Pair-wise additive

The initial cube measures 17.9 A along one edge, and contains 455 atoms. Accepting the approximation of pair-wise additivity of interactions, we are faced with the requirement for the evaluation of 455x454/2 - 103,285 pair interactions in order to once estimate the total energy of the system. Fortunately, only roughly 20% of these pairs interact significantly. 

In view of this and in line with the DFT-D approach described by Grimme [118], we have added an atom-atom pair-wise additive potential of the form Csemi-empirical energy [19-21] in order to account for dispersion effects [43], Thus the dispersion corrected semi-empirical energy ( Pm3-d) is now given by ... 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

An introduction to the modeling methods can be found in refs. [22,231. The classical MD simulations reported here were performed with the modified AMBER software/241 in which the potential energy consists of harmonic deformations of bond and angles, dihedral energies, plus non-bonded interactions represented by a sum of pair wise additive coulombic and van der Waals contributions ... 

Bader, Novaro, and Beltran-Lopez135 have calculated the potential surface for various geometries of three and four helium atoms near to the Hartree-Fock limit to determine the deviation of the Hartree-Fock energies of interaction from pair-wise additivity. 

Void [11] has established Hqn(I) for the attractive van dcr Waals interactions between two spherical particles of radius R covered with a shell of a different material of thickness 5, by assuming pair-wise additivity of molecular interactions. 

Assuming pair-wise additivity of the potential, the Helmholtz free energy F of a system of N particles of diameter d in a volume V can be expressed, to second order, as [21] ... 

To compute the dependence of the free energy on the thickness h of the film, pair-wise additivity is assumed and the 6-12 Lenard-Jones potential... 

The thermodynamic variables of liquid systems, such as energy, pressure, etc., can be expressed in terms of the radial distribution function if one makes the assumption that the total potential energy is pair-wise additive, i.e.,... 

Using the pair-wise additivity of U(R), it is possible to integrate Eq. (18) over the equilibrium configurations of (N — 2) particles. If one then uses the definition of the radial distribution function, an expression for E in terms of g( r) and u(r) is obtained, and it is referred to as the energy equation... 

The expressions for and E contain both pair-wise additive and three-body... 

Table 10. Intermolecular energies in unpolar systems using pair-wise additive potentials...

Specific polarization effects, beyond those modelled by a continuum dielectric model and the movement of certain atoms, are neglected in MIF calculations. Many-body effects are also neglected by use of a pair-wise additive energy function. Polarizable force fields are, however, becoming more common in the molecular mechanics force fields used for molecular dynamics simulations, and MIFs could be developed to account for polarizability via changes in charge magnitude or the induction of dipoles upon movement of the probe. 

In the vast majority of MD applications a further simplification is made by using effective pair-wise additive potentials for atomic interactions. In simulations which contain flexible molecules, it is common practice to add terms which represent chemical bonds, bond angles, improper torsions and dihedrals. Interactions between atoms of molecules are represented by effective pair-wise additive potentials. This empirical approach splits the total potential energy of the system into a bonded (inter-molecule) and non-bonded (intra-molecular) part. 

Using the MCY potential at constant pressure and temperature the system became structurally unstable as described in ref. [74], even though the first nearest neighbour distance was preserved at about 2.9 A. A considerable distribution was found for the local tetrahedral symmetry. This behaviour is reasonable since a simple 6-12 potential has no preference for a tetrahedral ly bonded structure. However, with a fixed cell volume the simulation became stable. Nearest neighbour molecules move within the energy minimum created by the pair-potential and the pair-wise additive electrostatic forces. At low temperatures, these molecules only sample the parabolic part of the potential... 

The influence of various substituents on 5 Sn for series of tetra-coordinate tin(I V) compounds SnX4 n Yn (n = 0 ) can be graphically depicted as U-shaped curves. These substituent effects are not linear, but approximately pair-wise additive. Some representative Sn NMR data are given in Table 2.1.3 (tin hydrides). Table 2.1.4 (tetraorgano. tin). Table 2.1.5 (tin halides). Table 2.1.6 (tin chalcogenides), and Table 2.1.7 (compounds with Sn—N, Sn—P, Sn—As, Sn—Sb, and Sn—Bi bonds). 

Henceforth, we consider a Lennard-Jones(12,6) (LJ) fluid between two plane parallel solid substrates. The ba.sic. setup of our model is schematically depicted in Fig. 4.3. We as.sume the fluid-substrate interaction to be a pair-wise additive sum of L.1 potentials. As the reference system we take a hard-sphere fluid (diameter hard-sphere substrates (diameter (Tg). Moreover, we smear the hard spheres of the surface layer of each... 

Here we consider a lattice model of a simple pure confined fluid, that is, a fluid composed of molecules having only translational degrees of freedom. The positions of theses molecules are restricted to M n Uyn sites of a simple cubic lattice of lattice constant f. Each site on the lattice can be occupied by one molecule at most which accounts for the infinitely repulsive hard core of each molecule. In addition to repulsion, pair-wise additive attractive interactions between the molecules exist. They are modeled according to square-well potentials where ff is the depth of the attractive well whose width equals t. 

The substrates are in registry meaning that slabs of the same type are exf tly opposite each other. Substrate atoms are assumed to be of the same diameter (substrate surfaces are taken to be (100) planes] having lattice constant C, which is taken to be the same for both species. Thus, substrate species are distinguished only by the strength of their interaction with film molecules. We assume the total potential energy to be a sum of pair-wise additive LJ (12,6) potentials, all of which have the form given in Eq. (5.24). For the interaction between a... 

In summary, induced interactions are weak and more short-ranged. Induced interactions are not pair-wise additive, and this fact adds complications to the calculation of molecular interactions. For example a molecule midway between two similar ions does not experience any electrical field, and the interaction potential from the ion-induced dipole term is zero. If we use Equation (55) and assume pair-wise additivity in such a case, we will get a large and quite erroneous result for the interaction energy. 

The interactions may be pair-wise additive or may not be additive due to the presence of other identical molecules of 1 and 2 nearby. 

To make further progress the form of 4> must be specified. Generally the potential energy will contain terms involving the coordinates of pairs, triplets, quadruplets, etc., of molecules. A few calculations of the equation of state of a fluid with such a general form for have been made. However, the common practice is to assume pair-wise additivity ... 

In Chapter 3 we discussed the calculation of the interaction between two particles by summing the intermolecular potential energies between all the molecules in one particle with all those in the other. This is often called the Hamaker or microscopic theory and assumes pair-wise additivity of the intermolecular potential energies. This implies that the potential associated with the interactions of a given molecule is independent of the presence of... 

The foregoing derivation of the transformation formula for converting potential energies into osmotic pressures was based upon the assumption of pair-wise additivity. Evans and Napper (1978) showed that if the barrier layer thickness was less than ca 15% of the particle radius, many-body interactions are distinctly unlikely and the assumption of pair-wise additivity introduces negligible error. 

Physically, measures the probability of finding a particle at a distance ri2 from a particle held at the origin. Note that for a pair-wise additive potential in a canonical ensemble... 

A next question could be is the interaction energy pair-wise additive That is. 

---