Nonadditive Terms Research Articles

Network thermodynamics: a simulation and modeling method based on the extension of thermodynamic thinking into the realm of highly organized systems

Network thermodynamic analysis is applied to the diffusion of a single nonelectrolyte through a system having the configuration of an epithelial membrane. The example is used to illustrate the methods of nodal, loop, and cut set analysis of the steady state as well as time dependent states of any linear network. The extension to the nonlinear case is also discussed. A discussion of the relation between network thermodynamics and compartmental analysis and other approaches using kinetics is given. Two examples of additional mathematical constraints inherent in the network thermodynamic approach are also given. These are the nonadditive terms in hybrid multiport representations of coupled systems (e.g., Kedem-Katchalsky) and the principle of detailed balance applied to systems containing cycles.

On the importance and problems in the construction of many-body potentials

A fundamental problem exists regarding the rigorous representation of the non-additive part of the many-body potential. This is illustrated by considering the contributions to the crystal binding energies at 0 K, E(0), of the rare gases (Ne, Ar, Kr and Xe) arising from reliable two-body potentials and a variety of non-additive terms including dispersion and first-order exchange energies. The non-additive contribution to the crystal binding energies, as defined by the difference ΔE(0) between the experimental results and those evaluated from the two-body potentials, is relatively large. It is shown that while this difference corresponds (in magnitude) to the contribution arising from the non-additive dispersion energies, there can be a significant cancellation between this contribution to ΔE(0) and that arising from the non-additive first-order three-body exchange energies with a resulting disagreement with experiment. Various possible non-additive interactions are considered in an attempt to resolve this discrepancy but, due to insufficient reliable data, no definite conclusions can be made on this point at the present time. The significance of these results regarding the understanding of the many-body potential and the evaluation of the properties of dense matter are discussed in a general way.

UHF study of multibody effects in small lithium clusters and their relevance to atom-surface interactions

An unrestricted Hartree-Fock (UHF) study of multibody non-additive effects on various lithium clusters, proposed as models of the Li (100) surface, is presented. Non-additive terms are used as a reliable guide to cluster stabilities and their adsorption capacities. Three- and four-body terms are shown to be of equal sign and comparable magnitude, thus suggesting a slow convergence of the multibody expansion of the interaction energy for the lithium surface.

Theoretical study of hydrogen-bonded trimers. Three-body nonadditive interactions between ammonia molecules

The ineraction energy of the system (NH3)3 was calculated for different geometrical arrangements and mutual orientations between the ammonia monomers. The calculations used the SCF LCAO MO method with a near-minimal set of Gaussian functions. The analysis of the three-body terms shows that although nonadditive contributions are comparatively small, they do play a role in the relative stability of the different structures for the trimer. The long-range induction nonadditive terms are evaluated, turning out to be substantially less dominant than for water trimers. The present results are related to existing experimental data on ammonia polymers and solid ammonia crystals.

The application of transformations to non-linear models

Abstract A transmuted model is introduced which allows non-additive error terms and is shown to be an extension of the Box-Cox transformation to intrinsically non-linear equations. Post-transformed models are used to analyze the underlying disturbance term structure with an empirical example presented to illustrate the increased flexibility provided by the transmuted model.

Use of pseudopotentials for the analysis of the non-additivity of short-range intermolecular interactions

The pseudopotential method of Barthelat . is shown to be quite adequate for obtaining short-range intermolecular potentials between closed-shell structures through the study of argon clusters. Not only the all-electron SCF pair interactions for the Ar 2 system are well reproduced but the more rigorous test of obtaining the non-additive few-body terms for small Ar clusters (e.g. Ar 3 and Ar 4) is also met by this approach giving quite good agreement with previous modified effective electorn studies for the same system.

Nonadditive effects in small beryllium clusters

The SCF analysis of few-body contributions to the energy of Be3, Be4, and Be5 clusters is presented. It is shown that nonadditive terms are decisive for the stability of such clusters. An explicit evaluation of the nonadditive five-body interaction is given.

