Force field Lennard-Jones parameters

Finally, the parametrization of the van der Waals part of the QM-MM interaction must be considered. This applies to all QM-MM implementations irrespective of the quantum method being employed. From Eq. (9) it can be seen that each quantum atom needs to have two Lennard-Jones parameters associated with it in order to have a van der Walls interaction with classical atoms. Generally, there are two approaches to this problem. The first is to derive a set of parameters, e, and G, for each common atom type and then to use this standard set for any study that requires a QM-MM study. This is the most common aproach, and the derived Lennard-Jones parameters for the quantum atoms are simply the parameters found in the MM force field for the analogous atom types. For example, a study that employed a QM-MM method implemented in the program CHARMM [48] would use the appropriate Lennard-Jones parameters of the CHARMM force field [52] for the atoms in the quantum region. 

Table 5.1 Parameters of the united atom force field for polyethylene used as the atomistic input for the coarse-graining procedure. The Lennard-Jones parameters pertain to CH2-group interaction, since chain ends were not considered in the coarse-graining. 

Here, avaw is a positive constant, and and ctJ are the usual Lennard-Jones parameters found in macromolecular force fields. The role played by the term avdw (1 — A)2 in the denominator is to eliminate the singularity of the van der Waals interaction. Introduction of this soft-core potential results in bounded derivatives of the potential energy function when A tends towards 0. 

The solute charge distribution obtained from the quantum calculation is then used as input in the molecular dynamics calculation. The solute-solvent Lennard-Jones parameters and the complete solvent-solvent force field are obtained from the literature. 

For the interaction potential between hydrogen and carbon, we introduce a new procedure to derive the Lennard-Jones parameters from existing parameters that are appropriate for carbon atoms with sp2 and sp3 hybridizations. These parameters may come from existing force fields, and may have been obtained using either experimental or ab initio results. The L-J parameters a and s are made explicitly dependent on the radius of the nanotube, r, using the following equations ... 

In the molecular mechanical force fields, the intermolecular interactions are most often described by the electrostatic and Van der Waals interactions. The MM atoms are normally represented by point charges and Lennard-Jones parameters (usually centered on atoms) in the calculation of intermolecular interactions. Therefore, a simple coupling can be established as shown schematically in Figure 2 and the corresponding Hamiltonian as in eq.(7). [12]... 

Define the interaction Hamiltonian. For a Hamiltonian of the form of equation 24 and for ab initio QM methods the only parameters that need to be defined are the Lennard-Jones parameters for the QM atoms as the charges and Lennard-Jones parameters for the MM atoms are determined once the MM force field is specified. The QM atom parameters can be obtained from the MM force field or from a parameterization procedure. For semiempirical methods it is necessary to specify parameters for the evaluation of the one-electron electrostatic interaction integrals. 

Koddermann et al. calculated the heats of vaporization for imidazolium-based ILs [Cnmim][NTf2] with n = 1, 2, 4, 6, 8 by means of MD simulations [81], The authors applied a force field which they had developed recently. Within this force field the authors reduced the Lennard-Jones parameters in order to reproduce experimental diffusion coefficients [81], The refined force field also led to absolute values of heats of vaporization as well as their increase with the chain length of the imidazolium cation such that this quantities were described correctly. The overall heats of vaporization were split in several contributions and discussed in detail. The authors observed that with increasing alkyl chain length, the Coulomb contribution to the heat of vaporization remained constant at around 80 kJ mol 1, whereas the van der Waals interaction increased continuously. The calculated grow of about 4.7 kJ mol"1 per CH2-group of the van der Waals contribution in the IL exactly matched the increase in the heats of vaporization for n-alcohols and n-alkanes, respectively. The results support the importance of van der Waals interactions even in systems completely composed of ions [81],... 

Short-range repulsions and London dispersion attractions are balanced by a shallow energy minimum at the van der Waals distance (Eq. (8)), describing the Lennard Jones potential, used by most force fields. Here the parameters A and B are calculated based on atomic radii and the minimum found at the sum of the two radii. 

Based on DFT calculations on chlorophylls and, additionally, on ubiquinone and the RC main detergent, lauryl dimethylamine oxide or LDAO, we have then developed a force field for their classical modelization. Our approach to this undertaking was straightforward. We initially use the DFT optimized structures and the vibrational analysis to determine the bonded part of the potential parameters described by the AMBER potential function. Then, atomic ab initio partial charges on the chromophore are used to account for electrostatic effects. At a later stage, experimental data from X-ray crystallography are used to check the structural properties of the molecule in the condensed state and to refine the intermolecular Lennard-Jones parameters. 

