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Downhill

A geographical analogy can be a helpful way to illustrate many of the concepts we shall encounter in this chapter. In this analogy minimum points correspond to the bottom of valleys. A minimum may be described as being in a long and narrow valley or a flat and featureless plain. Saddle points correspond to mountain passes. We refer to algorithms taking steps uphill or downhill. ... [Pg.273]

Fig. 5.1. (a) An automobile moving downhill can do work. It is this free work that drives the process. (b) In the simplest situation the free work can be calculated from the change in potential energy, mgh, that takes place during the process. [Pg.47]

What do we do when there are other ways of doing free work As an example, if our car were initially moving downhill with velocity v but ended up stationary at the bottom of the hill, we would have... [Pg.47]

A drop of water that is placed on a hillside will roll down the slope, following the surface curvature, until it ends up in the valley at the bottom of the hill. This is a natural minimization process by which the drop minimizes its potential energy until it reaches a local minimum. Minimization algorithms are the analogous computational procedures that find minima for a given function. Because these procedures are downhill methods that are unable to cross energy barriers, they end up in local minima close to the point from which the minimization process started (Fig. 3a). It is very rare that a direct minimization method... [Pg.77]

Figure 4 A representative step m the downhill simplex method. The original simplex, a tetrahedron in this case, is drawn with solid lines. The point with highest energy is reflected through the opposite triangular plane (shaded) to form a new simplex. The new vertex may represent symmetrical reflection, expansion, or contractions along the same direction. Figure 4 A representative step m the downhill simplex method. The original simplex, a tetrahedron in this case, is drawn with solid lines. The point with highest energy is reflected through the opposite triangular plane (shaded) to form a new simplex. The new vertex may represent symmetrical reflection, expansion, or contractions along the same direction.
Perform the IRC calculation (requested with the IRC keyword). This job will help you to verify that you have the correct transition state for the reaction when you examine the structures that are downhill from the saddle point. In some cases, however, you will need to increase the number of steps taken in the IRC in order to get closer to the minimum the AAoxPoinb option specifies the number of steps to take in each direction as its argument. You can also continue an IRC calculation by using the lRC=(ReStorhMaxPointe=n) keyword, setting n to some appropriate value (provided, of course, that you have saved the checkpoint file). [Pg.174]

Heat or energy, like water, of itself, will flow only downhill. [Pg.633]

The dynamics, which consists of two operations, is most conveniently expressed in terms of this local slope value. Start with an arbitrary initial pile of sand, r] t = 0). If the local slope at any site i exceeds an arbitrarily chosen threshold value r/c the system is allowed to relax by continually applying a relation rule 4>-re ax — Areiaxl ) that effectively slides one unit of sand downhill (i.e. to the right) according to (see figure 8.17)... [Pg.438]

Strategy Once you deduce the formula of the ionic compound, it s all downhill. Remember, though, that you have to show the relative numbers of cations and anions. [Pg.37]

Once the masses of the elements are obtained, it s all downhill follow the same path as in Example 3.5. [Pg.59]

In the event that the CO and NO2 molecules do not have sufficient energy to attain the summit, they reach a point only part way up the left side of the barrier. Then, repelling one another, they separate again, going downhill to the left. [Pg.134]

Fig. 9-8. Comparison of a chemical reaction to golf balls rolling downhill. Fig. 9-8. Comparison of a chemical reaction to golf balls rolling downhill.
The changes from initial to final state proceed spontaneously toward lowest potential energy, the direction corresponding to rolling downhill ... [Pg.157]

Since golf balls always roll downhill spon-... [Pg.157]

This analogy solves the problems of the simpler golf balls roll downhill picture. The bumpy road model contains a new feature that gives a basis for expecting reaction in the... [Pg.157]


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Downhill period of life

Downhill pipe section

Downhill process

Downhill reactions

Downhill simplex method

Downhill simplex minimization

Minimization downhill simplex method

Nelder-Mead downhill simplex

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