Energy computer simulation

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

This is Camalian and Starling s (CS) equation of state for hard spheres it agrees well with the computer simulations of hard spheres in the fluid region. The excess Hehnholtz free energy... 

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... 

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. 

Ghrayeb R, Purushotham M, Hou M and Bauer E 1987 Estimate of repulsive interatomic pair potentials by low-energy alkalimetal-ion scattering and computer simulation Phys. Rev. B 36 7364-70... 

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. 

Prenkel, D. Pree energy computation and first order phase transitions. In Molecular Dynamic Simulation of Statistical Mechanical Systems, Enrico Fermi Summer School, Varenna 1985, G. Ciccotti and W. Hoover, eds. North Holland, Amsterdam (1986) 43-65. 

Pree energy via molecular simulation A primer. In Computer Simulations of Biomolecular Systems, Vol 2, W.F. van Gunsteren, P.K. Weiner and A.J. Wilkinson, eds. Escom, Leiden (1993) 267-314. 

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... 

D. E. Smith and A. D. J. Haymet. Free energy, entropy and internal energy of hydrophobic interactions computer simulations. J. Chem. P/iys., 98 6445-6454,... 

T.P. Lybrand, Computer simulations of biomolecular systems using molecular dynamics and free energy perturbation methods, in Reviews in Computational Chemistry, Vol. 1, K.B. Lipkowitz, D.B. Boyd (Eds.), VCH, New York, 1990, pp. 295-320. 

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. 

Beveridge D L and F M DiCapua 1989. Free Energy via Molecular Simulation A Primer. In van Gunsteren W F and P K Weiner (Editors) Computer Simulation of Biomolecular Systems. Leiden, ESCOM, pp. 1-26. 

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. 

Mezei M and D L Beveridge 1986. Free Energy Simulations. In Beveridge D L and W L Jorgens (Editors) Computer Simulation of Chemical and Biomolecular Systems. Annals of the New Y Academy of Sciences 482 1-23. 

Jorgensen W L, J K Buckner, S Boudon and J Tirado-Reeves 1988. Efficient Computation of Absoluti Free Energies of Binding by Computer Simulations - Applications to the Methane Dimer ir Water. Journal of Chemical Physics 89 3742-3746. 

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