Path, reaction

The minimum energy path (MEP) in mass weighted Cartesian coordinates is the path through configuration space traced out by a hypothetical trajectory initiated at the saddle point with all inertia effects removed. It is the path that molasses would follow flowing downhill. The MEP satisfies the differential equation. 

The most computationally intensive step in statistical or dynamical studies based on reaction path potentials is the determination of the MEP by numerical integration of Eq. (2) and the evaluation of potential energy derivatives along the path, so considerable attention should be directed toward doing this most efficiently. Kraka and Dunning [1] have presented a lucid description of many of the available methods for determining the MEP. Simple Euler integration of Eq. 

The calculation of force constants at points along the MEP is often done separately from the determination of the path by numerical integration of Eq (2), but these two problems can profitably be combined. Methods recently have been proposed [6,7] that efficiently use the available force constants to better follow the path. To understand these methods and the relationship between them, consider two different Taylor series expansions about a point on the MEP. The first is the familiar expansion of the energy in the mass-weighted Cartesian coordinates, 

Here go, Fo, and Go are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at xQ. The square brackets 

Given partial third energy derivative information, further contributions to the coefficients in Eq (6) can be determined [7]. For example, the third order coefficient in Eq (6) requires the derivative of the force constant matrix with respect to s. This third derivative information can be estimated by a simple finite difference procedure if successive force constant matrices have been determined along the path. 

Once the product specifications have been fixed, some decisions need to be made regarding the reaction path. There are sometimes different paths to the same product. For example, suppose ethanol is to be manufactured. Ethylene could be used as a raw material and reacted with water to produce ethanol. An alternative would be to start with methanol as a raw material and react it with synthesis gas (a mixture of carbon monoxide and hydrogen) to produce the same product. These two paths employ chemical reactor technology. A third path could employ a biochemical reaction (or fermentation) that exploits the metabolic processes of microorganisms in a biochemical reactor. Ethanol could therefore also be manufactured by fermentation of a carbohydrate. 

Reactors can be broadly classified as chemical or biochemical. Most reactors, whether chemical or biochemical, are catalyzed. The strategy will be to choose the catalyst, if one is to be used, and the ideal characteristics and operating conditions needed for the reaction system. The issues that must be addressed for reactor design include  

As already noted, given that the objective is to manufacture a certain product, there are often a number of alternative reaction paths to that product. Reaction paths that use the cheapest raw materials and produce the smallest quantities of byproducts are to be preferred. Reaction paths that produce significant quantities of unwanted byproducts should especially be avoided, since they can create significant environmental problems. 

However, there are many other factors to be considered in the choice of reaction path. Some are commercial, such as uncertainties regarding future prices of raw materials and byproducts. Others are technical, such as safety and energy consumption. 

The lack of suitable catalysts is the most common reason preventing the exploitation of novel reaction paths. At the first stage of design, it is impossible to look ahead and see all of the consequences of choosing one reaction path or another, but some things are clear even at this stage. Consider the following example. 

Example 2.1 Given that the objective is to manufacture vinyl chloride, there are at least three reaction paths which can be readily exploited.  

The market values and molecular weights of the materials involved are given in Table 2.1. Oxygen is considered to be free at this stage, coming from the 

If an internal or cartesian coordinate can be identified with the reaction coordinate, dien, by monotonically increasing or decreasing the coordinate, the energy profile of die reaction path can be mapped. The transition state is, of course, the highest point. An example of such a reaction would be the formation of an addition complex, e.g, BF3 + NH3 BF3NH3, in which the B— N distance could be identified with the reaction coordinate. In another example, the H—C— N angle could be used as the reaction coordinate for the isomerization of HCN to HNC. 

Sometimes no simple reaction coordinate exists. In the rare cases in which the transition state is symmetric, say hydrogen abstraction from methane by a methyl group, symmetry constraints can be applied. 

