NMR Relaxation Rates of Quadrupolar Aqueous Ions from Classical Molecular Dynamics Using Force-Field Specific Sternheimer Factors

Abstract

The nuclear magnetic resonance (NMR) relaxation of quadrupolar nuclei is governed by the electric field gradient (EFG) fluctuations at their position. In classical molecular dynamics (MD), the electron cloud contribution to the EFG can be included via the Sternheimer approximation, in which the full EFG at the nucleus that can be computed using quantum density functional theory (DFT) is considered to be proportional to that arising from the external, classical charge distribution. In this work, we systematically assess the quality of the Sternheimer approximation as well as the impact of the classical force field (FF) on the NMR relaxation rates of aqueous quadrupolar ions at infinite dilution. In particular, we compare the rates obtained using an ab initio parametrized polarizable FF, a recently developed empirical FF with scaled ionic charges and a simple empirical nonpolarizable FF with formal ionic charges. Surprisingly, all three FFs considered yield good values for the rates of smaller and less polarizable solutes (Li + , Na + , K + , Cl − ), provided that a model-specific Sternheimer parametrization is employed. Yet, the polarizable and scaled charge FFs yield better estimates for divalent and more polarizable species (Mg 2+ , Ca 2+ , Cs + ). We find that a linear relationship between the quantum and classical EFGs holds well in all of the cases considered; however, such an approximation often leads to quite large errors in the resulting EFG variance, which is directly proportional to the computed rate. We attempted to reduce the errors by including first order nonlinear corrections to the EFG, yet no clear improvement for the resulting variance has been found. The latter result indicates that more refined methods for determining the EFG at the ion position, in particular those that take into account the instantaneous atomic environment around an ion, might be necessary to systematically improve the NMR relaxation rate estimates in classical MD.

Quadrupolar $^{23}$Na$^{+}$ NMR Relaxation as a Probe of Subpicosecond Collective Dynamics in Aqueous Electrolyte Solutions

“…The widely-used approximation (4) permits to calculate the quadrupolar spinlattice relaxation rate (1) using the stochastic dynamics of V ext from classical MD [32,[42][43][44][45]47] to facilitate the long-time sampling of EFG fluctuations. Yet, a consistent value of γ that reflects the charge density representation in a given classical force field (FF) at hand (that is, specific partial water point charges and induced dipoles) needs to be derived using auxiliary ab initio calculations [20]. Despite the straightforward nature of the Sternheimer approximation, in Sec.…”

“…The scaled ionic charges aim at taking into account the electronic contribution to the dielectric constant at high frequencies in a mean-field fashion [62]. At a moderate computational cost in comparison to fully polarizable models, the EFG relaxation within the Madrid-2019 FF [60] has recently been shown to accurately describe the quadrupolar NMR relaxation rates of alkali metal ions at infinite dilution [20], in particular that of Na + . Solutions comprised of N = 1000 water molecules and N p ion pairs were initialized at different salt concentrations c in molal (denoted with mol•kg -1 or m) between 0.06 m (N p = 1) and 4 m (N p = 72) in a cubic box at the equilibrium solution density ρ(c, T ) obtained in N P T simulations at P = 1 bar.…”

“…The relaxation rate is determined from a combination of the strength of the electric field gradient (EFG) around the ion, quantified by means of the quadrupolar coupling constant (QCC) C Q and the characteristic correlation time τ c with which the memory of fluctuations is lost. While absolute values of τ c can provide fundamental insights into the molecular mechanisms behind the quadrupolar relaxation and can be potentially correlated with useful dynamic properties, such as diffusion coefficients, solution viscosity, and conductivity [16][17][18], their unambiguous determination from the experimentally measured rates has remained essentially impossible due to the limited knowledge on the QCC of 23 Na in aqueous electrolytes that is sensitive to the local ionic hydration shell structure [19,20]. In principle, the correlation time can be extracted independently from the maximum of 1/T 1 that develops with reducing the system temperature [21][22][23][24].…”

“…Both ab initio [37][38][39][40][41] and classical [20,32,33,[42][43][44][45][46][47] molecular dynamics (MD) simulations have become indispensable tools in assessing predictions of the abovementioned theories. In particular, the isotropic monoexponential character of the quadrupolar relaxation dynamics that is often assumed in theories with a continuous solvent description was challenged both at the ab initio and classical levels.…”

“…Carof et al [33] underlined a major role of collective fluctuations in ionic solvation shells on the EFG dynamics. Compared to classical MD approaches that rely on approximations for the electron cloud contribution to the EFG at the ion position [20,45,48,49], ab initio methods provide the best accuracy of the computed QCC [37,39,[50][51][52]. However, the associated high computational cost often impedes the long-time sampling of EFG fluctuations [37,38] and, hence, the accuracy of the correlation time estimates, even in the case of infinitely diluted aqueous ions [37,39].…”