The importance of non-additive terms in the interaction energy of tetrahedral Be 4 clusters

A study of three- and four-body interaction energies in the Be 3 (equilateral triangle) and Be 4 (tetrahedral) clusters at several internuclear distances is presented. The SCF LCAO MO method with a 9s2p gaussian basis set for each Be atom has been used. At this level of approximation it is concluded that the two- and four-body terms give a repulsive contribution to the interaction energy, whereas the three-body terms are attractive. Although the latter are not sufficiently strong to stabilize the Be 3 clusters they do imply a bound state for Be 4 in the region that is close to the nearest neighbour distance in solid beryllium.

Theoretical evaluations of the intermolecular interaction energy of a crystal: application to the analysis of crystal geometry

Simple formulae for the interaction energy between two molecules have been used for writing a program which evaluates the total interaction energy of the molecules in a crystal. These formulae appear as sums of atom-atom and, eventually, atom-bond and bond-bond contributions. The non-additivity of the polarization energy is taken into account, and a rough estimate of the third-order non-additive terms ('triple dipole') is introduced. A suitable modification of the formulae for short interatomic distances allows us to treat hydrogen-bond interactions as well. We present results for the crystals of CH4, CO2, C6H6, and C6H5NO2. The energies calculated for the experimental geometry are in good agreement with experiment. For CO2 and C6H5NO2 minimizations of the computed energy (with respect to unit cell parameters and orientation and position of one molecule in the cell) were performed and it was found that the experimental configuration actually was very close to a minimum. The configurations of neighbour molecules in the crystal are compared with the optimal configuration of a binary complex, and it appears that, for non-hydrogen-bonded molecules, significant differences between these configurations may occur. Finally, for nitrobenzene several local minima seem to exist on the energy hypersurface; the minimum corresponding to the known experimental geometry appears to be the lowest, but only by a small amount.

Charge-overlap effects in the non-additive triple-dipole interaction

A formal partial wave analysis of the third-order energy for the interaction of three non-degenerate S-state atoms is used to identify those non-additive terms which, in the limit of negligible charge overlap between the interacting atoms, reduce to the usual long-range expanded form of the triple-dipole energy. Pseudo-state techniques are used to evaluate these terms, both with and without the inclusion of charge-overlap effects, over a range of interatomic separations and angles for the interaction of three ground-state hydrogen atoms. These results are used as a model to discuss the way in which charge-overlap effects modify the usual long-range result for the triple-dipole energy. In addition to the familiar modification of the radial dependence of the interaction energy, charge-overlap effects are found to give rise to a modification of the angular dependence; the latter effect has no analogue in two-atom interactions. The significance of these results is discussed briefly.

Localized orbitals and short-range molecular interactions. II. Nonadditivity in He3 and He4

Short-range interatomic interactions in singlet He3 and He4 are discussed in terms of a single determinant of doubly occupied nonorthogonal localized orbitals through the formalism described in Paper I [V. Magnasco and G. F. Musso, J. Chem. Phys. 60, 3744 (1974)]. Ab-initio calculations using a minimal set of Slater 1s orbitals are reported for several nuclear configurations. Nonadditive terms can be very well represented as resulting from penetration of pairs of nonorthogonal localized electron groups into the core provided by nuclei and ``unperturbed'' electrons of all other groups. Even in the small overlap region it is essential to include all the penetration effects, and this is easily done by the present formalism.

The accurate numerical determination of two- and three-body dispersion forces

A new computational method for the evaluation of dispersion coefficients between atoms is described. Results are given for the two-body coefficients through R-10 for a representative selection of one and two-electron atoms, the uncoupled Hartree approximation being employed in the latter case, and also the leading non-additive three-body terms are calculated for triplets of hydrogen and helium. Particular refined computations are described for atomic hydrogen, values of 6.499 026 705 and 21.642 464 54 au being obtained for the dipole-dipole and triple-dipole coefficients respectively.

