1. Summer Project: Further developing the parameterisation of the
solvent model in ONETEP for recent exchange correlation
functionals
Alexander Gheorghiu, Supervised by: C. Skylaris
August 2016
Abstract
Three different types of species; neutral (ammonia), an anion (nitrate) and a cation (methylammo-
nium) have been investigated at an ab initio level using the ONETEP solvation model. The parameters
for this solvation model need to be re-examined and potentially reparametrised, as they were established
five years ago with outdated approximations by J. Dziedzic et al. in 2011 [1]. This paper also addresses
how more recent exchange correlation (XC) functionals eg. VV10 and B97M-V compare with LDA
(one of the crudest XC functionals)[2] and PBE (one of the most currently used generalised gradient
approximation methods).[3] In this paper, we have found that changing the XC functional makes little
difference to free energy of solvation calculation accuracy. We come to the conclusion that parameters
may need minor modifications and further research must be done. In addition, we confirm the ONETEP
solvation model specifically requires further development for anions - as they have been proved tough to
accurately simulate.
1 Introduction
For the past 30 years, density functional theory (DFT) continues to be the established method for quantum
mechanical simulations of periodic systems. Further research enabled simulations to describe molecular
interactions in solution - whereby the solution is approximated as a continuous solvent (first reviewed by
Tomasi and Persico et al. in 1994).[4] Due to the abundance of reviews addressing liquid solutions, related
theoretical and computational reviews have vastly developed in the past two decades.
Solvents play a key role in many biological, chemical and physical processes. It is therefore important to
accurately simulate such processes by carefully taking solvation effects into account. The explicit inclusion
of solvent (in full atomic detail) is very costly due to the vast increase in number of simulated atoms. The
explicit model can also be seen as unnecessary - typically the long range electrostatic interactions in the
solvent are the most significant, whereby only a small proportion are involved chemically. When using an
implicit solvent approach, a self-consistent reaction field (SCRF) is implemented as the solvent, approxi-
mated as a dielectric continuum with permittivity, ε. The atomic detail of the solute is retained and the
1

2. molecule is placed in a suitably defined cavity.
A modern approach for calculating the free energy of solvation (ΔGsolvation) with implicit solvation is
known as the Polarisable Continuum Model (PCM).[5] This model treats the continuum as a polarisable
dielectric. It calculates the free energy of solvation as the summation of the following terms shown in
equation (1).
∆Gsolvation = ∆Gelectrostatic + ∆Gdispersion−repulsion + ∆Gcavitation (1)
The cavity generated in PCM is approximated by a set of overlapping spheres, defined by the van der
Waals radii of the individual atoms. Comparatively, this approximation does not accurately account for the
cavitation energy or dispersion-repulsion energy.
This is where the Solvation Model based on Density (SMD) comes in - as a more accurate SCRF alterna-
tive. It separates the solvation free energy into two main components; The electrostatic energy (calculated
primarily from IEF-PCM interaction), and the cavity-dispersion-solvent-structure term (the energy which
arises form the short range interactions between solute and solvent molecules in the first solvation shell.)[6].
Using electron density to estimate the solvent surface accessible area and atomic surface tensions, the SMD
model can calculate the dispersion and repulsion energies.
A more recent model proposed by Fattebert and Gygi [7] and later developed by Scherlis et al. [8] (hence
called the FGS model), utilises a dielectric cavity built directly from the electronic density of the solute.
(r) = 1 +
∞ − 1
2
1 +
1 − (ρ(r)/ρ0)2β
1 + (ρ(r)/ρ0)2β
(2)
Equation (2) is used to determine the smooth transition of relative permittivity, (r), where ρ(r) is the
electronic density of the solute, ∞ is the bulk permittivity. The parameter β, controls the smoothness of
transition (r) from 1 to ∞, and parameter ρ0 is the density value for which the permittivity drops to
∞/2. [7]
The cavitation free energy, is approximated to be proportional to the surface area, S, of the cavity
(calculated when ρ = ρ0).
∆Gcav = γS(ρ0) (3)
Equation (3) describes this relationship, where γ is the solvent surface tension. Rather than using the
known value (γ = 74.16 mNm−1
), we rescale this value by a factor of 0.281. This crude approximation takes
the dispersion repulsion effects into account, drastically improving the accuracy of the model.
ONETEP includes the non-polar cavitation term using a solvent-accessible surface-area (SASA) approx-
imation. This assumes the cavitation energy to be proportional to the surface area of the cavity, and the
2

