Approximation, two-body

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

Approximating the intermoleculai interactions to only include two-body effects, e.g. electrostatic forces are only calculated between pairs of fixed atomic chai ges in force field techniques. Or the discrete interactions between molecules may be treated only in an average fashion, by using Langevin dynamics instead of molecular dynamics. 

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

Let us now improve our two-body model by allowing the molecule of water to vibrate. A rather straightforward way to achieve the goal is simply to consider the potential energy between the two molecules as a sum of two contributions, one arising from the intermolecular and the second from the intramolecular motions an approximate interaction potential has been reported by Lie and dementi rather recently, where the intramolecular potential was simply taken over from the many body perturbation computation by Bartlett, Shavitt, and Purvis. The potential will henceforth be referred to as MCYL. 

The intracellular fluid (ICF) represents the water contained within cells and is rich in electrolytes such as potassium, magnesium, phosphates, and proteins. O The ICF is approximately two-thirds of TBW regardless of gender. For a 70-kg man, this would mean that the TBW is 42 L and the ICF is approximately 28 L. For a 70-kg woman, these values would be 35 L and 24 L, respectively. Note that ICF represents approximately 40% of total body weight in men and approximately 33% of total body weight in women. 

The remaining four energy contributions depend on the reference density only. This is an important observation, which allows to combine these contribution into the so called repulsive energy term Erep pu, which is treated in a simplified way by approximating it by a sum of two-body potentials [44], Erep[po] = Hap Uap Rap)... 

From this point of view it is of interest to examine the consequences of full ther-malization of the classical Drude oscillators on the properties of the system. This is particularly important given the fact that any classical fluctuations of the Drude oscillators are a priori unphysical according to the Bom-Oppenheimer approximation upon which electronic induction models are based. It has been shown [12] that under the influence of thermalized (hot) fluctuating Drude oscillators the corrected effective energy of the system, truncated to two-body interactions is... 

Cieplak, P., Caldwell, J. W., Kollman, P. A., Molecular mechanical models for organic and biological systems going beyond the atom centered two body additive approximation aqueous solution free energies of methanol and IV-methyl acetamide, nucleic acid base, and amide hydrogen bonding and chloroform/water partition coefficients of the nucleic acid bases, J. Comput. Chem. 2001, 22, 1048-1057... 

The vibration of a diatomic molecule, or any vibrational mode in a polyatomic molecule, may be approximated by two atoms of mass m and m2 joined by a Hooke s law bond that allows vibration relative to the centre of mass. The frequency of such a two-body oscillator is given by... 

As it will be explained in section 6, the usual way to evaluate the potential energy of a system simulated by Monte Carlo techniques, makes use of the pair potential approximation (although, as it will also be reviewed, several works have already appeared where nonadditivity corrections to the interaction potential have been included). In the pair potential approximation only two body interactions are taken into account. We will briefly explain here how to apply this approximation for the calculation of the potential energy, to the periodic system just described. The interaction potential energy under the pair potential approximation can be written as ... 

It should be noted that the results of VCS and BHF calculations using the same NN interaction disagree in several aspects. For instance for SNM the VCS and BHF calculations saturate at different values of density [5]. As for the SE using only two-body interactions the VCS approach yields a smaller value for 04 than BHF (see Table 1), and as a function of density in VCS the SE levels off at p 0.6 fm-3, whereas the BHF result continues to increase. Therefore it seems natural to ask whether the inclusion of more correlations by extending the BHF method (which is basically a mean field approximation) will lead to results closer to those of VCS. 

Fet us now confront the EOS predicted by the phenomenological TBF and the microscopic one. In both cases the BHF approximation has been adopted with same two-body force (Argonne uis). In the left panel of Fig. 4 we display the equation of state both for symmetric matter (lower curves) and pure neutron matter (upper curves). We show results obtained for several cases, i.e., i) only two-body forces are included (dotted lines), ii) TBF implemented within the phenomenological Urbana IX model (dashed lines), and iii) TBF treated within... 

We have described our most recent efforts to calculate vibrational line shapes for liquid water and its isotopic variants under ambient conditions, as well as to calculate ultrafast observables capable of shedding light on spectral diffusion dynamics, and we have endeavored to interpret line shapes and spectral diffusion in terms of hydrogen bonding in the liquid. Our approach uses conventional classical effective two-body simulation potentials, coupled with more sophisticated quantum chemistry-based techniques for obtaining transition frequencies, transition dipoles and polarizabilities, and intramolecular and intermolecular couplings. In addition, we have used the recently developed time-averaging approximation to calculate Raman and IR line shapes for H20 (which involves... 

The nucleosome is the fundamental repeating structural unit of chromatin. It is composed of two molecules of the core histones H2A, H2B, H3, H4, approximately two superhelical turns of double-stranded DNA, and linker histone HI (H5). In addition to biochemical studies, the existence of the nucleosome was established in electron micrographs (Fig. la) [1,2], and the name nucleosome, coined to incorporate the concept of the spherical nu-bodies [3]. Micrococcal nuclease limit digestion of chromatin established the nucleosome core particle (NCP) as the portion of the nucleosome containing only the core histones surrounded by 1.75 superhelical turns of double-stranded DNA [4,5]. 

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