Calculations and estimates of the ground state energy of helium trimers

Variational calculations of the ground state energy of 4He trimers are reported and are compared with the Hall-Post-Stenschke lower bound on the ground state energy. The variational result for the ground state energy for a range of model pair potentials is in the range −0.05 to −0.2 K; the effect of the triplet nonadditive term is estimated to be less than 1% of the contribution of the pair potentials. A study of the variation of the ground state energy of 3 bosons with pair potential coupling constant is also reported for a Lennard-Jones and a Morse potential model. There is no bound spin 3/2 3He trimer; a lower bound is given for the ground state energy of the spin 1/2 trimer and analogy with the boson results is used to argue that the spin 1/2 trimer is probably not bound. This paper contains computational applications of formal results derived elsewhere.

Exp-6 Cell Model Calculations for the Inert Gas Solids. II. Thermodynamic Properties—The Effect of the Triple-Dipole Term

Using the potential parameters developed in Part I of this series of papers we now present a comparison of the theoretically predicted and experimentally measured thermodynamic properties of the four inert gas solids, neon, argon, krypton, and xenon. It is shown that without the inclusion of any three-body term the exp-6 potential gives a good prediction of most available data. The predicted data are sensitive to the inclusion of the three-body term, but its inclusion in the calculations removes the good agreement with experiment. Some discussion is given to the usefulness of the “effective pair additive” potential, for which the exp-6 is obviously a better choice than is the commonly accepted Lennard-Jones (12–6). The calculations further show that the prediction of the solid state properties using a cell model, and including the triple-dipole nonadditive term, can indicate the unacceptability of a pair potential which nevertheless is able to accurately predict second virial gas data. The exp-6 potential, when parameters are characterized from solid state data with the inclusion of the three-body term in the calculation, is only partially successful in the prediction of either second virial or solid state data but at least shows the likelihood of a pair potential being successful in these two greatly disparate tasks.

Reduction Parameters for an Unspecified Intermolecular Potential. II. Core Potential Calculations Including Nonadditivity

The zero-point properties of the inert-gas solids are used to determine the parameter values for a range of m:6 Kihara core pair potential functions. Calculations are made with the inclusion of a long-range non-additivity term in the static lattice potential. It is found that when the experimental properties are satisfied this nonadditivity term becomes zero. The resulting pair potential parameters are then shown to give “best” reduction parameter values for Ar, Kr, and Xe, which allow the second virial and viscosity, two-body, gas-phase properties to be reduced to a pattern of corresponding-states behavior.

Deviations from Pairwise Additivity in Intermolecular Potentials

The various contributions to the interaction energy as defined in the theory of intermolecular forces in the region of small orbital overlap [J. N. Murrell, M. Randic, and D. R. Williams, Proc. Roy. Soc. (London) A284, 566 (1965)] are generalized to many-body systems. Each contribution is examined in a qualitative manner for deviations from pairwise additivity. Numerical calculations of the exchange energy for three helium atoms in triangular and linear arrays are made utilizing the available expansions of Slater-type orbitals in terms of a Gaussian set. The nonadditive terms for these energies are then calculated over a range of interacting distances. It is concluded that for weakly interacting nonpolar molecules nonpairwise additive terms in the intermolecular potential are likely to be small.

An expansion method for calculating atomic properties VII. The correlation energies of the lithium sequence

If the energy of an atomic system is expanded in inverse powers of the nuclear charge Z, the leading term of the correlation energy after degeneracies have been removed is a constant which can be expressed as a weighted sum of electron pair energies and certain non-additive terms. The pair energies may be obtained from direct two-electron variational calculations and the non-additive terms may be evaluated exactly. Calculations are carried out for the lithium 2 S sequence with the result that the non-relativistic eigenvalues are given by E(Z) = − 1.125Z 2 + 1.02280521Z−0.40814899 + O (Z -1 ).