Classical molecular simulation methods

Classical molecular simulation methods such as MC and MD represent atomistic/molecular-level modeling, which discards the electronic degrees of freedom while utilizing parameters transferred from quantum level simulation as force field information. A molecule in the simulation is composed of beads representing atoms, where the interactions are described by classical potential functions. Each bead has a dispersive pair-wise interaction as described by the Lennard-Jones (LJ) potential, ULj(Ly) ... 

Increases in computer power and improvements in algorithms have greatly extended the range of applicability of classical molecular simulation methods. In addition, the recent development of Internal Coordinate Quantum Monte Carlo (ICQMC) has allowed the direct comparison of classical simulations and quantum mechanical results for some systems. In particular, it has provided new insights into the zero point energy problem in many body systems. Classical studies of non-linear dynamics and chaos will be compared to ICQMC results for several systems of interest to nanotechnology applications. The ramifications of these studies for nanotechnology applications will be discussed. 

However, theories that are based on a basis set expansion do have a serious limitation with respect to the number of electrons. Even if one considers the rapid development of computer technology, it will be virtually impossible to treat by the MO method a small system of a size typical of classical molecular simulation, say 1000 water molecules. A logical solution to such a problem would be to employ a hybrid approach in which a chemical species of interest is handled by quantum chemistry while the solvent is treated classically. 

Contemporary computer-assisted molecular simulation methods and modern computer technology has contributed to the actual numerical calculation of solvent effects on chemical reactions and molecular equilibria. Classical statistical mechanics and quantum mechanics are basic pillars on which practical approaches are based. On top of these, numerical methods borrowed from different fields of physics and engineering and computer graphics techniques have been integrated into computer programs running in graphics workstations and modem supercomputers (Zhao et al., 2000). 

Figure 2. Illustration of simulation techniques available at various time and length scales. QC means first principles, quantum chemical methods. MD refers to classical molecular dynamics methods. (Monte Carlo methods are useful in roughly the same range of time and distance.) Methods for connecting QC, MD, and continuum methods are indicated in parentheses.

The success of any molecular simulation method relies on the potential energy function for the system of interest, also known as force fields [27]. In case of proteins, several (semi)empirical atomistic force fields have been developed over the years, of which ENCAD [28,29], AMBER [30], CHARMM [31], GRO-MOS [32], and OPLSAA [33] are the most well known. In principle, the force field should include the electronic structure, but for most except the smallest systems the calculation of the electronic structure is prohibitively expensive, even when using approximations such as density functional theory. Instead, most potential energy functions are (semi)empirical classical approximations of the Born-Oppenheimer energy surface. 

By far the major computational quantum mechanical method used to compute the electronic state in Car-Parrinello simulations is density-functional theory (DFT) (Hohenberg and Kohn, 1964 Kohn and Sham, 1965 Parr and Yang, 1989). It is the method used originally by Roberto Car and Michele Parrinello in 1985, and it provides the highest level of accuracy for the computational cost. For these reasons, in this section the only computational quantum mechanical method discussed is DFT. Section A consists of a brief review of classical molecular dynamics methods. Following this is a description of DFT in general (Section B) and then a description of practical DFT computations of chemical systems using the plane-wave pseudopotential method (Section C). The section ends with a description of the Car-Parrinello method and some basic issues involved in its use (Section D). 

Because many details of the dynamics and structure of chemical systems cannot be directly observed, molecular simulation methods such as molecular dynamics (MD) [1-31, molecular mechanics (MM) [4], and classical and quantum Monte Carlo [5,6] are extremely valuable tools for making sense of experimental results. In the context of nanotechnology, molecular simulation is crucial for studying the feasibility of proposed directions of research and development [7], With the rapid improvement in computing power and algorithms, the capabilities and range of applicability of molecular simulation have dramatically increased over the past decade. 

Norman, G.E., Stegailov, V.V. Stochastic theory of the classical molecular d5mamics method. Math. Models Comput. Simul. 5, 305-333 (2013)... 

Statistical mechanical Monte Carlo as well as classical molecular dynamic methods can be used to simulate structure, sorption, and, in some cases, even diffusion in heterogeneous systems. Kinetic Monte Carlo simulation is characteristically different in that the simulations follow elementary kinetic surface processes which include adsorption, desorption, surface diffusion, and reactivity . The elementary rate constants for each of the elementary steps can be calculated from ab initio methods. Simulations then proceed event by event. The surface structure as well as the time are updated after each event. As such, the simulations map out the temporal changes in the atomic structure that occur over time or with respect to processing conditions. 

Over the past 10-15 years computational methods have been developed which permit the study of protein and nucleic acid motions and structure, as well as some aspects of their reactivity. These techniques, known as biopolymer dynamics and mechanics [1,2], evolved from pioneering work by Alder and Wainwright [3] and Rahman [4] on the classical simulation of condensed phase systems. They were solidified by the first application of classical molecular dynamics to proteins by McCammon, Gelin and Karplus in 1977 [5]. Today a broad range of biophysical processes are explored using molecular simulation methods [1, 2]. 

At present the best method for calculation is the ab-initio molecular-dynamics method allowing simultaneous calculation of the evolution of the atomic system and electron subsystem. In this chapter, however, the classical molecular d5Uiamics method in combination with semiempirical potentials of atomic interaction is used in the fiamework of the embedded-atom method (EAM) [10]. On the one hand, the EAM-approach proved to be good for the simulation of the metal atomic stmcture in crystalline and liquid states. On the other hand, the EAM-approach is a reasonable compromise between the calculation complexity and physical validity, which allows to conduct the simulation of a system consisting of a larger munber of atoms than that in Refs. [6-9]. In addition, it will allow to establish to what extent the results of the local cluster stmcture simulation are sensitive to the model describing interatomic bonds. 

A method is proposed to predict the IR spectra of amorphous polymers. Based on classical molecular simulation and Kramers-Kronig relations, it allows the computations of absorption and transmittance spectra of polymer films in near and middle infra-red domains with good agreement with experimental data. 22 refs. 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

Bala, P., Grochowsky, R, Lesyng, B., McCammon, J.A. Quantum-classical molecular dynamics. Models and applications. In Quantum mechanical simulation methods for studying biological systems, D. Bicout and M. Field, eds. Springer, Berlin (1996) 119-156. 

The principal idea behind the CSP approach is to use input from Classical Molecular Dynamics simulations, carried out for the process of interest as a first preliminary step, in order to simplify a quantum mechanical calculation, implemented in a subsequent, second step. This takes advantage of the fact that classical dynamics offers a reasonable description of many properties of molecular systems, in particular of average quantities. More specifically, the method uses classical MD simulations in order to determine effective... 

P. Bala, P. Grochowski, B. Lesyng, and J. A. McCammon Quantum-classical molecular dynamics. Models and applications. In Quantum Mechanical Simulation Methods for Studying Biological Systems (M. Fields, ed.). Les Houches, France (1995)... 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

Molecular mechanics simulations use the laws of classical physics to predict the structures and properties of molecules. Molecular mechanics methods are available in many computer programs, including MM3, HyperChem, Quanta, Sybyl, and Alchemy. There are many different molecular mechanics methods. Each one is characterized by its particular/orce eW. A force field has these components ... 

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