Long range force

Ways to circumvent the above-mentioned problems have been to simply increase the cutoff distance to larger values, to use more than one cutoff value with different update frequencies, or to define more sophisticated cutoff schemes. In the last case, a truncation of the non-bonded interactions was replaced by shifting the interaction energies to zero or by additionally applying a switched sigmoidal func- 

In periodic boimdary conditions, one possible way to avoid truncation of electrostatic interaction is to apply the so-called Particle Mesh Ewald (PME) method, which follows the Ewald summation method of calculating the electrostatic energy for a number of charges [27]. It was first devised by Ewald in 1921 to study the energetics of ionic crystals [28]. PME has been widely used for highly polar or charged systems. York and Darden applied the PME method already in 1994 to simulate a crystal of the bovine pancreatic trypsin inhibitor (BPTI) by molecular dynamics [29]. 

They compared the PME method with equivalent simulations based on a 9 A residue-based cutoflF and found that for PME the averaged RMS deviations of the nonhydrogen atoms from the X-ray structure were considerably smaller than in the non-PME case. Also, the atomic fluctuations calculated from the PME dynamics simulation were in close agreement with those derived from the crystallographic temperature factors. In the case of DNA, which is highly charged, the application of PME electrostatics leads to more stable dynamics trajectories with geometries closer to experimental data [30]. A theoretical and numerical comparison of various particle mesh routines has been published by Desemo and Holm [31]. 

The theoretical understanding of the interaction between molecules at distances where the overlap is negligible has been well established for some years. The application of perturbation theory is relatively straightforward, and the recent work in this area has consisted in the main of the application of well-known techniques. In the case of neutral molecules, the first non-zero terms appear in the second order of perturbation, so that some method of obtaining the first-order wavefunction, or of approximating the infinite sum in the traditional form of the second-order energy expression, is needed. 

An alternative, and more recent, idea is the pseudo-spectral expansion,67 in which ffo is diagonalized in some large basis of convenient functions and the summation in the second-order energy is evaluated exactly over the resulting eigenfunctions. This method has been used most recently by Bukta and Meath, who use it to obtain third- 

The same pseudo-spectral expansion has also been applied to the case of two lithium atoms.70 In this, owing to the more complicated nature of Ho, difficulties arise from the finite size of the basis set. The authors use an extrapolation procedure to surmount this difficulty. Because the summation in the second-order energy is dominated by the first term, whose magnitude can be obtained experimentally from the excitation energy and oscillator strength of the corresponding electronic transition, the authors plot the calculated value of Co against the calculated first term for various sizes of basis set, and choose for Co that value which corresponds to the experimental value of the first term. In this way they obtain 

Both these problems have been attacked recently by other authors, the ground-state-excitcd-state hydrogen case by Deal and Young,71 the lithium-lithium case by Caves,72 who gives values for C and Co for two ground-state atoms, and also C values for the interaction of a ground-state atom with atoms in the 3s, 4s, 5s, and 6s states. 

Another approach to the evaluation of higher-order energies is to obtain the perturbed wavefunction, or some approximation to it, explicitly. This method has 

In general, for a box which is positioned at a cubic lattice point n (= (n -L, WyL, tizL) with n,. Uy, n-z being integers)  

The prime on the first summation indicates that the series does not include the interaction 

There is thus a contribution to the total energy from the interactions in the central box together with the interactions between the central box and all image boxes. There is also a contribution from the interaction between the spherical array of boxes and the surrounding 

The trick when calculating the Ewald sum is to convert the summation into two series, each of which converges much more rapidly. The mathematical foundation for this is the following identity  

The sum over point charges is now converted to a sum of the interactions between the charges plus the neutralising distributions This dual summation (the real space  

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The discussion focuses on two broad aspects of electrical phenomena at interfaces in the first we determine the consequences of the presence of electrical charges at an interface with an electrolyte solution, and in the second we explore the nature of the potential occurring at phase boundaries. Even within these areas, frequent reference will be made to various specialized treatises dealing with such subjects rather than attempting to cover the general literature. One important application, namely, to the treatment of long-range forces between surfaces, is developed in the next chapter. 

All the long-range forces discussed in this chapter play a role in biological processes. Interactions between membranes, proteins, ligands, antibodies... 

C. Long-Range Forces as a Factor in Emulsion Stability... 

The immediate site of the adsorbent-adsorbate interaction is presumably that between adjacent atoms of the respective species. This is certainly true in chemisorption, where actual chemical bond formation is the rule, and is largely true in the case of physical adsorption, with the possible exception of multilayer formation, which can be viewed as a consequence of weak, long-range force helds. Another possible exception would be the case of molecules where some electron delocalization is present, as with aromatic ring systems. 

It is usefiil to classify various contributions to intennolecular forces on the basis of the physical phenomena that give rise to them. The first level of classification is into long-range forces that vary as inverse powers of the distance r , and short-range forces that decrease exponentially with distance as m exp(-ar). 

There are tliree important varieties of long-range forces electrostatic, induction and dispersion. Electrostatic forces are due to classical Coulombic interactions between the static charge distributions of the two molecules. They are strictly pairwise additive, highly anisotropic, and can be either repulsive or attractive. 

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

Dalgarno A and Lewis J T 1956 The representation of long-range forces by series expansions. I. The divergence of the series Proc. Phys. Soc. A 69 57... 

A fine text suitable for both graduate students and researchers. Emphasizes theory of long-range forces. 

Stell G 1977 Fluids with long-range forces towards a simple analytic theory Statistical Mechanics part A, Equilibrium Techniques ed B Berne (New York Plenum)... 

Hemmer P C 1964 On van der Waals theory of vapor-liquid equilibrium IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75... 

Sharma R D and Brau C A 1969 Energy transfer in near-resonant molecular collisions due to long-range forces with application to transfer of vibrational energy from the mode of CO2 to N2 J. Chem. Phys. 50 924-30... 

A double-timestep algorithm with short- and long-range forces is obtained by applying the propagator [36] ... 

Tuckerman, M.E., Berne, B.J., Rossi, A. Molecular dynamics algorithm for multiple time scales systems with long range forces. J. Chem. Phys. 94 (1991) 6811-6815. 

Loncharich, R.J., Brooks, B.R. The effects of truncating long-range forces on protein dynamics. Proteins 6 (1989) 32 5. 

Perera, L., Essmann, U., Berkowitz, M. Effect of treatment of long-range forces on the dynamics of ions in aqueous solutions. J. Chem. Phys. 102 (1995) 450-456. 

M. E. Tuckerman and B. J. Berne. Molecular dynamics in systems with multiple time scales Systems with stiff and soft degrees of freedom and with short and long range forces. J. Comp. Chem., 95 8362-8364, 1992. 

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