One-dimensional potential energy

For eaeh of the one-dimensional potential energy graphs shown below, determine ... 

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... 

The reaction of hydrogen gas with a metal is called the absorption process and can be described in terms of a simplified one-dimensional potential energy curve (onedimensional Lennard-Jones potential) [30] (Figure 5.22). 

Figure 7.5 A (reasonably accurate) one-dimensional potential energy diagram for 238U indicating the energy and calculated distances for a decay into 234Th. Fermi energy Rs30 MeV, Coulomb barrier -28 MeV at 9.3 fm, Qa 4.2 MeV, distance of closest approach 62 fm. (Figure also appears in color figure section.)...

Fig. 1. STM images resolving (a) the hexagonal atomic structure of the close-packed fcc(lll) surface and (b) the anisotropic fcc(llO) surface of Ag. The surface unit cells and high symmetry directions are marked, (c) Schematic one-dimensional potential energy surface experienced by a simple individual adsorbate along a high-symmetry surface direction (Em migration energy barrier ICf, bonding energy a surface lattice constant).

Figure 9 Schematic presentation of the one-dimensional potential energy surface for oxygen adsorption on Pt(l 1 1) as derived from experiment (after [87]).

The situation again may be depicted in the form of a potential energy diagram, however we have to include the existence of different surface sites [84] Figure 10(a) [7] shows a one-dimensional potential energy diagram where the spatial coordinate extends parallel to the surface It has been assumed that every surface site provides identical binding conditions All identical... 

Figure 10 One-dimensional potential energy parallel to the surface (a) empty surface with a single particle bound with adsorption energy FjiK (b) superposition of the potential energy in (a) with a pairwise interaction potential of partitles on the surface (/>, ) which nia be either attractive or repulsive...

An external strain appHed to the vibronic Hamiltonian can be shown to be equivalent to a low symmetry distortion [7]. Within the approximation of the angular potential given by Eq. (3), the tetragonal (Sg) and orthorhombic (S ) components of the strain modify the one-dimensional potential energy surface according to... 

This situation is illustrated schematically for a one-dimensional potential-energy surface in Figure B.5, from which it can be inferred that... 

Many previous studies using ab initio, - > >- 3.58 molecular simulations - > >24.2939 and experimental methods,2°-2i.6o-63 jjave reported first shell structures and ion-water binding energies. In the next sections we present DFT results corresponding to the one-dimensional potential energy curve for the interaction ion/water, when the ion interacts with one and with six water molecules. Li, Na Be", Mg, Al" and were considered, and the results compared to previous calculations and experiments. 

Figure 2. A typical example of a one-dimensional potential energy curve where there are the two local points and the saddle point.

Fig. 7.21. One-dimensional potential energy profile for a diffusing atom. This schematic illustrates the type of energy that would be encountered by an adatom wandering on a rigid substrate.

Figure 1. Various one-dimensional potential energy curves for the proton motion in AH- -B...

The one-dimensional potential energy curve describes precisely the electronic term of a system only in the simple case of diatomic molecules. With A H bonds, where A is part of a multiatom molecule, the diatomic approximation is valid. But in the general case the potential energy as a function of the A H bond length, F(r), should be expressed using a polynomial expansion... 

Fig. 8. One-dimensional potential energy diagram for the adsorption and dissociation of oxygen on an Ag catalyst after Dean and Bowker, 1989).

It can be the case that both adsorption channels are important for a particular system. Examples of this are given here for O2 adsorption on Ag and Cu and for N2 dissociation on Fe. In these cases we can generalise and say that the precursor mediated route tends to dominate at low substrate and gas temperatures, while direct activated adsorption dominates at high gas temperatures. Furthermore, in all these cases, molecular chemisorbed states of adsorption can exist which complicate the pathway of adsorption. A one dimensional potential energy profile is shown in fig. 8 for the case of O2 adsorption on Ag taken from the work of Dean and Bowker (1988/89, 1989) and of Campbell (1985), although this is likely to be a general representation for this type of adsorption system with other adsorbate/metal combinations. 

Fig. 6. One-dimensional potential energy diagram depicting kinetic and thermodynamic information of 4NBT hydrogenation to the aniline.

The discussion up till now has been concerned with two colliding atomic particles, the motion of which can be described on one-dimensional potential energy curves. If however, one or both of the colliding particles are molecules, the collision process takes place on multidimensional potential surfaces. If N > 2 atoms are involved, we have s 3N — 6 independent relative... 

Figure 25.1 One-dimensional potential energy diagram illustrating the changes in the potential energy of a hydrogen molecule which approaches a metal (the location of the surface is indicated by the hatched area). The following processes may occur ...

Figure 6.16. One-dimensional potential-energy diagrams showing the possible transition from molecular physisorption to dissociative chemisorption.

Let us consider the hypothetical one-dimensional potential energy profile depicted in Figure 1. It features two equivalent minima (reactants and products) separated by an energy barrier (a transition state). This is a very well known problem in quantum physics as a huge number of systems falls in the category of the so called sjmimetric double well profile. 

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