Born-Oppenheimer level models

The study of McMillan-Mayer level models, in which the solvent coordinates have been averaged over so that only solvent-mediated ion-ion forces need be treated, is relatively well developed. However the real forces at this level are even more poorly known than the forces at the Born-Oppenheim level referred to above. It is found that McMillan-Mayer level models can be brought into good agreement with solution thermodynamic data. 

In this section we consider the possibility of applying the ion association concept to the description of the properties of electrolyte solutions in the ion-molecular or Born-Oppenheimer level approach. The simplest ion-molecular model for electrolyte solution can be represented by the mixture of charged hard spheres and hard spheres with embedded dipoles, the so-called ion-dipolar model. For simplification we consider that ions and solvent molecules are characterized by diameters R and Rs, correspondingly. The model is given by the pair potentials,... 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

The most extensive potential obtained so far with experimental confirmation is that of Le Roy and Van Kranendonk for the Hj — rare gas complexes 134). These systems have been found to be very amenable to an adiabatic model in which there is an effective X—Hj potential for each vibrational-rotational state of (c.f. the Born Oppenheimer approximation of a vibrational potential for each electronic state). The situation for Ar—Hj is shown in Fig. 14, and it appears that although the levels with = 1) are in the dissociation continuum they nevertheless are quasi bound and give spectroscopically sharp lines. 

The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born-Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as ... 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

An essential feature of MM methods for the treatment of biomole-cular systems is their computational efficiency. The inclusion of polarizability into the model increases the computational demand due to the addition of dipoles or additional charges centers and, in the context of MD simulations, the requirement for shorter integration time steps. In addition, for every energy or force evaluation it is necessary to solve for all the polarizable degrees of freedom in a self-consistent manner. Traditionally, this is performed via a self-consistent field (SCF) calculation based on the Born-Oppenheimer approximation in which the induced polarization is solved iteratively until a satisfactory level of convergence is achieved. With the Drude model, this implies that the Drude... 

In general, molecular dynamics simulations, in the framework of the Born-Oppenheimer or Car-Parrinello approximation, are of great importance for the understanding of materials dedicated to proton transport. Especially for materials, where interactions are dominated by covalent or hydrogen bonds, ab initio molecular dynamics provide a proper description. The results obtained by such methods give details at the atomic level, which are not accessible by experimental investigations. Nevertheless, the choice of the model system has to be done in a very careful way in order to consider the manifold possibilities of structures and mechanisms. 

The ionization potentials and electron affinities of the H, C, N, O and F atoms have been computed by means of state-of-the-art electronic structure methods. The conventional coupled-cluster calculations were performed up to the connected pentuple excitation level. For the purpose of the basis set truncation correction the implementation of the CCSD(F12) model in Turbomole was applied. Final results were supplemented with relativistic and diagonal Born-Oppenheimer corrections. Estimated values of the IPs and EAs are in good agreement with the experimental values and the deviations do not exceed 0.7 meV, in the cases of H, C and N atoms and the IP of O atom. The results obtained for fluorine differ by ca. 1 and 5 meV from the experiment, respectively for the IP and EA. The EA of oxygen is plagued with discrepancy that amounts to ca. 4 meV. 

Hamiltonian models are classified according to then-level of approximation. The features of Schroedinger (S), Born-Oppenheimer (BO), and McMillan-Mayer (MM) level Hamiltonian models are exemplified in Table I by a solution of NaCl in H2O. The majority of investigations on electrolyte solutions are carried out at the MM level. BO-Level calculations are a precious tool for Monte Carlo and molecular dynamics simulations as well as for integral equation approaches. However, their importance is widely limited to stractural investigations. They, as well as the S-level models, have not yet obtained importance in electrochemical engineering. S-Level quantum-mechanical calculations mainly follow the Car-Parinello ab initio molecular dynamics method. 

A more fundamental approach is to attempt to model electrolyte solutions using statistical mechanical methods, of which there are two kinds of models (reviewed extensively elsewhere ° ) Born-Oppenheimer (BO) level models in which the solvent species as well as the ionic species appear explicitly in the model for the solution and McMillan-Mayer (MM) level models in which the solvent species degrees of freedom are integrated out yielding a continuum solvent approximation. Thus, for a BO level model, in addition to the interionic pair potentials one must specify the ion-solvent and solvent-solvent interactions for all of the ionic and solvent species. In this case, the interionic potentials do not contain the solvent dielectric constant in contrast to the MM-level models. Kusalik and Patey carefully discuss the distinction between these two approaches. 

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