The Hill potential was originally developed to enable the more realistic exponential term to be written in terms of Lennard-Jones parameters. The coefficients 2.25,8.25 x 10 and 0.0736 in Equation (4.71) were determined by fitting to data for the rare gases and were assumed to be applicable to other non-polar gases. A Morse potential may also be used to model the van der Waals interactions in a force field, with appropriate parameters. 

To solve equation (18), both Gradient VF and Hessian H of the merit function F(P) should be derived first. For each of the systems for which the CHARMm force field is required to be reparameterized, only the two Lennard-Jones parameters (e , and of a specific atom pair a-b)... 

The existing CHARMm force field severely overestimates the stabilization of the structures around the global energy minimum structure. With the optimized Lennard-Jones parameters between the aromatic carbon and oxygen (C6T-OT atom pair), the optimized CHARMm is capable of producing the IPES for the interaction in excellent agreement with that from MP2. 

Baker, C, M., et al., Accurate calculation of hydration free energies using pair-specific Lennard-Jones parameters in the CHARMM Drude polarizable force field. /. Chem. Theory Comput, 2010. 6(4), 1181-1198. 

In the last two decades, substantial research occurred for the optimization of intermolecular force-field parameters [39-54]. In most cases, intermolecular parameters, especially Lennard-Jones parameters, cannot be strictly derived via physical considerations since they parameterize semiempirical models (i.e., based on classical mechanics) whom themselves only approximate reality. Hence, they are usually adjusted so that the resulting model is able to reproduce physical or chemical experimental target properties as accurately as possible. 

For the MM system and QM/MM interaction, Lennard-Jones parameters (Table 1) for argon were drawn from Maitland et al. [47] and those for the ethylene atoms were drawn from the AMBER ff99 force field [48]. For interactions between distinct atom types, the Lennard-Jones a parameter was chosen to be the arithmetic average of the values for the two atom types, while the parameter was the geometric mean of the two values. 

Electrostatic interactions can be most simply modeled as the Coulomb interaction between partial atomic charges, while the repulsion-dispersion part is usually described by a Lennard-Jones or, more accurately, an exp-6 form, each of which contains parameters that must be fixed. High-quality empirically fitted parameter sets have been developed, where the atom-atom interactions are parameterized to reproduce the structures, sublimation enthalpies and, sometimes, further observable properties of organic molecular crystals [73,74]. Their use has been very effective in CSP. Nonempirical approaches to fitting intermolecular force fields, where the parameters are derived from quantum mechanical calculations, have occasionally been applied for CSP [75-78], but these are currently limited to small molecules, so currently lack relevance for typical pharmaceutical molecules. 

The intramolecular parameters kf,ro, kg, and 0q from the AMBER force field [21, 22] were used. The oxygen atoms of alcohols and ethers were described by the OPLS parameters [17,18]. The Lennard-Jones parameters for halogens and the atoms of the aliphatic groups were optimized to reproduce the experimentally measured solubilities of methane and its halogenated derivatives in water and hexane, separately [23]. The resulting parameters do not follow the standard combination rules because we were unable to find a set of parameters consistent with the rules which would yield sufficiently accurate solubilities of the solutes in both phases. A similar difficulty was encountered in other studies of interfacial systems, in which the united atom representation was used to describe nonpolar molecules [24, 25]. 

A force field that focusses on intramolecular interactions, and particularly on the force constants that determine vibrational frequencies, is the spectroscopically determined force field of Krimm and co-workers (50-52). They advocate very high level quantum calculations to fix the geometries, force constants, and electrostatic terms while using OPLS nonbond Lennard-Jones parameters. The recent focus of this work has been on highly polar molecules, as are discussed later. While this force field reproduces gas-phase IR data very accurately, it does not appear to have been tested on condensed phases. 

The way the potential energy function is split into various kinds of interactions is quite arbitrary and may vary from one force field to another. Two different force fields may give very similar energy for the same molecule in the same conformation, but the individual contributions torsional, nonbonded, bond angles, etc. may be quite different in different force fields. This is why it is generally dangerous to mix force fields, i.e., take, say, the Lennard-Jones parameters from one force field and bonded parameters from another. 

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