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Article
Soft-sphere continuum solvation in electronic-structure calculationsGiuseppe Fisicaro, Luigi Genovese, Oliviero Andreussi, Sagarmoy
Mandal, Nisanth N. Nair, Nicola Marzari, and Stefan GoedeckerJ. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00375 • Publication Date (Web): 19 Jun 2017
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Soft-sphere continuum solvation in
electronic-structure calculations
G. Fisicaro,∗,† L. Genovese,‡ O. Andreussi,¶,§ S. Mandal,‖ N. N. Nair,‖ N.
Marzari,§ and S. Goedecker†
†Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland
‡Laboratoire de simulation atomistique (L_Sim), SP2M, INAC, CEA-UJF, F-38054
Grenoble, France
¶Institute of Computational Science, Universita’ della Svizzera Italiana,Via Giuseppe Buffi
13, CH-6904 Lugano, Switzerland
§Theory and Simulations of Materials (THEOS) and National Centre for Computational
Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de
Lausanne, Station 12, CH-1015 Lausanne, Switzerland
‖Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur 208016, India
E-mail: [email protected]
Abstract
We present an implicit solvation approach where the interface between the quantum-
mechanical solute and the surrounding environment is described by a fully continuous
permittivity built up with atomic-centered “soft” spheres. This approach combines
many of the advantages of the self-consistent continuum solvation model in handling
solutes and surfaces in contact with complex dielectric environments or electrolytes
in electronic-structure calculations. In addition it is able to describe accurately both
1
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neutral and charged systems. The continuous function, describing the variation of the
permittivity, allows to compute analytically the non-electrostatic contributions to the
solvation free energy that are described in terms of the quantum surface. The whole
methodology is computationally stable, provides consistent energies and forces, and
keeps the computational efforts and runtimes comparable to those of standard vacuum
calculations. The capabilitiy to treat arbitrary molecular or slab-like geometries as
well as charged molecules is key to tackle electrolytes within mixed explicit/implicit
frameworks. We show that, with given, fixed atomic radii, two parameters are sufficient
to give a mean absolute error of only 1.12 kca/mol with respect to the experimental
aqueous solvation energies for a set of 274 neutral solutes. For charged systems, the
same set of parameters provides solvation energies for a set of 60 anions and 52 cations
with an error of 2.96 and 2.13 kcal/mol, respectively, improving upon previous literature
values. To tackle elements not present in most solvation databases, a new benchmark
scheme on wettability and contact angles is proposed for solid-liquid interfaces, and
applied to the investigation of the stable terminations of a CdS (1120) surface in an
electrochemical medium.
1 Introduction
The computational study of matter in various environments is a continuously growing field
in solid state physics and chemistry. Systems of interest are for instance molecules, clusters
or surfaces in contact with solvents.1 A possibility is to include explicitly in the simulation
domain all the atoms and molecules of the solution. Such an explicit treatment is in principle
the natural way to account for solvent effects in a first-principles scheme. However, this
approach enormously increases the computational cost and limits at the same time the size
of the system contained in the explicit dielectric medium.2 As a consequence, the study of
the solute-solvent interactions at length scales larger than molecular sizes, or the generation
of long molecular dynamics trajectories, require big computational efforts.
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Numerically expensive investigations in material science such as structure predictions3
or reaction path determinations4 would become very time-consuming if not altogether unaf-
fordable. During an exploration of the potential energy surface, most of the minima would
arise from rearrangements of solvent molecule. For an hydrated surface, a lot of time would
consequently be spent moving water molecules rather than sampling the solid-water interface
that is of interest. The explicit description of water also suffers from some well-known lim-
itations of current state-of-the-art ab-initio methods, in particular regarding its structural
and dielectric properties.5–8
The alternative is an implicit description of the solvent effects, while still treating the
other parts of the system explicitly on an atomic level. Such an explicit/implicit treatment
requires three main ingredients:
• A dielectric cavity represented by a proper function ǫ(r) mimicking the surrounding
solvent of a solute as a continuum dielectric;
• A solver for the generalized Poisson equation9
∇ · ǫ(r)∇φ(r) = −4πρ(r), (1)
where φ(r) is the potential generated by a given charge density ρ(r);
• A model for the non-electrostatic terms to the total free energy of solvation.
The dielectric function ǫ(r) has to take the value of one where the solute is placed to
solve a vacuum-like quantum problem, and the bulk dielectric constant ǫ0 outside.
Several implicit solvation models have been reported in the quantum chemistry literature,
starting from the earliest work of Onsager10 to the widespread polarizable continuum model
(PCM) developed by Tomasi and co-workers.11–13
In the PCM formulation, the cavity surrounding the solute is described by a sharp and
discontinuous dielectric function ǫ(r), and a polarization charge density is exactly local-
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ized at the interface between the vacuum and the dielectric. In this way, the dielectric
environment is represented by an effective surface polarization charge, reducing the three-
dimensional dielectric problem into a two-dimensional one. In its original formulation this
approaches suffers from discontinuities in the atomic forces and numerical singularities, that
are a consequence of sharp cavities. This drawback hindered a PCM extension to ab-initio
molecular dynamics (MD). Modified versions have been proposed in the literature,14,15 as
well as advanced definitions of the dielectric cavity in terms of an isodensity of the solute
charge density.16,17
Another possibility is to represent the interface between the region inside the cavity
and the implicit solvent region outside the cavity by a smooth function of the solute elec-
tronic density ρ(r).18–20 Non-electrostatic contributions to the solvation free energy can be
expressed as function of the cavity volume and surface area.19,21,22 For a charge-dependent
approach ǫ[ρ](r) the functional derivative with respect to ρ(r) of the Hamiltonian delivers
new additional terms to the Kohn-Sham (KS) potential, both from the electrostatic energy
and the non-electrostatic contributions.22 These formulations are elegant and easy to be
parametrized. The cavity changes self-consistently during the wave function optimization
and forces do not depend explicitly on the shape of the cavity. However, there are a number
of drawbacks. The smooth transition between the quantum and continuum regions gives rise
to an inherently three-dimensional problem, while the self-consistent nature of the approach
requires solving a new problem at each step of electronic optimization: thus the approach is
intrinsically computationally more demanding than the rigid 2D alternatives such as PCM.
Moreover, accuracy might be affected due to the low number of parameters; neutral and
charged molecules cannot be treated simultaneously, as these require different parametriza-
tions,23 which lead to different Hamiltonians; furthermore, it is not possible to locally modify
cavity sizes looking only to the electronic density.
Greater flexibility can be added to these smooth functional approaches by defining them
in terms of a fictitious electronic density, which is kept rigid and defined in terms of pa-
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rameterized functions centered at atomic positions.24 Although this approach may overcome
some of the computational limitations of self-consistent alternatives, its original formulation
only focused on the electrostatic terms of the solvation free energy.
To make an mixed explicit/implicit treatment attractive to material science, it is neces-
sary to fulfill several requirements:
• Accurate forces and a numerical cost comparable to standard vacuum calculations to
make molecular dynamics or extensive potential energy surface (PES) explorations
feasible;
• A small number of cavity parameters, in order to easily fit a new environment or find
the best solvent conditions in a given range;
• Exact treatment of molecular or slab-like geometries, typical of systems plunged in a
solvent;
• Ability to treat neutral and charged molecules simultaneusly, in order to tackle complex
interfaces (e.g. a double layer).
The self-consistent continuum solvation (sccs) model developed by Andreussi et al.22
goes in this direction, satisfying the first three conditions. In the present work an improved
implicit solvation model is proposed. It preserves the flexibility of PCM to locally vary
the cavity’s size around each atom as well as the straightforward definition of the non-
electrostatic contributions to the total energy of the charge-dependent solvation models,
function of the cavity volume and surface.22 The dielectric cavity is based on analytic smooth
spheres centered on each atom, and it is fully continuous from the vacuum-like inner regions
to the external bulk solvent. Because of these features, we named our approach “soft-sphere”
model. The proposed formulation satisfies all the requirements listed above. It gives the same
accuracy for the ionic forces as an ordinary vacuum calculation. Compared to the similar
approache of Sanchez et al.,24 that is based on functionals of a fictitious solute density, our
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definition of the cavity in terms of atom-centered smooth functions simplifies significantly
the calculation of derivatives. In addition, the continuous function, describing the variation
of the permittivity, allows us to compute analytically all the non-electrostatic contributions
to the free energy. Once we fix the atomic radii, a given environment can be fitted with two
parameters in a straightforward way. Details of the model will be presented on Sec. 2.
Our implementation has a small numerical overhead, thanks to an efficient solver for
the generalized Poisson equation,9 based on a preconditioned conjugate gradient (PCG)
algorithm.
The method has been tested within Kohn-Sham density functional theory (DFT) as
implemented in the BigDFT package.25,26 Thanks to the underlining electrostatic solver of
this package,27 the solvation problem for slab or isolated systems is solved with the correct
boundary conditions. An extensive parametrization study has been carried out both for
neutral molecules and ions in water and two non-aqueous solvents. Results are reported in
Sec. 3.
Our tests suggest that our approach can well handle slab-like boundary conditions. In Sec.
