The Combined Influence of Nuclear Quantum Effects and van der Waals Interactions on the Structure of Ambient Water
Jeffrey M. McMahon
Miguel A. Morales
Brian Kolb
Timo Thonhauser
David M. Ceperley
Path-integral molecular dynamics simulations based on density functional theory employing exchange--correlation density functionals capable of treating nonlocal van der Waals (vdW) interactions self-consistently provide a remarkably accurate description of ambient water. Moreover, they suggest that water's structure may be impacted by a \jmmemph{combined influence between nuclear quantum effects and vdW interactions. The latter strongly favor the formation of a high-density liquid, whereas the inclusion of the former mitigates this by decreasing the mean hydrogen-bond (H-bond) distance. Examining the structure of water reveals that while the major fraction of molecules do in fact exhibit the traditional picture of near-tetrahedral coordination, the liquid considerably softer than previously simulations have suggested, including a much lower proportion of molecules double-donating H-bonds as well as a much larger distribution of their angles.
H2O_PIMD_ms_to_arXiv.tex
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% ! typically, I do not like to do a division of results and discussion, but it is necessary and I think in this manuscript things work out best this way
% ! for Nature Chemistry, I removed ``(Color online)'' (as used in APS journals) from all figure captions
% *****
% ! all of the figures are laid out the same way, and the tables follow this as well
% ! because I moved the temperature-rescaled results (the optB88 at 330 K) to the SI, I removed the statements of things that are at 300 K
% ! be sure to update tables/figures in sync with each other (and SI)
% ! note that the units in most figures are in parenthesis, but in the beta plots they are in brackets (because parenthesis are already used in the y-axis label), I think this is better than putting brackets in everything, for example
% ! I debated many times on the use of PCF vs RDF, and looking through papers, in some cases I see PCF (e.g., Galli 2011) and in others I see RDF (e.g., Car 2008). I ?think? that for a multi-component liquid it would be easier to use PCF and for a single-component and homogeneous/isotropic liquid RDF would be equally good if not better

% ! information/details on the simulations are on the file 'README_ALL-DETAILS_ETC' on my laptop
% ! ... and keep in mind --> the shift inward of the first peak (technically its how it is stationary during the entire PIMD simulation), is perhaps the strongest indication of NO collapse and indicative of convergence of the simulations as well

% !!! JMM
% ! perhaps I should mention something about the quantum kinetic energy vs classical KE
% ! I can't seem to make the beta in text (in relation to fbeta, for example) non-italic
% ! note that the y-axes of figures have g_xx with g non-italic, but in text it is always italic (interestingly, this appears to be consistent with many other papers too)
% ! remove footnotes?
% !!! : recall that there is the driving paper for the 2009 vdW studies that came from ~2000 saying such interactions could explain a number of water's anomalous properties
% !!! : highlight that you could get things right for the wrong reasons and that we are trying to develop a first-principles description of a liquid where small energy differences are very important as the liquid is only 25K from melting, so discuss energy scales, and how you have to include all effects, NQEs, vdW, etc.
% !!! : ? how much stronger are H-bonds vs (outer-neighbor) vdW interactions? in my head I have a number of ~20
% ! ??? note also that I think the vdW interactions should be relatively uniform over a molecule ???
% ! keep in mind that the mixture model is not new, the original "Mixture Model" was that proposed by [W. K. Roentgen, Annu. Phys. 45, 91 (1892)] to explain the thermodynamic anomaly of temperature of max density at 4 deg C -- there is really no need to reference that here (first paragraph) though
% !: could reference when I bring up atomistic simulations: almost exclusively density functional theory (DFT), pioneered in Ref.\ \onlinecite{laasonen:9080}.
% ! maybe elaborate on what type of information the PCF gives
% ! perhaps comment on the slight disagreement with beta, which is commented out in the discussion
% ! is there something I can say about the following, perhaps in the conclusions --> These results allows us to address the intriguing possibility of the anomalous NQEs recently observed in ice [find and re-read this paper]. ] 
% ! also keep in mind the recent calculations that have shown that NQE can result in weak $\text{H}$ bonds getting weaker and strong $\text{H}$ bonds getting stronger [JMM: PNAS 2009]. While water lies slight on the weaker side of this ...
% ! note that both models of water are consistent with the PCFs
% While integration of the second peak in $g_\text{OH}(r)$ indicates that water is near-tetrahedrally coordinated (see the SI), 
% ! maybe work in --> one somewhat important point of the remarkable agreement is that such simulations may prove immensely useful for parametrization empirical models in general environments. 

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\title{The Combined Influence of Nuclear Quantum Effects and van der Waals Interactions on the Structure of Ambient Water}

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% Authors / affiliations
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\author{Jeffrey M. McMahon}
\email[]{mcmahonj@illinois.edu}
% \affiliation{The Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, IL 61801}
\affiliation{Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA}

\author{Miguel A. Morales}
%\email[]{moralessilva2@llnl.gov}
\affiliation{Lawrence Livermore National Laboratory, Livermore, California 94550, USA}

\author{Brian Kolb}
\affiliation{Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, USA}

\author{Timo Thonhauser}
\affiliation{Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, USA}

% ! decided that he did not contribute enough to be included 
% \author{David M. Ceperley}
% %\email[]{ceperley@ncsa.uiuc.edu}
% \affiliation{Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA}
% \affiliation{NCSA, University of Illinois at Urbana-Champaign, Illinois 61801, USA}


