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The CANDLE benchmarks deliver working examples of large-scale deep learning applied to different cancer research challenges. To provide a performance baseline to measure progress and improvement and implement deep learning architectures relevant to scientific challenges in cancer research and opportunities in exascale computing, the developed benchmark models were developed.

Solvents play a critical role in determining chemical reaction mechanisms and rates, but the need for thermodynamic phase-space sampling renders direct treatment of the liquid in electronic structure calculations far too computationally intensive. The standard solution to this problem is to use continuum solvation models which empirically describe the dominant effects of the solvent within a single electronic structure calculation of the solute alone. This enables rapid estimations of free energies of reaction intermediates, allow ing for a theoretical screening of reaction mechanisms, and providing insight into the mechanisms involved in catalysis required for the development of more efficient catalysts.

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For the cavity formation free energy Gcav, we adopt the parameter-free weighted density approximation from Ref. 11 without modification. Briefly, this model for the cavity formation free energy begins with a weighted-density ansatz motivated from an intuitive microscopic picture of surface tension and completely constrains the functional form to bulk properties of the solvent including the number density, surface tension and vapor pressure. The resulting functional accurately describes the free energy of forming microscopic cavities of arbitrary shape and size in comparison to classical density-functional theory and molecular dynamics results.16 (See Ref. 11 for a full specification of Gcav[s].)

Here, Zval is the number of valence electrons in the solvent molecule and the Gaussian width lq is selected so that the overlap of the model electron densities of two solvent molecules crosses n c at a separation equal to twice the vdW radius RvdW of the solvent. This condition reduces to the transcendental equation in lq,

For x > 0 (anions), fsat(x) saturates to Dmax for large x so that the modulation of n c eff is limited to a factor of eDmax. This provides numerical stability. We set Dmax = 3 which is just sufficient to cover the parameter changes observed in the SCCS fits. In principle, we could fit Dmax to solvation energies as well, but this parameter is strongly covariant with pcav, and hence, for simplicity, we hold it fixed at the above value. Fitting Dmax could marginally improve the accuracy of the model, but does not seem to be necessary for the solvents considered so far.

This parametrization is of course not unique, but it is one of the simplest choices that capture the observed charge asymmetry and remains numerically stable. Note that a similar dependence on the solute electric field would be extremely unstable in a conventional isodensity model that depends on the local electronic density. Here, the nonlocality introduced by the convolutions with wlq(r) is critical to the success of the present model.

The cavity of conventional density-based solvation models represents the shape of an effective continuum dielectric that reproduces solvation energies. In contrast, the cavity of the SaLSA model (and hence, the one determined above) corresponds to the physical distribution of solvent molecule centers because the model directly captures the nonlocal dielectric response of the solvent molecules. However, this nonlocal dielectric response requires an expansion in angular momentum that is computationally intensive and practically applicable only for solvents involving small, approximately spherical and rigid molecules.

The CANDLE solvation model restores the standard local response approximation to achieve computational expediency and generality, but this, in turn, then requires an empirical description of the dielectric as in other local solvation models. We use the dielectric shape function,

The fit parameter controls the distance between the solute and the solvent dielectric boundary, analogous to the primary fit parameters of conventional solvation models such as electron-density thresholds or atomic radii scale factors. However, compared to conventional iso-density solvation models7,10 which only employ an empirical dielectric cavity, the present approach uses the physical cavity from the SaLSA approach for the cavity-formation and dispersion terms. This enables the use of physical models for those terms that capture the correct cavity shape and size dependence instead of empirical surface tension models.

Convergence of energy and forces for an acetic acid molecule solvated in water using the CANDLE model and a simpler iso-density model (LinearPCM from Ref. 10). Part (a) shows the energy at each SCF cycle collected over all ionic steps. Part (b) shows the forces at each ionic step.

Fit parameters and physical properties that constrain the CANDLE solvation model. We obtain vdW radii from Ref. 17 and all other physical properties from Ref. 29 (at standard conditions, T = 298 K and p = 101.3 kPa).