4 we propose a benchmarking database for slab like geometries, that is based on the wetting
properties and solvent contact angles of solvated surfaces. Standard protocols parametrize
implicit solvation models on a database of experimental free solvation energies for neutral
and ionic solutes in several solvents.28 However, if applied to solid-liquid interfaces in ma-
terial science, several issues arise. The molecules of the standard benchmark sets have a
limited representation of some elements of the periodic table. Many of these underrepre-
sented elements, such as transition metal atoms, play however an important role in many
materials science problems. Solar-energy harvesting in dye-sensitized29 and hybrid cells30,31
or electro-catalytic water splitting32,33 are just two simple examples where such materials
are extensively exploited.
Finally, as an illustrative application of the approach we present a test case of relevance
in material science, namely the prediction of stable terminations of the CdS (1120) surface
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in an electrochemical medium.
2 Soft-sphere model
The electrostatic solvation energy is defined as the difference between the total energy of a
given atomic system in the presence of the dielectric environment Gel and the energy of the
same system in vacuum G0
∆Gel = Gel −G0. (2)
A full comparison with experimental solvation energies needs the inclusion of non-electrostatic
contributions to obtain the total free energy of solvation ∆Gsol. In this case the main terms
in the solute Hamiltonian, as introduced by PCM,12 are
∆Gsol = ∆Gel +Gcav +Grep +Gdis +∆Gtm + P∆V, (3)
where ∆Gel is the electrostatic contribution, Gcav the cavitation energy, i.e. the energy
necessary to build up the solute cavity inside the solvent medium. Grep is a repulsion term
representing the continuum counterpart of the short-range interactions induced by the Pauli
exclusion principle, whilst the dissociation energy Gdis reflects van der Waals interactions.
The thermal term ∆Gtm accounts for the vibrational and rotational changes and, finally,
P∆V includes volume changes in the solute Hamiltonian.
In the case of a charge density dependent cavity, it has been shown that the non-
electrostatic terms Gcav, Grep, and Gdis can be expressed as linear functions of the “quantum
surface” S and “quantum volume” V of the dielectric cavity.19,22 In particular, Scherlis et
al.19 extended the work of Fattebert and Gygi18 including a cavitation term expressed as a
product of the experimental surface tension of the solvent γ and the quantum surface S of
the solute cavity
Gcav = γS. (4)
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Both the quantum surface and volume can be expressed in terms of the dielectric function
ǫ(r). Neglecting as in other models the thermal contribution ∆Gtm to the total energy and
considering that for simulations at standard pressure the term P∆V does not play any role,
the total solvation free energy can be modeled by the following expression22
∆Gsol = ∆Gel + (α + γ)S + βV, (5)
where α and β are solvent-specific parameters which can be fitted to experimental data
such as solvation energies of neutral molecules and ions. The linear relation between the
repulsion energy and the quantum surface follows the idea of traditional PCM approaches,
since in PCM models Grep is proportional to the solute electronic density lying outside the
dielectric cavity34 and this last quantity can be linearly correlated to the cavity surface. The
linear model for the non-electrostatic terms follows the same lines as the solvation models
developed by Cramer and Truhlar,13,35 i.e. the SMx family, where these contributions are
evaluated as a sum over all the atoms, each one proportional to the solvent-accessible surface
area, albeit they use more variables. Although Eq. (5) has been introduced within a charge-
dependent model,22 we will use the same expression for the total solvation free energy for the
soft-sphere model after a suitable generalization of the quantum surface and volume. These
last non-electrostatic contributions do not change during the wavefunction optimization as
they do not depend anymore on the electronic charge density (now the dielectric cavity is
a function of the atomic coordinates). As a consequence only the electrostatic contribution
Gel needs to be computed via an optimization of the electronic wavefunction at the ab-initio
level. Appendix A outlines the main equations that have to be implemented in a DFT code
for the wavefunction optimization procedure in presence of an implicit environment.
The spatially varying dielectric function ǫ[ρ](r) or ǫ(r, Ri) has to meet several condi-
tions:
• Go monotonically from a value of 1 inside the cavity (mimicking vacuum) to the bulk
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dielectric constant ǫ0;
• Be smooth in the transition region to guarantee a proper discretization on a three
dimensional grid;
• Have as many continuous derivatives as possible to avoid convergence and numerical
issues during the iterative wavefunction optimization loop.
The soft-sphere cavity, which satisfies all these requirements, will be presented in Sec. 2.1.
Definitions of the quantum surface and volume will be given in Sec. 2.2.
For a dielectric cavity where ǫ(r, Ri) is an explicit function of the atomic coordinates
Ri, additional contributions arise to the atomic forces with respect to their gas-phase for-
mulation. Formulas for the forces are presented in Appendix A.3. A test for the energy and
force accuracy is reported on Sec. 2.3.
The methodology described in this paper as well as a complementary method based on a
charge density dependent continuum solvation model, developed by Andreussi et al.,22 have
been implemented in the BigDFT software package.
2.1 Definition of the dielectric cavity
We define the soft-sphere cavity by means of atomic-centered interlocking spheres described
by continuous and differentiable h functions which smoothely switch from 0 inside the sphere
to the value 1 outside. Their product, which is still continuous and differentiable in the whole
domain, reproduces the analytic cavity
ǫ(r, Ri) = (ǫ0 − 1)
∏
i
h(ξ; ‖r − Ri‖)
+ 1, (6)
where ǫ0 is the dielectric constant of the surrounding medium, ξ a set of parameters
describing the spheres and ‖r−Ri‖ the distance from the sphere center Ri. The derivative
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of the dielectric function with respect to Ri is
∂ǫ(r, Ri)∂Ri
=
[
ǫ(r, Ri)− 1
h(ξ; ‖r − Ri‖)
]
∂h(ξ; ‖r − Ri‖)∂Ri
(7)
for a given function h. This function can be expressed in terms of the error function
h(ri,∆; ‖r − Ri‖) =1
2
[
1 + erf
(‖r − Ri‖ − ri∆
)]
, (8)
where ri are the sphere radii which depend on the particular atomic species, and ∆ a pa-
rameter (fixed for all atoms) which controls the transition region (≈ 4∆ wide) where the
polarization charge is located. This choice for h guarantees a “soft” dielectric function in
Eq. (6) that is analytic everywhere except at the point r = Ri. With our choice of ∆
the discontinuities in the derivatives are however exponentially small and do not have any
negative effect on our calculations.
Following the lines of all PCM implementations,11,12,35,36 ri are taken to be equal to the
van der Waals radii rvdWi of the corresponding elements scaled by a constant multiplying
factor f (i.e. ri = frvdWi ). Finally, the soft-sphere cavity for a system of atoms with
coordinates Ri is uniquely defined by the set of parameters ǫ0, rvdWi ,∆, f. The solvation
model needs three additional parameters α, β and γ for the non-electrostatic terms to the
total solvation energy [Eq. (5)]. The parametrization of all parameters will be reported on
Sec. 3.
2.2 Geometrical contributions
Eq. (3) allows us to compute the total free energy of solvation. Thanks to Eq. (5) the inte-
gration at DFT level of the non-electrostatic terms becomes straightforward. The evaluation
of the free energy and its functional derivative can be expressed in term of the quantum
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surface and volume. Introducing the function
θ[ǫ](r) =ǫ0 − ǫ(r)
ǫ0 − 1, (9)
which takes a value of one inside the dielectric cavity and zero externally, the quantum
volume V of the solute can be defined as
V [ǫ] =
∫
drθ[ǫ](r). (10)
The integral has to be evaluated in the whole simulation domain. The associated quantum
surface S can be defined from the gradient of θ[ǫ]:21
S[ǫ] =
∫
dr|∇θ[ǫ](r)| = 1
ǫ0 − 1
∫
dr|∇ǫ(r)|. (11)
The functional derivatives of the quantum volume and surface with respect to the dielectric
function are:
δV [ǫ]
δǫ(r)= − 1
ǫ0 − 1, (12)
δS[ǫ]
δǫ(r)=
1
ǫ0 − 1
∫
dr′δ∂iǫ(r
′)
δǫ(r)
∂iǫ(r′)
|∇ǫ(r′)|
=1
ǫ0 − 1
(
∂iǫ(r)∂jǫ(r)∂i∂jǫ(r)
|∇ǫ(r)|3 − ∇2ǫ(r)
|∇ǫ(r)|
)
, (13)
where repeated indices are considered as summed.
Thanks to the continuous and differentiable permittivity ǫ(r), the function θ[ǫ](r) as well
as the quantum volume and surface can be analytically computed. This holds also for their
derivatives of Eq. (11-13).
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2.3 Energy and force accuracy
Accurate energies and forces are fundamental, especially for molecular dynamics and struc-
ture prediction calculations. Smooth and analytical cavities can guarantee accuracy levels
comparable to the ones in vacuum. Appendix A.3 shows the additional terms to the forces
for the soft-sphere model. These can be analytically computed, [see Eq. (48)]. In addition,
it is worth noting that all these analytic contributions to the forces can be computed nonce
once during a wave function optimization.
The extent of the transition region (≈ 4∆ wide) can affect the energy and force accuracy.
A large ∆ can lead to a not-physical superposition of large dielectric cavities, while a small
value can require a high resolution mesh which becomes computationally expensive. As
a consequence a compromise is needed, considering that the more mesh points fall in the
transition region, the more accurate is the description of solvent effects over the solute and
the polarization charge there located. In the following we report a test of the energy and
force accuracy for a water molecule in implicit water. Details of the model’s parameter
setting will be given in Sec. 3. In the present test we used the parametrization reported in
row 3 of Table 1.