3. constant of proportionality being the physical surface tension of the solvent, γ.
The original FGS model did not include dispersion-repulsion effects. This results in some questionable
results for larger, neutral molecules, as the electrostatic contribution to solvation is insignificant compared
to the non-polar terms. For example, the deficiency is evident in the case of dichloroethane, where the
model predicts ΔG solvation of +4.0 kcal/mol [8] compared to the experimental value of -1.4 kcal/mol.
The current model can only approximate the dispersion repulsion energy, using the SASA approximation
used by the cavitation term. This is simply achieved by using a modified value for solvent surface tension,
γ, rather than the physical value of solcent surface tension. This allows for an approximate inclusion of
dispersion repulsion effects, leading to vastly improved results. [1, 9]
Proposed by Fattebert and Gygi, [8] total potential of a solute in a dielectric, φ(r) can be obtained by
solving the non-homogeneous Poisson equation (NPE) shown in equation (4).
.( [ρ] φ) = −4πρtot (4)
Because the NPE is solved for molecular density and produces a molecular potential, a trick termed
’smeared-ion formalism’ is used to combine this with the typical DFT method in terms of separating valence
and core electronic densities. In this formalism, nuclear cores are modelled as narrow positive Gaussian
distributions with the usual energy terms re-allocated. In general, the total energy of a system is unaltered
by applied smeared-ion formalism, however, due to some numerical inaccuracy, some variation may be ob-
served. These variations may be cancelled out if smeared-ion formalism is turned on for both solvated and
in vacuo systems. [9]
For smeared-ion formalism, the molecular Hartree energy is obtained by solving Poisson equation (homo-
geneous in vacuo, heterogeneous in solution) in real space under open boundary conditions. The remaining
of the energy terms must therefore use open boundary conditions for consistency.
2 Aim
To first replicate the parameters density threshold ρ0, and solvation β , produced by J. Dziedzic et al. using
the same functional (PBE) and identical settings.[1] To then observe if the parameters change with different
exchange correlation functionals such as: BLYP, VV10 and the recent B97M-V. In addition to changing
the functional, various options in the input file were modified to the recent default. Should the parameters
require adjustment, solvation energies for 60 molecules [1] previously studied will be recalculated. If the
parameters are proved adequate, solvation energies for the 60 molecules will still be recalculated, but with
different functionals to observe how accuracy depends on functional.
3

4. 3 Method
3.1 Replication of previous results
This experiment focused on the design and simulation of three different types of molecules; a neutral species
(ammonia), and cationic species (methylammonium) and an anionic species (nitrate). The molecules were
drawn with GaussView 5.0 and the atomic coordinates were transferred to ONETEP input files. Once
optimised with their respective exchange correlation functionals, the solvation energy (ΔGsolvation) was cal-
culated using auto SMD in ONETEP. For each molecule, values of β were modified between the ranges
of (1.0 - 2.0) in intervals of 0.1 and values of ρ0 were modified between (0.00015 - 0.00110) in intervals of
0.0005. All simulations were completed in a cube cell with edge length (L =40.5 a0).
Two previously used options were modified, as the the new default settings had changed:
• Smeared ion width of 0.8 bohr was used as it is the most recent proposed default [9] (0.6 bohr was
used originally)
• Finite difference order 8 was used (10 was used originally)
It is also worth noting, updated pseudopotential files taken from the CASTEP on-line repository ’Opium
Rappe Benett recpots 2012’ were used in the new calculations.
To reproduce these results, three separate environments were tested:
• A replica of results obtained by J.D with occ mix = 1
• A replica of results obtained by J.D with default occ mix = 0.25
• A replica of results obtained by J.D with NGWF radius 8 a0, new pseudopotentials and occ mix = 1.
Occ mix specifies the fraction of NGWF gradient to which occupancy preconditioning is applied (where
occ mix = 1 is a fully preconditioned gradient).
3.2 Reparameterisation attempt
To observe the dependence of four different exchange functional with parameters ρ0, and β (Using the
following revised values; NGWF radius = 8 a0, updated pseudopotentials and occ mix = 1):
• PBE
• BLYP
• VV10
• B97M-V
4

