Many-neighbor interactions

In actual polymer calculations it is necessary to decide how many-neighbor interactions must be taken into account to obtain satisfactory results. This is by no means a trivial question. Experience gained in numerous calculations (see below) indicates that the number of neighbors to be taken into account explicitly is smaller if one performs a band-structure calculation, than if one is interested in the total energy per unit cell. To imderstand the reason for this we now examine the latter quantity for a linear chain obtained by straightforward generalization of the Hartree-Fock-Roothaan expression for molecules, namely... 

In Section 5.5, we shall return in detail to the gap problem of alternating trans-PA. We point out here only that the gap obtained from every ab initio Hartree-Fock calculation is usually too large (though it is essentially decreasing with increase in the basis set). Besides geometry optimization and correct treatment of the many-neighbor interactions in the Hartree-Fock case (see Section 1.3) it is also absolutely necessary to include correlation effects to obtain reasonable agreement with experiment. ... 

DilP recently discussed the merits and limitations of models that assume thermodynamic additivity and independence (of energy types, of neighbor interactions, of conformational freedom, of monomer contact pairing frequencies, etc.). He states that biological molecules may achieve stability in the face of thermal uncertainty, as polymers do, by compounding many small interactions this summing can stump modelers because application of the additivity principle leads to accumulated error. Entropies and free energy may not be additive to describe weak interactions that are ensembles of states. He concludes that additivity principles appear to be few and limited in scope in biochemistry. 

The difference in energy between subshells in multielectron atoms results from electron-electron repulsions. In hydrogen, the only electrical interaction is the attraction of the positive nucleus for the negative electron, but in multielectron atoms there are many different interactions to consider. Not only are there the attractions of the nucleus for each electron, there are also the repulsions between every electron and each of its neighboring electrons. 

Covalent Solids. Interatomic potentials are the most difficult to derive for covalent solids. The potential must predict the directional nature to the bonding (i.e. the bond angles). Most covalent solids have rather open crystal stmctures, not close packed ones. Pair potentials used with diatomic molecules, such as the Lennard-Jones and Morse potentials, are simply not adequate for solids because atoms interacting via only radial forces prefer to have as many neighbors as possible. Hence, qualitatively wrong covalent crystal stmctures are predicted. 

This means that a particle of mass M interacts linearly with a chain of infinitely many particles with masses mj, and so on, which also interact with each other via a linear coupling (only nearest-neighbor interactions are considered). The dynamics of the particle of mass M is defined by its space coordinate x and velodty v. The space coordinates and velocities of the particles of mass m,- are denoted by the symbols y, and w, respectively. More proper variables are the relative distances... 

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

On the other hand, in the condensed phases the concept of supermolecules is not useful because every atom or molecule interacts simultaneously with many neighbors. The many-body nature of the induction process, combined with the statistical mechanics of liquids and a complex local field problem have been serious difficulties for a quantitative description of CILS in dense matter. Furthermore, for an accurate modeling, irreducible (i.e., the pairwise nonadditive) contributions of the intermolecular interaction and induction mechanisms may be significant, which complicate the problem even more. Most treatments of CILS in the dense phases have been undertaken in the time domain, based on correlation functions of the type... 

The present paper is focused on the role of the cooperativity (or many-body interactions) on the structure of ordinary ice and liquid water. For this purpose, the energy of a H bond was expressed as a function of the number of H-bonded neighbors (see eq 7). Then, the probabilities of breaking various types of... 

Many problems in D-dimensional statistical mechanics with nearest-neighbor interactions can be converted into quantum mechanics problems in (D — 1) dimensions of space and one dimension of time [84]. The quantum theory arises here in a Feynman path integral formulation [85]. 

Many interesting problems in physical organic chemistry have been clarified by numerical calculations based on next-neighbor interaction and zero-differential-overlap approximations, especially in the field of... 

---