Forces can be tested by means of the test-forces tool of BigDFT, an internal program
which compares the variation of the total energy ∆E with the path integral of the forces
−Fa · dra. As consequence of Eq. (46), their difference δ = ∆E +∑
a Fa · dra should be
zero. Due to the finite discretization and the numerical integration, such value depends on
the 3 dimensional spatial grid [hx, hy, hz] of the simulation domain (where the KS problem
is solved). It reaches δ ∼ 10−6 Ha in gas phase for very accurate calculations. Typical values
of hi (with i = x, y, z) are in most cases between 0.3 and 0.6 bohr. Present test runs make
use of an uniform grid spacing in all three directions, i.e. hgrid := hx = hy = hz. Fig. 1
shows the accuracy δ of the force computation as function of ∆. Solid lines represent the
value of δ for a test-forces run of the H2O molecule in vacuum for a given hgrid. The accuracy
of the forces is comparable to that of vacuum runs, for ∆ values between 0.2 and 0.8 a.u.;
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deviations start to take place at ∆ = 0.1 a.u., while smaller values affect the convergence
of the generalized Poisson solver due to the increased sharpness of the transition region. In
Fig. 1b we reported the accuracy of the solvation free energy ∆Gsol, i.e. Eq. (5), for the
same set of calculations. The accuracy is defined as the difference between values extracted
for a given hgrid and the most accurate grid we tested at hgrid = 0.2 a.u.. Varying ∆ the
same accuracy is recovered for hgrid ≥ 0.4 a.u.. An hgrid = 0.4 a.u. guarantees an accuracy
of 10−2 kcal/mol. Similar results obtained with other small organic molecules lead to similar
conclusions. Performance of the soft-sphere model is discussed in Appendix B for the same
system.
3 Parametrization
The soft-sphere solvation model is defined by the parameter set ζ = ǫ0, rvdWi ,∆, f ;α, β, γ,
i.e. six parameters plus an atomic radius rvdWi for each distinct atomic species of the quan-
tum system in contact with the implicit solvent. Only the first four parameters are related
to the dielectric cavity and the electrostatic contribution ∆Gel. To parametrize the solvation
model we need to map the ζ = ǫ0, rvdWi ,∆, f ;α, β, γ parameter space, and minimize the
mean absolute error (MAE) over a given collection of experimental free solvation energies.
In the following we will report the parametrization for a water environment, both for neutral
molecules and ions. In addition, we have parametrized two non-aqueous solvents, namely
mesitylene and ethanol. Structures and experimental data have been retrieved from the
Minnesota Solvation Database, version 2012.28,37–39 Solutes used as benchmark contain at
most the following elements: H, C, N, O, F, Si, P, S, Cl, Br, and I.
Soft norm-conserving pseudopotentials including a non-linear core correction40,41 along
with PBE functional were used to describe the core electrons and exchange-correlation for
all calculations of the present section. The Libxc42 library was exploited for the calculation
of the functionals. Solvation free energies were computed via Eq. (5) using atomic structures
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Figure 1: H2O molecule in implicit water described by the soft-sphere model: (a) force accuracy δ
obtained by means of the test-forces tool of BigDFT (see text) as function of the cavity parameter∆ for different grid spacing hgrid (colored solid lines represent the accuracy in vacuum runs for eachhgrid); (b) accuracy of the solvation free energy ∆Gsol with respect to the spacial grid hgrid fordifferent ∆.
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optimized both in vacuum and in the presence of the implicit solvent. BigDFT represents
the wavefunction in a wavelet basis set. These basis functions are centered on a cartesian
mesh of resolution [hx, hy, hz].25 The generalized Poisson equation, i.e. Eq. (1), has been
solved via the recently developed generalized Poisson solver based on Interpolating Scaling
Functions.9 Its preconditioner, based on the BigDFT vacuum Poisson solver, allows to solve
the electrostatic problem in some ten iterations with an exact treatment of free, surface and
periodic boundary conditions.27,43 A new version of the algorithm which includes a proper
use of an input guess is reported in Appendix B. All simulations reported in the present
study use an uniform grid of hx = hy = hz = 0.40 a.u. and free boundary conditions.
3.1 Neutral molecules
The soft-sphere model has been benchmarked on a collection of 274 aqueous solvation free
energies to reproduce a water environment. At first, to speed up the exploration of the
parameter space, we trained the model on a representative subset, which consists of 13
molecules spanning the main functional groups of the entire set.44
3.1.1 Cavity parameters
In order to reduce the size of the parameter space (in principle a MAE should be extracted
for each ζ setup), we proceeded fixing the largest possible number of parameters. Since
we are benchmarking the model on aqueous solvation free energies, the experimental value
at low frequency and ambient conditions of ǫ0 = 78.36 has been used. A critical point in
PCM approaches is the choice of shape and size of the cavity, which should reflect the charge
distribution of the solute. In the first formulation of this model,45 PCM atomic radii ri were
chosen to be proportional to the van der Waals radii.46,47 Importing a consistent set of van
der Waals radii rvdWi which spans the whole periodic table is more meaningful than fitting
radii for all the elements which would presumably lead to some overfitting. We decided to
import rvdWi from the Unified Force Field (UFF) parameterization48 for all elements of the
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periodic table. Hydrogen atoms have been considered explicit in the present work. ∆ has
been fixed to 0.5 a.u. for all atoms, generating a transition region of ≈ 2 a.u.. Different
choices of ∆ will be also investigated and discussed at the end of the paragraph. Finally, only
the multiplying factor f for the atomic radii are left free to vary, reducing the four-parameter
problem for the cavity definition to a single-parameter optimization.
The flexibility to locally change atomic radii has also another advantage. Considering
a wet surface, the radii for atoms which are in the bulk region may be further enlarged to
ensure that the dielectric permittivity is exactly 1 for all the domain which is inaccessible
by the solvent. This condition is not always satisfied by an implicit solvation model and has
to be checked.
Since we imported the non-electrostatic contributions from the sccs approach, we started
the present parametrization from the same non-electrostatic proportional factors (α + γ =
50.0 dyn/cm, β = −0.35 GPa). Finally, we ended up with a single-parameter optimization
for f to define the model. A value of f = 1.12 minimizes the prediction error of the
experimental aqueous solvation energies over the representative set of 13 molecules, giving a
MAE of 1.39 kcal/mol. Exporting the same ζ setup (with the optimized f) to the whole
set of 274 neutral molecules, we obtained a MAE of 1.10 kcal/mol. This demonstrates that
the soft-sphere model gives about the same accuracy as the charge density dependent sccs
model (MAE of 1.14 kcal/mol on the same set of 274 neutrals, see Table 1) and that it
is possible to export the linear approach for the non-electrostatic terms (implemented and
parametrized for the sccs model).
A careful analysis of the error distribution over all functional groups, allowed us to fine
tuning some van der Waals atomic radii rvdWi . We found that a slight variation of the
Nitrogen radius from the UFF48 value 1.83 Å to the one of Bondi46 1.55 Å improves the
whole parametrization. This trend reflects the interaction of the Nitrogen lone pair with
Hydrogen atoms of water, making their reciprocal distances smaller. The parametrization
curve varying f over the 13 molecules has been reported in Fig. (2) (black dots, full line).
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Table 1: Mean Absolute Errors in aqueous solvation free energies (kcal/mol) for a set of 274neutral molecules obtained with the soft-sphere solvation model.
atomic radii rvdWi cavity multiplying factor f α+ γ [dyn/cm] β [GPa] MAE [kcal/mol]
UFF48 1.12 50.0 -0.35 1.10
UFFa 1.16 11.5 0.00 1.12
UFFa 1.12 50.0 -0.35 1.03
Pauling47 1.36 50.0 -0.35 1.17
Bondi46 1.32 50.0 -0.35 1.06
a Van der Waals atomic radii from the Unified Force Field setup48 with Bondi’s radius46 for Nitrogen.
The MAE minimum is located again at f = 1.12 with a MAE of 1.15 kcal/mol for the 13
molecules. Applying the new setup to the full set of 274 molecules gives a MAE of 1.03
kcal/mol (see Table 1). This remarkable result has been achieved with the crude import of
the linear non-electrostatic model from sccs. The inset of Fig. 2 reports a global comparison
between experimental and ab-initio aqueous solvation free energies for the full set of 274
neutrals. All dots stay close to the diagonal, reflecting the accuracy of the soft-sphere model
over all functional groups.
PCM formulations in the literature mainly use van der Waals radii from Pauling47 or
Bondi46’s work. Even though these sets work well for organic molecules, values for the full
periodic table are not available, making the UFF setup more tempting for general ab-initio
calculations. Nevertheles we explored the performance of our solvation model with the van
der Wall radii of Pauling and Bondi’s. Following the same procediure as in our the previous
optimization, we first performed an optimization of the cavity multiplying factor f over the
representative set of 13 molecules, and only afterwards for the entire ζ setup of 274 neutral
molecules. Both f and MAEs for the full set are reported in Table 1. The optimized f for
both sets is larger than the one obtained with the UFF implementation. This is related to
Hydrogen radius, being 1.20 Å in the Pauling and Bondi’s work, and 1.443 Å in the UFF. The
overall accuracy does not improve with respect to the UFF parametrization, where MAEs
of 1.17 and 1.06 kcal/mol have been found for Pauling and Bondi’s radii, respectively.
