OrthoBoXY: A Simple Way to Compute True Self-Diffusion Coefficients from MD Simulations with Periodic Boundary Conditions without Prior Knowledge of the Viscosity

Computing the frequency-dependent NMR relaxation of 1H nuclei in liquid water

We present a computational framework for reliably determining the frequency-dependent intermolecular and intramolecular nuclear magnetic resonance (NMR) dipole-dipole relaxation rates of spin 1/2 nuclei from Molecular Dynamics (MD) simulations. This approach avoids the alterations caused by the well-known finite-size effects of translational diffusion. Moreover, a procedure is derived to control and correct for effects caused by fixed distance-sampling cutoffs and periodic boundary conditions. By construction, this approach is capable of accurately predicting the correct low-frequency scaling behavior of the intermolecular NMR dipole-dipole relaxation rate and thus allows for the reliable calculation of the frequency-dependent relaxation rate over many orders of magnitude. Our approach is based on the utilization of the theory of Hwang and Freed for the intermolecular dipole-dipole correlation function and its corresponding spectral density [L.-P. Hwang and J. H. Freed, J. Chem. Phys. 63, 4017-4025 (1975)] and its combination with data from MD simulations. The deviations from the Hwang and Freed theory caused by periodic boundary conditions and sampling distance cutoffs are quantified by means of random walker Monte Carlo simulations. An expression based on the Hwang and Freed theory is also suggested for correcting those effects. As a proof of principle, our approach is demonstrated by computing the frequency-dependent intermolecular and intramolecular dipolar NMR relaxation rates of 1H nuclei in liquid water at 273 and 298 K based on the simulations of the TIP4P/2005 model. Our calculations are suggesting that the intermolecular contribution to the 1H NMR relaxation rate of the TIP4P/2005 model in the extreme narrowing limit has previously been substantially underestimated.

Computing Accurate True Self-Diffusion Coefficients and Shear Viscosities Using the OrthoBoXY Approach

In a recent paper [Busch, J.; Paschek, D. J. Phys. Chem. B 2023, 127, 7983-7987], we have shown that for molecular dynamics (MD) simulations using orthorhombic periodic boundary conditions with "magic" box length ratios of Lz/Lx = Lz/Ly = 2.7933596497, the self-diffusion coefficients Dx and Dy in x- and y-directions are independent of the system size. They both represent the true self-diffusion coefficient D0 = (Dx + Dy)/2, while the shear viscosity can be calculated from diffusion coefficients in x-, y-, and z-directions, using η = kBT·8.1711245653/[3πLz(Dx + Dy - 2Dz)]. In this contribution, we test this "OrthoBoXY" approach by its application to a variety of different systems: liquid water, dimethyl ether, methanol, triglyme, water/methanol mixtures, water/triglyme mixtures, and imidazolium-based ionic liquids. The chosen systems range from small-sized molecular liquids to complex mixtures and ionic liquids, while spanning a viscosity range of almost 3 orders of magnitude. We assess the efficiency of the method for computing true self-diffusion and viscosity data and provide simple formulas for estimating the required MD simulation lengths and sizes for delivering reliable data with targeted uncertainty levels. Our analysis of the system size dependence of statistical uncertainties for both the viscosity and the self-diffusion coefficient leads us to the conclusion that it is preferable to extend the simulation length instead of increasing the system size. MD simulations consisting of 768 molecules or ion pairs seem to be perfectly adequate.

An OrthoBoXY-method for various alternative box geometries

We have shown in a recent contribution [Busch and Paschek, J. Phys. Chem. B, 2023 127, 7983-7987] that for molecular dynamics (MD) simulations of isotropic fluids based on orthorhombic periodic boundary conditions with "magical" box length ratios of Lz/Lx = Lz/Ly = 2.7933596497, the computed self-diffusion coefficients Dx and Dy in x- and y-direction become system size independent. They thus represent the true self-diffusion coefficient D0 = (Dx + Dy)/2, while the shear viscosity can be determined from diffusion coefficients in x-, y-, and z-direction, using the expression η = kBT·8.1711245653/[3πLz(Dx + Dy - 2Dz)]. Here we present a more generalized version of this "OrthoBoXY"-approach, which can be applied to any orthorhombic MD box of any shape. In particular, we would like to test, how the efficiency is affected by using a shape more akin to the cubic form, albeit with different box length ratios Lx/Lz ≠ Ly/Lz and Lx < Ly < Lz. We use NVT and NpT simulations of systems of 1536 TIP4P/2005 water molecules as a benchmark and explore different box geometries to determine the influence of the box shape on the computed statistical uncertainties for D0 and η. Moreover, another "magical" set of box length ratios is discovered with Ly/Lz = 0.57804765578 and Lx/Lz = 0.33413909235, where the self-diffusion coefficient in x-direction becomes system size independent, such that D0 = Dx.

System Size Dependence of the Diffusion Coefficients in MD Simulations: A Simple Correction Formula for Pure Dense Fluids

A practical correction formula relating the self-diffusion coefficient of dense liquids from molecular dynamics (MD) simulations with periodic boundary conditions to the self-diffusion coefficient in the thermodynamic limit is discussed. This formula applies to pure dense fluids and has a very simple form D = D0(1 - γN-1/3), where D0 is the self-diffusion coefficient in the thermodynamic limit and N is the number of particles in the simulation. The numerical factor γ is dependent on the geometry of the simulation cell. Remarkably, γ ≃ 1.0 is the most popular cubic geometry. The success of this formula is supported by results from MD simulations, including very recent simulations with a "magic" simulation geometry.