Potential energy description

Figure 6.23 Potential energy description of an electron-transfer reaction. The parabolic curves intersect at the transition state (if)...

A quantitative treatment of the Jahn-Teller effect is more challenging (46). A major issue is that many theoretical models explicitly or implicitly assume the Bom—Oppenheimer approximation which, for octahedral Cu(II) systems in the vibronic coupling regime, cannot be correct (46,51). Hitchman and co-workers solved the vibronic Hamiltonian in order to model the temperature dependence of the molecular structure and the attendant spectroscopic properties, notably EPR spectra (52). Others, including us, take a more simphstic approach (53,54) but, in either case, a similar Mexican hat potential energy description of the principal features of the Jahn-Teller effect in homoleptic Cu(II) complexes emerges (Fig. 13). 

Mackay has considered larger icosahedral assemblies obtained by adding further complete layers of atoms to theA = 55 icosahedron shown in Figure 1 such structures contain N atoms where = lOn — 15 + 11 — 3) = 147, 309, 561, etc., where n is an integer greater than or equal to 4. Baker and Hoare have devised structures containing numbers of atoms intermediate between the A, -values quoted above which are close competitors for the absolute minimum potential-energy description. ... 

Early efforts were essentially static calculations of the interaction energy between a rigid protein and a surface (112,114,121,179-181). Pairwise atomistic potential energy descriptions were used to calculate the van der Waals and electrostatic contributions. In the case of hydrophobic surfaces, solvation energies... 

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

The use of isotopic substitution to detennine stmctures relies on the assumption that different isotopomers have the same stmcture. This is not nearly as reliable for Van der Waals complexes as for chemically bound molecules. In particular, substituting D for H in a hydride complex can often change the amplitudes of bending vibrations substantially under such circumstances, the idea that the complex has a single stmcture is no longer appropriate and it is necessary to think instead of motion on the complete potential energy surface a well defined equilibrium stmcture may still exist, but knowledge of it does not constitute an adequate description of the complex. 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Rather than using transition state theory or trajectory calculations, it is possible to use a statistical description of reactions to compute the rate constant. There are a number of techniques that can be considered variants of the statistical adiabatic channel model (SACM). This is, in essence, the examination of many possible reaction paths, none of which would necessarily be seen in a trajectory calculation. By examining paths that are easier to determine than the trajectory path and giving them statistical weights, the whole potential energy surface is accounted for and the rate constant can be computed. 

In the chapter on reaction rates, it was pointed out that the perfect description of a reaction would be a statistical average of all possible paths rather than just the minimum energy path. Furthermore, femtosecond spectroscopy experiments show that molecules vibrate in many dilferent directions until an energetically accessible reaction path is found. In order to examine these ideas computationally, the entire potential energy surface (PES) or an approximation to it must be computed. A PES is either a table of data or an analytic function, which gives the energy for any location of the nuclei comprising a chemical system. 

Molecular mechanical force fields use the equations of classical mechanics to describe the potential energy surfaces and physical properties of molecules. A molecule is described as a collection of atoms that interact with each other by simple analytical functions. This description is called a force field. One component of a force field is the energy arising from compression and stretching a bond. 

The researchers established that the potential energy surface is dependent on the basis set (the description of individual atomic orbitals). Using an ab initio method (6-3IG ), they found eight Cg stationary points for the conformational potential energy surface, including four minima. They also found four minima of Cg symmetry. Both the AMI and PM3 semi-empirical methods found three minima. Only one of these minima corresponded to the 6-3IG conformational potential energy surface. 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

The internal (potential) energy is a direct sum of energies, which is normally given as a sum over pairwise interactions (i.e. van der Waals and electrostatic contributions in a force field description). 

Interest at Brookhaven was stirred by the contrast between the excellent agreement between theory and experiment for the D2+ + D2 reaction and the rather poor description provided for the H2-He and H2-Ne systems. The H2-He system is particularly interesting because of the relatively few particles involved in the reaction and its potential for accurate theoretical treatment. The reactions of H2 + or HD+ with He will be among the first to be treated in terms of a theoretically computed potential energy surface comparison of experiment and theory in this system is therefore of prime importance. 

Potential energy hypersurfaces form the basis for the complete description of a reacting chemical system, if they are throughly researched (see also part 2.2). Due to the fact that when the potential energy surface is known and therefore the geometrical and electronical structure of the educts, activated complexes, reactive intermediates, if available, as well as the products, are also known, the characterizations described in parts 3.1 and 3.2 can be carried out in theory. 

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