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Figure 2: Mean Absolute Error in aqueous solvation free energies for the representative set of 13neutral molecules as function of the cavity multiplying factor f (UFF radii rvdWi
48 with Bondi’sradius46 for Nitrogen, α + γ = 50.0 dyn/cm and β = −0.35 GPa). The inset reports experimentaland ab-initio aqueous solvation free energies for the full set of 274 neutral molecules with the sameparameters setup (optimized f = 1.12).
So far, all parameters related to the dielectric cavity have been optimized, except ∆. A
window of 0.5 < ∆ < 0.65 does not change the results in term of accuracy (increasing ∆
request a proportional increase of f to recover the same MAE).
3.1.2 Non-electrostatic parameters
The last optimization step treats the non-electrostatic parameters α, β and γ. Assuming a
surface tension of γ = 72 dyn/cm for water at room temperature, only α and β need to be
optimized. The sum α + γ can be considered as a single parameter since both variables are
multiplying factors of the quantum surface S in Eq. (5). We extrapolated all the β and α+γ
couples which minimize the MAE over the representative set of 13 molecules (determining
also the optimized f for each fixed couple). As noticed in Ref. [ 22], we found an almost
perfectly linear behavior between the error, i.e. the MAE, as function of β and α+ γ. This
was probably due to the small variation in size of the molecules considered in the training
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set, which makes the volume and the surface terms to be linearly related. All MAE minima
lie approximately on the same line β = c(α + γ) + d, where c = −9.1 nm−1 and d = 0.1045
GPa. The best parametrization still correspond to the sccs optimal setup with α+ γ = 50.0
dyn/cm and β = −0.35 GPa.
However, given the strong correlation between the two tunable parameters, it is important
to stress that the optimal parametrization reported above may be strongly linked to the type
of systems (small isolated molecules) under study. Whether these parameters could be used
in systems with very different sizes, e.g. large molecular and supramolecular systems or
solids described using a slab geometry, is not obvious. For this reason, unless a more refined
optimization that takes into account experimental data on large systems is performed, a
parametrization only relying on the surface term, thus putting β = 0 GPa, should be more
transferable and preferable for the above applications. For surface calculations, β = 0 GPa
is mandatory to avoid an unphysical dependence of the energy on the slab volume.
3.1.3 Results
The final MAEs for both surface (row 2) and small cluster (row 3) parametrizations over
the full set of 274 experimental solvation energies are 1.12 and 1.03 kcal/mol, respectively
(see Table 1). Aqueous solvents are widely used and of great relevance in wet chemistry.
Investigations may require a mixed explicit/implicit treatment of a water environment. The
soft-sphere model predicts a solvation free energy for H2O and (H2O)2 in implicit water of
−5.74 and −10.19 kcal/mol for the surface parametrization, and −5.60 and −10.34 kcal/mol
for the small clusters one. Experimental values are −6.31 and −11.27 kcal/mol, respec-
tively.28 Predictions are within 1.1 kcal/mol and both parametrizations perform pretty well.
Marenich et al. in Ref. [ 38] explored performances of several implicit solvation models.
These approaches hold as common feature the use of dozen of parameters to model all
contributions to the solvation free energy. Among others, they tested the SM838 model of the
SMx family, the Integral Equation Formalism Polarizable Continuum Model45 of Gaussian
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03 (IEF-PCM),49,50 the Conductor-like PCM model51 in GAMESS52 (C-PCM/GAMESS),
Jaguar’s Poisson-Boltzmann self-consistent reaction field solver53,54 (PB/Jaguar) and the
Generalized Conductor-like Screening Model in NWChem55 (GCOSMO/NWChem). MAEs
of these methods for aqueous solvation free energies are been reported in Table 2. for the
set of 274 neutrals Data from the recent SM12 implementation39 have been also included.
In addition we tested the charge-dependent sccs model implemented within the BigDFT
package. Using only two parameters (surface parametrization), the soft-sphere model applied
to the same dataset lies in the same range of accuracy. A carefully tuning of atomic radii
and/or a functional dependence of delta over the periodic table could further lower the MAE
and improve the whole accuracy.
Table 2: Mean Absolute Errors in aqueous solvation free energies (kcal/mol) for several solvationmodels (MAEs from Ref. [ 38]). Model benchmarks refer to same set of 274 neutrals, 60 anionsand 52 cations of the Minnesota Solvation Database, version 2012.28
Method neutrals cations anions
soft-spherea 1.12 2.13 2.96
sccs23 1.14b 2.27c 5.54c
SM838 0.55 2.70 3.70
SM1239 0.59 2.90 2.90
PB/Jaguar38 0.86 3.10 4.80
IEF-PCM38 1.18 3.70 5.50
C-PCM/GAMESS38 1.57 7.70 8.90
GCOSMO/NWChem38 8.17 11.00 7.00
a parametrization of row 2 Table 1; b sccs implemented in BigDFT; c sccs for ions corresponds to a
reduced set of 55 anions and 51 cations23 of the same Minnesota data set.
3.2 Anions and cations
In order to strictly validate the soft-sphere model, we benchmarked it on a set of aqueous free
energies of solvation for 112 singly-charged ions (60 anions and 52 cations). The Minnesota
Solvation Database, version 2012,28,37,39 contains two sets of them, i.e. the unclustered and
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the selectively clustered data set. The last is identical to the first, except for 31 ions which are
clustered with an explicit water molecule. More details can be found in the article discussing
the SM6 parametrization.37 As done in the last articles of the SMx family,38,39 and in order
to make comparisons with others solvation models, we benchmarked the soft-sphere model on
the selectively clustered data set. The two main model parametrizations have been explored
in row 2 and 3 of Table 1, except for the radii multiplying factor f which was left free to be
optimized.
Due to the nonlinear effects in the polarization of the surrounding dielectric with high
fields, the factor f , or alternatively all the radii of atoms bearing the ionic charge, need
to be reconsidered.11,12 Furthermore, ab-initio gas-phase optimizations and Monte Carlo
simulations of systems in aqueous solution suggested that solvent molecules of the first
solvation shell come closer to a charged molecule than to a neutral.56 Although most PCM
models implement a charge dependency on the atomic radii,11,57 we decided to reproduce
experimental solvation energies optimizing the global cavity factor f both for anions and
cations. A similar approach has been also investigated by others authors.56 This procedure
is less accurate than modifying single atomic radii, especially when the solute molecule sizes
become larger or for extended interfaces. However, it is still a good approximation for small
size ions. A more detailed approach with a fine tuning of the individual van der Waals radii
and their dependence on the local atomic charges is beyond the scope of the present paper.
Table 3 reports MAEs both for anions and cations. Fixing the non-electrostatic parame-
ters to α+γ = 50.0 dyn/cm and β = −0.35 GPa, the optimization procedure on f suggested
an f = 0.98 as optimal multiplying factor for anions and f = 1.10 for cations, with MAEs
of 3.05 and 2.07 kcal/mol, respectively. These results are in perfect agreement with similar
findings obtained in the parametrization of self-consistent cavities.23 Fig. 3 reports experi-
mental and ab-initio aqueous solvation free energies both for anions (empty blue circles) and
cations (full black circles).
Concerning cations, dots belonging to the diagonal are mainly cations containing Nitro-
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gen. On the other side, the model tends to underestimate free solvation energies for four
different types of alcohols, two types of ketons and the water cation. This trend suggests
that smaller cavities are needed for these cations due to the stronger electrostatic interaction
between the solvent and the solute. Excluding these last seven solutes, the overall MAE for
the remaining 45 cations decreases from 2.07 to 1.33 kcal/mol. A factor of f = 1.04 allows
to improve the prediction for the 7 units (alcohols, ketons and the water cation) where their
MAE is 2.07 kcal/mol. This fine analysis has been conducted to identify margins for im-
provements. Similar arguments are valid for anions, but a systematic study of functional
groups is less straightforward since oxygen comes into play in the majority of the units. An
optimization of f keeping fixed α + γ = 11.5 dyn/cm and β = 0 GPa, leads to a similar
accuracy for the prediction of solvation free energies (see Table 3). MAEs are 2.96 and 2.13
kcal/mol for anions and cations, respectively.
We see that the soft-sphere model performs very well for ions in water, considering that
the estimated experimental average uncertainty for solvation free energies of ionic solutes
is 3 kcal/mol.37 It is worth noting the high accuracy has been reached, by only optimizing
one parameter, i.e. f , with respect to the parametrization for neutral solutes. Table 2
reports MAEs for the same set of ions in water obtained with others solvation models. All
data refer to the same selectively clustered data set of 60 anions and 52 cations of the
Minnesota Solvation Database, version 2012, except sccs which corresponds to a reduced set
of 55 anions and 51 cations23 of the same database. The soft-sphere model outperforms all
current implicit solvation models, with a MAE of 2.57 kcal/mol over the full ion dataset. The
possibility to use the same α, β and γ and to change only the radii upon ionization allows
to tackle systems which contains simultaneously neutral and charged complexes. This is not
possible in charge density dependent approaches which can not distinguish between atoms
and need different non-electrostatic parametrizations (and then different Hamiltonians) for
neutrals and ions.
