nonlocal polarizable continuum models from joint density functional
TRANSCRIPT
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum modelsfrom joint density functional theory
Ravishankar Sundararaman Kendra Letchworth-WeaverDeniz Gunceler Kathleen Schwarz T.A. Arias
Department of Physics, Cornell University
June 14, 2013
Nonlocal polarizable continuum models from joint density functional theory
Outline
Ab initio studies of solvated systems
Joint density functional theory (JDFT) overview
Nonlocal polarizable continuum models from JDFT
Demonstration: underpotential deposition of Cu on Pt(111)
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Why study solvated systems?
Technological applications
Electrochemical systems: fuel cell catalysts, battery materials (EMC2, DOE)Biological systems: structure and mechanism prediction
Rich physics
Fundamental processes at solid-liquid interfacesComplex equilibria sensitive to electron chemical potential
Theory development
Constructing free energy functionals for liquidsInteractions between ‘classical’ liquids and quantum electrons
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Why study solvated systems?
Technological applications
Electrochemical systems: fuel cell catalysts, battery materials (EMC2, DOE)Biological systems: structure and mechanism prediction
Rich physics
Fundamental processes at solid-liquid interfacesComplex equilibria sensitive to electron chemical potential
Theory development
Constructing free energy functionals for liquidsInteractions between ‘classical’ liquids and quantum electrons
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Why study solvated systems?
Technological applications
Electrochemical systems: fuel cell catalysts, battery materials (EMC2, DOE)Biological systems: structure and mechanism prediction
Rich physics
Fundamental processes at solid-liquid interfacesComplex equilibria sensitive to electron chemical potential
Theory development
Constructing free energy functionals for liquidsInteractions between ‘classical’ liquids and quantum electrons
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Solvent effects
Ignore solvent altogether
Single monolayer of water with hydrogen-bonded geometries chosen manually
One snapshot need not be representative of the liquid effect
Thermodynamic sampling with molecular dynamics significantly more expensive
Additional complexity detracts from intuition towards the system of interest
Capture and abstract away equilibrium effect of solvent
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Solvent effects
Ignore solvent altogether
Single monolayer of water with hydrogen-bonded geometries chosen manually
One snapshot need not be representative of the liquid effect
Thermodynamic sampling with molecular dynamics significantly more expensive
Additional complexity detracts from intuition towards the system of interest
Capture and abstract away equilibrium effect of solvent
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Solvent effects
Ignore solvent altogether
Single monolayer of water with hydrogen-bonded geometries chosen manually
One snapshot need not be representative of the liquid effect
Thermodynamic sampling with molecular dynamics significantly more expensive
Additional complexity detracts from intuition towards the system of interest
Capture and abstract away equilibrium effect of solvent
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Solvent effects
Ignore solvent altogether
Single monolayer of water with hydrogen-bonded geometries chosen manually
One snapshot need not be representative of the liquid effect
Thermodynamic sampling with molecular dynamics significantly more expensive
Additional complexity detracts from intuition towards the system of interest
Capture and abstract away equilibrium effect of solvent
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Solvent effects
Ignore solvent altogether
Single monolayer of water with hydrogen-bonded geometries chosen manually
One snapshot need not be representative of the liquid effect
Thermodynamic sampling with molecular dynamics significantly more expensive
Additional complexity detracts from intuition towards the system of interest
Capture and abstract away equilibrium effect of solvent
Nonlocal polarizable continuum models from joint density functional theory
Ab initio studies of solvated systems
Solvent effects
Ignore solvent altogether
Single monolayer of water with hydrogen-bonded geometries chosen manually
One snapshot need not be representative of the liquid effect
Thermodynamic sampling with molecular dynamics significantly more expensive
Additional complexity detracts from intuition towards the system of interest
Capture and abstract away equilibrium effect of solvent
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Joint density functional theory (JDFT)
Exact variational principle after integrating out electrons of part of the system:
Φ = minn,{Nα}
[ΦJDFT[n, {Nα}] +
∫V · n +
∑α
∫Vα · Nα
]
Decompose into physical pieces that can be approximated independently:
ΦJDFT[n, {Nα}] = FHKM[n]︸ ︷︷ ︸Electrons
+ ΦCDFT[{Nα}]︸ ︷︷ ︸Liquid
+ ∆Φ[n, {Nα}]︸ ︷︷ ︸Coupling
S. A. Petrosyan et al, Phys. Rev. B 75, 205105 (2007)
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Joint density functional theory (JDFT)
Exact variational principle after integrating out electrons of part of the system:
Φ = minn,{Nα}
[ΦJDFT[n, {Nα}] +
∫V · n +
∑α
∫Vα · Nα
]
Decompose into physical pieces that can be approximated independently:
ΦJDFT[n, {Nα}] = FHKM[n]︸ ︷︷ ︸Electrons
+ ΦCDFT[{Nα}]︸ ︷︷ ︸Liquid
+ ∆Φ[n, {Nα}]︸ ︷︷ ︸Coupling
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Approximation schemes
Kohn-Sham electronic DFT with exact non-interacting energy
FHK[n]→ FKS[{φa}] =∑a
∫|∇φa|2
2+
∫n(~r)n(~r ′)
2|~r −~r ′|+ EXC[n].
