MEP

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

Information about critical points on the PES is useful in building up a picture of what is important in a particular reaction. In some cases, usually themially activated processes, it may even be enough to describe the mechanism behind a reaction. However, for many real systems dynamical effects will be important, and the MEP may be misleading. This is particularly true in non-adiabatic systems, where quantum mechanical effects play a large role. For example, the spread of energies in an excited wavepacket may mean that the system finds an intersection away from the minimum energy point, and crosses there. It is for this reason that molecular dynamics is also required for a full characterization of the system of interest. 

Bonaccorsi ct al. [204 defined for the first time the molecular electrostatic potential (MEP), wdicli is dearly tfie most important and most used property (Figure 2-125c. The clcctro.static potential helps to identify molecular regions that arc significant for the reactivity of compounds. Furthermore, the MEP is decisive for the formation of protein-ligand complexes. Detailed information is given in Ref [205]. 

The MEP at the molecular surface has been used for many QSAR and QSPR applications. Quantum mechanically calculated MEPs are more detailed and accurate at the important areas of the surface than those derived from net atomic charges and are therefore usually preferable [Ij. However, any of the techniques based on MEPs calculated from net atomic charges can be used for full quantum mechanical calculations, and vice versa. The best-known descriptors based on the statistics of the MEP at the molecular surface are those introduced by Murray and Politzer [44]. These were originally formulated for DFT calculations using an isodensity surface. They have also been used very extensively with semi-empirical MO techniques and solvent-accessible surfaces [1, 2]. The charged polar surface area (CPSA) descriptors proposed by Stanton and Jurs [45] are also based on charges derived from semi-empirical MO calculations. 

This quantity is found to be related to the local polarization energy and is complementary to the MEP at the same point in space, making it a potentially very useful descriptor. Reported studies on local ionization potentials have been based on HF ab-initio calculations. However, they could equally well use semi-empirical methods, especially because these are parameterized to give accurate Koopmans theorem ionization potentials. 

In order to define how the nuclei move as a reaction progresses from reactants to transition structure to products, one must choose a definition of how a reaction occurs. There are two such definitions in common use. One definition is the minimum energy path (MEP), which defines a reaction coordinate in which the absolute minimum amount of energy is necessary to reach each point on the coordinate. A second definition is a dynamical description of how molecules undergo intramolecular vibrational redistribution until the vibrational motion occurs in a direction that leads to a reaction. The MEP definition is an intuitive description of the reaction steps. The dynamical description more closely describes the true behavior molecules as seen with femtosecond spectroscopy. 

The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

An ensemble of trajectory calculations is rigorously the most correct description of how a reaction proceeds. However, the MEP is a much more understandable and useful description of the reaction mechanism. These calculations are expected to continue to be an important description of reaction mechanism in spite of the technical difficulties involved. 

MEP (IRC, intrinsic reaction coordinate, minimum-energy path) the lowest-energy route from reactants to products in a chemical process MIM (molecules-in-molecules) a semiempirical method used for representing potential energy surfaces... 

V. T. Morgan, Porous Metal Bearings and Their Application, MEP-213, Mechanical Engineering Pubhcations, Ltd., Workiagton, UK, 1984. 

A newer technology for the manufacture of chromic acid uses ion-exchange (qv) membranes, similar to those used in the production of chlorine and caustic soda from brine (76) (see Alkali and cm ORiNE products Chemicals frombrine Mep rane technology). Sodium dichromate crystals obtained from the carbon dioxide option of Figure 2 are redissolved and sent to the anolyte compartment of the electrolytic ceU. Water is loaded into the catholyte compartment, and the ion-exchange membrane separates the catholyte from the anolyte (see Electrochemical processing). 

The power of the engine is expressed by die mean effective pressure, mep, where... 

Computations have shown that in the quantum region it is possible to have various most probable transition paths (ranging from the classical minimum energy path (MEP) to the straight-line one-dimensional tunneling of early models), depending on the PES geometry. 