Since both parametrizations of Table 1, i.e. row 2 and 3, are able to accurately reproduce
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solvation energies of neutrals and ions, and taking into account that surface calculations
require β = 0 GPa and that reducing the parameter space is always advantageous, we can
just use the quantum surface S in the model of non-electrostatic contributions in Eq. (5).
The whole solvation model is thus uniquely defined by two parameters, i.e. f and α.
Figure 3: Experimental and ab-initio aqueous solvation free energies for a set of 112 single-chargedions, of which 60 anions (empty blue circles) and 52 cations (full black circles), obtained with thesoft-sphere model (UFF radii rvdWi
48 with Bondi’s radius46 for Nitrogen, α+ γ = 50.0 dyn/cm andβ = −0.35 GPa, f = 0.98 for anions and 1.10 for cations).
Table 3: Mean Absolute Errors in aqueous solvation free energies (kcal/mol) for a set of 112 ions(60 anions and 52 cations) obtained with the soft-sphere modela .
cavity multiplying factor f α+ γ [dyn/cm] β [GPa] MAE [kcal/mol]
Anions 0.98 50.0 -0.35 3.051.00 11.5 0.00 2.96
Alcohols and ketons cations 1.04 50.0 -0.35 2.071.04 11.5 0.00 1.88
Cations without alcohols and ketons 1.10 50.0 -0.35 1.331.11 11.5 0.00 1.46
All Cations 1.10 50.0 -0.35 2.071.10 11.5 0.00 2.13
Total ions - 50.0 -0.35 2.60- 11.5 0.00 2.57
a
UFF radii rvdWi
48 with Bondi’s radius for Nitrogen.
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3.3 Non-aqueous solvents
The study of chemical reactions within non-aqueous solvents is as also important. Therefore
it is worth investigating how the parametrization has to be modified for a non-aqueous
solvent. As test cases, we took two solvents, namely mesitylene and ethanol, which have low
dielectric constants ǫ0 of 2.2658 and 24.85, respectively at 20 C.
To optimize the soft-sphere parameters ζ = ǫ0, rvdWi ,∆, f ;α, β, γ, experimental sol-
vation free energies for a set of seven (mesitylene) and eight (ethanol) organic molecules
have been taken from the Minnesota Solvation Database.28 Changing only ǫ0 to the exper-
imental value of the non-aqueous solvent and taking all others parameters from the water
parametrization (row 2 or 3 of Table 1) does not reproduce the experimental data, underesti-
mating all solvation free energies with an overall MAE of 3.42 and 2.84 kcal/mol for mesity-
lene and ethanol, respectively. Turning off the non-electrostatic contributions (α + γ = 0
dyn/cm, β = 0 GPa), and using the proper ǫ0 has the same effect with MAEs of 3.42 and 1.74
kcal/mol. As a consequence each solvent needs its own parametrization. PCM approaches
usually have solvent-dependent parameters, like the multiplying factor for the atomic radii
or several collections of parameters for the non-electrostatic energetic.
We therefore set in addition the values rvdWi for the radii to the UFF values48 with the
exception of Nitrogen where we use the Bondi’s radius and used again ∆ = 0.5 as we did for
water. The surface tension at 20 C of mesitylene is γ = 28.80 dyn/cm,59 whilst for ethanol
is γ = 22.10 dyn/cm. Since solvation energies are well reproduced with only the surface
term in Eq. (5), we end up with a two-parameters optimization, i.e. f and α. First we
optimized α, keeping f to the water value of f = 1.16. This provides the major correction
to the overall MAE. Then f and α have been fine tuned concurrently. A value of f = 1.22
and α+ γ = −12.0 dyn/cm provide a MAE of 0.71 kcal/mol for mesitylene. Doing the same
for ethanol, a f = 1.22 and α+ γ = −4.0 dyn/cm allow to reproduce solvation free energies
with a MAE of 1.28 kcal/mol. Accuracies are comparable to the SM8 model, which gives a
MAE of 0.40 (mesitylene) and 1.53 (ethanol) kcal/mol for the same data set of neutrals.38
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Hence our approach has an accuracy comparable to similar state-of-the-art methods, despite
the reduced number of parameters. Nonetheless, it is important to stress that the reported
parameterizations may be affected by the reduced size of the fitting set: the limited numbers
and the narrow ranges of experimental solvation free energies used for the fit may give rise
to a parameterization which is not transferable to a broader set of compounds.
This is particularly striking in a model which only relies on two tunable parameters, for
which the optimized non-electrostatic terms end up being always negative. Thus, similarly
to the electrostatic term, these terms contribute to a stabilization of the solute, excluding
the possibility of modeling insoluble compounds with positive solvation free energies. On
the other hand, the optimization procedure gave a prefactor for the atomic radii f greater
than the corresponding value in water. This trend is in agreement with the larger sizes
of mesitylene and ethanol molecules with respect to H2O, and meets previous studies of
solvated molecules where the cavity is defined as a convolution of both solute and solvent
electronic densities.60 Including an optimization on β or additional dependences for the non-
electrostatic terms on specific solvent descriptors like the refractive index and the Abraham’s
hydrogen bond acidity and basicity parameters,38 can further improve the prediction power
on non-aqueous solvents and make the soft-sphere model universal, where “universal” means
applicable to all solvents.
4 Solid-liquid interfaces and contact angle
As a further benchmark system for implicit solvation models we consider the contact angle
θC that a liquid drop on a surface forms with the surface plane. It is experimentally accessi-
ble quantity and a collection on several substrates can be used as test set. Sessile-drop and
captive-bubble techniques are common experimental reference methods.61 This benchmark-
ing approach is alos useful for the parametrization of elements of the periodic table that are
not represented in standard databases of experimental solvation energies.
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Although the idea of a direct comparison of predicted and experimental contact angles
seems straightforward, its is not technically challenging. Clean surfaces do not exist in na-
ture where reconstructions, adsorbed molecules or radicals, and defects usually take place.
An implicit solvation approach gives results for ideal surfaces whereas experimental measure-
ments may be influenced by all these deviations from the ideal behaviour. Other simulation
techniques could be applied where both the surface and the liquid are treated explicitly (i.e.
molecular dynamics, Monte Carlo methods or others62). The proposed benchmark provides
necessary conditions that an implicit scheme has to fulfill if applied to a solid-liquid interface.
Simple conditions have to be verified like partial or complete wetting, and the degree of the
solid-liquid interaction by approximately comparing θC.
The contact angle quantifies the wettability of a surface and reflects the equilibrium
between liquid (drop), solid (surface) and vapor-phase interactions. Looking to the contact
line at the surface and imposing the thermodynamic equilibrium between surface tensions of
the three phases, a relation can be deduced for the contact angle in terms of the interfacial
energies solid-vapor γSG, solid-liquid γSL, and liquid-vapor γLG (i.e. the surface tension)
cos θC =γSG − γSL
γLG
. (14)
This relation is known as the Young’s equation.63,64 Since the three surface energies at equi-
librium form the side of a triangle, partial wetting occurs only when the triangle inequalities
γij < γjk + γik is satisfied.
θC can easily be related to the work of adhesion of the solid and liquid phases in contact,
i.e. WLS = γSG + γLG − γSL, leading to the Young-Dupré equation65 WLS = γLG(cos θC + 1).
The work of adhesion is the work which must be done to separate two adjacent materials at
their phase boundary (liquid-liquid or solid-liquid) from one another. Conversely, it is the
energy which is released in the process of wetting and is a measure of the strength of the
contact between two phases. γSG and γLG are the energies required to create a new surface
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whilst γSL represents the work that has to be done in order to form the interface. These
quantities are difficult to determine experimentally and only θC and γLG can be directly
determined. Contact angle measurements can give access to γSG and γLG indirectly via the
Young equation. In this case γSL is assumed to be a function of γSG and γLG. Several
functional relations are present in the literature,66–69 taking into account the polar or non-
polar character of the solvent as well as its acid or basic nature.70 In order to classify
hydrophobic and hydrophilic surfaces, van der Oss exploited the concept of free energy of
solvation ∆GSL both for molecules and condensed matter systems.71 It can be related to the
work of adhesion being ∆GSL = −WLS
Another parameter, especially useful to characterize hydrophilic surfaces, is the spreading
coefficient S defined as S = γSG − γLG − γSL. It represents the work performed to spread
a liquid over a unit surface area of a clean and non-reactive solid (or another liquid) at
constant temperature and pressure and in equilibrium with liquid vapor. Coupled with the
Young’s equation it leads to S = γLG(cos θC − 1), which means that a liquid drop partial
wets the surface only for S < 0, while a total wetting occurs for S > 0. Zero wetting takes
place when WLS < 0. These relations can be made explicit by rewriting cos θC as
cos θC =WLS
γLG
− 1 =S
γLG
+ 1. (15)
It is also clear that WLS > γLG implies that θC < 90, i.e. the surface has an hydrophilic
character, whereas it is hydrophobic in the opposite case. Furthermore, both WLS and
S should be similar in magnitude to γLG to achieve partial-wetting conditions (i.e. 0 ≤
WLS/γLG ≤ 2, −2 ≤ S/γLG ≤ 0). Once θC is experimentally determined, these conditions
need to be fulfilled, providing useful fixed points to benchmark a solvation model.
