Simulation of Molecules

Iterate steps 3 through 5 until the system has equilibrated. 

There are a few variations on this procedure called importance sampling or biased sampling. These are designed to reduce the number of iterations required to obtain the given accuracy of results. They involve changes in the details of how steps 3 and 5 are performed. For more information, see the book by Allen and Tildesly cited in the end-of-chapter references. 

The size of the move in step 3 of the above procedure will affect the elhciency of the simulation. In this case, an inefficient calculation is one that requires more iterations to obtain a given accuracy result. If the size is too small, it will take many iterations for the atom locations to change. If the move size is too large, few moves will be accepted. The efficiency is related to the acceptance ratio. This is the number of times the move was accepted (step 5 above) divided by the total number of iterations. The most efficient calculation is generally obtained with an acceptance ratio between 0.5 and 0.7. 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra- 

---

The simulation of molecules in solution can be broken down into two categories. The first is a list of elfects that are not defined for a single molecule, such as diffusion rates. These types of effects require modeling the bulk liquid as discussed in Chapters 7 and 39. The other type of effect is a solvation effect, which is a change in the molecular behavior due to the presence of a solvent. This chapter addresses this second type of effect. 

Simulation of molecules can be done at the quantum mechanical level, as is necessaiy to determine the electronic properties of molecules, to analyze covalent bonds or simulate bond formation and breaking. However, quantum mechanical simulation is extremely computationally intensive and is too time-consuming for all but the smallest molecular systems. 

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 the last few years, the improvements in computer hardware and software have allowed the simulation of molecules and materials with an increasing number of atoms. However, the most accurate electronic structure methods based on N-particle wavefunctions, for example, the configuration interaction (Cl) method or the coupled-cluster (CC) method, are computationally too expensive to be applied to large systems. There is a great need for treatments of electron correlation that scale favorably with the number of electrons. 

A significant issue widi modem force fields is that it can be difficult to simultaneously address both generality and suitability for use in condensed-phase simulations. For example, the MMFF94 force field is reasonably robust for gas-phase conformational analysis over a broad range of chemical functional groups, but erroneously fails to predict a periodic box of n-butane to be a liquid at —0.5 °C (Kaminski and Jorgensen 1996). The OPLS force field, on the other hand, is very accurate for condensed-phase simulations of molecules over which it is defined, but it is an example of a force field whose parameterization is limited primarily to functionality of particular relevance to biomolecules, so it is not obvious how to include arbitrary solutes in the modeling endeavor. 

The first computational investigations of phospholipids were undertaken using purely statistical methods to describe overall properties. In the 1980s the simulation of molecules and their behavior using an explicit atomic description was extended to large systems such as proteins and lipids. To simulate the static and dynamic behav-... 

It will be certainly worthwhile to meet all these challenges. Dynamical simulations of molecule-surface interactions from first principles have been very successful in the past, and will continue to be so in the future. 

Semiempirical techniques are the next level of approximation for computational simulation of molecules. Compared to molecular mechanics, this approach is slow. The formulations of the self-consistent field equations for the molecular orbitals are not rigorous, particularly the various approaches for neglect of integrals for calculation of the elements of the Fock matrix. The emphasis has been on versatility. For the larger molecular systems involved in solvation, the semiempirical implementation of molecular orbital techniques has been used with great success [56,57]. Recent reviews of the semiempirical methods are given by Stewart [58] and by Rivail [59],... 

Figure 16 BO-MD simulation of molecule 13. Relative potential energy (A/ilxlt) (in a.u.) vs. time (in ps)...

Xantheas and co-workers [159,160] have incorporated polarization in a model scheme and have used that to provide a clear basis for the enhancement of water s dipole in ice. A model potential with polarization has been reported for the formaldehyde dimer [161]. It is an example of a carefully crafted potential, which is system-specific because of its application to pure liquid formaldehyde, but which has terms associated with properties and interaction elements as in the above models. As well, some of the earliest rigid-body DQMC work, which was by Sandler et al. [162] on the nitrogen-water cluster, used a potential expressed in terms of interaction elements derived from ab initio calculations with adjustment (morphing). Stone and co-workers have developed interaction potentials for HF clusters [163], water [164], and the CO dimer [165], which involve monomer electrical properties and terms derived from intermolecular perturbation theory treatment. SAPT has been used for constructing potentials that have enabled simulations of molecules in supercritical carbon dioxide [166]. There are, therefore, quite a number of models being put forth wherein electrical analysis and/or properties of the constituents play an essential role, and some where electrical analysis is used to understand property changes as well as the interaction energetics. 

Linse, R and Soderman, O. The validity of the short-gradient-pulse approximation in NMR studies of restricted diffusion. Simulations of molecules diffusing between planes, in cylinders and spheres, /. Magn. Reson. Ser. A, 116, 77, 1995. 

Molecular dynamics simulations can be done on molecules in the gas phase (in vacuo), in the liquid phase as a pure liquid or dilute solution, and in the solid phase. In the simulation of molecules in the liquid and solid phase, periodic boundary conditions are used to reduce the surface effects because of the limited number of molecules that can reasonably be studied. The main principle is that as an atom or molecule leaves the main box, its image from one of the adjacent boxes enters. A natural consequence of periodic boundary conditions is the concept of minimum image convention. That is, a molecule will interact with all the N-1 molecules whose centers lie within a region of the same size and shape as that of the original box (see Figure 4). ... 

Recently we developed a quantum/molecular mechaiucs (QM/MM) method for the accurate statistical simulation of molecules in polar environments such as in aqueous solutions [30, 31], which speeds up the calculations by 4 or more orders of magnitude or more, even for relatively small solutes and at the inexpensive DFT level. The generalized polarizabilities in our protocol can be used for the efficient... 

In a perturbation approach the interaction energy between two molecules is treated as a small perturbation to the ground state wavefunction of the isolated molecules. For weak perturbations, such as intermolecular interactions, one can separate the interaction terms into components that depend only on the properties of isolated molecules, a primary approximation in virtually all fields of simulations of molecules. The general functions of these interaction terms will also give the appropriate functional forms of the potential surface used in most molecular simulations. A division of the total molecular interaction can schematically be written... 

In the simulation of molecules in aqueous solution, treatment of the solvent and solvent ions is particularly demanding both theoretically and computationally. Continuum or implicit solvent models provide a computationally efficient alternative to the inclusion of explicit solvent molecules. These models typically treat electrostatic and nonpolar contributions separately. A popular approach is to treat, respectively, the former with a continuum electrostatics model and the latter with a surface area model. These models often provide free energies and enthalpies to an accuracy that is comparable to, or better than, that of explicit solvent models, but with two or more orders of magnitude less computation. Other properties such as solvent induced forces, solvent structural features and specific hydrogen-bonding interactions are better treated with explicit solvent models... 

---