Optimizing the Accuracy and Efficiency of Fast Hierarchical Multipole Expansions for MD Simulations

Abstract

Based on p'th order Cartesian Taylor expansions of Coulomb interactions and on hierarchical decompositions of macromolecular simulation systems into hierarchies of nested, structure-adapted, and adaptively formed clusters of increasing size, fast multipole methods are constructed for rapid and accurate calculations of electrostatic interactions. These so-called SAMMp algorithms are formulated through totally symmetric and traceless tensors describing the multipole moments and the coefficients of local Taylor expansions. Simple recursions for the efficient evaluation and shifting of multipole moments are given. The required tensors are explicitly given up to order p = 4. The SAMMp algorithms are shown to guarantee the reaction principle. For systems with periodic boundaries, a reaction field (RF) correction is applied, which introduces at distances beyond the "minimum image convention" boundary a dielectric continuum surrounding each cluster at the top level of coarse graining. The correctness of the present SAMMp implementation is demonstrated by analyzing the scaling of the residuals and by checking the numerical accuracy of the reaction principle for a pair of distant molecular ions in vacuum. Molecular dynamics simulations of pure water and aqueous solutions containing artificial ions, which are enclosed by periodic boundaries, demonstrate the stability and low-noise behavior of SAMMp/RF.

Coupling density functional theory to polarizable force fields for efficient and accurate Hamiltonian molecular dynamics simulations

“…Due to a balanced combination of mth order multipole moments with nth order local Taylor expansions, which is expressed by the equation p = n + m, the electrostatic forces calculated with SAMM p exactly obey Newton's reaction principle. 47,64 Furthermore, this choice additionally guarantees a minimal computational effort for a predefined level p of accuracy (p = 4 is the standard of the current implementation available in IPHIGENIE). A predecessor version called SAMM had been used in Eichinger's DFT/MM approach, 34 whose fully Hamiltonian DFT/(P)MM extension will be explained below.…”

“…The symbol denotes the inner contraction product of two tensors (Ref. 47 thoroughly explains the employed tensorial notation). Finally, the class specific expansion coefficients The charges and induced dipoles of the PMM atoms j ∈ C 1 μ generate the coefficients T n,p (r μ | C 1 μ ) (lower dashed arrow).…”

“…The PMM atoms in C 2 μ are collected into structural units, whose electrostatic signatures are represented by multipole expansions. 47 For the PMM unit u , for instance, such an expansion is symbolized by three black dotted arrows pointing toward its reference point "×". The multipole potentials originating from u are expanded into a Taylor series at the reference point of the DFT unit u (upper dashed arrow), from which the additional contributions T n,p (r μ | C 2 μ ) to the atom-centered expansion coefficients are inherited (dotted arrow) by a simple shifting procedure.…”

“…The multipole potentials originating from u are expanded into a Taylor series at the reference point of the DFT unit u (upper dashed arrow), from which the additional contributions T n,p (r μ | C 2 μ ) to the atom-centered expansion coefficients are inherited (dotted arrow) by a simple shifting procedure. 47 are nth rank tensors generated by the nth order partial derivatives of the potentials T (r | C l μ ) at r μ , which originate from point-like electrostatic moments (cf. the discussion of Fig.…”

“…We give analytical expressions for the calculation of the forces and, therefore, are able to employ the new DFT/PMM scheme for molecular dynamics simulations. For the implementation, the program packages of choice are the parallelized PMM-MD program IPHIGENIE 47 and the parallelized grid-based plane wave DFT program CPMD.…”