For a slab, the DFT surface energy γSG is calculated according to
γSG =1
2Alim
N→+∞
(
ENslab-SG −NEbulk
)
(16)
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where N is the number of atoms within the slab, A is the slab’s cross section area, ENslab-SG
is the total energy of the N -atoms relaxed slab, and Ebulk the energy of the bulk system per
atom. The solid-liquid interface energy γSL can be easily computed by means of a similar
equation
γSL =1
2Alim
N→+∞
(
ENslab-SL −NEbulk
)
, (17)
considering ENslab-SL as the total energy of the slab in contact with the implicit solvent. This
approach gives direct access to γSL and it is not necessary to specify an analytical functional
dependence of γSL in terms of γSG and γLG in order to estimate the contact angle via Eq.
(14).
We determined spreading the coefficient S, the work of adhesion WLS and the contact
angles θC for a collection of surfaces in contact with water. Table 4 reports computed data for
a silver (001) surface, a cleaved and reconstructed (001) surface of SiO2 α-quartz72 and two
carbon-based surfaces, i.e. diamond (001) and graphene. Thanks to the implemented UFF
tabulation of van der Waals radii, our soft-sphere model is able to handle such materials. We
used the parametrization reported in the second row of Table 1, valid for surface calculations
(β = 0 GPa). It is worth noting that additional benchmark procedures have to be carried out
for the cavity parameters ζ = ǫ0, rvdWi ,∆, f ;α, β, γ to guarantee a consistent physical
description of the solid-liquid interface. In particular van der Waals radii rvdWi for elements
not present in the regular set of organic molecules have to be monitored. Small values can
lead to no-physical dissolution of surface atoms.
The computation of Ebulk is not necessary because θC, WLS and S are all functions of
the difference γSG − γSL, and Ebulk of Eq. (16,17) is cleared. From this side, cos θC can be
seen as the solvation energy per unit area ∆Gsolslab = (EN
slab-SL −ENslab-SG)/2A divided by γLG,
namely
cos θC = −Gsolslab
γLG
. (18)
For water at room temperature γLG has the value 72 dyn/cm. The parameters for the DFT
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calculations (grid spacings, Monkhorst-Pack k -point mesh) have been chosen in BigDFT
to reach a surface energy accuracy of 0.01 J/m2. All systems have been relaxed both in
vacuum and in the presence of the implicit aqueous environment until all the forces were less
than 5 meV/Å. BigDFT allows to use exact surface boundary conditions, avoiding spurious
interactions in the direction orthogonal to the surface. As an additional check, we compared
vacuum surface energies [Eq. (16)] with DFT literature data, recovering a good agreement
for all systems.
Table 4: Simulated work of adhesion WLS, spreading coefficient S and contact angle θC for severalsurfaces in contact with implicit water described by the soft-sphere model.
slab layers WLS S θC
[mJ/m2] [mJ/m2] [degree]
silver (001) 8 479 33 -
SiO2 α-quartz cleaved (001) 18 181 35 -
SiO2 α-quartz reconstructed (001) 27 76 -69 87
Diamond (001) 12 69 -76 92
Graphene 1 62 -82 97
Unless an oxide layer is present, clean metal surfaces are hydrophilic meaning that wa-
ter completely or nearly completely wets them (gold, silver, copper and others),61,73 The
positive value of S for clean (001) silver surface (see Table 4) predicted by the soft-sphere
model agrees with its hydrophilic character, allowing for a complete wetting of its surface.
A similar validation can be extended to other metallic materials and represents a prelimi-
nary checkpoint if implicit solvation approaches need to be integrated in such solid-liquid
investigations.
Furthermore, as a consequence of their bulk bonding strength, surfaces can also be divided
into high- and low-energy surfaces, The stronger the chemical bonds are, i.e. the higher the
surface energy is, the more easily complete wetting is reached. In our set of surfaces, the
cleaved SiO2 α-quartz surface falls in this last class. Among the different SiO2 highly-reactive
surfaces, the (001) was chosen because it is the most stable.72 Applying Eq. (16) we recovered
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a surface energy of γSG = 2.22 J/m2. This value is consistent to the one reported in Ref. [
72], from where the cleaved and reconstructed SiO2 structures come from. Implicit solvation
correctly predicts its hydrophilic character with a spreading coefficient S > 0 as reported in
table 4. Conversely, the reconstructed SiO2 α-quartz (001) has a γSG o 0.39 J/m2. Because of
its negative spreading coefficient, implicit solvation predicts a partial wetting for this surface
with a contact angle of θC = 87 and a work of adhesion of WLS = 76 mJ/m2. Therefore
the implicit procedure is able to predict different θC for high and low energy surfaces of
the same material. A direct comparison with experimental contact angles is difficult since
real surfaces undergo reconstruction as well as hydroxylation, contamination and patterning
concurrently.74
Similar considerations hold for the clean (001) diamond surface. Molecular dynamics and
Monte Carlo simulations predicted an hydrophobic character for an hydrogen-terminated
diamond slab.62,75 In our case a work of adhesion of 69 mJ/m2 has been obtained, together
with a θC = 92 degree. Remaining within the carbon-material class, we applied the implicit
solvation scheme to a graphene sheet. Graphene is a two dimensional arrangement of carbon
atoms, bonded covalently to form an honeycomb lattice due to the sp2 orbital hybridization.
Weak van der Waals interactions take place with water molecules once graphene is in contact
with an aqueous solution, allowing for a partial wetting of the surface. Several experimental
and simulation studies investigated the wetting features of such carbon allotropes. For
graphene the measured a contact angle is 127,76 although a theoretical study demonstrated
a discrepency with the contact angle of graphite,77,78 suggesting a θC of the order of 95-100.79
For the partial wetting behavior, data extracted with our soft-sphere solvation scheme is in
agreement with these experimental and theoretical studies, predicting a negative spreading
coefficient, a contact angle of 97 and a work of adhesion of 62 mJ/m2. A lowering of the
cavity prefactor to f = 1.12 does not affect the whole wetting predictions for all studied
solid-liquid interfaces in terms of the hydrophobic/hydrophilic character.
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5 Structure of CdS (1120) surface in electrochemical me-
dia
As an application, we investigated the structure of the CdS (1120) surface in contact with
water at different pH and bias potentials, mimicking electrochemical conditions. CdS is
among the few promising photocatalytic materials for the water splitting reaction, due to
its appropriate band gap and band positions.80–82 Recently, it was reported that nanorods
and other nanoporous materials based on CdS show very high photon-to-hydrogen conversion
efficiency when used together with hole-scavenger molecules.83–85 It was observed experimen-
tally that a high pH is necessary for the efficient photo-catalytic activity of CdS.84 However,
photocorrosion was seen after prolonged photo-exposure at such pH levels.84,85 Understand-
ing the structure and reactivity of the CdS surfaces at electrochemical conditions can provide
valuable informations on improving the CdS-based photocatalytic systems.
We employed here the ab-initio thermodynamics based “computational hydrogen elec-
trode” (CHE) approach86,87 to study the structure of the most exposed CdS (1120) surface
at different pH and bias potentials. Solvent effects have been included using the implicit
soft-sphere solvation model. Most of the CHE computations in literature were carried out
under in vacuo conditions,87–92 and a qualitative understanding of solvent effects on the
results of such computations remain unclear.93–96 Recently the CHE approach was used to
study the structure of CdS surfaces in contact with vacuum.97
Here we have computed the formation free energies of various surface terminations (O,
OH, OOH), varying pH and bias potentials. The following reactions were considered for the
formation of these terminations:
2H2O∗ → 2O∗ + 4H+ + 4e−, (19)
2H2O∗ → 2HO∗ + 2H+ + 2e−, (20)
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2H2O∗ + 2H2O → 2HOO∗ + 6H+ + 6e−. (21)
The above equations include two moles of H2O∗ as there are two adsorption sites in the
surface model that we have considered. Here O∗, HO∗, HOO∗ and H2O∗ represent O, HO,
HOO and H2O terminated surfaces, respectively. Following Ref. [ 86], the free energy
change ∆G of the above reactions with reference to a standard hydrogen electrode (SHE)
at pH2= 1 bar and T = 298 K as a function of bias potential U and pH can be calculated,
respectively, as
∆G2O∗ = ∆E2O∗ +∆EZPE2O∗ − T∆S2O∗
+4∆G(pH)− 4eU, (22)
∆G2HO∗ = ∆E2HO∗ +∆EZPE2HO∗ − T∆S2HO∗
+2∆G(pH)− 2eU, (23)
∆G2HOO∗ = ∆E2HOO∗ +∆EZPE2HOO∗ − T∆S2HOO∗
+6∆G(pH)− 6eU. (24)
Here ∆EZPE and T∆S are the zero-point energy correction and the vibrational entropic
contribution to the free energy, respectively, and ∆G(pH) = −kBTpH ln 10. ∆E2O∗ , ∆E2HO∗
and ∆E2HOO∗ are calculated from DFT calculations as follows:
∆E2O∗ = E2O∗ + 2EH2− E2H2O∗ , (25)
∆E2HO∗ = E2HO∗ + EH2− E2H2O∗ , (26)
∆E2HOO∗ = E2HOO∗ + 3EH2− E2H2O∗ − 2EH2O. (27)
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The energies of the surface-adsorbed structures were calculated with and without the implicit
solvation model.
