Pairwise potential

P(r,i) is the pairwise potential, which, depending upon the model, can be considered tc include electrostatic and repulsive contributions. The second term is a function of th< electron density, and varies with the square root, in keeping with the second-momen approximation. The electron density for an afom includes contributions from the neigh bouring atoms as follows ... 

If the gas particles interact through a pairwise potential, then the contribution to the viriai from the intermolecular forces can be derived as follows. Consider two atoms i and j separated by a distcmce r. ... 

The pressme is related to the product of positions and forces (for pairwise potentials). Here the first part is for an ideal gas. 

Molecular dynamics calculations have been performed (35-38). One ab initio calculation (39) is particularly interesting because it avoids the use of pairwise potential energy functions and effectively includes many-body interactions. It was concluded that the structure of the first hydration shell is nearly tetrahedral but is very much influenced by its own solvation. 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

When the interaction is expressed in terms of pairwise atom-atom potential functions, all three components of the van der Waals interactions are grouped together because of their common r 6 distance dependence. A repulsive term is added, while Coulombic interactions may be accounted for separately. In the expression due to Lennard Jones, the repulsion has an r 12 dependence, to give the pairwise potential function... 

Such a decomposition has physical meanings in surface science. For example, the potential between an atom and a crystalline surface can be calculated as the sum of pairwise potentials the tunneling current from a tip state to a crystalline surface can be calculated as the sum of the tunneling currents to the... 

Cabrera, N., and Goodman, F. O. (1972). Summation of pairwise potentials in gas-surface interaction calculations. J. Chem. Phys. 56, 4899-4902. 

In the pairwise-potential approximation and the shell model, the lattice energy can be written as... 

Calculations have been performed on atom clusters by Hoare and Pal (55) and Burton (39) to determine their geometrical and thermodynamic properties using empirical potential functions. In this technique the pairwise potentials V(Rjj) between atoms, a distance Ry apart, are assumed additive to give a potential energy... 

For the PMF (m>0, = q3m), becomes simply Z, a configurational partition function of a system of N particles. In a special case, if m = 2, then q3m for the pairwise potential V may be denoted as R - the distance between two chosen particles. Using this for calculating the intergral of Eq. (3-12) denoted in parentheses, leads to the following form ... 

An approximation to the packing energy can be given by the sum of terms due to all the interactions among atoms belonging to adjacent molecules. This approach is the so-called atom-atom potential method [6] where pairwise potentials are empirically determined and the molecular conformation is kept fixed. This method enables is to estimate the packing energy and to justify crystal properties, such as the enthalpy of formation. 

Finally, substitute the numbers into Eq. 10.53 to obtain 75.2 GPa (1 GPa = 10 Pa). This result is noticeably larger than the experimental value (percent error = 88.2 percent). The difference implies that a pairwise potential is not entirely adequate for describing NaCI, even though the Cauchy relation holds. 

The various parameters needed in these equations come from other work and are given in Table 5.12, taken from Woodcock and Singer. The pairwise potential for potassium chloride, first calculated by Tosi and Fumi in 1964, is reproduced in Fig. 5.12. Typical computational data for an MC calculation are given in Table 5.13. The software of the day generated only 80,000 steps per hour. 

A pairwise potential widely used in both Monte Carlo and molecular dynamics computations is given as... 

Recently, detailed molecular pictures of the interfacial structure on the time and distance scales of the ion-crossing event, as well as of ion transfer dynamics, have been provided by Benjamin s molecular dynamics computer simulations [71, 75, 128, 136]. The system studied [71, 75, 136] included 343 water molecules and 108 1,2-dichloroethane molecules, which were separately equilibrated in two liquid slabs, and then brought into contact to form a box about 4 nm long and of cross-section 2.17 nmx2.17 nm. In a previous study [128], the dynamics of ion transfer were studied in a system including 256 polar and 256 nonpolar diatomic molecules. Solvent-solvent and ion-solvent interactions were described with standard potential functions, comprising coulombic and Lennard-Jones 6-12 pairwise potentials for electrostatic and nonbonded interactions, respectively. While in the first study [128] the intramolecular bond vibration of both polar and nonpolar solvent molecules was modeled as a harmonic oscillator, the next studies [71,75,136] used a more advanced model [137] for water and a four-atom model, with a united atom for each of two... 

Moliere central potential — an analytical expression to approximate the repulsive potential between two atoms at short range. The universal pairwise potential requires only two parameters — the atomic numbers of the interacting atoms. 

The simplest model uses atom-atom pairwise potentials, as was done for clusters of molecules with rare gas atoms [45] and with other molecules [46]. The Haas group suggested such models for the anthracene- and perylene ammonia adducts, in which an electrostatic interaction [47] was added to account for the charge distribution in the molecules [40], and also as for the larger donors (dimethylaniline). The potential is written as... 

---