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\begin{abstract}
Path-integral molecular dynamics simulations based on density functional theory employing exchange--correlation density functionals capable of treating nonlocal van der Waals (vdW) interactions self-consistently provide a remarkably accurate description of ambient water. Moreover, they suggest that water's structure may be impacted by a \jmmemph{combined} influence between nuclear quantum effects and vdW interactions. The latter strongly favor the formation of a high-density liquid, whereas the inclusion of the former mitigates this by decreasing the mean hydrogen-bond (H-bond) distance. Examining the structure of water reveals that while the major fraction of molecules do in fact exhibit the traditional picture of near-tetrahedral coordination, the liquid considerably softer than previously simulations have suggested, including a much lower proportion of molecules double-donating H-bonds as well as a much larger distribution of their angles. 
\end{abstract}

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\keywords{liquid water; hydrogen bonding; van der Waals interactions; nuclear quantum effects; first-principles simulations; path-integral molecular dynamics}


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%                    ******************************
%                    *            INTRO           *
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% ********************
%\section{Introduction}
%\label{sec:intro}
% ********************

% INTRODUCTION TO H2O AND STRUCTURE CONTROVERSY
% ! note the 3.5 H-bonds may come from the 1980 study, but the traditional view is the 1933 study, but the way this is referenced below seems the most appropriate
% ! note that many of waters thermodynamic, kinetic, and dynamic anomalies have fairly reasonable physical explanations (at least people think this)
% ! in the traditional view of water: water retains features of hexagonal ice ($\text{I}_\text{h}$), displaying features associated with a low-density liquid (LDL)
Water plays a central role in many scientific fields \cite{water_Franks}; consider, for example, its involvement in nearly all chemical, biological, and geophysical processes. Despite such broad importance, water's most basic property, its local structure at ambient conditions, characterized by the geometry of its underlying hydrogen-bond ($\text{H}$-bond) network, has remained a matter of debate for over a century \cite{XAS_water-structure_overview_Ball-Nature-2008, water-structure_review_THG-MolPhys-2010, water-struct_perspective_Nilsson-ChemPhys-2011}. The traditional view asserts that H bonds form a macroscopically-connected random network, where molecules on average exhibit near-tetrahedral coordination ($\jmmapprox 3.5$ H bonds / molecule) \cite{continuum-model_Fowler-JCP-1933} with thermal motion causing continuous topological reformations via straining and breaking \cite{H2O-struct_thermal-fluct_Stillinger-Science-1980}.
% ! the following is unnecessary as the results are superseded -->: An interpretation of recent X-ray absorption (XAS) and Raman spectroscopy measurements, for example, suggests that $\jmmapprox 80\%$ of tetrahedral $\text{H}$ bonds are broken (compared to only $\jmmapprox 20\%$ in the traditional view \cite{water-structure_review_THG-MolPhys-2010}) forming strongly $\text{H}$-bonded chains or rings, displaying features related to a high-density liquid (HDL), where individual clusters are connected by weak $\text{H}$ bonds \cite{H2O-struct_SD-H-bond-model_Wernet-Science-2004}. 
% ! but, in line with the earlier results above, recall that there is also X-ray emission spectroscopy results [see 7 in the Huang 2009 PNAS] that I did not mention
Recent small-angle X-ray scattering (SAXS) data \cite{H2O-struct_HDL-LDL-model_Huang-PNAS-2009}, however, suggests the possibility that structural polyamorphism between interconverting high-density liquid (HDL) and low-density liquid (LDL) polyamorphs may exist not only in the supercooled region \cite{HDA-LDA_HDL-LDL_review_Stanley-Nature-1998}, but at ambient conditions as well. 
% ! the Car 2010 work shows that the tetrahedral picture is still consistent with the XAS and XRS, but does not rule out the possibility of the two-phase scenario
% ! the following SAXS study by THG may not be entirely correct though, as Nilsson's study showed by focusing on the isothermal compressibility
% However, the traditional view remains consistent with the XAS \cite{H2O_XAS-reanalysis_Car-PRL-2010} and SAXS \cite{H2O_SAXS-reanalysis_THG-PNAS-2010} as well.

% ATOMISTIC SIMULATIONS AND IMPORTANCE OF VARIOUS EFFECTS (H-BONDING, vdW, & NQEs)
% ! note that I now do not focus on empirical potentials
% ! note the pioneering MC and MD studies employed different intermolecular potentials (this is discussed in the MD paper)
% ! I originally commented on empirical models giving better experimental description than ab initio, but I think the comparison is more mixed 
% ! note the previous AIMD sims:
%      - the 2005 Parrinello J Chem Phys has a number of references previous AIMD sims
%      - the 2011 vdW Galli paper also has a number of references (for AIMD, vdW functionals, etc.)
%      ?- I think the Marzari 2005 paper that I use the O PP from is probably the goto PBE reference for liquid water
% ! note in the SI under constraints for simulations, some other approximations are outlined

% !!! ???: one thing I noticed, and I am not sure if Galli goes into this, but:: Secondly, by looking at water dimers [JMM: Galli], vdW-DF exhibits an attraction for non-bonded configurations and underestimates the strength of the H-bond). However, this latter point is surprising for vdW-DF2, as the dimers are actually described quite well [JMM: Galli].

