Point-charge simulation

The observation that point-charge simulation reproduces the exact same result as a Heitler-London calculation, but only for first-order bonds, confirms the previous conclusion that covalent interactions are mediated by the sharing of a maximum two electrons per bond, allowed by the exclusion principle. In view of this stipulation the conventional assumption, that several pairs of electrons contribute to the formation of high-order bonds, should therefore... 

In point-charge simulation this electronic rearrangement is of no immediate consequence except for the assumption of a reduced interatomic distance, which is the parameter needed to calculate increased dissociation energies. However, in Heitler-London calculation it is necessary to compensate for the modified valence density, as was done for heteronuclear interactions. The closer approach between the nuclei, and the consequent increase in calculated dissociation energy, is assumed to result from screening of the nuclear repulsion by the excess valence density. Computationally this assumption is convenient and effective. 

In this instance the point-charge simulation is precise, but unlike the covalent case, the primary interaction is not shielded against secondary interference. Each ion is surrounded by six nearest neighbours of opposite charge, at a distance d = a/2, equivalent to an electrostatic coulomb interaction,... 

The simplest model of a covalent bond is based on an electrostatic point-charge simulation of overlapping spherical valence-electron charge clouds that surround monopositive atomic cores. For a homonuclear pair of atoms with radius r and internuclear distance d, the dissociation energy D is calculated from... 

Heitler-London simulation of general covalence depends on a set of characteristic atomic radii, assumed to describe a single electron in the valence state. Such radii were obtained empirically [17], in the first instance, by point-charge simulation of covalent interaction [1]. A more satisfactory derivation of atomic radii was discovered in the simulated compression of atoms in Hartree-Fock calculations, resulting in ionization at a characteristic compression, closely related to the empirical radii [18]. 

With reference to the point-charge simulation of covalent interaction, we note that the common volume between two overlapping spheres of radius r, with centers at a distance d apart, is calculated as... 

Corresponding values obtained by point-charge simulation [8] are shown in parentheses... 

All of the values are in the expected range (see other force fields, such as MM3, Amber and Momec, e.g. in [9,10,13,18,29,45]), but by no means are they refined and do not define an accurate force field. The major difference between the number-theory parameters and the alkane force field derived by point-charge simulation occurs in the strain-free C-C bond length with = 1.53 A compared to 1.51 A. The optimized structural parameters of a series of aliphatic hydrocarbons, shown in Table 2, although less accurate than with a properly optimized force field, reflect the... 

Fig. 9.2 EOM CCSD calculated shifts (in cm ) of Br2 inside point charge simulated and 5 6 clathrate cages. 0 and (in degrees) correspond to the orientation of the dihalogen with respect to the main symmetry axis of the cage...

Specific solute-solvent interactions involving the first solvation shell only can be treated in detail by discrete solvent models. The various approaches like point charge models, siipennoleciilar calculations, quantum theories of reactions in solution, and their implementations in Monte Carlo methods and molecular dynamics simulations like the Car-Parrinello method are discussed elsewhere in this encyclopedia. Here only some points will be briefly mentioned that seem of relevance for later sections. 

Alper H E and R M Levy 1989. Computer Simulations of the Dielectric Properties of Water - Studies of the Simple Point-Charge and Transferable Intermolecular Potential Models. Journal of Chemical Physics 91 1242-1251. 

It is sometimes desirable to include the effect of the rest of the system, outside of the QM and MM regions. One way to do this is using periodic boundary conditions, as is done in liquid-state simulations. Some researchers have defined a potential that is intended to reproduce the effect of the bulk solvent. This solvent potential may be defined just for this type of calculation, or it may be a continuum solvation model as described in the next chapter. For solids, a set of point charges, called a Madelung potential, is often used. 

In this model of electrostatic interactions, two atoms (i and j) have point charges q and qj. The magnitude of the electrostatic energy (Veel) varies inversely with the distance between the atoms, Ry. The effective dielectric constant is 8. For in vacuo simulations or simulations with explicit water molecules, the denominator equals eRij. In some force fields, a distance-dependent dielectric, where the denominator is eRy Rjj, represents solvent implicitly. 

Another difference between the force fields is the calculation of electrostatic interactions. AMBER, BIO+, and OPLS use point charges to model electrostatic interactions. MM+ calculates electrostatic interactions using bond dipoles. The bond dipole method may not adequately simulate very polar or charged systems. 

It thus appears that point charges provide a much better simulation of electrochemical promotion than uniform electric fields. 

Later calculations showed that the defect binding energies were invariant to the values chosen for the point charges. As those calculated for the fully-ionic system my be directly compared to those obtained using classical simulation, geometry optimizations were carried out using the fully-ionic point-ions. 

When the MM subsystem is being optimized, or a molecular dynamics simulation is being carried out on the MM subsystem, the QM/MM electrostatic interactions are approximated with fixed point charges on the QM atoms which are fitted to reproduce the electrostatic potential (ESP) of the QM subsystem [37],... 

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

In addition to the described above methods, there are computational QM-MM (quantum mechanics-classic mechanics) methods in progress of development. They allow prediction and understanding of solvatochromism and fluorescence characteristics of dyes that are situated in various molecular structures changing electrical properties on nanoscale. Their electronic transitions and according microscopic structures are calculated using QM coupled to the point charges with Coulombic potentials. It is very important that in typical QM-MM simulations, no dielectric constant is involved Orientational dielectric effects come naturally from reorientation and translation of the elements of the system on the pathway of attaining the equilibrium. Dynamics of such complex systems as proteins embedded in natural environment may be revealed with femtosecond time resolution. In more detail, this topic is analyzed in this volume [76]. 

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