Non bonded interactions

As a general rule, the best approach when developing a force field for a metal complex is to use, without modification, the non-bonded interaction terms developed for organic compounds. The best-known and most popular are the MM2 and MM3 force fields, and these have been extended and adapted for modeling metal complexes by a number of groups [164-167]. Force fields developed for organic molecules do not have parameters for some of the elements present in metal complexes but reasonable estimates are available for most of these [95, 104, 125, 130, 140, 171, 173, 193]. 

One of the main reasons why non-bonded interactions involving metal ions have not been included in most force fields is a lack of good estimates for the parameters. As discussed in Section 3.2.5, values for the van der Waals radius and the polarizability (e) are required. In the case of metal complexes it is difficult to obtain estimates 

In any host-guest calculation therefore, the form of the equation and the parameters used for E b and E are of crucial importance. The form of the equation for E b is usually either a Lennard-Jones or Buckingham potential. 

This energy is summed over all pairs of atoms i,j in the structure. Subscripts m and n represent the atom types of the i th and j th atoms in the structure thus atoms i and j have van der Waals radii (rand (rj respectively (r Jo is the sum of these van der Waals radii (r and (r X for foe interacting pair of atoms which are r apart. is the potential well depth and is calculated as the geometric mean of for atom types m and n. 

In several molecular mechanics programs equation (4) is written as 

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 

Other force fields offer slight variations of either the Lennard-Jones or Buckingham equation. For example the latest force field in INSIGHT/DISCOVER [4] introduces a 9-6 rather than a 12-6 potential. 

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In addition, the non-bonded forces can be divided into several regions according to pair distances. The near region is normally more important than the distant region because the non-bonded forces decay with distance. Since most of the CPU time in a MD simulation is spent in the calculation of these non-bonded interactions, the separation in pair distance results in valuable speedups. Using a 3-fold distance split, the non-bonded forces are separated in 3 regions near, medium, and fax distance zones. Thus, the Liouville operator can be express as a sum of five terms... 

In the next step, the suggested models are translated into 3D space by subsequently combining the templates. Again, each model is assessed and ranked according to various structural criteria, such as the geometric fit of the 3D templates and non-bonding interactions (steric clashes). If none of the solu-... 

Figure 2-99. Elimination of non-bonded interactions (close contacts).

Vn is often called the barrier of rotation. This is intuitive but misleading, because the exact energetic barrier of a particular rotation is the sum of all V components and other non-bonding interactions with the atoms under consideration. The multiplicity n gives the number of minima of the function during a 360° rotation of the dihedral angle o). The phase y defines the exact position of the minima. 

It is noteworthy that it is not obligatory to use a torsional potential within a PEF. Depending on the parameterization, it is also possible to represent the torsional barrier by non-bonding interactions between the atoms separated by three bonds. In fact, torsional potentials and non-bonding 1,4-interactions are in a close relationship. This is one reason why force fields like AMBER downscale the 1,4-non-bonded Coulomb and van der Waals interactions. 

N is the number of point charges within the molecule and Sq is the dielectric permittivity of the vacuum. This form is used especially in force fields like AMBER and CHARMM for proteins. As already mentioned, Coulombic 1,4-non-bonded interactions interfere with 1,4-torsional potentials and are therefore scaled (e.g., by 1 1.2 in AMBER). Please be aware that Coulombic interactions, unlike the bonded contributions to the PEF presented above, are not limited to a single molecule. If the system under consideration contains more than one molecule (like a peptide in a box of water), non-bonded interactions have to be calculated between the molecules, too. This principle also holds for the non-bonded van der Waals interactions, which are discussed in Section 7.2.3.6. 

The raie gas atoms reveal through their deviation from ideal gas behavior that electrostatics alone cannot account for all non-bonded interactions, because all multipole moments are zero. Therefore, no dipole-dipole or dipole-induced dipole interactions are possible. Van der Waals first described the forces that give rise to such deviations from the expected behavior. This type of interaction between two atoms can be formulated by a Lennaid-Jones [12-6] function Eq. (27)). 

A second idea to save computational time addresses the fact that hydrogen atoms, when involved in a chemical bond, show the fastest motions in a molecule. If they have to be reproduced by the simulation, the necessary integration time step At has to be at least 1 fs or even less. This is a problem especially for calculations including explicit solvent molecules, because in the case of water they do not only increase the number of non-bonded interactions, they also increase the number of fast-moving hydrogen atoms. This particular situation is taken into account... 

Figure 7-14. All-atom and united-atom representation of the amino acid isoleucine. In this example, 13 atoms, which are able to form explicit non-bonding interactions, are reduced to only four pseudo-atoms,...

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-... 

Although in teraetion s between vicinal I 4 atom s arc n om in ally treated as non bonded interactions, triost of the force fields treat these somewhat differently from normal 1 5 and greater non-bonded interactions. HyperCbern allows each of these 1 4 non-bonded interactions to be scaled down by a scale factor < 1.0 with AMBHR or OPI-S. bor HIO+ the electrostatic may be scaled and different param eters rn ay be ti sed for I 4 van dcr Waals interactions, fh e. AMBHR force field, for exam pie, n orrn a lly uses a seal in g factor of 0.5 for both van der Waals an d electrostatic interactions. 

A typical force field model for propane contains ten bond-stretching terms, eighteen angle-bending terms, eighteen torsional terms and 27 non-bonded interactions. 

In a Urey-Bradley force field, angle bending is achieved using 1,3 non-bonded interaction rather than an explicit angle-bending potential. The stretch-bond term in such a forci field would be modelled by a harmonic function of the distance between the 1,3 atoms ... 

Independent molecules and atoms interact through non-bonded forces, which also play an important role in determining the structure of individual molecular species. The non-bonded interactions do not depend upon a specific bonding relationship between atoms, they are through-space interactions and are usually modelled as a function of some inverse power of the distance. The non-bonded terms in a force field are usually considered in two groups, one comprising electrostatic interactions and the other van der Waals interactions. 

Fhe van der Waals and electrostatic interactions between atoms separated by three bonds (i.c. the 1,4 atoms) are often treated differently from other non-bonded interactions. The interaction between such atoms contributes to the rotational barrier about the central bond, in conjunction with the torsional potential. These 1,4 non-bonded interactions are often scaled down by an empirical factor for example, a factor of 2.0 is suggested for both the electrostatic and van der Waals terms in the 1984 AMBER force field (a scale factor of 1/1.2 is used for the electrostatic terms in the 1995 AMBER force field). There are several reasons why one would wish to scale the 1,4 interactions. The error associated wilh the use of an repulsion term (which is too steep compared with the more correct exponential term) would be most significant for 1,4 atoms. In addition, when two 1,4... 

The computational effort is significantly increased if three-body terms are included in the model. Even with a simple pairwise model, the non-bonded interactions usually require by far the greatest amount of computational effort. The number of bond, angle and torsional terms increases approximately with the number of atoms (N) in the system, but the number of non-bonded interactions increases with N. There are N(N —l)/2 distinct pairs of... 

By itself, the use of a cutoff may not dramatically reduce the time taken to compute the number of non-bonded interactions. This is because we would still have to calculate the distance between every pair of atoms in the system simply to decide whether they are dose enough to calculate their interaction energy. Calculating all the N N — 1) distances takes almost as much time as calculating the energy itself. 

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