Scheme of inner distance classes for the FMM evaluation of the DFT/PMM electrostatics: A water molecule (left) representing a structural unit u (dotted circle) of the SAMM p hierarchy is embedded in a rectangular DFT box, which is discretized by a grid (dots). Only those points γ ∈ G μ are shown which belong to a selected DFT atom μ (large black dot) through the Voronoi tessellation of the box. This tessellation defines the index sets G μ and is indicated by the dashed gray lines. Two dashed gray segments of circles [radii , ] around μ and the reference point “×” of the structural unit u indicate the outer limits of the distance classes and , respectively. Representative atoms and of PMM water molecules belonging to these two classes and of a structural unit are drawn as black dots.
Evaluation of Φext at a grid point γ ∈ G μ: Contributions from Gaussian charges and induced dipoles of a PMM atom are evaluated directly (solid gray arrow), whereas the contributions from more distant atoms, like the one indicated by the dashed black arrow for a PMM atom , are calculated by a Taylor expansion around the position of the DFT atom μ. The dotted gray arrow marks the connection r γμ of the points μ and γ used in the Taylor expansion.
Computation of Taylor expansion coefficients: The charges and induced dipoles of the PMM atoms generate the coefficients (lower dashed arrow). The PMM atoms in 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 to the atom-centered expansion coefficients are inherited (dotted arrow) by a simple shifting procedure. 47
Energy conservation in reference simulations of the water dimer. (a) PMM dynamics at close contact (exact electrostatics, l = 0) and (b) softly restrained to a distance d OO ≈ 10 Å (SAMM4 electrostatics, l = 2). (c) DFT Born-Oppenheimer dynamics at close contact.
Energy conservation in DFT/PMM hybrid simulations of the water dimer with the electrostatics treated at different distance class levels l. (a) Close contact (l = 0), (b) softy restrained to d OO ≈ 7 Å (l = 1), and (c) to d OO ≈ 10 Å (l = 2).
Influence of the algorithm for DFT box movement on the energy conservation as exemplified by the DFT/PMM water dimer at close contact. The black and gray lines show the trajectories of the total energy in a simulation using a naive and our refined algorithm, respectively (see the text for explanation). The gray line represents the data of Fig. 5(a) on a different energy scale.
The absolute value |p(t)| of the DFT fragment's dipole moment during a MD simulation of the aqueous DFT/PMM system described in Sec. IV . A short (100 fs) section of a trajectory was chosen to visualize the fluctuations of |p(t)| at a high time resolution.
Average computing times (walltimes) spent for our liquid water sample system per MD integration step on the various parts of a DFT/PMM (black) or DFT/MM (gray) calculation. Here, the DFT part was executed in parallel on eight core and the (P)MM part sequentially on one core. The time T MM spent on the MM part in the DFT/MM setting is taken as the reference. For explanation see the text.
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