The obtained PES forms the basis for the subsequent dynamical calculation, which starts with determining the MEP. The next step is to use the vibrationally adiabatic approximation for those PES degrees of freedom whose typical frequencies a>j are greater than a>o and a>. Namely, for the high-frequency modes the vibrationally adiabatic potential [Miller 1983] is introduced,... 

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

Fig. 17. Contour plots for a

Physically the cutting-corner trajectory implies that the particle crosses the barrier suddenly on the time scale of the slow -vibration period. In the literature this approximation is usually called sudden , frozen bath and fast flip approximation, or large curvature case. In the opposite case of small curvature (also called adiabatic and slow flip approximation), coj/coo < sin tp, which is relevant for transfer of fairly heavy masses, the MEP may be taken to a good accuracy to be the reaction path. 

In the antisymmetric case the possible reaction path ranges from the MEP (when cui and coq are comparable) to the sudden path (coi <4 coq), when the system waits until the q vibration symmetrizes the potential (the segment of path with Q = Qo) and then instantaneously tunnels in the symmetric potential along the line q = 0. All of these types of paths are depicted in fig. 17. ... 

Fig. 33. Three-dimensional instanton trajeetories of a partiele in a symmetrie double well, interaeting with symmetrieally and antisymmetrieally eoupled vibrations with eoordinates and frequeneies q, to, and ru, respeetively. The curves are 1, ru, ru, P ojq (MEP) 2. to, (u, < (Oq (sudden approximation) 3. ru, < cOq, oj, P ojo 4. to, > (Oq, < (Oq.

When both vibrations have high frequencies, Wa, coq, the transition proceeds along the MEP (curve 1). In the opposite case of low frequencies, rUa.s the tunneling occurs in the barrier, lowered and reduced by the symmetrically coupled vibration q, so that the position of the antisymmetrically coupled oscillator shifts through a shorter distance, than that in the absence of coupling to qs (curve 2). The cases (0 (Oq, < (Oo, and Ws Wo, (Oq, characterized by combined trajectories (sudden limit for one vibration and adiabatic for the other) are also presented in this picture. 

A number of empirical tunneling paths have been proposed in order to simplify the two-dimensional problem. Among those are MEP [Kato et al. 1977], sudden straight line [Makri and Miller 1989], and the so-called expectation-value path [Shida et al. 1989]. The results of these papers are hard to compare because slightly different PES were used. As to the expectation-value path, it was constructed as a parametric line q(Q) on which the vibration coordinate q takes its expectation value when Q is fixed. Clearly, for the PES at hand this path coincides with MEP, since is a harmonic oscillator. 

Fig. 38, Contour plot, MEP and instanton trajectory for isomerization of malonaldehyde (6.4). The instanton is drawn for large but finite in the limit = oo it emanates from the potential minimum.

Fig. 39. Contour plot, MEP and instanton trajectory for isomerization of hydrogenoxalate anion (6.5).

Fig. 43. Contour plot of potential (6.13) with two transition states. F(Q) = (Q -2Q ) Fo, C = 20Fo, /4=90Fo, Qc = 0.5. MEPs are shown.

The rate constants and ko were equal to 3 x 10 and lO s respectively [Shian et al. 1980 Bratan and Strohbusch 1980]. There are two equivalent ways of stepwise transfer, and, therefore, the transition state and MEP are two-fold, if the stepwise transfer is energetically preferable. On the other hand, there is only one way of concerted transfer, which lies between the saddle points. Based on this reasoning, de la Vega et al. [1982] have found that the barrier for stepwise transfer (25kcal/mol) is 3.1 kcal/mol lower than that for concerted transfer. These authors have proposed a model two-dimensional PES,... 

One-dimensional motion along Q corresponds to concerted transfer, while the two MEPs, corresponding with stepwise transfer, are... 

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