Loop rule

The equal valence rule or loop rule equation (3) is less rigorously obeyed, and does not apply to the enviromnents of atoms with electronically driven anisotropies arising from, for example, lone electron pairs see Lone Pair, Electronic Structure of Main-group Compounds) or Jahn Teller distortions see Jahn-Teller Effect, Copper Inorganic Coordination Chemistry). 

The above-mentioned rule, which we would like to call the loop rule", is the most general formulation of the (Am + 2)-type rules. The Hiickel rule is obviously a consequence of it. 

As an important application of the loop rule, approximate formulae can be derived to show the dependence of E on several graph parameters. The problem of finding the dependence of En on molecular... 

The valence sum rule (equation (7)) is the same as equation (1), but the loop rule (equation (8)), has been modified by the addition of bond weights, Cy. The effect of giving a bond a large weight is to increase its predicted valence. The problem then reduces to choosing the weights, Cy, to be assigned to each bond. 

F.T. Brown. Direct Application of the Loop Rule to Bond Graphs. Journal of Dynamic Systems, Measurement and Control, 94(3) 253-261, September 1992. 

According to the circuit loop rule, the sum of the potentials through each element must be equivalent to the driving potential applied ... 

In 1953, Cohen and Coon [2] developed a set of controller tuning recommendations that correct for one deficiency in the Ziegler-Nichols open-loop rules. This deficiency is the sluggish closed-loop response given by the Ziegler-Nichols rules on the relatively rare occasion when process dead time is large relative to the dominant open-loop time constant. 

C, Anchors, Molecules and Independent Quantum Species II. The Phase-Change Rule and the Construction of Loops... 

In this section, the systematic search for conical intersections based on the Longuet-Higgins phase-change rule is described. For conciseness sake, we limit the present discussion to Hiickel-type systems only, unless specifically noted otherwise. The first step in the antilysis is the determination of the LH loops containing a conical intersection for the reaction of interest. 

We have seen (Section I) that there are two types of loops that are phase inverting upon completing a round hip an i one and an ip one. A schematic representation of these loops is shown in Figure 10. The other two options, p and i p loops do not contain a conical intersection. Let us assume that A is the reactant, B the desired product, and C the third anchor. In an ip loop, any one of the three reaction may be the phase-inverting one, including the B C one. Thus, the A B reaction may be phase preserving, and still B may be attainable by a photochemical reaction. This is in apparent contradiction with predictions based on the Woodward-Hoffmann rules (see Section Vni). The different options are summarized in Figure 11. 

Accepting the Longuet-Higgins rule as the basis for the search of conical intersection, it is necessary to look for the appropriate loop. The -type degeneracy of a Jahn-Teller system is removed by a nonsymmetric motion. 

In this section, we apply the phase-change rule and the loop method to some representative photochemical systems. The discussion is illustiative, no comprehensive coverage is intended. It is hoped that the examples are sufficient to help others in applying the method to other systems. This section is divided into two parts in the first, loops are constructed and a qualitative discussion of the photochemical consequences is presented. In the second, the method is used for an in-depth, quantitative analysis of one system—photolysis of 1,4-cyclohexadiene. 

An alternative starting network is one without stream spHts. The networks from the TI method maximize energy recovery and may introduce heat-load loops. Stream spHts ate not made in the initial steps of network invention. The ED method is proposed to be one in which heuristic rules and strategies would be used to improve the networks developed by the TI method. The importance of a thermodynamic base for evolutionary rules is stressed in this proposal, but there is no expHcit guidance for the evolutionary process. 

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

---