Atomistic simulations have the potential to resolve these issues. Unfortunately, many challenges exist to simulating ambient water, even at an \textit{ab initio} level through density-functional theory (DFT) calculations. In fact, providing an accurate theoretical description has remained a central topic and open challenge in physical chemistry for many decades. 
% ! I want to focus on ab initio simulations now, so this is commented out -->: Pioneering simulations \cite{Barker1969144, rahman:3336} employed empirically parametrized intermolecular potentials. While these initial potentials were simple, the development of rather sophisticated models has become an active area of research. However, extracting reliable information from simulations based on such models outside of the parametrization range must always be approached with caution. This caveat does not apply when the intermolecular interactions are obtained from \textit{ab initio} calculations, almost exclusively density functional theory (DFT), pioneered in Ref.\ \onlinecite{laasonen:9080}.
Challenges arise because water is only $\jmmapprox 25$ K from the melting temperature of ice, where a variety of subtle and complex effects become important. While the structure is dominated by $\text{H}$ bonds between neighboring molecules, both van der Waals (vdW) interactions (which, in this context, refers to dispersion forces resulting from dynamical nonlocal electron correlations) and nuclear quantum effects (NQEs) influence the topology of the $\text{H}$-bond network. In fact, as discussed below, recent simulations have demonstrated that each of these effects has a \jmmemph{profound} influence on the underlying H-bond network \cite{water_DFT-D_DCACPs_Rothlisberger-JPCB-2009, water_DFT-D_NpT_Hutter-JPCB-2009, H2O_vdW-DF1_Fernandez-Serra-JCP-2011, H2O_vdW-DF2_Nilsson-JPCB-2011, H2O_NQE_Martyna-ChemPhysLett-2005, H2O_NQE_Car-PRL-2008}. 
%%%%%
%Even including such effects though has still resulted in discrepancies with experiment, such as either an over-structured liquid or the formation of a HDL. These studies, however, have considered such effects in isolation. It is precisely a combination of these seemingly subtle effects (compared to $\text{H}$ bonding) though that we suggest in this Letter are key to accurately describing ambient water, but have been (until recently) difficult or impossible to model.
%%%%%
% While such simulations are in principle without approximations, many are often made in practice, as will be discussed below. Understandably, this has resulted in \textit{ab initio} simulations being met with limited success. 

% vdW INTERACTIONS
% ! note there is an erratum to vdW-DF1 (the proposal of the DF) that I don't think is necessary to put here
% ! note that at one point I was calling vdW interactions long ranged, but they really aren't that long ranged when you consider that 1/r^6 decays much faster than 1/r^3, where long range is typically given by a decay slower than 1/r^d with d being the dimensionality
vdW interactions are particularly important in water, due to the high polarizability of oxygen ($\text{O}$). While much weaker than $\text{H}$ bonds, they can bend those in the first coordination-shell via outer-neighbor interactions. Including vdW interactions in DFT simulations though has traditionally been problematic, as semilocal exchange--correlation density functionals (DFs) (obviously) lack the ability to describe them. As highlighted by recent simulations including empirical vdW corrections \cite{water_DFT-D_DCACPs_Rothlisberger-JPCB-2009, water_DFT-D_NpT_Hutter-JPCB-2009}, their neglect may be one of the primary reasons for such simulations to typically result in an over-structured (or supercooled) liquid with a slow self-diffusion and an overestimation of the number of $\text{H}$ bonds per molecule \cite{grossman:300, schwegler:5400}. 
%While these results are encouraging, the transferability of such methods is questionable as these are empirical corrections that are fit against reference systems (much like the classical potentials). 
Recently, however, DFs capable of calculating nonlocal correlations self-consistently \cite{PhysRevB.76.125112} have been proposed; standard ones so-called vdW-DF \cite{vdW-DF_Lundqvist-PRL-2004} and vdW-DF2 \cite{vdW-DF2_Langreth-PRBR-2010}. The form of each is similar, combining a reasonable approximation to semilocal exchange with a plasmon-pole model for correlation. Unfortunately though, while such DFs significantly improve the DFT description of small water clusters and hexagonal ice (ice $\text{I}_\text{h}$) \cite{PhysRevB.84.045116}, their application to liquid water \cite{H2O_vdW-DF1_Fernandez-Serra-JCP-2011, H2O_vdW-DF2_Nilsson-JPCB-2011} has led to the somewhat puzzling result of the formation of a HDL.

% NQEs
% ! one other AIMD PIMD sims there is of course the Parrinello 2003 paper, however this just confuses things as Car 2008 supersedes it (shows it to be wrong)
% ! note that the competing effects quantum effects (Manolopoulos 2009) are quite small (well, the result of the interaction), seen by the rate of translational diffusion and orientational relaxation being only increased by $15\%$ by nuclear quantum effects [Manolop 2009]. 
% Further, the primary indication of water structure, $\text{O}$--$\text{O}$ correlations, from classical simulations appear to be consistent with the inclusion of NQEs with a temperature rescaling of $\jmmapprox 30$ K \cite{H2O_NQE_Car-PRL-2008}. 
% While this approximation is easily avoidable through the path-integral (PI) formalism of quantum statistical mechanics, it is often invoked out of necessity to mitigate the greatly increased computational cost (by a factor of $\jmmapprox 32$ at room temperature\cite{H2O_NQE_Car-PRL-2008}).
The problem with overstructuring has also been partly attributed to the lack of including NQEs \cite{H2O_NQE_Martyna-ChemPhysLett-2005, H2O_NQE_Car-PRL-2008}. These have been shown to have a large impact on the $\text{H}$-bond network, due to the fact that water is primarily in its ground vibrational-state at ambient conditions and the light $\text{H}$ mass causes zero-point motion (ZPM) to be large. These studies, however, still resulted in discrepancies with experiment, presumably because they too did not include other relevant effects, such as, perhaps, vdW interactions.

