An 𝒪(n) framework for internal coordinate molecular dynamics applicable to molecules with arbitrary constraints and geometries

For molecules with constraints such as fixed lengths and angles, it is more efficient to consider the molecular movement in the space of generalised internal coordinate than in Cartesian coordinate. This paper presents a new framework in the simulation of molecular movement, especially for macro-molecules with massive length and angle constraints. The generalised forces are calculated to invert the dense mass matrix for integrating the constraints equation. The inverting of the mass matrix was made based on distance descending ordering method. The method does an reordering of the internal variables to make the Cholesky decomposition need no fill-in, which promises the O(n) time complexity in doing mass matrix inverting. The method was extended for application to loop structures. It is found that the mass matrix would become singular when the bond angles approach 0 or π; thus a rotation convention switch method was proposed to resolve the singularity. The O(n) time complexity has been demonstrated and the length and angle constraints can be arbitrarily applied. The long-time energy conservation in NVE ensemble was compared for results from symplectic and non-symplectic time integrators. The non-symplectic fourth-order Runge–Kutta method still has satisfactory long-time energy conservation if using small time step.