Monte Carlo Simulation of Molecules

For rigid, non-spherical molecules, the orientations of the molecules must be varied as well as their positions in space. It is usual to translate and rotate one molecule during each Monte Carlo step Translations are usually described in terms of the position of the centre of mass. There are various ways to generate a new orientation of a molecule. The simplest approach is to choose one of the three Cartesian axes x, y or z) and to rotate about the chosen axis by a randomly chosen angle 5a , chosen to lie within the maximum angle variation, [Barker and Watts 1969]. The rotation is achieved by applying routine trigonometric relationships. For example, if the vector (xi,yj,zk) describes the orientation of a molecule then the new vector (x i,y j,z k) that corresponds to rotation by 6w about the X axis is calculated as follows  

The alternative is to sample in 0 and to modify the acceptance or rejection criteria as follows  

This second approach may give problems if oid equals zero. 

A disadvantage of the Euler angle approach is that the rotation matrix contains a total of six trigonometric functions (sine and cosine for each of the three Euler angles). These trigonometric functions are computationally expensive to calculate. An alternative is to use quaternions. A quaternion is a four-dimensional vector such that its components sum to 1 0 + 1 + 72 + = 1- quaternion components are related to the Euler angles as follows  

The Euler angle rotation matrix can then be written 

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Kaznessis YN, Snow ME, Blankley CJ (2001) Prediction of blood-brain partitioning using Monte Carlo simulations of molecules in water. J Comput Aided Mol Des 15 697-708. 

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

Figure C2.3.6. Illustration of micelle stmcture obtained by Monte Carlo simulations of model octanoate amphiphiles. There are 30 molecules simulated in this cluster. The shaded spheres represent headgroups. Reproduced by pennission from figure 2 of [35].

R. Dickman, C. K. Hall. High density Monte Carlo simulations of chain molecules Bulk equation of state and density profile near walls. J Chem Phys 59 3168-3174, 1988. 

Monte Carlo Simulations of Isolated Liquid Crystal Molecules.. . 51... 

Jorgensen et al. has developed a series of united atom intermolecular potential functions based on multiple Monte Carlo simulations of small molecules [10-23]. Careful optimisation of these functions has been possible by fitting to the thermodynamic properties of the materials studied. Combining these OPLS functions (Optimised Potentials for Liquid Simulation) with the AMBER intramolecular force field provides a powerful united-atom force field [24] which has been used in bulk simulations of liquid crystals [25-27],... 

Fig. 6.11. The error in the free energy measured by several NEW implementations. Results are from Monte Carlo simulations of ion charging in water at 298 K. System 0 consists of a single Lennard-Jones atom with charge of +le and 216 SPC water molecules, and system 1 is the same but with the charge turned off. One work cycle contains 100 nonuniform steps in 7 from 0 to 1 and back. For a detailed description of the simulation, see [43]...

In principle, the ideal description of a solution would be a quantum mechanical treatment of the supermolecule consisting of representative numbers of molecules of solute and solvent. In practice this is not presently feasible, even if only a single solute molecule is included. In recent years, however, with the advances in processor technology that have occurred, it has become possible to carry out increasingly detailed molecular dynamics or Monte Carlo simulations of solutions, involving hundreds or perhaps even thousands of solvent molecules. In these, all solute-solvent and solvent-solvent interactions are taken into account, at some level of sophistication. 

One way to include these local quantum chemical effects is to perform ab initio calculations on an HOD molecule in a cluster of water molecules, possibly in the field of the point charges of the water molecules surrounding the cluster. In 1991 Hermansson generated such clusters from a Monte Carlo simulation of the liquid, and for each one she determined the relevant Bom Oppenheimer potential and the vibrational frequencies. The transition-dipole-weigh ted histogram of frequencies was in rough agreement with the experimental IR spectrum for H0D/D20 [130],... 

Mitchell et al. (390) using nuclear reaction analysis (NRA), found 6 = 0.25 for the saturation adsorption of C2H4 on Pt(lll) at 100 K (also see 391). This result has been confirmed by a combined study done with NRA and XPES (392) and by STM (393-395). Furthermore, the value of 8 = 0.25 for C2H4 saturation coverage at low temperature is in agreement with a Monte Carlo simulation of QH4 adsorption on Pt(lll) by Windham et al. (396), who showed that an ensemble of four Pt surface atoms is required to absorb one C2H4 molecule. 

The explicit approach uses a Monte Carlo simulation of the solute immersed in a box containing a large number of solvent molecules. Results are usually good but the computational cost is rather high. 

The RSA model received renewed attention after Feder [12] observed that the adsorption on the surface of apo-ferritin molecules (large iron-storage proteins with a diameter of about 10 nm), which adsorb irreversibly, reached saturation at a coverage (k = 0.518. Monte Carlo simulations of Random Sequential Adsorption of disks on a surface last prohibitively long in the vicinity of the jamming point however Feder [12] noted that in the vicinity of the jamming coverage, 9 has a power-law dependence on time ... 

A Monte-Carlo simulation of gas chromatography at vacuum conditions follows the same scheme as of Figure 1 [16]. The lifetime of each atom or molecule is calculated using Equation 28. The mean residence time of the atom in the adsorbed state at a defined temperature is calculated using Equation 34. The residence time of the atom in the adsorbed state, which is distributed logarithmically, can be calculated ... 

This method avoids the convergence and accuracy problems of molecular dynamics or Monte Carlo simulations of systems containing explicit solvent molecules, by evaluating the electrostatic free energy of just one solute conformation surrounded by a dielectric continuum, and by adding the surface term and an estimate of the loss of the configurational entropy upon binding.77... 

Figure 8. Probability map of the water positions obtained from the Monte Carlo simulation of cyclo(L-Ala-L-Pro-D-Phe)t. In this section the density for the two water molecules with two alternate positions 3 and 4 are shown. The x-ray positions are indicated by the circles within the density. Note the density spanning these, indicating that the water molecules oscillate between these positions during the...

Grand Canonical Monte Carlo Simulations of Adsorption of Mixtures of Xylene Molecules in Faujasite Zeolites. 

More complicated 3D effects were studied in Refs. 6 and 7 with the help of 3D Monte Carlo digital simulation performed with a rather powerful computer (RISK System/6000). Sedimentation FFF with different breadth-to-width channel ratios and both codirected and counterdirected rotation and flow were studied. Secondary flow forming vortexes in the y-z plane is generated in the sedimentation FFF channel, both due to its curvature, and the Coriolis force caused by the centrifuge rotation. The exact structure of the secondary flow was calculated by the numerical solution of the Navier-Stokes equations and was used in the Monte Carlo simulation of the movement of solute molecules. 

The Monte Carlo method can also be used to study the effect of solvent on the position of tautomeric equilibrium (Metropolis et al., 1953). This method models a dilute solution as N solvent molecules (N — 102) and 1 solute molecule. The fundamental requirement for a Monte Carlo simulation of tautomer solvation is a set of pair potentials for the solvent-solvent and solvent-solute interactions because approximately 106 config-... 

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