Statistical approximations, potential energy

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. 

Here, we review an adiabatic approximation for the statistical mechanics of a stiff quantum mechanical system, in which vibrations of the hard coordinates are first treated quantum mechanically, while treating the more slowly evolving soft coordinates and momenta for this purpose as parameters, and in which the constrained free energy obtained by summing over vibrational quantum states is then used as a potential energy in a classical treatment of the soft coordinates and momenta. 

It is not uncommon for a single molecule to have multiple populations. At non-zero temperatures, the population of different conformations will be dictated by Boltzmann statistics. If we make the approximation that we may neglect the continuous character of conformational space and simply work with discrete potential energy minima, we can replace a statistical mechanical probability integral with a discrete sum, and the equilibrium fraction F of any given conformer A at temperature T may be computed as... 

Statistical theories, such as those just described, are currently the only practical approach for many ion-neutral reactions because the fine details of the collision process are unknown all the information concerning the dynamics of collision processes is, in principle, contained in the pertinent potential-energy surfaces. Although a number of theoretical groups are engaged in accurate ab initio calculations of potential surfaces (J. J. Kaufman, M. Krauss, R. N. Porter, H. F. Schaefer, I. Shavitt, A. C. Wahl, and others), this is an expensive and tedious task, and various approximate methods are also being applied. Some of these methods are listed in Table VI, for example, the diatomics-in-molecules method (DIM). 

There exists a whole number of approximate expressions for Vl(r) (see, for example [139]). The simplest, called the Thomas-Fermi potential, follows from the statistical model of an atom. Unfortunately, it leads to results of very low accuracy. More accurate is the Thomas-Fermi-Dirac model, in which an attempt is made to account for the exchange part of the potential energy of an electron in the framework of the free electron gas approach. Various forms of the parametric potential method are fairly widely utilized, particularly for multiply charged ions. Such potentials may look as follows [16] ... 

Have the required statistical mechanical considerations been taken into account In many cases, converting potential energies to free energies can be done approximately reasonably easily, and it is often free energies that are more relevant to experiment. For very large systems with multiple conformers, this aspect can be challenging—and a simpler truncated model may be more informative. 

A wide variety of dynamical approximations have been applied to cluster dynamics and kinetics. Most calculations to date are based on simplified potentials and classical mechanics or statistical methods. In the near future, we can expect to see more work with detailed potential energy surfaces (both analytic and implicitly defined by electronic structure calculations) and progress in sorting out quantum effects and treating them more accurately. 

The approximation in Eq. 6.72 requires that d be large compared to 1 Ik ( diffuse double layer thickness ) but small compared to the particle dimension. See Chaps. 4 and 6 in R. J. Hunter, op. cit.1 It should be noted in passing that V(r) is, speaking strictly, not a potential energy but is instead a potential of mean force, a statistical thermodynamic quantity (hence the dependence of statistical mechanics of the electrical double layer, Adv. Chem. Phys. 56 141 (1984). 

The wave eigenfunctions and energy eigenvalues were obtained by real space electronic state calculations, which were performed by the use of the program code SCAT of the DV-Xx molecular orbital (MO) cluster method with the Hartree-Fock-Slater approximation [8,9]. In the method, the exchange-correlation term Vxc in the one-electron Hamiltonian was expressed in terms of the statistical local potential (1),... 

---