Preliminary calculations showed that a slab built up from an 1×1 supercell with 8 atomic
layers, i.e. Cd16S16, and a k-point sampling 5×1×5 adequately converged the energy (see
the Supporting Information). Zero-point energy and vibrational entropic contributions were
computed from the harmonic frequencies, extracted by means of a finite difference method.
For these last two contributions, a 2×1×2 k-points mesh guarantees reasonable convergence.
Further, we found that both ∆EZPE and T∆S were nearly the same with and without con-
tinuum solvent (see the Supporting Information). Thus, we have not included solvent effects
while computing these quantities. All computations were performed with spin-polarized
DFT in a wavelet basis uising the PBE98 exchange-correlation functional as implemented
in the BigDFT program.25,26,42 Surface boundary conditions were employed for solving the
Poisson equations.43 The real space grids along all the three directions were set to 0.45 Bohr.
Soft norm-conserving pseudopotentials including non-linear core correction40,41 were used to
describe the core electrons. For the dielectric cavity, a van der Waals radius for Cd of 1.5846
Å has been used, since the UFF value 1.424 Å leads to the surface dissolution. The implicit
model predicts a spreading coefficient S of 28 mJ/m2 for the CdS (1120) surface in contact
with water, confirming its hydrophilic character (S > 0).
Different contributions to ∆G at standard conditions (with reference to SHE) are given in
Table 5. The zero-point energy and the entropy of H2(g) and H2O(g) were taken from an ex-
perimental database.99 GH2O(l) was computed asGH2O(g)(298 K, 1 bar)−∆Gv(298 K, 1 bar),100
where ∆Gv is the free energy of vaporization, equal to 0.09 eV computed based on an exper-
imental database99 (see also the Supporting Information). We have considered adsorption
at both surface Cd and S sites for all the adsorbates. It was found that except for oxygen,
all the other adsorbates prefer to bind on Cd sites, consistent with previous works.97
In Fig. 4 we show the free energy change ∆G for different surface terminations as a
function of bias potential U at pH = 0 and pH = 14. We report results both for pure gas-
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phase conditions (solid lines) and in the presence of an implicit water environment (dotted
lines). For small bias potentials, the most stable surface is H2O terminated, where molecular
water is adsorbed on surface Cd atoms at both pH values. The HOO terminated surface
is stable above 3.12 V (2.53 V) bias potential with (without) implicit water at pH = 0,
whilst in the range from 0.93 V (1.24 V) to 3.12 V (2.53 V) the O terminated surface is
thermodynamically preferred under these conditions. Similar conclusions hold for pH = 14.
The inclusion of the solvent interaction results in a substantial shift of the relative stability
with respect to the bias potentials. For e.g., the threshold bias for observing HOO terminated
surface has increased by nearly 0.6 V by including the implicit solvent in the calculations.
Table 5: Contributions to the free energy change ∆G computed at 298 K and 1 bar with referenceto the standard hydrogen electrode for O, HO and HOO terminated CdS (1120) surfaces. ∆ESolvent
and ∆E are the change in energy calculated with and without implicit solvent, respectively [Eqs.(25-27)].
∆E [eV] ∆ESolvent [eV] ∆EZPE [eV] T∆S [eV]
2O∗ (S) 6.39 5.14 -0.63 0.79
2HO∗ (Cd) 5.38 4.73 -0.40 0.56
2HOO∗ (Cd) 10.96 10.87 -0.82 0.10
6 Conclusions
A soft-sphere solvation model for ab-initio electronic-structure calculations has been devel-
oped, parametrized and tested. The transition from the dielectric cavity to outside is contin-
uous and differentiable, allowing for the analytical calculation of the additional terms to the
forces as well as the cavity-dependent non-electrostatic contributions to the total energy that
are described in terms of the quantum surface. As consequence we could obtain very high
accuracy on energies and forces. The computational cost does not increase significantly with
respect to standard gas-phase calculations. We expect that these features make our methods
useful for numerous applications in material science. Fixed the atomic radii, only two fitting
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Figure 4: Free energy change ∆G as a function of the bias potential U for O, HO, HOO andH2O terminated CdS (1120) surfaces at pH = 0 and pH = 14. Results obtained in pure gas-phaseconditions (solid lines) and with the presence of an implicit water environment (dotted lines) areshown. Various threshold bias potentials are marked.
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parameters are needed to define the model, independently of the cluster or slab geometry.
Mean absolute errors with respect to experimental solvation energies for neutral molecules
and ions are close to the performances of others solvation approaches with a larger number
of parameters. In particular for ions the implemented approach outperforms in accuracy all
existing implicit solvation models. The flexibility of the model to locally change atomic radii
allows to simultaneously simulate neutrals and ions, conditions typical of a double layer of
an electrolyte in contact with a surface. A new benchmark protocol for solid-liquid inter-
faces is proposed in terms of wettability and contact angles. This allows to find parameters
for elements not represented in current solvation databases. Results are shown for several
metallic and insulating surfaces in contact with water. As a showcase application we studied
the terminations of CdS (1120) surface in an electrochemical medium as a function of bias
potential and pH. We observed that the inclusion of solvent effects introduces a substantial
shift in the bias potentials at which surface terminations change. The relative stabilities
of several adsorbed species are affected by the presence of solvent. As a consequence, its
inclusion in computations which are based on the CHE approach is crucial.
Acknowledgement
G. Fisicaro thanks Dr. D. Tomerini for continuous useful discussions. This work was done
within the PASC and NCCR MARVEL projects. Computer resources were provided by
the Swiss National Supercomputing Centre (CSCS) under Project ID s707. L. Genovese
acknowledges also support from the EXTMOS EU project.
Supporting Information
Convergence tests are reported varying the k -point sampling for the total energy as well as
the zero-point energy EZPE and the vibrational entropic contribution TS of the clean CdS
surface. Formulas are presented for the free energy of liquid water GH2O(l)(298 K, 1 bar).
This information is available free of charge via the Internet at http://pubs.acs.org.
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A Set up of KS Self-Consistent cycle under implicit en-
vironments
In this section we will outline the main equations that have to be implemented in a DFT code
for the wavefunction optimization procedure in presence of implicit environments. Formulas
reported are general and do not depend on the particular functional form of the dependence
on the atomic coordinates Ri or on the electronic charge density ρ(r) for the cavity.
A.1 Electrostatic Energy
For an implicit environment the Hartree Potential might be seen as a functional of both
the electronic density and the dielectric function, i.e. φ = φ[ρ, ǫ; Ω]. Here Ω represents
the real-space domain (together with the appropriate boundary conditions) choosen to solve
Eq. (1). Clearly, linearity is preserved with respect to ρ, namely φ[ρ1+ρ2, ǫ; Ω] = φ[ρ1, ǫ; Ω]+
φ[ρ2, ǫ; Ω]. A given electrostatic potential is related to an electrostatic energy
Eel[ρ, ǫ; Ω] ≡1
2
∫
Ω
drρ(r)φ[ρ, ǫ; Ω](r) , (28)
that by construction is quadratic in the charge density ρ, which implies
Eel[ρ1 + ρ2, ǫ; Ω] = Eel[ρ1, ǫ; Ω] + Eel[ρ2, ǫ; Ω] +
∫
Ω
drφ[ρ1, ǫ; Ω](r)ρ2(r) . (29)
It can be easily demonstrated that the functional derivative of the electrostatic energy with
respect to the dielectric function is:
δEel[ρ, ǫ; Ω]
δǫ(r)= − 1
8π|∇φ[ρ, ǫ; Ω](r)|2 . (30)
As the electrostatic environment is modified, also the terms describing the ionic energy
and potential must be defined accordingly. In the definition of the external potential con-
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tribution, we have to treat consistently the contribution of v(r) coming from the solution of
the generalized Poisson equation where ionic charges ρaion can be given as input. In other
terms, we can define
v(r) = vion(r) + v∆(r) (31)
where vion = φ[ρion, ǫ; Ω], and ρion =∑
a ρaion. Here v∆ contains extra terms, localised around
the atoms, coming for instance from the definition of the pseudopotentials. In this framework,
the ionic energy of the system can equivalently be defined as the electrostatic energy of the
above generalized Poisson equation minus the self-energy contribution of each of the ions in
vacuum.
Eion[ǫ] = Eel[ρion, ǫ; Ω]− Eself , (32)
where
Eself =∑
a
Eel[ρaion, 1;R3] (33)
A.2 Calculating the Ground-State energy in the Kohn-Sham for-
mulation
The total energy within the Kohn-Sham formalism and the soft-sphere solvation model might
be defined as:
E[ρ, ǫ] = Eion[ǫ] + Ts[ρ] +
∫
v(r)ρ(r)dr + Eel[ρ, ǫ; Ω] + Exc[ρ] + (α + γ)S[ǫ] + βV [ǫ]
= Ts[ρ] + E∆[ρ] + Exc[ρ] + Eel[ρ+ ρion, ǫ; Ω]− Eself + (α + γ)S[ǫ] + βV [ǫ] . (34)
where Ts[ρ] is the electron kinetic energy, Eel[ρ+ ρion, ǫ; Ω] the total electrostatic energy re-
lated to the ionic charges ρion and electronic charge density ρ, Exc[ρ] the exchange-correlation
term. The additional terms E∆[ρ] and Eself are both related to the ionic charges. In par-
ticular E∆ =∫
drv∆(r)ρ(r). The non-electrostatic terms to the total energy come from Eq.
