Potential energy curve surface

Quite frequently, potential energy curves (surfaces) cross and these crossings have very important consequences. One of the simplest examples for potential curve interaction is predissociation of a diatomic molecule. The key points are ... 

Figure Bl.26.21. Potential energy curves for an electron near a metal surface. Image potential curve no applied field. Total potential curve applied external field = -E. ...

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

The potential energy curve in Figure 6.4 is a two-dimensional plot, one dimension for the potential energy V and a second for the vibrational coordinate r. For a polyatomic molecule, with 3N — 6 (non-linear) or 3iV — 5 (linear) normal vibrations, it requires a [(3N — 6) - - 1]-or [(3A 5) -F 1]-dimensional surface to illustrate the variation of V with all the normal coordinates. Such a surface is known as a hypersurface and clearly cannot be illustrated in diagrammatic form. What we can do is take a section of the surface in two dimensions, corresponding to V and each of the normal coordinates in turn, thereby producing a potential energy curve for each normal coordinate. 

The nuclei move under the influence of a potential that is generated by the electrons, so once again we meet the concept of a potential energy curve (or surface, for more complicated systems). 

Potential energy curves for a reaction proceeding homogenously (full curve) or on a surface (dotted line). 

If a piece of metal, such as silver, is dipping into a solvent, and a positive atomic core is taken from the surface into the solvent, the ion is again surrounded by its electrostatic field but free energy has been lost by the dielectric, and a relatively small amount of work has had to be done. The corresponding potential-energy curve (Fig. 96) is therefore much less steep and has a much shallower minimum than that of Fig. 9a. For large distances d from a plane metal surface this curve is a plot of — c2/4td where t is the dielectric constant of the medium at the temperature considered The curve represents the work done in an isothermal removal of the positive core. 

Figure 6.3. Schematic potential energy curve describing the interactions between colloidal particles. The overall potential is a sum of an electrostatic repulsive term which arises due to any charged groups on the surface of the particle and the attractive van der Waals term.

Figure 6.34 shows potential energy curves for a hypothetical diatomic molecule X2, which approaches a surface, coming from the right-hand side of the diagram. First... 

Experimentally derived potential energy curves are shown in Figures 10 and 11. (Note that only one particle size is illustrated, namely, 10 pm.) The shape of these potential energy curves as a function of ionic strength, solution pH, particle and surface composition, etc. may be used to explain the effect of some of these variables on particle capture and... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

The presence or absence of a homoaromatic interaction is often based solely on the distance between the non-bonded atoms. Distances greatly over 2.0 A are thought to lead to a p-p overlap that is too small to make any significant contribution. This simplistic approach is not necessarily reliable as shown by Cremer et al. (1991). Their calculations on the homotropylium cation [12] indicate a double-minimum potential energy surface with respect to variations of the C(l)-C(7) distance at the Hartree-Fock level of theory. At the MP4(SDQ) level of theory, only a single-minimum curve was found with the minimum at 2.03 A. The calculated potential energy curves are quite flat in this region. 

Fig. 6-3S. Potential energy curves for water adsorption on metal surface in the states of molecules and hydrozjd radicals c = energy r = reaction coordinate solid curve = adsorption as water molecules and as partially dissociated hydroxj4 and hydrogen radicals broken curve = adsorption of completely dissociated oxygen and hydrogen radicals.

Figure 9-1 illustrates the energy barrier to the transfer of metallic ions across the electrode interface these energy barriers are represented by two potential energy curves, and their intersection, for surface metal ions in the metallic bond and for hydrated metal ions in aqueous solution. As described in Chaps. 3 and 4, the energy level (the real potential, a. ) of interfadal metal ions in the metallic bonding state depends upon the electrode potential whereas, the energy level (the real potential, of hydrated metal ions is independent of the electrode potential. 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

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