# A QM/MM Derived Polarizable Water Model for Molecular Simulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Computational Details for QM/MM Based Polarizability Fitting

^{3}filled with pre-equilibrated SPC water (1000 SPC molecules in this work) [10]. Note that use of different solvent models was previously shown to lead to similar QM/MM fitted values for the polarizability [25,26]. In the subsequent energy minimization and molecular dynamics (MD) simulations, the solute is positionally constrained to the QM-optimized geometry in vacuum (aligned according to Figure 1). The leap-frog algorithm is used during MD to integrate Newton’s equations of motions using a time step of 2 fs. Following a steepest-descent energy minimization, the simulation system is equilibrated during 100 ps under NVT conditions. Production simulations lasted for 2 ns under NVT conditions, where coordinates are written out every 4 ps (500 frames). The temperature was weakly coupled to an external bath using a Berendsen thermostat with a coupling constant of 0.1 ps and a target temperature of 298.15 K [30]. Interactions were handled using a twin-range pairlist with a short-range cutoff of 0.8 nm updated every time step and a long-range cutoff up to 1.4 nm, where interactions are updated every fifth time step. A reaction field long range correction was added [31,32], with a cutoff distance of 1.4 nm equal to the long range cutoff (${r}_{c,rf}={r}_{c,lr}$). The homogeneous medium outside of the cutoff was assigned a dielectric constant of 78.4, equal to the experimental value for water [33]. The SPC molecules in the simulations were kept rigid and were constrained using SHAKE with a relative tolerance of 0.0001 [34]. From each of the 500 frames, a unique set of solvent coordinates is extracted and gathered around the solute molecule. Only water molecules of which the oxygen is within an interaction distance (1.4 nm) of any of the solute atoms are considered for the combined QM/MM calculations. The point charges from the considered water molecules are introduced as Bq (i.e., MM partial) charges in a quantum calculation at the B3LYP/QZ4P level of theory. These QM/MM calculations are used to evaluate effective electrostatic potentials ($\varphi $) at grid points around the solute molecule. A suitable Connolly grid [35,36,37] for this analysis is generated using GAMESS-US 2014 [38,39], using four incremental layers with a point density of five points per bohr

^{2}as described in detail in Ref. [25].

#### 2.2. Electrostatic Model

#### 2.3. Dispersion Calculations

#### 2.4. Pure-Liquid MD Simulations

#### 2.5. Pure-Liquid Property Analysis

## 3. Results and Discussion

^{−1}nm

^{−1}e

^{−1}) for the local electric field in one or multiple directions. As a result, we found that, in some cases, even the sign of the fitted multipole can change. Here, we omitted the use of a cutoff in determining the electric fields at the solute atoms, in order to be consistent with the MM solvent shells entering our QM/MM calculations. These improved electric field calculations led to a decrease in the standard deviations for the calculated polarizabilities. This is demonstrated when determining the effective molecular polarizability located at the water oxygen atom, in particular in the y- and z-directions (with relatively small values for the corresponding values of the local electric field), with standard deviations decreasing from 17.8 $\times {10}^{-3}$ nm

^{3}and 38.1 $\times {10}^{-3}$ nm

^{3}(data not shown) to 8.1 $\times {10}^{-3}$ nm

^{3}and 10.4 $\times {10}^{-3}$ nm

^{3}, respectively, Figure 2a. However, negative values for the polarizability were still observed (Figure 2a) and these standard deviations still imply a significant uncertainty in the calculated polarizabilities, which may well become more pronounced in cases in which net electric fields are even lower (e.g., alkanes or other hydrophobic compounds). Note that, while the original method is noisy for small electric fields, the overall fitted induced dipoles in each dimension (x, y and z) do show linear response to the external electric fields. The correlation coefficient (${R}^{2}$) of approximately 0.9 indicates a linear response for the range of external electric field strengths in a hydrated environment (Figure 3a–c). Based on this finding, our model does not include a damping factor of the polarizability at high electric fields, as is applied, for example, in the recently published COS/D2 model [8], which is also based on charge-on-spring polarization.