(5).
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Within this formula, the Kohn-Sham hamiltonian can be defined asHKS = −1/2∇2+VKS,
where
VKS[ρ](r) = Vxc(r) + v∆(r) + φtot(r) + Vextra(r) , (35)
Vextra(r) = − 1
8π
∫
dr′δǫ(r′)
δρ(r)
|∇φtot(r′)|2 + (α + γ)
δS[ǫ]
δǫ(r′)
− β
ǫ0 − 1
δǫ(r)
δρ(r). (36)
Here we called the total solute charge ρtot = ρ+ ρion and the corresponding potential φtot =
φ[ρtot, ǫ; Ω]. In Eqs. (35) and (36) an explicit dependence of the dielectric function from
the charge density has been considered, namely ǫ = ǫ[ρ], and δS[ǫ]/δǫ(r′) is explicited in
Eq. (13). When ǫ is a local function of the density,18,22 we have δǫ(r′)/δρ(r) = ǫ(r)δ(r− r′)
and ∂iǫ = ǫ∂iρ. Therefore
δS
δρ=
ǫ
ǫ0 − 1
ǫ
|ǫ|
(
∂iρ∂jρ∂i∂jρ
|∇ρ|3 − ∇2ρ
|∇ρ|
)
(37)
= − |ǫ|ǫ0 − 1
∂i∂iρ
|∇ρ| (38)
Vextra(r) = −ǫ(r)(
1
8π|∇φtot(r)|2 +
α + γ
ǫ0 − 1(39)
[
∂iρ(r)∂jρ(r)∂i∂jρ(r)
|∇ρ(r)|3 − ∇2ρ(r)
|∇ρ(r)|
]
+β
ǫ0 − 1
)
As the dielectric function ǫ might evolve with the charge density, it is important to re-
evaluate the ionic contribution to the total energy for each of the density optimisation steps.
In the above equation we have also assumed that ǫ/|ǫ| = −1 as the dielectric function should
increase with the decreasing of the density.
The DFT energy can be alternatively obtained from the band structure term of the KS
Hamiltonian:
EBS =∑
j
fj〈ψj|HKS|ψj〉 = Ts[ρ] + Epot (40)
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where the potential energy is defined as the expectation value of the local potential:
Epot =
∫
drVKS(r)ρ(r) = Etotel − Eion − Eself + Eel + E∆ + EVxc + EVex . (41)
Here we called Etotel = Eel[ρtot, ǫ; Ω], and
EVex =
∫
drVextra(r)ρ(r) , (42)
EVxc =
∫
drVxc(r)ρ(r) , (43)
By defining the Hartree energy of the cavity as
EcH ≡ Epot − Etot
el − E∆ − EVxc , (44)
we can recover the formal expression for the KS energy:
E = EBS − EcH − EVxc − Eself . (45)
The above expression might be interesting as it does not require an explicit evaluation of
the double-counting term EVex.
A.3 Forces for the soft-sphere model
Forces are defined as minus the derivative of the total energy E[ρ, ǫ] [Eq. (34)] with respect
to the atomic positions:
Fa = − dE
dRa
. (46)
Here the subscript a points to a particular atom.
The Hellmann-Feynman theorem states that, at self-consistency, the total derivative be-
comes the derivative with respect to the explicit dependence on the atomic positions. Since
the electronic kinetic energy Ts[ρ] and the exchange-correlation contribution Exc[ρ] do not
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depend explicitly on the atomic coordinates, the only terms contributing to the forces are
related to the total electrostatic energy and to the non-electrostatic contributions of the
surrounding dielectric related to the quantum surface and volume. Contributions also come
from the additional terms E∆[ρ] and Eself.
Being φtot = φ[ρtot, ǫ; Ω] the electrostatic potential generated by ρ+ρion, the contribution
coming from the total electrostatic energy Eel[ρ+ ρion, ǫ; Ω] can be split in different parts:
∂Eel
∂Ra
=
∫
drφtot(r)∂ρion(r)
∂Ra
− 1
8π
∫
dr∂ǫ(r)
∂Ra
|∇φtot(r)|2 , (47)
which gives rise to the following expression
Fa = −∂E∆
∂Ra
−∫
drφtot(r)∂ρion(r)
∂Ra
−∫
dr∂ǫ(r)
∂Ra
(α + γ)δS[ǫ]
δǫ(r)+ β
δV [ǫ]
δǫ(r)− 1
8π|∇φtot(r)|2
. (48)
By symmetry ∂Eself/∂Ra = 0. The first two terms can be calculated as in vacuum. Consid-
ering the particular case of our soft-sphere cavity defined by Eq. (6), all non-electrostatic
additional terms can be analytically evaluated. To compute the nabla of φtot, central, for-
ward and backward finite difference filters of order 16 have been used, which match the
accuracy of the underlying vacuum Poisson solver.27
B Input guess implementation for the PCG algorithm
In our previous work9 we reported a preconditioned conjugate gradient (PCG) algorithm to
solve the generalized Poisson equation, i.e. Eq. (1), to compute the electrostatic potential
φ(r) generated by a given charge density ρ(r). We demonstrated that it is able to solve the
electrostatic problem with some ten iterations reaching an accuracy of ∼ 10−10 arb. units
with respect to analytic functions.
During a self-consistent field (SCF) loop, the generalized Poisson equation has to be
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solved several times as much as the number of iterations needed to reach the wavefunction
convergence. Solving the problem repeatedly from scratch is computationally inefficient,
especially when the wavefunction is approaching its final configuration. A proper use of
an input guess for the potential φ(r) from previous calculations can considerably speed up
the solution of the electrostatic problem and, by consequence, the whole SCF loop making
implicit solvation and gas phase runtime closer. In our previous paper we implemented a
standard input guess approach for the electrostatic potential, where the generalized Poisson
operator
A = − 1
4π∇ · ǫ(r)∇ (49)
was applied to the potential input guess φ0 and the minimization procedure started from the
initial gradient r0 = ρ − Aφ0. Here ǫ(r) is the continuum dielectric function describing the
dielectric cavity. This approach makes use of a finite difference filter for the nabla operator
∇, which could affect the overall converge especially when the mesh of the simulation box
is too coarse or numbers become small with respect to the numerical noise. Such an effect
could definitely corrupt the overall PCG performances.
In order to prevent that, a solution has been found and reported in Algorithm 1. The
implemented idea follows the same mathematical argument which allowed us to avoid the use
of finite difference filters within the PCG scheme (see Ref. [ 9] for details). The minimization
procedure starts from the initial gradient r0. P is the preconditioner which inverse has to
be applied to the residual vector rk returning the preconditioned residual vk, and, finally, φk
is the solution of the generalized Poisson equation. The convergence criterion is imposed on
the Euclidean norm of the residual vector rk.
Both performances and accuracy of the suggested PCG scheme (with input guess) has
been tested for a single-point DFT energy evaluation of a water molecule in implicit water
described by the soft-sphere solvation model. Fig. 5 report the gradient norm during the
wave function optimization. The SCF loop converged in 15 iterations. Red vertical dots
represent the number of PCG iterations needed to solve the generalized Poisson equation on
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Algorithm 1 Preconditioned conjugate gradient (PCG) algorithm with input guess
1: compute q = 14π
√ǫ ∇2√ǫ
2: r0 = ρ− φ0q
3: v0 = P−1r0
4: r1 = (φ0 − v0)q
5: set p0 = s0 = 0, φ1 = v0
6: for k = 1, ... do
7: if ‖rk‖2 ≤ τ exit
8: vk = P−1rk
9: pk = vk + βkpk−1 (where βk = (vk,rk)(vk−1,rk−1)
, k 6= 0 )
10: sk = Apk = vkq + rk + βksk−1 (since Avk = vkq + rk)
11: αk = (vk,rk)(pk,sk)
12: φk+1 = φk + αkpk
13: rk+1 = rk − αksk
14: end for
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each SCF step. Beside the input guess, a dynamic exit from the PCG loop has been also used
as accurate potentials are not needed during the first stages of SCF optimization. Hence the
exit threshold τ from the PCG loop on the residual norm ‖rk‖2 is made dynamic, set equal
to the norm of the Kohn-Sham energy gradient from the current wave function optimization
and allowed to vary between τmin and τmax (setting τmin = 10−4 and τmin = 10−6 guarantees
a correct SCF convergence while lowering the overall electrostatic calculation time for the
water molecule in implicit water). Fig. 5 shows that a mean of 4 PCG iterations are needed
for each SCF iteration, which ultimately consists of a standard Poisson application and a
mere vector multiplication. Tests on a larger system have been reported in our previous
work9 for a protein PDB ID: 1y49 (122 atoms) and compared to an optimized version of
the COSMO51 Poisson Solver for dielectric environments.101 Applying the soft-sphere cavity
with the new input guess implementation (Algorithm 1), we recovered a ratio of the wave
function optimization runtime in solvent and gas phase of 1.16.
Figure 5: SCF convergence during a single-point DFT energy evaluation for water molecules inimplicit water. Red vertical dots represent the number of PCG iterations to solve the generalizedPoisson equation. Input guess and dynamic exit from the PCG loop have been used.
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