Short-range terms

Here the orientational dependence is expressed in terms of scalar products between the unit intersite vector R and unit bond vectors Zj and Z2, defined to point from the center of molecules 1 and 2, respectively, toward the atoms whose interactions are being calculated (see Table 1). The combination of anisotropy of these two sets of symmetry-related S functions shifts the repulsive wall outward at the end of the molecule and close to the intramolecular bond, allowing closer approach in the plane through the atom perpendicular to the intramolecular bond. Thus this picture of the effective shape of the N2 molecule is compatible with its charge distribution. 

Several strategies are available to enhance the orbital density in regions far from the nuclei (we only discuss the case of Gaussian type orbitals (GTOs)). 

An alternative basis combining moderate size with the accuracy of the 

Finally, entry 10 illustrates the extended-zeta (EZ) basis of strategy 4, a 6s basis in this example. It is seen to give better results than the equally large 

GTO basis Monomer energy (hartree) Most diffuse exponent MBS DBS Ref 5( uhartree) from Eq. (22) 

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The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

Wesolowski and Warshel197 introduced a DFT based approach in which all short-range terms in the effective Hamiltonian (Eq. 4.25) were derived entirely from density functional theory and were involved in the construction of the Fock matrix195 196. In this approach, the H croEnv is expressed using explicit functionals of the electron density ... 

Classical molecular dynamics (MD) implementing predetermined potentials, either empirical or derived from independent electronic structure calculations, has been used extensively to investigate condensed-matter systems. An important aspect in any MD simulation is how to describe or approximate the interatomic interactions. Usually, the potentials that describe these interactions are determined a priori and the full interaction is partitioned into two-, three-, and many-body contributions, long- and short-range terms, etc., for which suitable analytical functional forms are devised. Despite the many successes with classical MD, the requirement to devise fixed potentials results in several serious problems... 

A number of techniques have been employed to model the framework structure of silica and zeolites (Catlow Cormack, 1987). Early attempts at calculating the lattice energy of a silicate assumed only electrostatic interactions. These calculations were of limited use since the short-range interactions had been ignored. The short-range terms are generally modelled in terms of the Buckingham potential,... 

With such an elaborate form for the helium wave function, the matrix elements of the operator L between any two short-range terms in the trial function of the product form (I>,u.(/)t and j may be more conveniently expressed, after integrating by parts, as... 

The electronic coupling is the driving force of EET processes, and accounts for the dependence of the rates on the interchromophoric separation and mutual orientation. In the last decade, there has been a lot of research effort aimed at the development of theoretical methods able to accurately estimate this quantity. [1] We shall start by noting that the electronic coupling can be partitioned into a long-range Coulombic contribution, VCoul, and a short-range term which depends on the orbital overlap between D and A, Vshort ... 

In Forster theory, [76] the short-range term Vshort is neglected, and the Coulomb contribution is approximated as a dipole-dipole interaction between the transition dipole moments of D and A ... 

From the point of view of the Buckingham formula (Equation (2.23)) only the effect of long-range electrostatic and induction interactions crE of the solvent molecule with the reaction field is included in the traditional methods of the (n) group (continuum models). Contrary to that, the supermolecular approach (I) or combined MD/QM methods (III) includes the short-range term cra and the long-range crw and some of the [Pg.132]

The electrostatic part, Wg(ft), can be evaluated with the reaction field model. The short-range term, i/r(Tl), could in principle be derived from the pair interactions between molecules [21-23], This kind of approach, which can be very cumbersome, may be necessary in some cases, e.g. for a thorough analysis of the thermodynamic properties of liquid crystals. However, a lower level of detail can be sufficient to predict orientational order parameters. Very effective approaches have been developed, in the sense that they are capable of providing a good account of the anisotropy of short-range intermolecular interactions, at low computational cost [6,22], These are phenomenological models, essentially in the spirit of the popular Maier-Saupe theory [24], wherein the mean-field potential is parameterized in terms of the anisometry of the molecular surface. They rely on the physical insight that the anisotropy of steric and dispersion interactions reflects the molecular shape. 

We showed previously that a simple model for the ion-hydration interactions, which separates the ion-hydration forces in a long-range term due to the behavior of water as a continuous dielectric (the screened image force) and a short-range term due to the discreetness of the water molecules (SM/SB), can explain almost quantitatively a number of phenomena related to the electrolyte interfaces.6 In this article, we examined the limitations of the model in predicting the distributions of ions near the air/water interface, by comparison with molecular dynamics simulations. It is clear that the real ion-hydration forces are more complicated than the simple model employed here however, the interfacia] phenomena (including specific ionic effects) can be understood, at least qualitatively, in terms of this simple approach. 

Equations 8-34 and 8-35 are more suited for calculations based on simulation data as require a constrained simulation, considering only the vibrational ground state condition, where the energy shift does not include the short-range term which is likely to be not properly sampled in a constrained simulation with a fixed subset of classical coordinates. 

Waals (1/rfj dependence) and short range terms. The latters result from Pauli s principle, which impedes the overlap of closed electronic shells. It may be taken under a Lennard-Jones ( jr variation), or a under a Born Mayer form (By exp(—r /p)). The parameters entering By are fitted, so that the properties of the perfect lattice are reproduced. In the applications to surfaces, they are kept equal to their bulk values, in particular the ionic charges qi. 

In this chapter we examine the ion-water properties of mono and hexahydrates obtained from DFT calculations. Ion-water short-range repulsive and attractive parameters are obtained by fitting the DFT results to a functional form containing Coulombic and short-range terms. Polarization effects in the ion-water interactions are investigated. The systems studied include Na, Li, Mg, Be ", and MD simulations are then used to test the... 

For the ion-water potential Uiw(r) we use a similar combination of electrostatic and short-range terms ... 

In the RISM methods (there are several variants) attention is paid to site-site interactions from which site-site correlation functions are derived. There are no formal limits on the number and quality of such sites. In the RISM-SCF model a difference is introduced between sites belonging to the solute and to the solvent. All the site-site potentials are expressed as a sum of a short-range term and of a Coulombic long-range term ... 

The energy components are most naturally divided into long-range and short-range terms. The long-range terms may be defined in terms of London s perturbation theory leading to electrostatic, induction and... 

Both of the short-range terms may be transformed into the divergence of a local pressure tensor (cf. Appendix II), so that... 

In addition, the energy of interaction between any two ions will contain a contribution from the mirror potential of the second ion u r.) is now given by a short-range term and a term of the form... 

When si is expanded in powers of a, and the polarizabilities of the ions are again neglected, only short-range terms appear. For the first term, in a , we find from Eq. (88) ... 

In an ionic crystal the interaction between ions can be divided into two parts a short-range term and coulomb interactions. The potential given by equation (2) is then... 

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