Illustrating Example

In the following, we will briefly illustrate the application of nonequilibrium free energy calculations for a simple ID model system. Shown in Fig. 5.1 are the potential energy surfaces 

We have performed Brownian dynamics simulations (i.e., overdamped Langevin dynamics) on the potential surface V(x, A) with a time-dependent A. In the simulations, the position x was updated as 

The work distribution is bimodal, with a dominant ( 99%) peak at high work values of (3W m 6, and a small second peak at low work values of ftW —6. The high-work peak is mainly due to trajectories that start in the low-energy well near 

Shown in Fig. 5.3d are free energies estimated from the same forward and backward simulation runs using Bennett s optimal estimator, obtained by solving (5.50) using a Newton-Raphson method. Unlike the direct exponential estimator (which 

In summary, using work collected from forward and backward paths greatly improves the accuracy of the estimates, and for the symmetric system studied here eliminates the bias. In our particular example, the cumulant estimators using forward and backward work data produce the most precise free energy estimates, followed by Bennett s optimal estimator. However, this somewhat poorer performance of the optimal estimator is caused in part by the high degree of symmetry of the system studied. 

---

Desulfurization will become mandatory when oxidizing catalysts are installed on the exhaust systems of diesel engines. At high temperatures this catalyst accelerates the oxidation of SO2 to SO3 and causes an increase in the weight of particulate emissions if the diesel fuel has not been desulfurized. As an illustrative example, Figure 5.22 shows that starting from a catalyst temperature of 400°C, the quantity of particulates increases very rapidly with the sulfur content. 

We consider first some experimental observations. In general, the initial heats of adsorption on metals tend to follow a common pattern, similar for such common adsorbates as hydrogen, nitrogen, ammonia, carbon monoxide, and ethylene. The usual order of decreasing Q values is Ta > W > Cr > Fe > Ni > Rh > Cu > Au a traditional illustration may be found in Refs. 81, 84, and 165. It appears, first, that transition metals are the most active ones in chemisorption and, second, that the activity correlates with the percent of d character in the metallic bond. What appears to be involved is the ability of a metal to use d orbitals in forming an adsorption bond. An old but still illustrative example is shown in Fig. XVIII-17, for the case of ethylene hydrogenation. 

A few illustrative examples are the following. Photohydrogenation of acetylene and ethylene occurs on irradiation of Ti02 exposed to the gases, but only if TiOH surface groups are present as a source of hydrogen [319]. The pho-toinduced conversion of CO2 to CH4 in the presence of Ru and Os colloids has been reported [320]. Platinized Ti02 powder shows, in the presence of water, photochemical oxidation of hydrocarbons [321,322]. Some of the postulated reactions are ... 

An illustrative example is provided by investigating the possible momenta for a single particle travelling in the v-direction, p First, one writes the equation that defines the eigenvalue condition... 

A marvellous and rigorous treatment of non-relativistic quantum mechanics. Although best suited for readers with a fair degree of mathematical sophistication and a desire to understand the subject in great depth, the book contains all of the important ideas of the subject and many of the subtle details that are often missing from less advanced treatments. Unusual for a book of its type, highly detailed solutions are given for many illustrative example problems. 

The field of gas phase reaction dynamics has been extensively reviewed elsewhere [1, 2 and 3] in considerably greater detail than is appropriate for this chapter. Here, we begin by simnnarizing the key theoretical concepts and experimental teclmiques used in reaction dynamics, followed by a case study , the reaction F + H2 HF + H, which serves as an illustrative example of these ideas. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

As an illustrative example, consider the vibrational energy relaxation of the cyanide ion in water [45], The mechanisms for relaxation are particularly difficult to assess when the solute is strongly coupled to the solvent, and the solvent itself is an associating liquid. Therefore, precise experimental measurements are extremely usefiil. By using a diatomic solute molecule, this system is free from complications due to coupling... 

In this section, I present a few illustrative examples of applications of NMR relaxation studies within different branches of chemistry. The three subsections cover one story each, in order of increasing molecular size and complexity of the questions asked. 

G) ILLUSTRATIVE EXAMPLES OF THE ELECTRONIC AND OPTICAL PROPERTIES OF MODERN MATERIALS... 

In other words, (n VQ ) is imaginary, making real. As an illustrative example, n) may assumed to be given by Eq. (24), in which case... 

Essential Dynamics In most applications details of individual MD trajectories are of only minor interest. An illustrative example due to Grubmuller [10] is documented in Figure 3. It describes the dynamics of a polymer chain of 100 CH2 groups. Possible stepsizes for numerical integration are confined... 

Having found a place (the sp -sp bon d t to establish the boundary between classical atom s and quantum atoms, the next cpiesiion is how to cap the quantum atoms. Let s first of all look at an illustrative example of the problem. ... 

A simple illustrative example of reciprocal space is that of a 2D square lattice where the vectors a and b are orthogonal and of length equal to the lattice spacing, a. Here a and b are directed along the same directions as a and b respectively and have a length 1/a... 

As an illustrative example we consider the Galerkin finite element solution of the following differential equation in domain Q, as shown in Figure 2.20. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

---