Calculation computational simulation

Figure 6. Comparison of calculated (computer simulation results for the OPLS model) and experimental heats of vaporization in kcal mol- for the liquids in Table 1 and TIP4P water. Reprinted with permission from W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc. 110, 1657 (1988) [8], Copyright 1988 American Chemical Society.

Figure A2.3.10 compares the virial and pressure equations for hard spheres with the pressure calculated fonu the CS equations and also with the pressures detemiined in computer simulations.

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

The themiodynamic properties calculated by different routes are different, since the MS solution is an approximation. The osmotic coefficient from the virial pressure, compressibility and energy equations are not the same. Of these, the energy equation is the most accurate by comparison with computer simulations of Card and Valleau [ ]. The osmotic coefficients from the virial and compressibility equations are... 

Perturbation theory is also used to calculate free energy differences between distinct systems by computer simulation. This computational alchemy is accomplished by the use of a switching parameter X, ranging from zero to one, that transfonns tire Hamiltonian of one system to the other. The linear relation... 

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

Classical ion trajectory computer simulations based on the BCA are a series of evaluations of two-body collisions. The parameters involved in each collision are tire type of atoms of the projectile and the target atom, the kinetic energy of the projectile and the impact parameter. The general procedure for implementation of such computer simulations is as follows. All of the parameters involved in tlie calculation are defined the surface structure in tenns of the types of the constituent atoms, their positions in the surface and their themial vibration amplitude the projectile in tenns of the type of ion to be used, the incident beam direction and the initial kinetic energy the detector in tenns of the position, size and detection efficiency the type of potential fiinctions for possible collision pairs. 

Neumann M, Steinhauser O and Pawley G S 1984 Consistent calculation of the static and frequency-dependent dielectric constant in computer simulations Mol. Phys. 52 97-113... 

Swope W C and Andersen H C 1995 A computer simulation method for the calculation of chemical potentials of liquids and solids using the bicanonical ensemble J. Chem. Phys. f02 2851-63... 

The complexity of polymeric systems make tire development of an analytical model to predict tlieir stmctural and dynamical properties difficult. Therefore, numerical computer simulations of polymers are widely used to bridge tire gap between tire tlieoretical concepts and the experimental results. Computer simulations can also help tire prediction of material properties and provide detailed insights into tire behaviour of polymer systems. A simulation is based on two elements a more or less detailed model of tire polymer and a related force field which allows tire calculation of tire energy and tire motion of tire system using molecular mechanisms, molecular dynamics, or Monte Carlo teclmiques 1631. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Neumann, M., Steinhauser, O. On the calculation of the frequency-dependent dielectric constant in computer simulations. Chem. Phys. Lett. 102 (1983) 508-513. 

Hummer, G., Szabo, A. Calculation of free energy differences from computer simulations of initial and finial states. J. Chem. Phys. 105 (1996) 2004-2010... 

Gerber, P. R., Mark, A. E., van Gunsteren, W. F. An approximate but efficient method to calculate free energy trends by computer simulation Application to dihydrofolate reductase-inhibitor complexes. J. Comp. Aid. Mol. Desgn 7 (1993) 305-323... 

Many problems in force field investigations arise from the calculation of Coulomb interactions with fixed charges, thereby neglecting possible mutual polarization. With that obvious drawback in mind, Ulrich Sternberg developed the COSMOS (Computer Simulation of Molecular Structures) force field [30], which extends a classical molecular mechanics force field by serai-empirical charge calculation based on bond polarization theory [31, 32]. This approach has the advantage that the atomic charges depend on the three-dimensional structure of the molecule. Parts of the functional form of COSMOS were taken from the PIMM force field of Lindner et al., which combines self-consistent field theory for r-orbitals ( nr-SCF) with molecular mechanics [33, 34]. 

T. Fox, C. Chipot, A. PohorUle, The development/application of a minimalisf organic/biochemical molecular mechanic force field using a combination of ab-initio calculations and experimental data, in Computer Simulation of Biomolecular Systems. 

Cillan M J 1991. Calculating the Properties of Materials from Scratch, In Meyer M and V Pontikis (Editors). Computer Simulation, NATO ASI Series E 205 (Computer Simulations in Materials Science) pp. 257-281. 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

Another way to improve the error in a simulation, at least for properties such as the energy and the heat capacity that depend on the size of the system (the extensive properties), is to increase the number of atoms or molecules in the calculation. The standard deviation of the average of such a property is proportional to l/ /N. Thus, more accurate values can be obtained by running longer simulations on larger systems. In computer simulation it is unfortunately the case that the more effort that is expended the better the results that are obtained. Such is life ... 

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

Catlow C R A and W C Mackrodt 1982. Theory of Simulation Methods for Lattice and Defect Energy Calculations in Crystals. In Lecture Notes in Physics 166 (Comput. Simul. Solids), pp. 3-20. 

The vibrational states of a molecule are observed experimentally via infrared and Raman spectroscopy. These techniques can help to determine molecular structure and environment. In order to gain such useful information, it is necessary to determine what vibrational motion corresponds to each peak in the spectrum. This assignment can be quite difficult due to the large number of closely spaced peaks possible even in fairly simple molecules. In order to aid in this assignment, many workers use computer simulations to calculate the vibrational frequencies of molecules. This chapter presents a brief description of the various computational techniques available. 

Many molecular mechanics potentials were developed at a time when it was computationally impractical to add large numbers of discrete water molecules to the calculation to simulate the effect of aqueous media. As such, techniques came into place that were intended to take into account the effect of solvent in some fashion. These techniques were difficult to justify physically but they were used nevertheless. 

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