% MANUSCRIPT PURPOSE/ORGANIZATION
It is the purpose of this Letter to present results from simulations incorporating both vdW interactions and NQEs. It is important to note that it is not \textit{a priori} obvious how these effects should together behave, as there can be a complex interplay between intramolecular and intermolecular ZPM \cite{H2O_competing-quant-effects_Manolopoulos-JCP-2009}. The simulations were performed using  path-integral molecular dynamics (PIMD) simulations based on DFT employing vdW-DF \cite{vdW-DF_Lundqvist-PRL-2004} and vdW-DF2 \cite{vdW-DF2_Langreth-PRBR-2010} DFs; see the Methods section. These results are presented in order to provide insight into the underlying structure of water. 


%                    ******************************
%                    *           RESULTS          *
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%\section{Results}
%\label{sec:results}

% ******************************
% PCFs
% ******************************

%\subsection{Pair-correlation Functions}
%\label{subsec:PCFs}

% ! note the Fernandez-Serra JCP 2011 does not see a complete collapse of gOO with vdW-DF1, as a strongly shifted OO peak inwards is seen (like HDL water), which is similar to our vdW-DF2 results
% ! note that in the previous vdW-DF2 study, the inward shift of correlations move them to $3.3$ -- $3.7$\AA{}. 
Standard benchmarks by which the accuracy of ambient-water simulations can be judged, but, moreover, provide important insight into its angular-averaged structure, are the pair-correlation functions (PCFs); in particular, that providing insight into $\text{O}$--$\text{O}$ correlations, $g_\text{OO}(r)$; Fig.\ \ref{fig:gOO}.
% The PCF $g_\text{AB}(r)$ describes the probability of finding a particle of type $\text{B}$ a distance $r$ from a particle of type $\text{A}$, the latter assumed to be at the origin, averaged over all angles. $g_\text{OO}(r)$, which gives the angular-averaged distribution of molecules, has become a standard benchmark to compare to experimental neutron scattering data \cite{H2O-liq_neutron-data_Soper-ChemPhys-2000} [! If I were to keep this sentence, I should also include x-ray data].
% ! note that in this FIRST PCF plot, I explicitly state "classical (nuclei) simulations", but below I drop the "nuclei" and just say "classical simulations"
% ! note that I say "Error bars from PCFs" and NOT "Error bars for PCFs", because I don't actually show the PCFs (otherwise I would use the latter)
%
\begin{figure}
%  \includegraphics[scale=0.3]{./gOO_vdW-DF_w-schem.png}
  \includegraphics[scale=0.5]{PCF_OO_vdW-DF_new-exp-data}  
  \caption{Calculated $\text{O}$--$\text{O}$ PCFs, $g_\text{OO}(r)$. Error bars from PCFs derived from experimental x-ray diffraction data \cite{OO_PCF_Nilsson_JCP-2013} are also shown. The error bars for simulation results are very small, on the order of line-thickness or symbol-size, and thus are not shown. Results for classical (nuclei) simulations are denoted using ``Cl''. These latter two points are also relevant for the remaining figures, and will not be repeated.}
  \label{fig:gOO}
\end{figure}
%
Consistent with recent calculations \cite{H2O_vdW-DF1_Fernandez-Serra-JCP-2011, H2O_vdW-DF2_Nilsson-JPCB-2011, H2O_optB88-vdW_Galli-JCTC-2011}, the classical (nuclei) simulations demonstrate that vdW interactions cause $\text{O}$--$\text{O}$ correlations near $4$ -- $5$ \AA{} to shift inward (leading to a complete collapse of the second coordination-shell in the case of vdW-DF), relative to results derived from experimental x-ray diffraction measurements \cite{OO_PCF_Nilsson_JCP-2013}, compensated for by a slight outward shift of those in the first coordination-shell. These directions are marked in Fig.\ \ref{fig:gOO}, and taken together are indicative of the formation of (or tendency toward) a HDL \cite{PhysRevLett.84.2881}. The remarkable agreement between the vdW-DF2 and experimentally-derived results should also be noted.

Surprisingly, the inclusion of NQEs seems to mitigate these shifts, directions also marked in Fig.\ \ref{fig:gOO}. As this behavior is not seen with semilocal functionals \cite{H2O_NQE_Martyna-ChemPhysLett-2005, H2O_NQE_Car-PRL-2008}, nor with NQEs and \jmmemph{not} vdW interactions, this result strongly suggests that not only are both vdW interactions and NQEs important in describing ambient water, but there may be a heretofore unappreciated combined influence between them. This is further supported by the fact that this effect is apparent irrespective of the choice of DF, as long as election correlations are treated nonlocally \jmmemph{and} self-consistently -- see also the Supplementary Information (SI). In passing, we note that while it may appear at first that these results imply a ``hardening'' of water structure with NQEs, below it is demonstrated that this is in fact not the case.

