Section of potential energy surfaces

Subsequent transformations of the above H complexes are affected by a way of temperature or reactant concentration enhancement as well as by addition of the third donor or acceptor component (catalyst) [4,47,48]. Quantum chemical computations of a section of potential energies surface along the reaction pathway for some Sgi and SNi processes with silica OH group reveal that the linear H-complex with this group formed in the first reaction stage reactions thereafter transform to a cyclic donor-acceptor complex. The second minimum on the potential curve (Fig. 1) is due to formation of a cyclic donor-acceptor complex (DAr) which is recognized for SeJ and SnI reactions [49]. 

Fig. 11. Sections of potential energy surfaces along the reaction path. Dotted lines represent sections for zero approximation and full lines adiabatic surface sections...

Fig. 5.12. Sections of potential energy surfaces of the reaction of Eq. (5.17) for various distances of F from the molecule SCIF obtained by the CNDO/2 method [167]. The numbers in the breaks of the curves are relative energies (in kcal/mol)...

Figure 3 A section of potential energy surfaces for coordinate calculated with R-2.2 A and r=1.13A. (Adopted with permission fwm reference 8. Copyright 1998 American Institute of Physics)...

Figure 6 A section of potential energy surface along 6 with R=2.7 A and r=U3A.

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

Theoretical predictions of potential energy surfaces and reaction paths can sometimes yield quite surprising results. In this section, we ll consider an example which illustrates the general approach toward and usefulness of studying potential energy surfaces in detail. 

Experiments have also played a critical role in the development of potential energy surfaces and reaction dynamics. In the earliest days of quantum chemistry, experimentally determined thermal rate constants were available to test and improve dynamical theories. Much more detailed information can now be obtained by experimental measurement. Today experimentalists routinely use molecular beam and laser techniques to examine how reaction cross-sections depend upon collision energies, the states of the reactants and products, and scattering angles. 

Fig. 4. A section of the energy surface, analogous to Fig. 3, for an isomeric transition. The values of the configurational coordinates about a and b correspond to the two isomeric forms of the molecule which are stable against small atomic displacements. The electronic energies for the two forms, namely Ea and E0, are assumed to be the same, although this need not necessarily be the case. This section of the energy surface is assumed to pass through the saddle point in the potential range separating the two minima a and b. The energy of the saddle point is E. ...

Chapter 1 introduces Potential Energy Surfaces as the connection between structure and energetics, and shows how molecular equilibrium and transition-state geometry as well as thermodynamic and kinetic information follow from interpretation of potential energy surfaces. Following this, the guide is divided into four sections ... 

We will not discuss the actual construction of potential energy surfaces. This monograph deals exclusively with the nuclear motion taking place on a PES and the relation of the various types of cross sections to particular features of the PES. The investigation of molecular dynamics is — in the context of classical mechanics — equivalent to rolling a billiard ball on a multi-dimensional surface. The way in which the forces i fc(Q) determine the route of the billiard ball is the central topic of this monograph. In the following we discuss briefly two illustrative examples which play key roles in the subsequent chapters. 

The concept of potential-energy surface (or just potentials) is of major importance in spectroscopy and the theoretical study of molecular collisions. It is also essential for the understanding of the macroscopic properties of matter (e.g., thermophysical properties and kinetic rate constants) in terms of structural and dynamical parameters (e.g., molecular geometries and collision cross sections). Its role in the interpretation of recent work in plasmas, lasers, and air pollution, directly or otherwise related to the energy crisis, makes it of even greater value. 

The structure of this chapter is as follows. In Section II, after the concept of potential-energy surface and the coordinate systems in which the potential can be represented have been introduced, we describe the most important topographical characteristics of the molecular potential function. The general aspects, which refer to the calculation of the potential energy by ab initio methods, are analyzed in Section III. The need to develop efficient methods for the calculation of the potential function and the corresponding gradient in... 

Before a detailed presentation of the ab initio dynamics simulations, first the fundamental difference between atomic and molecular adsorption on the one hand and dissociative adsorption on the other hand has to be addressed. Then I will briefly discuss the question whether quantum or classical methods are appropriate for the simulation of the adsorption dynamics. This section will be followed by a short introduction into the determination of potential energy surfaces from first principles and their continuous representation by some analytical or numerical interpolation schemes. Then the dissociative adsorption and associative desorption of hydrogen at metal and semiconductor surfaces and the molecular trapping of oxygen on platinum will be discussed in some detail. 

The following discussion makes frequent use of such one-dimensional cross sections through potential energy surfaces. The reaction coordinate Q used as the abscissa often remains unspecified in schematic representations. Caution is required in interpreting such cross sections. What appears as a minimum, barrier, or saddle point in one cross section may look quite different in another one. A typical example is a maximum of a reaction profile, which appears as a minimum in a cross section perpendicular to the reaction coordinate. 

Figure 4.2. Conical intersection of potential energy surfaces with different symmetries with respect to a symmetry element which is present only along the X, axis a) cross section along this x, axis b) three-dimensional representation, with symmetry lowering along the x axis (adapted from Lorquet et al..

Quantum chemical computations of potential energies surface sections along the reaction pathway (PEES) for interaction of typical electrophiles (halogensilanes HaSiX (X = F, Cl, Br, I), trimethylchlorosilane [48,49], acetyl chloride [51]) and nucleophiles (hydrogen halides HX (X = F, Cl, Br, I), water, aliphatic amines, aliphatic alcohols [52], amino acids [53] and substituted phenols [54]) with the silica OH group in a cluster approach using semiempirical AMI, NDDO, MNDO and MNDO/H methods were performed. Representative PEES is shown in Fig.l. 

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