmetry control in olefinic hydrocarbons. [114]. 

Rule 6 None of the bonds which are part of a cyclic system linking a pair of common atoms (i.e., fused, bridgedheads or spiro-hng junction atoms) may be considered to be "strategic" if this cyclic system contains chiral centres. In this way retrosynthetic cleavages which would leave chiral centres on side chains are avoided... 

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