Sometimes the reaction path is very complicated. In that case the transition state can be approached from two directions simultaneously. This method, known as the saddle technique, would then be used for locating the transition state. 

Once a rough approximation to the transition state has been obtained, gradient minimization techniques can be used for refining the system. The most commonly used techniques are Bartels and the Mclver-Komornicki methods. 

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Example 2.2 Devise a process from the three reaction paths in Example 2.1 which uses ethylene and chlorine as raw materials and produces no byproducts other than water. Does the process look attractive economically ... 

In summary, path 2 from Example 2.1 is the most attractive reaction path if there is a large market for hydrogen chloride. In practice, it tends to be difficult to sell the large quantities of hydrogen chloride produced by such processes. Path 4 is the usual commercial route to vinyl chloride. 

Having made a choice of the reaction path, we need to choose a reactor type and make some assessment of the conditions in the reactor. This allows assessment of reactor performance for the chosen reaction path in order for the design to proceed. 

Reducing waste from primary reactions which produce waste byproducts. If a waste byproduct is formed from the reaction, as in Eq. (10.1) above, then it can only be avoided by different reaction chemistry, i.e., a different reaction path. 

Changing the reaction path to reduce or eliminate the formation of unwanted byproducts. 

Dealing with NO emissions. There are two main reaction paths for NO,r formation ... 

In many cases, however, well-designed catalysts provide intrinsically different reaction paths, and the specific nature of the catalyst surface can be quite important. This is clearly the case with unimolecular reactions for which the surface concentration effect is not applicable. 

Multidimensionality may also manifest itself in the rate coefficient as a consequence of anisotropy of the friction coefficient [M]- Weak friction transverse to the minimum energy reaction path causes a significant reduction of the effective friction and leads to a much weaker dependence of the rate constant on solvent viscosity. These conclusions based on two-dimensional models also have been shown to hold for the general multidimensional case [M, 59, and 61]. 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

As a multidimensional PES for the reaction from quantum chemical calculations is not available at present, one does not know the reason for the surprismg barrier effect in excited tran.s-stilbene. One could suspect diat tran.s-stilbene possesses already a significant amount of zwitterionic character in the confomiation at the barrier top, implying a fairly Tate barrier along the reaction path towards the twisted perpendicular structure. On the other hand, it could also be possible that die effective barrier changes with viscosity as a result of a multidimensional barrier crossing process along a curved reaction path. 

Larson R S 1986 Simulation of two-dimensional diffusive barrier crossing with a curved reaction path Physica A 137 295-305... 

To calculate N (E-Eq), the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The fomier approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Hamionic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by perfomiing an appropriate nomial mode analysis as a fiinction of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to detemiine anliamionic energy levels for die transitional modes [27]. 

Fehrensen B, Luckhaus D and Quack M 1999 Inversion tunneling in aniline from high resolution infrared spectroscopy and an adiabatic reaction path Hamiltonian approach Z. Phys. Chem., NF 209 1-19... 

The close-coupling approach works readily and simply if the reaction is purely melastic . The method can also be made to work very simply for a single product arrangement (as in collinear reactions), by using a twisted coordinate system, most conveniently reaction path coordinates [37, 38 and 39] as shown in figure B3.4.3. 

Figure B3.4.3. A schematic figure showing, for the DH2 collinear system, a reaction-path coordmate Q coimecting continuously the reactants and the single products asymptote. Also shown are the cuts denoting the coordinate perpendicular to Q.

Heidrich D (ed) 1995 The Reaction Path In Chemistry Current Approaches and Perspectives (Boston Kluwer Academic)... 

Miller W H, Handy N C and Adams J E 1980 Reaction path Hamiltonian for polyatomic molecules J. Chem. Phys. 72 99... 

Fast P L and Truhlar D G 1998 Variational reaction path algorithm J. Chem. Phys. 109 3721 Billing G D 1992 Quantum classical reaction-path model for chemical reactions Chem. Phys. 161 245... 

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