% ! the O--H correlations I took from the minima in the PIMD vdW-DF2 results surrounding the second shell (precise numbers 1.325 -- 2.425 AA)
% ! the shift with vdW interactions can bee seen when comparing PBE to PBE with nonlocal vdW correlation -- as shown in the SI
More insight into the possibility of a combined influence between vdW interactions and NQEs is obtained via the $\text{O}$--$\text{H}$ PCF, $g_\text{OH}(r)$; Fig.\ \ref{fig:gOH}. 
%
\begin{figure}
%  \includegraphics[scale=0.3]{./gOH_vdW-DF_w-schem.png}
  \includegraphics[scale=0.5]{PCF_OH_vdW-DF_new-data}  
  \caption{Calculated $\text{O}$--$\text{H}$ PCFs, $g_\text{OH}(r)$. Error bars from PCFs derived from experimental neutron scattering data \cite{H2O-liq_neutron-data_Soper-ChemPhys-2000} are also shown.}
  \label{fig:gOH}
\end{figure}
%
Correlations near $1.3$ -- $2.4$ \AA{}, which are directly indicative of $\text{H}$ bonding, demonstrate that when vdW interactions are included, $\text{H}$ bonds increase (weaken), consistent with prior work \cite{water_DFT-D_DCACPs_Rothlisberger-JPCB-2009, water_DFT-D_NpT_Hutter-JPCB-2009, H2O_vdW-DF1_Fernandez-Serra-JCP-2011, H2O_vdW-DF2_Nilsson-JPCB-2011}. When NQEs are included, however, these correlations shift in the opposite direction (as marked in Fig.\ \ref{fig:gOH}), suggesting that the mean $\text{H}$-bond distance is decreased, or the mean strength of intermolecular interactions is increased. While this strengthening is rather small, it is apparently nonetheless enough to overcome the much weaker outer-neighbor vdW interactions \footnote{Compare also the vdW-DF and vdW-DF2 results in Figs.\ \ref{fig:gOO} and \ref{fig:gOH}, where the slight relative outward shift of $\text{O}$--$\text{H}$ correlations when using the former is likely the reason for the relatively soft second coordination-shell of $g_\text{OO}(r)$.}. It is prudent to note that Fig.\ \ref{fig:gOH} also shows there is also a greater distribution of $\text{H}$-bond distances suggesting a ``softening'' of the $\text{H}$-bond network, consistent with intuitive expectation and also previous simulations employing semilocal DFs \cite{H2O_NQE_Martyna-ChemPhysLett-2005, H2O_NQE_Car-PRL-2008}. While the effects of strengthening intermolecular interactions, yet a softer $\text{H}$-bond network may seem at odds, they are consistent with the recent findings of competing quantum effects in water \cite{H2O_competing-quant-effects_Manolopoulos-JCP-2009}, mentioned above, where intermolecular ZPM and tunneling effects soften the $\text{H}$-bond network, while intramolecular ZPM results in strengthened intermolecular interactions, which arise from an increased dipole moment. Again note the level of agreement between the vdW-DF2 results and those experimentally-derived.

% ! the softening in the first minimum was taken from the PIMD simulation (the precise number falls between 1.825 and 1.875 AA)
While $g_\text{OO}(r)$ and $g_\text{OH}(r)$ provide the primary insight into the combined influence between NQEs and vdW interactions, for a complete description of the angular-averaged structure, we show the $\text{H}$--$\text{H}$ PCF, $g_\text{HH}(r)$, in Fig.\ \ref{fig:gHH}.
%
\begin{figure}
%  \includegraphics[scale=0.3]{./gHH_vdW-DF.png}
  \includegraphics[scale=0.5]{PCF_HH_vdW-DF_new-data}  
  \caption{Calculated $\text{H}$--$\text{H}$ PCFs, $g_\text{HH}(r)$. Error bars from PCFs derived from experimental neutron scattering data \cite{H2O-liq_neutron-data_Soper-ChemPhys-2000} are also shown.}
  \label{fig:gHH}
\end{figure}
%
From this, insight into both the intramolecular potential energy surface of an isolated molecule in the condensed-phase environment and the relative orientations of neighboring molecules (given by the first and second peaks, respectively), can be inferred. For example, it can be seen that the classical simulations poorly describe intramolecular motion, which is significantly improved with NQEs, as expected, because water is primarily in its ground vibrational-state at ambient conditions and NQEs are large, also mentioned above. 


%                    ******************************
%                    *       MICROSTRUCTURE       *
%                    ******************************
% ! I also analyzed the number of O, OH, H2O, H3O, and H4O units -- but in most cases the dominant form was H2O (>>99%) (although, for optB88-vdW does have only ~99.4% of H2O and minor amounts of OH and H3O+)
% ! the format in this subsection follows similar to that of Voth 2007 -- first the probability distributions are given then the ND, SD, and DD characteristics

