Simulation technique

The MC and MD simulation approaches have become viable only after the introduction of fast computers. Starting from the pioneering works of Metropolis etal. [101] and Alder and Wainwright [102], the basic algorithms on which computer simulations are based were developed in the ensuing 20-30 years. They are now well established and described in standard textbooks [95,96], and able to provide a useful link between experiment and theory. Nowadays MC simulations are typically used for lattice and simple off-lattice models, while MD models are largely employed for atomistic systems (which are tricky to sample with MC) but also for coarse-grained models. 

A last increasingly popular family of MC simulations, not to be confused with the Metropolis method, exploits sampling from non-Boltzmann distributions to simulate kinetic events which are extremely rare compared to the typical molecular timescales, e.g., as is the case of charge transfer (dynamic or Kinetic MC [105], KMC) or reactive events (Gillespie s stochastic simulation algorithm [106]). 

A standard MD computer simulation consists in the computation of the trajectory in the phase space of a system of N interacting bodies. The time evolution is determined by solving Newton s equations of motion of classical mechanics with finite difference methods. Such a model system corresponds to the microcanonical ensemble (NVE) of statistical mechanics with a constant number of particles N, volume V, and total energy E. In MD simulations the collective properties are then determined from the trajectory of all particles, i.e., from the time evolution of positions r = r, and momenta p = p,. The method relies on the assumption that stationary values of every average observable A can be defined as time integrals over the trajectory in the phase space  

In more detail, an MD simulation performs a finite difference integration of the equations of motion usually cast into the Hamilton s form 

The momenta at half time step t + AtH are computed from those at time t and the forces at time t  

There are two major techniques for generating an ensemble Monte Carlo and molecular dynamics. 

In Monte Carlo (MC) methods, a sequence of points in phase space is generated from an initial geometry by adding a random kick to the coordinates of a randomly chosen particle (atom or molecule). The new configuration is accepted if the energy decreases and with a probability of e if the energy increases. This Metropolis procedure ensures that the configurations in the ensemble obey a Boltzmann 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

Alternatively stated, the ergodic hypothesis implies that no matter where a system is started, it is possible to get to any other point in phase space. MC techniques perform an ensemble average, while MD performs a time average. 

A simulation can be characterized by quantities such as volume (V), pressure (F), total energy ( ), temperature (T), number of particles (A), chemical potential (jx), etc., but not all of these are independent. For a constant number of particles, either the volume or the pressure can be fixed, but not both. Similarly, either the total energy or the temperature can be fixed, but not both, and a constant chemical potential is incommensurable with a constant number of particles. The ensemble is labelled according to the fixed quantities, as shown in Table 14.1, with the remainder being derived from the simulation data, and thus displaying a statistical fluctuation. 

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For several years, the French Atomic Energy Commission (CEA) has developed modelling tools for ultrasonic NDT configurations. Implemented within the CIVA software for multiple technique NDT data acquisition and processing [1,2], these models are not only devoted to laboratory uses but also dedicated to ultrasonic operators without special training in simulation techniques. This approach has led us to develop approximate models carrying out the compromise between as accurate as possible quantitative predictions and simplicity, speed and intensive use in an industrial context. 

The solution adopted by us is the use of computer simulations of mathematical models of the process and the mock-up situations. Eventually, simulation techniques will become so accurate, that the mock-up step can be discarded. For the time being it is reasonable to use such models to generate corrections for smaller differences between mock-up and process. 

The first term represents the forces due to the electrostatic field, the second describes forces that occur at the boundary between solute and solvent regime due to the change of dielectric constant, and the third term describes ionic forces due to the tendency of the ions in solution to move into regions of lower dielectric. Applications of the so-called PBSD method on small model systems and for the interaction of a stretch of DNA with a protein model have been discussed recently ([Elcock et al. 1997]). This simulation technique guarantees equilibrated solvent at each state of the simulation and may therefore avoid some of the problems mentioned in the previous section. Due to the smaller number of particles, the method may also speed up simulations potentially. Still, to be able to simulate long time scale protein motion, the method might ideally be combined with non-equilibrium techniques to enforce conformational transitions. 