5. 3.3 Validation of solvation model for different XC functionals for a test set of
molecules
A selection of 60 Gaussian input files (20 neutral, 20 anions, 20 cations) were obtained from the Minnesota
database and converted to ONETEP input files (the geometries were optimised in gas phase, taken from ref.
[10]). Using the parameters previously established and the same adjustments as earlier in the document,
the free energy of solvation for the molecules were calculated with the following XC functionals.
• PBE
• VV10
• BLYP
• B97M-V
• B3LYP
• LDA
The results were then compared to experimental values from the Minnesota solvation database.[10]
4 Results and Discussion
4.1 Replication of previous results
Figure 1: Reproduction of original results: Red - old results from 2010, Orange - reproduction of old results,
no occ mix, Blue - reproduction of old results occ mix 1.0, Green - NGWF radius 8a0, new pseudopotentials
and occ mix 1.0. Dashed lines indicates error of ±1 kcal/mol
Figure (1) displays the attempt of reproducing results produced by J. Dziedzic et al.[1]. The solid curved
lines represent the ideal combinations of parameters β and ρo. It is important to select parameter combi-
nations that cover all three types of molecules (neutral, anions and cations) as equal as possible. Upon
first inspection of the graphs, one may notice the relatively varied spread between the nitrate lines. This
is because the error boundary ±1 kcal/mol (indicated by the dashed lines), is far wider than for ammonia
5

6. and methylammonium. Due to the close nature of methylammonium lines, it is very difficult to distinguish
between results on the graph - although the curves appear to be in close proximity of one another, it should
be noted that the error boundary is far tighter than the other two molecules - resulting in a smaller difference
on the graph translating to a larger error.
For the nitrate anion, the original results (shown in red) do not extend beyond β = 1.5 and ρo = 0.00075.
When the results were produced by J. Dziedzic et al. [1], a finite difference order of 10 was used. For this
experiment, a revised value of 8 was used, which allowed the calculation to converge past the aforementioned
values.
In general, the blue curve was expected to be the closest reproduction of the results produced by J.
Dziedzic et al. [1]. This is because the settings used in that simulation were identical (bar the modifications
described earlier). However, we can see the blue curve does not run exactly over the red one for none of
the species. There are many potential reasons for this irregularity. For example, there is a high chance that
the geometry of the molecules used in these simulations were not identical to the previous experiments. We
also note that during the original experiment, the physical value of water’s solvent surface tension, γ , was
used and rescaled after the calculations. In this repeated experiment, a rescaled value of γ was used from
the beginning - this modification only affects post processing. For the reproduction of results, the smeared
ion width was increased to 0.8 bohr. Previously, a smeared ion width of 0.6 bohr was used. A low value
of 0.6 bohr has the possibility to negatively impact the convergence of the multigrid whereas a value larger
than 1.0 bohr is likely to lead to non-physical results.
For all three graphs, one can see that the majority of reproduced lines have been raised by a small
amount. It appears the values proposed by J. Dziedzic et al. [1] in 2011 are optimal for the PBE XC
functional where, β = 1.3 and ρo = 0.0008.
Figure 2: Dependence of exchange functional with parameters (all done with NGWF radius = 8 a0, new
pseudopotentials and occ mix=1: Red - PBE, Orange - BLYP, Blue - VV10, Green - B97M-V. Dashed lines
indicates error of ±1 kcal/mol.
6