^{3}is slightly lower than the value of 1.1 $\times {10}^{-3}$ nm

^{3}determined in previous work by us and still close to the value of 1.08 $\times {10}^{-3}$ nm

^{3}reported by Schropp and Tavan for use in combination with a point-determined electric field [24,63]. Schropp and Tavan argued that the effective electric field $<E>$ due to the water solvent as averaged over the approximate molecular volume of a hydrated water solute (i.e., the field that causes the actual polarization in the QM/MM calculations) is significantly lower than the point-determined ${E}_{O}$ at the oxygen nucleus, which is commonly used when determining induced dipole moments in molecular simulations with the COS model. This argument can explain both the lower value of effective model values for the polarizability when compared to its gas-phase estimate e.g., from quantum calculations (1.44 $\times {10}^{-3}$ nm

^{3}) [64] and the overpolarization (in terms of too high dielectric permittivities) observed for early COS water models, for which the polarizability was set to 1.225 $\times {10}^{-3}$ nm

^{3}or higher [13,17]. We note that, if isotropic atomic polarizabilities are fitted on three sites (hydrogens and off-atom M site), the average molecular isotropic polarizability increases to 1.21 $\times {10}^{-3}$ nm

^{3}(data not shown). The fact that the molecular polarizability increases further to 1.39 $\times {10}^{-3}$ nm

^{3}when treating the electric field in a homogeneous manner (i.e., by applying a continuous electric field to the water solute instead of a field of explicit solvent point charges in order to determine ${\varphi}_{solv}$ in Equation (7)) also supports the argument of Schropp and Tavan. Thus, the inhomogeneous treatment of the response to the solvent electric field may well be on the basis of the observed decrease in molecular polarizability when comparing the gas-phase reference value with our QM/MM-determined (and other) condensed-phase estimates, also considering that point-charge inclusion of the external electric field in the QM Hamiltonian cannot lead to exchange repulsion with the solvent. Our QM/MM determined value for ${\alpha}_{iso}$ is also close to the empirically determined polarizability of 1.04 $\times {10}^{-3}$ nm

^{3}in the Drude-oscillator (DO) SWM-DP model [12]. In the more recent six-site and negative DO models, this polarizability was scaled down to values of 0.97825 $\times {10}^{-3}$ nm

^{3}and 0.88 $\times {10}^{-3}$ nm

^{3}[7,16], which are lower than our QM/MM determined value.

^{2}s

^{−1}, versus an experimental value of 2.3 × ${10}^{-5}$ cm

^{2}s

^{−1}[69]. However, Yeh and Hummer have shown that the diffusion constant depends on the simulation box size and should therefore be corrected with a term related to the shear viscosity of the liquid [61]. Correcting the diffusion coefficient based on the model’s shear viscosity ($\eta $ in Table 2) results in a diffusion constant of 2.17 × ${10}^{-5}$ cm

^{2}s

^{−1}, close to the experimental value.

^{3}versus ${\alpha}_{iso}$ = 1.05 $\times {10}^{-3}$ nm

^{3}). While the reference value for the pure-liquid molecular dipole moment of water has been debated for several years, there has been consensus that it is larger than the values that are typically employed in additive models [76]. Table 2 shows that our and other polarizable models also have a lower average molecular dipole moment than the value reported in Ref. [69]. Considering the treatment of electrostatics in terms of point charges and a single local inducible dipole moment in these models, their relatively low molecular dipole moment may be in line with the argument of Schropp and Tavan discussed above for the effective decrease in polarizability when going from the gas-phase reference to its QM/MM estimated condensed-phase value [63].

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Axis alignment of the QM water solute as applied during polarizability and charge fitting. The oxygen is placed at the origin of the axis system and the hydrogens are placed in the positive x-axis regime. Only the sign of the y-axis value differentiates between the first and second hydrogen. The molecule is placed in the x–y plane and the z-axis is defined as x × y.

**Figure 2.**(

**a**) box plots of the calculated polarizabilities in either x-, y- or z-dimension using the original fitting method per solvent configuration; (

**b**) box plots of the calculated polarizabilities in either x-, y- or z-dimension using the new consensus fitting approach for polarizabilities. In panels (

**a**,

**b**), data are partitioned after sorting into four quartiles, and boxes depict the inter-quartile-range (irq) with a middle line that denotes the median. The whiskers are placed at the minimum and maximum values considered, with a maximum deviation of 4.0 times the irq. Outliers are denoted by circles (for ${\alpha}_{z}$) and crosses (for ${\alpha}_{y}$); (

**c**) error in fit (${\chi}^{2}$) as obtained from the free unconstrained original fit (free) and compared to the constrained consensus-fit error in fit.

