Potential energy pairwise intermolecular

P17.15 Eqn. 17.42 relates the second virial coefficient to the pairwise intermolecular potential energy ... 

T term energy (and To), temperature u two-particle (pairwise) intermolecular potential energy U potential energy... 

Classical statistical mechanics views fluids (i.e., gases and liquids) as a collection of N mutually interacting molecules confined to a volume V at a temperature T and specifies the system by a total intermolecular potential energy U, U (xi,X2,..., xn) = U( 1,. , N), where Xi stands for a set of generalized coordinates of molecule i. Not only for convenience and simplicity, but as an utmost necessity if a tractable theory is to be ultimately applied, the assumption of pairwise additivity is made at this stage, and U is simplified to... 

We proceed then with a second assumption, that of pairwise additivity, according to which the potential energy of the fluid can be approximated by the sum of interactions between all molecular pairs. This assumption allows us to determine the thermodynamic properties of a real fluid in terms of the intermolecular potential energies and the radial distribution junction, which describes the position of the molecules in the space they occupy. 

The assumption of additivity that underlies many empirical intermolecular potentials can be stated more formally as follows. Suppose that A, B, C,... represent chosen molecules in a given spatial configuration. The potential-energy function V(A, B, C,...) (relative to isolated molecules61) will be said to be pairwise additive (and denoted Vpw) if... 

The second additivity problem concerns the question whether the interaction potential in Van der Waals trimers or multimers (or molecular solids or liquids) is a sum of pairwise intermolecular (A—B) potentials. This question can be considered for each component of the interaction energy. The (first order) electro-... 

Thus, we see that the general theme formulated above is also applicable to intermolecular interactions, namely that a specific second derivative of the energy extracts from the total potential energy the derivative of the pairwise interaction between the two atoms, i.e., the derivative of the isolated interatomic interaction. H - is of course a function of the coordinates through the distance between the oxygens and if we plot it against the intermolecular... 

This is the final form of the pressure equation for a system of spherical particles obeyingH he assumption of pairwise additivity for the total potential energy. Note that the first term is the ideal gas pressure (i.e., starting with the ideal gas partition function with 1/ = 0). The second term carries the effect of the intermolecular forces on the pressure. Note that, in general, g(R) is a function of density so that this term is not the second-order term in the density expansion of the pressure. There is a... 

The pairwise potential energy V(r) between two non-bonded intermolecular pairs (strictly speaking atoms) as a function of intermolecular separation, r, can be obtained by summing repulsive and attractive potentials which are often approximated by the 6-12 Lennard-Jones potential ... 

Consider the same N molecules at rest in a perfectly ordered infinite crystalline array with one molecule in the asymmetric unit. All molecules are equal, and surface or truncation effects are neglected, so the following discussion refers to a bulk crystal. Assuming for the moment that the intermolecular potential is pairwise additive, e.g. in the atom-atom potential approximation, the packing potential energy, PPE, or the interaction energy of any reference molecule m with the surrounding molecules n, is a sum of molecule-molecule terms, each of which is in turn a sum of atom-atom terms ... 

When intermolecular interaction energies are not pairwise additive, the total potential packing energy is not the sum of terms over separate molecule-molecule interaetions. For example, the polarization energy at molecule m is the result of the action of an electric field due to the simultaneous influence of all surrounding molecules. Consider a molecule A surrounded by two neighbors B and C. The electric field at molecule A,... 

The potential energy controls the outcome of the Monte Carlo simulations by equations (6) and (7). It is normally computed from the pairwise sum of the intermolecular interaction energies between all monomers, plus the sum of the monomers intramolecular energies (equation 8). 

The intermolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44. 50. 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

Atom-atom potential method The packing energy is expressed by summing all intermolecular interactions, the pairwise potentials are determined especially, and, again, the molecular structure is kept unchanged. 

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