Numerical integration of the equations of motion for rigid polyatomics: The matrix method

We present a new method for isothermal rigid body simulations using the quaternion representation and Langevin dynamics. It can be combined with the traditional Langevin or gradient (Brownian) dynamics for the translational degrees of freedom to correctly sample the NVT distribution in a simulation of rigid molecules. We propose simple, quasi-symplectic second-order numerical integrators and test their performance on the TIP4P model of water. We also investigate the optimal c...

A class of trigonometric integrator is proposed for the constrained ring polymer Hamiltonian dynamics, arising from the path integral molecular dynamics. The integrator is formulated by the composition of flows, thereby integrating the Cartesian equations of motions under normal mode representation and preserving the holonomic constraints by iterations. It is illustrated that the trigonometric method can preserve the symplectic structure and time-reversibility, and its near-c...

Classical molecular dynamics simulation is performed mostly using the established velocity Verlet integrator or other symplectic propagation schemes. In this work, an alternative formulation of numerical propagators for classical molecular dynamics is introduced based on an expansion of the time evolution operator in series of Chebyshev and Newton polynomials. The suggested propagators have, in principle, arbitrary order of accuracy which can be controlled by the choice of ex...

This is a book chapter soon to appear (2002) in the "Handbook for Numerical Analysis" volume dedicated to "Computational Chemistry" edited by Claude Le Bris. The series editors are P.G. Ciarlet and J. L. Lions. [North Holland/Elservier]. This review deals with some of the methods known under the umbrella term quantum Monte Carlo (QMC), specifically those that have been most commonly used for electronic structure.

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometr...

The transformation from angle-action variables to Cartesian coordinates is a crucial step of the (semi) classical description of bimolecular collisions and photo-fragmentations. The basic reason is that dynamical conditions corresponding to experiments are ideally generated in angle-action variables whereas the classical equations of motion are ideally solved in Cartesian coordinates by standard numerical approaches. To our knowledge, the previous transformation is available ...

Molecular Dynamics method is based on solution of Newtonian differential equations of motion. A new very accurate and efficient time-reversible explicit integrator was derived on the basis of second order Tailor expansion of force. There is good reason to think the new method will be easy-to-use for MD and, possibly, celestial mechanics applications.

The numerical integration plays a fundamental role in understanding the behaviour of many mechanical systems. In this paper some important aspects of the mechanical integrators on the dynamics of a mechanical system are studied. More specific, we have shown that if that the Lie-Trotter integrator is obtained, in case of Euler equations for the dynamics of symmetric free rigid body, then it is a Poisson integrator. At the end of the paper some important remarks are presented.

A recent article in J. Chem. Phys. argues that the two algorithms, the velocity-Verlet, and position-Verlet integrators, commonly used in Molecular Dynamics (MD) simulations, are different \cite{Ni2024}. But not only are the two algorithms just different formulations of the same discrete algorithm, but so are other simple discrete algorithms used in MD in the natural sciences. They are all reformulations of the discrete algorithm derived by Newton in 1687 in $\textit{Proposit...

We introduce a new Adaptive Integration Approach (AIA) to be used in a wide range of molecular simulations. Given a simulation problem and a step size, the method automatically chooses the optimal scheme out of an available family of numerical integrators. Although we focus on two-stage splitting integrators, the idea may be used with more general families. In each instance, the system-specific integrating scheme identified by our approach is optimal in the sense that it prov...