**Figure 3.**Fitted induced dipole moments (${\mu}_{ind}$) in the presence of an external electric field (E), plotted for each separate dimensional component (x, y or z). In (

**a–c**), the results obtained using our original method (with updated E calculation) are presented, where induced dipoles of the QM solute were fitted independently for each individual solvent configuration obtained from MD. The x, y and z components of these induced dipoles are presented in (

**a**–

**c**), respectively; (

**d**–

**f**) present the results of our consensus fitting approach, where 20 independent solvent configurations are used simultaneously in each constrained fit of the induced dipole moments. The x, y and z components of these induced dipoles are presented in (

**d**–

**f**), respectively.

**Figure 4.**Rigid water geometry after QM optimization in the gas phase at the B3LYP/QZ4P level of theory. All bonds are constrained during simulation and the bond angle is enforced by an extra bond between the hydrogens. The location of the off-site (M) particle and its offset are included in the figure. The COS particle is attached to the M-site.

**Figure 5.**Position ${r}_{M}$ of the off-atom (M) site along the x-axis in Figure 1 versus the root-mean-square deviation (RMSD) between the partial-charge fitted and QM (B3LYP/QZ4P) estimated molecular electrostatic potential in the gas phase.

**Figure 6.**(

**a**) Density and (

**b**) heat of vaporization ($\Delta {H}_{vap}$) over a range of temperatures (248 K–373 K) for water: comparison between experimental values (black crosses) and values calculated from simulation using the water model presented in this work (blue circles). Calculated values of $\Delta {H}_{vap}$ are corrected for changes in vibrational modes upon evaporization, which contribute significantly as documented by Horn et al. [14].

**Table 1.**Force-field parameters for dispersion and electrostatic interactions. Values obtained in this work are listed together with values used in SPC and in other charge-on-spring (COS) and Drude-oscillator (DO) water models. As the form of the van der Waals potential energy function used here is different from the other models (by means of inclusion of a ${C}_{8}$ and ${C}_{11}$ term in this work), the functional form of the repulsive term is listed as ${C}_{x}$. Van der Waals parameters are listed using homo-dimer dispersion parameters and repulsion is listed in terms of a van der Waals radius ${r}_{o}$. Electrostatic parameters are listed for the atomic site of the oxygen (${q}_{o}$), the atomic site of the hydrogen (${q}_{h}$), the offsite position (${q}_{m}$), oxygen lone pairs (${q}_{l}$) and the charge on the COS particle (${q}_{u}$). Damping power (p) and the damping field strength (${E}_{0}$) are only used in the COS/D2 model. All water models (except SPC) include a single polarizable site; therefore, a molecular polarizability ($\alpha $) is listed. Note that for charge assignments the values listed are the charges as written in the topology; GROMOS-style force fields (This work, SPC, COS) assign internally an effective charge of ${q}_{i}$-${q}_{COS}$ for a polarizable site, to balance out the introduction of a large COS charge. Reference values for ${C}_{6}$ and $\alpha $ were taken from Refs. [64,68], respectively. Values for SWM4-NDP and SWM6 DO water models were taken from Ref. [7] and values for SPC, COS/G2 and COS/D2 were taken from Ref. [8].

Expt. | This Work | SPC | COS/G2 | COS/D2 | SWM4-NDP | SWM6 | ||
---|---|---|---|---|---|---|---|---|

