The Angular Localization Function (ALF): a practical tool to measure solvent angular order with Molecular Density Functional Theory

Abstract

Molecular density functional theory is a powerful technique for efficiently computing the spatially and orientationally dependent equilibrium density of a molecular solvent around an arbitrary solute. This density encodes the detailed solvent structure, but contains so much information that it is difficult to interpret comprehensively. Although spatial dependence is frequently analyzed through orientationally integrated number density, angular information remains poorly exploited. The present work addresses this gap by introducing a function that provides a local measure of the angular order: the Angular Localization Function (ALF), derived from the ideal free energy functional, which quantifies the entropy. We discuss the connections between ALF and well known statistical functions. We illustrate the utility of ALF by discussing the solvent structure for three systems immersed in water: water as a solute, an octanol molecule, and three clay minerals (talc, fluorotalc and pyrophyllite) with small differences in their structure leading to subtle effects on their interactions with water. ALF provides information complementary to quantities such as the average polarization or charge density to characterize the local orientational distribution of solvent molecules around solutes and next to surfaces. It also offers a convenient visualization tool akin to the Electronic Localization Function (ELF) used to analyze bonding in quantum chemistry.

Figures (5)

• Figure 1: Oxygen density with respect to the homogeneous density $n({\boldsymbol{r}})/n_0$ (solid black), angular localization function ALF$({\boldsymbol{r}})$ defined by Eq. \ref{['eq:ALF']} (dashed red), Jensen-Shannon divergence $D_\text{JS}$ defined by Eq. \ref{['eq:JS']} (dashed blue) and parallel polarization (dotted orange) computed according to Eq. \ref{['eq:P_parallel']} around a water molecule. Results are computed from the equilibrium density $\rho({\boldsymbol{r}},{\boldsymbol{\Omega}})$ obtained by MDFT and plotted along two specific axes (indicated by black arrows) around a water molecule: (A) along an O-H bond and (B) along an oxygen lone-pair direction (see text). The distance $r$ is defined from the H or O atom in the former and latter case, respectively. The number density $n/n_0$ is a dimensionless quantity whose values can be read on the left axis. ALF and $D_\text{JS}$ are dimensionless quantities whose values can be read on the right axis. The polarization density is in arbitrary units to fit on the graph and since the sign of the projection $P_\parallel$ in the direction considered in panel (B) is negative, we report $-P_\parallel$ in that case.
• Figure 2: Slices of ALF (left) and of $S_{\Omega}$ (right) obtained for SPC/E water solvent in the plane containing the C (cyan), O (red) and H (white) atoms of an octanol molecule using MDFT. Both quantities have been normalized with respect to the maximum value to ease comparison. High value regions are in magenta while zero regions are in white.
• Figure 3: Maximum value of ALF along the O-H direction, for octanol molecules with a modified charge distribution within the alcohol group ($q_{\rm O}=q_{\rm O}^{ref}+\Delta q$ and $q_{\rm H}=q_{\rm H}^{ref}-\Delta q$). $\Delta q/e=0$ corresponds to the reference case discussed in Fig. \ref{['fig:octanol_maps']}. The parallel polarization profiles in the O-H direction, $P_\parallel(r)$, with $r$ the distance from the H atom are also shown as insets for some representative values of $\Delta q$.
• Figure 4: Top and side views of pyrophyllite (A,B) and talc (C,D). Oxygen atoms are in red, hydrogen atoms in white, silicon atoms in yellow, aluminium atoms in pink and magnesium atoms in blue. Fluorotalc has the same structure as talc except that the hydroxyl groups in the hexagonal cavities are replaced by fluorine atoms.
• Figure 5: Angular localization function (solid lines) and oxygen density (dashed lines) along the line perpendicular to the surface and passing through the center of the hexagonal rings. The curves for pyrophyllite, talc and fluorotalc are shown in green, blue and red, respectively.