7. 4.2 Reparameterisation attempt
The results of the second section of the experiment are displayed in Figure (2). The curve of PBE (shown
in red) is significantly below all of the other tested exchange functionals. This result indicates that when
using a XC functional other than PBE, the value of ρ0 should be raised by some degree.
4.3 Validation of solvation model for different XC functionals for a test set of
molecules
Table 1: The maximum error and root mean square (rms) in kcal/mol, with respect to experimentally cal-
culated free energies of solvation [10], and the corresponding coefficient of correlation, r, between calculated
and experimental values for the 60 neutral, cationic and anionic species studied.
LDA was used as a starting comparison to see how much of an improvement the more recent XC func-
tionals compare. As we can see from Table (4.3), there is little difference in total rms error between the
recent meta-GGA functional B97M-V, to the electron density approach that is LDA.
Interestingly, both B3LYP and B97M-V produced results with the lowest total rms error of 11.03. It is
worth mentioning due to B3LYP being a hybrid functional, the calculations were relatively computationally
extensive, taking some 8 to 10 times longer to complete. It is important to have an XC functional that
produces results in agreement with experiment for the three species
We also observe that the most accurate approximations are for neutral molecules, followed by cations and
anions. Most common DFT approximations suffer from electron self-interaction error.[11] It is believed to
be the likely cause to many of the failures of these approximations. Anions, species with naturally excessive
amounts electrons are particularly vulnerable to this error.
5 Conclusion and further thoughts
To conclude, it is likely to say that the deficiency may lie within the solvent model itself, rather than the
choice of XC functional used. The results clearly show that there is not that significant of a difference between
calculated free energies of solvation between XC functional. Although B3LYP and B97M-V produced
7

8. results with the lowest total rms error, they are still only 5% more accurate than BLYP (the overall total
least accurate XC functional). It is also worth mentioning that due to B3LYP being a hybrid functional,
calculations took some 10x longer to complete. In an attempt to improve the solvent model, parameters
ρ0 and β should be recalculated. However, rather than using just three molecules (ammonia, nitrate and
methylammonium), these parameters should be acquired from a test subject of 30 molecules from the
solvation database (10 neutral, 10 cations, 10 anions) and then test on the remaining 30 molecules.
8

9. References
[1] J Dziedzic, H H Helal, C.-K Skylaris, A A Mostofi, and M C Payne. Minimal parameter implicit solvent
model for ab initio electronic-structure calculations. EPL, 95(43001), 2011.
[2] W Kohn and L J Sham. Self-Consistent Equations Including Exchange and Correlation Effects. Physical
Review, 140(4A):1133, 1965.
[3] John P Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized Gradient Approximation Made
Simple. 77(19):3865 – 3868, 1996.
[4] Jacopo Tomasi ’ and Maurizio Persico. Molecular Interactions in Solution: An Overview of Methods
Based on Continuous Distributions of the Solvent. Chem. Rev, 94:2027–2094, 1994.
[5] J Tomasi, B Mennucci, and E Canc`es. The IEF version of the PCM solvation method: an overview of
a new method addressed to study molecular solutes at the QM ab initio level. Journal of Molecular
Structure (Theochem), 464:211–226, 1999.
[6] Aleksandr V. Marenich, Christopher J. Cramer, and Donald G. Truhlar. Universal solvation model
based on solute electron density and on a continuum model of the solvent defined by the bulk dielectric
constant and atomic surface tensions. Journal of Physical Chemistry B, (113):6378–6396, 2009.
[7] Jean-Luc Fattebert and Fran¸cois Gygi. Density Functional Theory for Efficient Ab Initio Molecular
Dynamics Simulations in Solution. J Comput Chem, 23:662–666, 2002.
[8] Gygi F. Cococcioni M. andMarzari N Scherlis D., Fattebert J. Density functional theory for efficient
ab initio molecular dynamics simulations in solution. The Journal of Chemical Physics, 124(074103),
2006.
[9] Jacek Dziedzic. Implicit solvation in ONETEP. 2011.
[10] Aleksandr V Marenich. Univeristy of Minnesota Minneapolis. Minnesota Solvation Database Version
2012, 2012.
[11] J. P. Perdew and Alex Zunger. Self-interaction correction to density-functional approximations for
many-electron systems. Physical Review B, 1981.
9