We calculate the gas-phase energy for each solute at the optimized vacuum geometry. We optimize the solution-phase geometry using an initial guess for the solvation model parameters, and at that optimum geometry, calculate the solvation energy and its analytical Hellman-Feynman derivatives with respect to the parameters on a coarse grid in the parameter space of the solvation model. Using the analytical derivatives, we interpolate the solvation energies to a finer grid in parameter space and then select the optimum parameters to minimize the mean absolute error (MAE) of all the solutes. We re-optimize the solution-phase geometries with these parameters, and repeat the above parameter sweep process till the optimum parameters converge. For both solvents considered here, the second sweep yields identical optimum parameters as the first, and we show the results of that final self-consistent parameter sweep.

Table II compares the accuracy of the CANDLE solvation model for water with that of the SCCS models and the integral-equation formalism (IEF)-PCM30,31 in GAUSSIAN32 on exactly the same set of solutes. The IEF-PCM model exhibits large errors for cations as well as anions, while the SCCS model fit to neutral molecules alone works reasonably well for cations but systematically undersolvates anions resulting in a large error of 17 kcal/mol. This error is reduced to 5.5 kcal/mol by fitting a separate set of parameters for anions alone. With charge asymmetry built in, the CANDLE solvation model with a single parameter set exhibits comparable accuracy to the individual SCCS models fit to each solute type.

MAEs of the CANDLE solvation model for water compared to various parametrizations of the SCCS model,7 and IEF-PCM30,31 in GAUSSIAN32 using identical sets of solutes. (SCCS and GAUSSIAN results from Ref. 8.)

As an independent test of accuracy, Figure 4 compares predicted acid dissociation constants of mostly inorganic acids (not present in the fit set) with experiment. The CANDLE model marginally increases the error in pKa of cationic acids compared to the local LinearPCM model, but significantly improves the predictions for neutral and anionic acids since it solves the anion under-solvation issue. Note, in particular, that CANDLE makes reasonable predictions even for the second and third dissociations of sulfuric and phosphoric acid, which require solvation of dianions and trianions, respectively. For the set considered here, the MAE is 4.7 pKa units for CANDLE compared to 8.4 pKa units for LinearPCM.10

Finally, we examine the predictions of the CANDLE solvation model for a class of relatively clean electrochemical systems: single crystalline noble metal electrodes in an aqueous non-adsorbing electrolyte. The surface charge on these electrodes depends on the electrochemical potential, and the surface becomes neutral at the potential of zero charge (PZC). Experimentally, these potentials are referenced against the standard hydrogen electrode (SHE). The absolute level of the SHE is difficult to determine experimentally and estimates range from 4.4 to 4.9 eV.34 Correlating the theoretical electron chemical potential of solvated neutral metal surfaces with the measured PZC provides a theoretical estimate of this absolute potential.9,10 Here, we reexamine this theoretical estimate with the nonlocal solvation models, CANDLE and SaLSA.

Figure 6 plots the calculated electron chemical potential of neutral metal surfaces using various solvation models against the experimental PZC, and Table III summarizes the absolute offset and error in the correlation so obtained. The absolute offsets predicted using various solvation models agree to within 0.1 eV and are well within the expected experimental range. The CANDLE model exhibits a marginally higher scatter but overall agrees well with the linear and nonlinear local models studied in Ref. 10. The nonlocality of the SaLSA and CANDLE models, therefore, does not significantly alter the predictions of the local solvation models for the absolute SHE potential.

Correlation of theoretical electron chemical potentials (e) with experimental potential of zero charge (relative to SHE) for various solvation models. The results are for single crystalline copper (squares), silver (triangles), and gold (circles) surfaces, with 111, 100, and 110 orientations from left to right, respectively. (LinearPCM and NonlinearPCM data from Ref. 10.) 2351a5e196