${C}_{6}$ | $a.u.$ | 45.4 | 43.44 | 45.40 | 56.27 | 56.28 | 63.80 | 50.34 |

${C}_{8}$ | $a.u.$ | 1201.3 | ||||||

${r}_{o}$ | $nm$ | 0.1605 | 0.1583 | 0.1598 | 0.1582 | 0.1592 | 0.1599 | |

${C}_{x}$ | ${C}_{11}$ | ${C}_{12}$ | ${C}_{12}$ | ${C}_{12}$ | ${C}_{12}$ | ${C}_{12}$ | ||

${q}_{o}$ | e | −0.82 | 1.71636 | 1.91589 | ||||

${q}_{h}$ | e | 0.539 | 0.41 | 0.5265 | 0.57 | 0.55733 | 0.53070 | |

${q}_{m}$ | e | −1.078 | −1.053 | −1.14 | −1.11466 | −1.13340 | ||

${q}_{l}$ | e | −0.10800 | ||||||

${q}_{u}$ | e | −8.0 | −8.0 | −8.0 | −1.71636 | −1.62789 | ||

$\alpha $ | 10^{−3} nm^{3} | 1.44 | 1.05 | 1.255 | 1.3 | 0.97825 | 0.88 | |

${E}_{0}$ | kJ mol^{−1} nm^{−1} e^{−1} | 140 | ||||||

p | 8 |

**Table 2.**Pure-liquid properties of water as calculated from a 10 ns $NpT$ simulation at 298.15 K and 1 atm (This work). Reference experimental data (Expt.) are listed for the density ($\rho $) [14,70], heat of vaporization ($\Delta {H}_{vap}$) [14,71], diffusion constant (corrected for box size, D) [72], static dielectric permittivity (${\u03f5}_{0}$) [33], the static (${\mu}_{static}$) [65] and averaged molecular dipole moment ($\langle \mu \rangle $) [69], constant-pressure heat capacity (${C}_{p}$) [73], thermal expansion coefficient (${\alpha}_{p}$) [14,70], shear viscosity ($\eta $) [74], and isothermal compressibility (${\kappa}_{T}$) [75]. Values for the diffusion constant under periodic boundary conditions (${D}_{pbc}$) and the self-polarization energy (${U}_{selfpol}$) are also listed. Values for SWM4-NDP and SWM6 water models were taken from Ref. [7] and values for SPC, COS/G2 and COS/D2 were taken from Ref. [8].

Expt. | This Work | SPC | COS/G2 | COS/D2 | SWM4-NDP | SWM6 | ||
---|---|---|---|---|---|---|---|---|

$\rho $ | kg m^{−3} | 997 | 993 | 973 | 999 | 999 | 994 | 996 |

$\Delta {H}_{vap}$ | kJ mol^{−1} | 44.01 | 43.81 | 43.9 | 43.7 | 44.08 | 43.7 | 44.0 |

${D}_{pbc}$ | ${10}^{-5}$ cm^{2} s^{−1} | 1.90 | 4.1 | 2.0 | 2.2 | 2.35 | 1.76 | |

D | ${10}^{-5}$ cm^{2} s^{−1} | 2.3 | 2.17 | 2.85 | 2.14 | |||

${\u03f5}_{0}$ | 78.4 | 77.6 | 64.7 | 96.6 | 78.9 | 78.0 | 78.1 | |

${\mu}_{static}$ | D | 1.855 | 1.856 | 2.27 | 1.85 | 1.855 | 1.85 | 1.85 |

$\langle \mu \rangle $ | D | 2.95 | 2.47 | 2.27 | 2.61 | 2.55 | 2.459 | 2.431 |

${U}_{selfpol}$ | kJ mol^{−1} | 12.5 | 15.9 | 14.4 | ||||

${C}_{p}$ | J mol^{−1} K^{−1} | 75.3 | 91.9 | 93.0 | 107.7 | 88.9 | ||

${\alpha}_{p}$ | ${10}^{-4}$ K^{−1} | 2.57 | 3.86 | 9.0 | 7.0 | 4.9 | ||

$\eta $ | cP | 0.89 | 0.72 | 0.66 | 0.87 | |||

${\kappa}_{T}$ | ${10}^{-6}$ atm^{−1} | 45.8 | 41.7 | 47.8 | 47.8 | 44.4 |

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**MDPI and ACS Style**

Visscher, K.M.; Swope, W.C.; Geerke, D.P. A QM/MM Derived Polarizable Water Model for Molecular Simulation. *Molecules* **2018**, *23*, 3131.
https://doi.org/10.3390/molecules23123131

**AMA Style**

Visscher KM, Swope WC, Geerke DP. A QM/MM Derived Polarizable Water Model for Molecular Simulation. *Molecules*. 2018; 23(12):3131.
https://doi.org/10.3390/molecules23123131

**Chicago/Turabian Style**

Visscher, Koen M., William C. Swope, and Daan P. Geerke. 2018. "A QM/MM Derived Polarizable Water Model for Molecular Simulation" *Molecules* 23, no. 12: 3131.
https://doi.org/10.3390/molecules23123131