Molecular mechanics Lennard-Jones 6-12 potential

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

In Eq. (10), s is the potential well depth (Fig. 1). The Lennard-Jones potential can be used as the nonbonded potential because it contains both an attractive (if n = 6) and a repulsive part. In theory, n > 6. But when n = 12, the potential arranges to the simple Lennard-Jones 6/12 potential often used in molecular mechanics ... 

As demonstrated in Table 5.3, the potential energy functions in side-chain prediction methods have varied tremendously from simple steric exclusion terms to full molecular mechanics potentials. In most cases, the potential energy function is a standard Lennard-Jones potential ... 

Many molecular modelling techraques that use force-field models require the derivatives of the energy (i e the force) to be calculated with respect to the coordinates. It is preferable that analytical expressions for these derivatives are available because they are more accurate and faster than numerical derivatives. A molecular mechanics energy is usually expressed in terms of a combination of internal coordinates of the system (bonds, angles, torsions, etc.) and interatomic distances (for the non-bonded interactions). The atomic positions in molecular mechanics are invariably expressed in terms of Cartesian coordinates (unlike quantum mechanics, where internal coordinates are often used). The calculation of derivatives with respect to the atomic coordinates usually requires the chain rule to be applied. For example, for an energy function that depends upon the separation between two atoms (such as the Lennard-Jones potential. Coulomb electrostatic interaction or bond-stretching term) we can write ... 

Force field-based scoring functions are based on the nonbonded terms of a classical molecular mechanics force field (e.g.. Amber and CHARMm). A Lennard-Jones potential describes van der Waals interactions, whereas the Coulomb energy describes the electrostatic components of the interactions. The nonbonded interaction energy takes the following form ... 

Rather, flie assignment is more serious wifli intermolecular interaction potential used. For simple molecules, empirical model potential such as fliose based on Lennard-Jones potential and even hard-sphere potential can be used. But, for complex molecules, potential function and related parameter value should be determined by some theoretical calculations. For example, contribution of hydrogen-bond interaction is highly large to the total interaction for such molecules as HjO, alcohols etc., one can produce semi-empirical potential based on quantum-chemical molecular orbital calculation. Molecular ensemble design is now complex unified mefliod, which contains both quantum chemical and statistical mechanical calculations. 

Lennard-Jones potential (p. 287) torsional potential (p. 288) molecular mechanics (p. 290) global optimization (p. 292) global minimum (p. 292)... 

The Buckingham potential which involves a 6-exp function is sometimes used in some molecular mechanics programs as an alternative to the Lennard-Jones potential. The form of this equation used in the MM2 program is... 

Knowledge of the adsorbate-adsorbent interaction is fundamental in any statistical mechanics theory of adsorption. As indicated earlier, the comparison between experimental Henry s constants or gas-solid virial coefficients and theory [8,33] permits one to test the validity of a given model for the gas-solid potential. As a first approximation, the potential f/sf( ,) is considered to be a function only of the perpendicular distance z for monolayer mobile adsorption on homogeneous surfaces [29,33,43,219]. The analytical forms used are similar to the Lennard-Jones potential, but replacing r by z and considering different (10-4 or 9-3, for example) powers than the 12-6 case expressed in Eq. (12). In each case, the gas-surface molecular parameters, Sjf and cTsf, can be determined by comparison with experimental results. This procedure must be considered as semiempirical and thus not fidly theoretical. 

In atomic and molecular systems the range and strength (inverse temperature) of the attraction is set by quantum mechanics. In many cases a Lennard-Jones potential describes the pair interaction quite well [7]. The phase behavior of atomic and molecular systems is often represented in a pressure versus temperature diagram. Quite often the distance between triple point (tp) and critical point (cp) is significant so that there is a wide region where a liquid exists the liquid window is then wide. 

I mentioned earlier that molecular-mechanics force fields have to be transferable from molecule to molecule, and that it was found many years ago that extra terms were needed apart from the pure valence ones. Non-bonded interactions are usually taken as the Lennard-Jones 12-6 potential... 

The Lennard-Jones (6-12) potential has served very well as an inter-molecular potential and has been widely used for statistical mechanics and kinetic-theory calculations. It suffers, however, from having only two adjustable constants, and there is no reason why it should not gradually be replaced by more flexible and more realistic functions. Recently a number of applications have been made of the Buckingham (6-exp) potential [Eq. (82)], which has three adjustable parameters. For this potential the first approximation to the coefficient of diffusion is written by Mason (M3) in the form... 

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. 

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