%\subsection{$\text{H}$-bond Topology}
%\label{subsec:H-bonding}

% ------------------------------
% PROBABILITY DISTRIBUTION OF H-BONDS
% ------------------------------

% ! the peak fbeta was taken from the experimental data that I have (which is precisely 11.58477284, but I think the true value is just slightly smaller)
% ! note that H2O_NQE_Martyna-ChemPhysLett-2005 also calculated this, but there really is no comparison to make here
While the PCFs give important insight into the angular-averaged structure of water, they do not provide information about angular correlations that are necessary to understand local geometries. One measure that does give such information though is the probability distribution function $\fbeta$ of $\text{H}$-bond angles, which quantifies the probability of finding a $\text{H}$ bond at an angle $\beta$, the latter considered to be that between the two $\text{O}$ atoms and the covalent $\text{O}$--$\text{H}$ direction (see the inset of Fig.\ \ref{fig:H-bond_prob_dist}).
%
\begin{figure}
  \includegraphics[scale=0.3]{fbeta_vdW-DF_w-schem}
  \caption{Calculated probability distribution functions $\fbeta$ of the $\text{H}$-bond angle $\beta$ (shown in the inset), compared to experimentally-derived results \cite{H-bond_geom_exp_Halle-PRL-2003}, shown as open black circles.}
  \label{fig:H-bond_prob_dist}
\end{figure}
%
Note that the definition of a $\text{H}$ bond is not rigorous, with many possibilities existing \cite{H-bond_def_Skinner-JCP-2007}, and so in order to make a comparison to existing results derived by experiment \cite{H-bond_geom_exp_Halle-PRL-2003}, in this case, we consider each $\text{H}$ atom to participate in a $\text{H}$ bond, the latter found by simply determining the two nearest $\text{O}$ atoms (a more sophisticated definition is utilized below). Figure \ref{fig:H-bond_prob_dist} shows that, on average, $\text{H}$ atoms are shifted slightly outward from the linear $\text{O}$--$\text{O}$ direction, positioned essentially in a ``cone'' of angles around the donating $\text{O}$ atom (peaked near $11.6$\textdegree). The inclusion of NQEs shifts the peak angle even farther outward, as well as increases the angular distribution, both demonstrative of a softening of the $\text{H}$-bond network, confirming some of the remarks made above.

% ------------------------------
% ND, SD, & DD CHARACTERISTICS
% ------------------------------

% ! note that the Skinner IR/Raman fitting used 999 randomly selected water clusters from MD simulations at 300 K (using SPC/E model); also note that one of their assumptions (which is probably in fact not a MINOR assumption) is that the fitting is independent of the underlying ab initio model used to calculate the spectra
% ! note that I mention the softness induced by NQE with reference to the empirical calculations, because the older Martyna work only gives the DD species in the quantum calculations
% ! note that while sometimes they do, different definitions of H bonds do not ALWAYS give similar results
In addition to angular correlations, it is also important to consider actual bonding configurations. A quantitative measure can be made by looking at the relative proportions of $\text{H}$ bonds donated by each $\text{O}$ atom, the latter each of which can be labeled as non-donating (ND), single-donating (SD), or double-donating (DD) (higher proportions, resulting from $\text{H}_3\text{O}^{+}$, for example, are negligibly small); Table \ref{Tb:no_H-bonds_microscopic}.
%
\begin{table}
  \caption{Calculated percentages of ND, SD, and DD $\text{O}$ atoms (in reference to $\text{H}$ bonds), in comparison to theoretical liquid-state spectroscopy results \cite{H-bond_dist_Skinner-PNAS-2007}, as described in the text.}
  \label{Tb:no_H-bonds_microscopic}
  \begin{ruledtabular}
    \begin{tabular}{c c c c}
                                     & ND         & SD         & DD         \\
      \hline
      \hline
      Theor.\ Liq.-state Spect.\                            & $3$        & $27$       & $70$      \\
      \hline
      %vdW-DF                         & $0.0581227$      & $0.387844$      & $0.553896$  ; td=0.000137079   \\
      vdW-DF                         & $6$        & $39$       & $55$       \\
      \hline
      %vdW-DF2                        & $0.0270337$      & $0.324141$      & $0.648579$  ; td=0.000247068   \\
      vdW-DF2                        & $3$        & $32$       & $65$       \\
      \hline
      %vdW-DF Cl                     & $0.0514349$      & $0.418336$      & $0.530195$  ; td=0  \\
      vdW-DF Cl                      & $5$        & $42$       & $53$      \\
      \hline
      %vdW-DF2 Cl                    & $0.0163118$      & $0.290426$      & $0.693228$  ; td=0  \\
      vdW-DF2 Cl                     & $2$        & $29$       & $69$      \\
    \end{tabular}
  \end{ruledtabular}
\end{table}
%
In what follows, a $\text{H}$-bond using the geometric criterion of Ref.\ \onlinecite{H2O-struct_SD-H-bond-model_Wernet-Science-2004}: one is considered to exist between two $\text{O}$ atoms if the separation of the latter ($\text{R}_\text{OO}$) satisfies $\text{R}_\text{OO} \leq -0.00044 \beta^2 + 3.3$ \AA{}, with $\beta$ in degrees. Note that this quantity is shown in the inset of Fig.\ \ref{fig:H-bond_prob_dist}, and in this case, precisely corresponds to the aforementioned cone of angles around the donating $\text{O}$ atom for which a $\text{H}$ atom may be found. Note also that Table \ref{Tb:no_H-bonds_microscopic} shows theoretical liquid-state spectroscopy results \cite{H-bond_dist_Skinner-PNAS-2007} as a reference, which are based on  calculated IR and isotropic Raman spectra fit to experiment (a procedure \jmmemph{assumed} to be independent of the underlying computational method), as definitive experimental conclusions (interpretations) are still matters of debate \cite{XAS_water-structure_overview_Ball-Nature-2008, water-structure_review_THG-MolPhys-2010, water-struct_perspective_Nilsson-ChemPhys-2011}. 

