Lennard-Jones attraction parameters

There exist several linkages of the SBUs which should be avoided, for example, the short separation of two metal atoms (M Mj), one ligand atom and one bridging atom (Lj Bj), and L Mj in SBUs unlinked. To prevent these undesirable linkages, other potential functions have to be considered, including the repulsive potential between Mj Mj pairs, the attractive potential between L, - - - Mj pairs, and the repulsive potential between Lj Bj pairs. A repulsive potential between Mr Mj pairs prevents SBUs from overlapping with each other. The distance between the Mr Mj pair is limited to 3.4 A for D4R. The Lennard-Jones potential parameters used in assembling the D4R are provided in Table 7.2. 

All protein atoms except hydrogens bonded to carbon were explicitly represented. Positions of polar hydrogens were calculated on the basis of coordinates of the heavier atoms (5) and standard geometric constraints (14). The non-bonded energy for the interaction of the water molecule and the protein was calculated as a sum of electrostatic and 6-12 Lennard-Jones attractive and repulsive contributions for the three atoms of the water molecule and all atoms of the protein within 6 X of the water oxygen s center. Empirical parameters were used for the 6-12 potentials (15,16,17), partial atomic charges were values obtained with molecular orbital calculations (1 ) for hydrogen-bonded interactions modified 6-12 parameters were used (18). 

Additionally to and a third adjustable parameter a was introduced. For a-values between 14 and 15, a form very similar to the Lennard-Jones [12-6] potential can be obtained. The Buckingham type of potential has the disadvantage that it becomes attractive for very short interatomic distances. A Morse potential may also be used to model van der Waals interactions in a PEF, assuming that an adapted parameter set is available. 

The above potential is referred to as a Lennard-Jones or 6-12 potential and is summed over all nonbonded pairs of atoms ij. The first positive term is the short range repulsion and the second negative term is the long range attraction. The parameters of the interaction are Aj and B... The convenient analytical form of the 6-12 potential means that it is often used, although an exponential repulsion term is usually considered to be a more accurate representation of the repulsive forces (as used in MM-t). 

Here the atoms in the system are numbered by i, j, k, l = 1,..., N. The distance between two atoms i, j is ry, q is the (partial) charge on an atom, 6 is the angle defined by the coordinates (i, j, k) of three consecutive atoms, and 4> is the dihedral angle defined by the positions of four consecutive atoms, e0 is the dielectric permittivity of vacuum, n is the dihedral multiplicity. The potential function, as given in equation (6), has many parameters that depend on the atoms involved. The first term accounts for Coulombic interactions. The second term is the Lennard-Jones interaction energy. It is composed of a strongly repulsive term and a van der Waals-like attractive term. The form of the repulsive term is chosen ad hoc and has the function of defining the size of the atom. The Ay coefficients are a function of the van der Waals radii of the... 

In reality, molecules each occupy some space, so the empty volume of the container decreases as the concentration N/ V increases. In addition, there is generally some attraction even at distances substantially larger than the nominal diameter of the molecules, and the repulsive part is somewhat soft so that collisions are not instantaneous. The exact form of this interaction must be calculated by quantum mechanics, and it depends on a number of atomic and molecular properties as discussed in Chapter 3. For neutral, nonpolar molecules, a convenient approximate potential is the Lennard-Jones 6-12 potential, discussed in Chapter 3 Table 3.5 listed parameters for some common atoms and molecules. 

Of somewhat more use here is an approximate description of the interactions in terms of the well-known Lennard-Jones interaction (Lennard-Jones, 1924, 1925). This is a two-parameter model that Lennard-Jones fitted to properties of the giiseous phase. This form contained an attractive term proportional to d, in accord with the Van dcr Waals form. The repulsive term rises more rapidly, and a term proportional to d is commonly taken, though most results are not highly sensitive to the exact exponent. The Lennard-Jones overlap interaction is then written... 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

The Lennard-Jones potential [4], [5], [9] (sixth-power attraction, twelfth-power repulsion) is quite realistic and appears to be the one most commonly used in practice. This potential contains two adjustable parameters (a size and a "strength ), which are defined and listed for various chemical compounds in [5], [6], and [9]. The collision integrals appearing in the first approximations to the transport properties are tabulated as functions of useful dimensionless forms of these two parameters in [5], [6], and [9]. Similar tabulations for other potentials may also be found in [5]. 

The interaction between water molecules and silica substrate is described in the framework of the PN-TrAZ model [15] which has proven to model successfiilly the adsorption of simple adsorbates on various zeolites [20]. In this model, the pair potential decomposes in two parts a repulsion term Ae" due to electronic clouds and the attractive dispersion terms. The repulsive parameters (A,b) for silica atoms (Si, O, H) are those obtained from studies of adsorption of simple gazes on various zeolites [20] and mesoporous glass [21]. Those for water oxygen are chosen to fit the repulsive part of Lennard-Jones from SPC model in the range around equilibriiun distance, and those for water hydrogen are taken equal to the parameters for surface hydrogen of vycor. The cross repulsive parameters A and b are obtained by Bohm and Ahlrichs [22] combination mles. The dispersion terms are calculated from polarizabilities and effective niunber of electron Neff according to the PN-TrAZ model up to order r °. Values are listed in table 1. 

One of the simplest and therefore computationally less expensive potential functions for ion-water consists of the sum of long-range Coulorabic electrostatic interactions plus short-range dispersion interactions usually represented by the Lennard-Jones potential. This last term is a combination of 6 and 12 powers of the inverse separation between a pair of sites. Two parameters characterize the interaction an energetic parameter e, given by the minimum of the potential energy well, and a size parameter a, that corresponds to the value of the pair separation where the potential energy vanishes. The 6-th power provides the contribution of the attractive forces, while repulsive forces decay with the 12-th power of the inverse separation between atoms or sites. 

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