Fig. 5. To generate an ensemble using Molecular Dynamics or Monte-Carlo simulation techniques the interaction between all pairs of atoms within a given cutoff radius must be considered. In contrast, to estimate changes in free energy using a stored trajectory only those interactions which are perturbed need be determined making the approach highly efficient.

In this chapter we shall discuss some of the general principles involved in the two most common simulation techniques used in molecular modelling the molecular dynamics and the Monte Carlo methods. We shall also discuss several concepts that are common to both of these methods. A more detailed discussion of the two simulation methods can be found in Chapters 7 and 8. 

Molecular simulation techniques can be used to predict how a compound will interact with a particular active site of a biological molecule. This is still not trivial because the molecular orientation must be considered along with whether the active site shifts geometry as it approaches. 

Of all the topics discussed in this text, mesoscale simulations are probably at the most infantile stage of development. The idea of the mesoscale calculations is very attractive and physically reasonable. However, it is not as simple as one might expect. The choice of bead sizes and parameters is crucial to obtaining physically relevant results. More complex bead shapes are expected to be incorporated in future versions of these techniques. When using one simulation technique to derive parameters for another simulation, very small errors in a low-level calculation could result in large errors in the final stages. 

Monte Carlo a simulation technique that incorporates a random movement of atoms or molecules... 

RIS (rotational isomeric state) a polymer simulation technique... 

Mathematical modeling, using digital computers, aids in performing a systems-type analysis for either the entire system or parts of it. By means of integer or linear-programming techniques, optimum systems can be identified. The dynamic performance of these can then be determined by simulation techniques. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

Normal mode analysis exists as one of the two main simulation techniques used to probe the large-scale internal dynamics of biological molecules. It has a direct connection to the experimental techniques of infrared and Raman spectroscopy, and the process of comparing these experimental results with the results of normal mode analysis continues. However, these experimental techniques are not yet able to access directly the lowest frequency modes of motion that are thought to relate to the functional motions in proteins or other large biological molecules. It is these modes, with frequencies of the order of 1 cm , that mainly concern this chapter. 

A number of studies have compared normal mode analysis predictions with results from more realistic simulation techniques or experiments. These studies shed light on the nature of the conformational energy surface and the effect of solvent. 

Computer simulation techniques offer the ability to study the potential energy surfaces of chemical reactions to a high degree of quantitative accuracy [4]. Theoretical studies of chemical reactions in the gas phase are a major field and can provide detailed insights into a variety of processes of fundamental interest in atmospheric and combustion chemistry. In the past decade theoretical methods were extended to the study of reaction processes in mesoscopic systems such as enzymatic reactions in solution, albeit to a more approximate level than the most accurate gas-phase studies. 

Molecular simulation techniques, namely Monte Carlo and molecular dynamics methods, in which the liquid is regarded as an assembly of interacting particles, are the most popular... 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

Essentially, the RISM and extended RISM theories can provide infonnation equivalent to that obtained from simulation techniques, namely, thermodynamic properties, microscopic liquid structure, and so on. But it is noteworthy that the computational cost is dramatically reduced by this analytical treatment, which can be combined with the computationally expensive ab initio MO theory. Another aspect of such treatment is the transparent logic that enables phenomena to be understood in terms of statistical mechanics. Many applications have been based on the RISM and extended RISM theories [10,11]. 

The simulations were carried out on a Silicon Graphics Iris Indigo workstation using the CERIUS molecular modeling and the associated HRTEM module. The multislice simulation technique was applied using the following parameters electron energy 400 kV (lambda = 0.016 A) (aberration coefficient) = 2.7 mm focus value delta/ = 66 nm beam spread = 0.30 mrad. 

Uncertainty - analyzes the uncertainty of a system, sequence, or end state using either the Monte Carlo or Latin Hypercube simulation technique. 

Design or trouble shooting During the design stage an on-site experiment with the actual equipment is not usually possible. Numerical prediction may then be simpler. However, for trouble shooting, field measurements are strongly recommended, ideally complemented by simulations techniques. 

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