% ! note that the mentioned empirical model suggests relative softness ($\jmmapprox 59\%$ DD species). 
Table \ref{Tb:no_H-bonds_microscopic} provides important insight into the actual structure of ambient water. It suggests that just under two-thirds of molecules actively exhibit the traditional picture of near-tetrahedral (DD) coordination (see the vdW-DF2 results). This is considerably softer than current thinking \cite{water-structure_review_THG-MolPhys-2010} (but which in large part has been based on previous \textit{ab initio} simulations -- e.g., Refs.\ \onlinecite{H2O_NQE_Martyna-ChemPhysLett-2005} and \onlinecite{H2O_NQE_Car-PRL-2008}) that suggest such coordination is closer to $\jmmapprox 80\%$. Nonetheless, these results agree quite well with the theoretical liquid-state spectroscopy results \cite{H-bond_dist_Skinner-PNAS-2007}, and in fact resolve discrepancies between the aforementioned previous \textit{ab initio} simulations \cite{H2O_NQE_Martyna-ChemPhysLett-2005} and sophisticated empirical models \cite{TTM2p1-F-model_quantum-effects_Voth-JCP-2007}. 

% ! note that consistent with $g_\text{OO}(r)$ (Fig.\ \ref{fig:gOO}), vdW-DF is considerably softer (less DD species) than vdW-DF2
While the percentage of DD species is relatively low, they do nonetheless continue to form the majority, indicative of average near-tetrahedral coordination. What is at first surprising is that the inclusion of NQEs is found to have only a small impact on the results, as a reduction in DD species is expected (e.g., empirical simulations \cite{TTM2p1-F-model_quantum-effects_Voth-JCP-2007} suggest a $\jmmapprox 15\%$ reduction at ambient temperature). However, this is further consistent with a scenario where there is a combined influence between NQEs and vdW interactions; while the former alone likely reduce the proportion of DD species, in the presence of vdW interactions, they mitigate the large reduction already caused, and in the end, seemingly having little overall impact.


%                    ******************************
%               * DISCUSSION/CONCLUSIONS *
%                    ******************************

%\section{Discussion}
%\label{sec:discussion}


% AGREEMENT WITH EXPERIMENT
% ! note that the level of agreement seems to parallel or outperform even empirical models, such as TTMF2.1 (Voth 2007) -- one question I have is how was TTMF2.1 fit (i.e.., to reproduce what)?
% ! even though I don't state it, the results really fall within or very close to any likely experimental uncertainties
In conclusion, the inclusion of both NQEs and vdW interactions into \textit{ab initio} DFT simulations provides a remarkably accurate description of ambient water (the vdW-DF2 results, in particular). Excellent agreement with experiment (or results derived therefrom) was obtained in all cases considered.
% While such extensions should be approached with caution, as discussed above, the accuracy obtained is consistent with the level of agreement obtained in analogous calculations on molecular complexes (e.g., the water dimer \cite{vdW-DF2_Langreth-PRBR-2010}).
% ! not entirely helpful here, perhaps move up to results --> The worst agreement arises in $\fbeta$ (Fig.\ \ref{fig:H-bond_prob_dist}), where the peak distribution near $11.5$\textdegree ~ is somewhat decreased relative to experiment \cite{H-bond_geom_exp_Halle-PRL-2003}, compensated for by an increased probability at large angles. However, it is quite possible that this is because the experimental results are fit to the product of a sinusoidal and exponential function centered at $\beta = 0$\textdegree.
The level of agreement should be especially compared to the plethora of results from previous simulations, which have resulted in severe over-structuring or the formation of a HDL without resort to \textit{ad hoc} or empirical corrections; see Refs.\ \onlinecite{grossman:300, schwegler:5400,H2O_vdW-DF1_Fernandez-Serra-JCP-2011, H2O_vdW-DF2_Nilsson-JPCB-2011}, for example.
% COMBINED INFLUENCE OF NUCLEAR QUANTUM EFFECTS AND vdW INTERACTIONS & OUTLOOK FOR FUTURE SIMULATIONS
Furthermore, the results suggest that a combined influence between NQEs and vdW interactions impacts the structure of ambient water; vdW interactions favor the formation of a HDL, while NQEs mitigates this by strengthening the mean intermolecular interaction. The importance and influence of such subtle effects (combined NQEs and vdW interactions) on the structure (and likely behavior) of ambient water suggests some clear caveats to future simulations, in particular \textit{ad hoc} or empirical corrections used to account for them (see the SI). Looking forward, it is quite possible that these subtle effects, and the competition that they exhibit, are not only important in water, but also other liquids containing $\text{H}$-bonds.


