Energy conservation in molecular dynamics

We want to derive an expression for the rate of change of the energy with time, dE/dt First, we differentiate the kinetic energy term with respect to time  

The potential energy is written as a series of pairwise interaction terms  

The derivative of the potential energy with respect to time can be written  

For a given atom i, there will be a total of N -1 terms of the form v rij) in the expression for the potential energy due to the interactions between i and all other atoms. Hence we can write diT/dt as follows  

The force on atom i due to its interaction with atom equals minus the gradient with respect to Ti, or —dv rij)/dii. Thus the total force on the atom is equal to 

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Teleman, O. An efficient way to conserve the total energy in molecular dynamics simulations boundary effects on energy conservation and dynamic properties. Mol. Simul. 1 (1988) 345-355. 

In order to conserve the total energy in molecular dynamics calculations using semi-empirical methods, the gradient needs to be very accurate. Although the gradient is calculated analytically, it is a function of wavefunction, so its accuracy depends on that of the wavefunction. Tests for CH4 show that the convergence limit needs to be at most le-6 for CNDO and INDO and le-7 for MINDO/3, MNDO, AMI, and PM3 for accurate energy conservation. ZINDO/S is not suitable for molecular dynamics calculations. 

When the cutoff is sharp, discontinuities in the forces and resultant loss of conservation of energy in molecular dynamics calculations can result. To minimize edge effects of a cutoff, often the cutoff is implemented with a switching or shifting function to allow the interactions to go smoothly to zero. 

Wang, J., Cai, Q., Li, Z.-L, Zhao, H.-K., and Luo, R. (2009). Achieving energy conservation in Poisson-Boltzmann molecular dynamics Accuracy and precision with finite-difference algorithms, Chem. Phys. Lett. 468, pp. 112-118. 

Smooth Cut-Off. In molecular dynamics simulations it is convenient to have a pair potential with compact support for efficiency reasons. While it is possible to simply use a split function (setting the potential—or force—to zero outside some finite range), this has undesirable consequences for energy conservation if the function is not sufficiently smooth. Consider the alternative [346] ... 

S-S annihilation phenomena can be considered as a powerful tool for investigating tire exciton dynamics in molecular complexes [26]. However, in systems where tliat is not tire objective it can be a complication one would prefer to avoid. To tliis end, a measure of suitably conservative excitation conditions is to have tire parameter a< )T < 0.01. Here x is tire effective rate of intrinsic energy dissipation in tire ensemble if tire excitation is by CW light, and T = IS tire... 

Kitchen, D.B., Hirata, F., Westbrook, J.D., Levy, R. Conserving energy during molecular dynamics simulations of water, proteins, and proteins in water. J. Comput. Chem. 11 (1990) 1169-1180. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

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