%                    ******************************
%                    *           METHODS          *
%                    ******************************

\

\noindent
\textbf{Methods}
%\section{Methods}
%\label{sec:methods}

% ! I believe that VASP was used for the equilibration, but this detail is relatively unimportant
% ! note that the use of USPPs have been shown to be nearly as accurate as norm-conserving ones in water \cite{H2O_O-PRB-PP_Marzari-JCP-2005}, but at a greatly reduced computational cost -- although, based on Miguel's recent study of vdW DFs, this may only hold without nonlocal correlations
% ! below I don't reference the PI+GLE paper when discussing that 8 slices converges quantum water (but note that this paper used an empirical model to show that 8 slices was good in water) 
Simulations were performed using the Quantum ESPRESSO \textit{ab initio} DFT code \cite{QE-2009}. Distribution versions of ultrasoft pseudopotentials generated with the Perdew-Burke-Ernzerhof (PBE)  \cite{PBE_exch-correl_PRL-1996} DF were used for both $\text{O}$ and $\text{H}$. A plane-wave basis set with kinetic-energy and charge-density cutoffs of $30$ and $240$ Ry, respectively, was found to adequately converge forces for the molecular dynamics. Periodic boundary conditions were enforced automatically by the basis set and $\Gamma$-point Brillouin-zone sampling. A number of simulations were performed with various DFs (semilocal, in some cases with an empirical vdW correction, as well as those capable of treating nonlocal correlations self-consistently), as discussed in the main text or SI. For classical simulations, statistics were collected in the canonical ensemble using an Andersen thermostat \cite{Andersen_thermostat_JCP-1980} to control the temperature.
% in the $NVT$ ensemble, where $N$, $V$, and $T$ are the number of particles, volume, and temperature, respectively. 
Quantum statistics were generated using an accelerated PIMD approach based on a generalized Langevin equation \cite{PI+GLE_Parrinello-JCP-2011}, with $8$ imaginary-time slices (which was found to be adequate to converge both the quantum kinetic and potential energies). For both types of simulation, a time step of $10$ a.u.\ was used to propagate the trajectories.

% ! note the equilibrium density of actual water is 0.997 g/cm^3, pressure of 1 atm (from wiki: 1 atm = 1.01325 bar), and temperature of 298.15 K
% ! note that the Parrinello JCP 2005 showed that slight density differences have little effect -- I may want to ref this in the future
% ! keep in mind here that 4000 steps is ~1 ps
% ! the classical vdW-DF calculations started from 12K steps of the quantum calcs
% ! also note that in general, the classical simulations were performed even longer time than the PI simulations
% ! note that without a CHANGE in DF, the further equilibration may be less (e.g., PBE, vdW-DF, and vdW-DF2 classical calculations from the quantum ones)
% ! the number of steps actually used was around 32k for initial equilibration, 2k per change in DF, and statistic collection for around 20k
% *****
% ! I obtained a *very crude* approximation (perhaps even conservative) of the total length of simulations by (which totals 63.5 ps): 
%      8 ps equilibration 
%      4 ps vdW-DF, 4 ps vdW-DF2, 2.5 ps optB88-vdW (quantum)
%      5.5 ps vdW-DF, 5.5 ps vdW-DF2, 5 ps optB88-vdW 300K, 5 ps optB88-vdW 330K (classical)
%      2 ps PBE 300 K, 2 ps PBE 330 K (classical)
%      4 ps PBE, 8 ps PBE-vdW, 8 ps PBE+Grimme (quantum) (I guessed PBE as 4, since the liquid was equilibrated already with PBE)
% ! on the other hand, in the following, I could probably actually say closer to 0.1 ns -- especially since the PBE simulations were probably each performed for 20 ps (?)
% Simulations were checked every couple of ps for convergence in PCFs. In total, $\jmmapprox 65$ ps of simulations were performed.
% *****
% The new vdW-DF2 results were 3 ps equilibration and 8 ps of statistics collection, but this was following the first vdW-DF2 simulations
% *****
To represent ambient water, $64$ $\text{H}_2\text{O}$ molecules (with atomic masses of $1.00794$ and $15.9994$ for $\text{H}$ and $\text{O}$, respectively) were simulated at a density of $1$ $\text{g} / \text{cm}^3$ and temperature of $300$ K. An estimate of the equilibrium density for vdW-DF2 to justify this choice (also used for other functionals) is provided in the SI. Equilibration of the liquid was performed by first (semi-)melting ice $\text{I}_\text{h}$ at $400$ K and performing classical simulations using PBE for $\jmmapprox 8$ ps. Further equilibration was then performed for a further few ps at the PIMD-level, followed by many ps of statistics collection. For vdW-DF2, for example, precise times were $\jmmapprox 7$ ps equilibration and a further $8$ ps of statistics collection, for a total simulation time of $23$ ps.


% ******************************
% ACKNOWLEDGMENTS
% ******************************

% If you have acknowledgments, this puts in the proper section head.
\begin{acknowledgments}
We thank David M.\ Ceperley and Eric Schwegler for insightful discussions and providing helpful comments on an early draft of this manuscript. J.\ M.\ M.\ was supported by DOE Grant DE-NA$0001789$. M.\ A.\ M.\ was supported by the DOE at Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This research was also supported in part by the NSF through XSEDE resources provided by NICS under grant number TG-MCA93S030.
\end{acknowledgments}


%                    ******************************
%                    *        SI DESCRIPTION       *
%                    ******************************

\

\noindent
\textbf{Associated Content}
%\section{Associated Content}
%\label{sec:SI_description}

\

\noindent
\textbf{Supporting Information.} Calculation of the equilibrium density of vdW-DF2; results for semi-local DFs, with and without empirical vdW corrections, as well as additional vdW-DFs; caveats to future simulations. This material is available free of charge via the Internet at http://pubs.acs.org.


% ******************************
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% ! I will just delete the URLs, sprints, etc. by hand once I hard-code them into here
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\end{thebibliography}


\end{document}

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