Classical DFT with exact rigid-molecular ideal gas free energy
ΦCDFT[{Nα}]→ ΦCDFT[{pω}] = T
∫pω(ln pω − 1) + Fex[{Nα}]
in terms of orientation probability density pω(~r)
Density-only density functional approximation to fluid-electron interactions
∆Φ[n, {Nα}] ≈∫
ρelρlq
|~r −~r ′|︸ ︷︷ ︸Coulomb
+ETXC[n + nlq]− ETXC[n]− ETXC[nlq]︸ ︷︷ ︸Kinetic−Exchange−Correlation
−∫
C6iαNα(~r)
f |~Ri −~r |6︸ ︷︷ ︸Dispersion
W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965)
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Approximation schemes
Kohn-Sham electronic DFT with exact non-interacting energy
FHK[n]→ FKS[{φa}] =∑a
∫|∇φa|2
2+
∫n(~r)n(~r ′)
2|~r −~r ′|+ EXC[n].
Classical DFT with exact rigid-molecular ideal gas free energy
ΦCDFT[{Nα}]→ ΦCDFT[{pω}] = T
∫pω(ln pω − 1) + Fex[{Nα}]
in terms of orientation probability density pω(~r)
Density-only density functional approximation to fluid-electron interactions
∆Φ[n, {Nα}] ≈∫
ρelρlq
|~r −~r ′|︸ ︷︷ ︸Coulomb
+ETXC[n + nlq]− ETXC[n]− ETXC[nlq]︸ ︷︷ ︸Kinetic−Exchange−Correlation
−∫
C6iαNα(~r)
f |~Ri −~r |6︸ ︷︷ ︸Dispersion
Systematically improvable- can replace KS DFT with QMC:
K. Schwarz et al., Phys. Rev. B 85, 201102(R) (2012)
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Approximation schemes
Kohn-Sham electronic DFT with exact non-interacting energy
FHK[n]→ FKS[{φa}] =∑a
∫|∇φa|2
2+
∫n(~r)n(~r ′)
2|~r −~r ′|+ EXC[n].
Classical DFT with exact rigid-molecular ideal gas free energy
ΦCDFT[{Nα}]→ ΦCDFT[{pω}] = T
∫pω(ln pω − 1) + Fex[{Nα}]
in terms of orientation probability density pω(~r)
Density-only density functional approximation to fluid-electron interactions
∆Φ[n, {Nα}] ≈∫
ρelρlq
|~r −~r ′|︸ ︷︷ ︸Coulomb
+ETXC[n + nlq]− ETXC[n]− ETXC[nlq]︸ ︷︷ ︸Kinetic−Exchange−Correlation
−∫
C6iαNα(~r)
f |~Ri −~r |6︸ ︷︷ ︸Dispersion
R. Sundararaman and T.A. Arias, arXiv :1302.0026
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Approximation schemes
Kohn-Sham electronic DFT with exact non-interacting energy
FHK[n]→ FKS[{φa}] =∑a
∫|∇φa|2
2+
∫n(~r)n(~r ′)
2|~r −~r ′|+ EXC[n].
Classical DFT with exact rigid-molecular ideal gas free energy
ΦCDFT[{Nα}]→ ΦCDFT[{pω}] = T
∫pω(ln pω − 1) + Fex[{Nα}]
in terms of orientation probability density pω(~r)
Density-only density functional approximation to fluid-electron interactions
∆Φ[n, {Nα}] ≈∫
ρelρlq
|~r −~r ′|︸ ︷︷ ︸Coulomb
+ETXC[n + nlq]− ETXC[n]− ETXC[nlq]︸ ︷︷ ︸Kinetic−Exchange−Correlation
−∫
C6iαNα(~r)
f |~Ri −~r |6︸ ︷︷ ︸Dispersion
K. L.-Weaver, R. Sundararaman and T.A. Arias, (under preparation)
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Hydration energy of microscopic cavities
R
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6 7 8 9
Su
rfa
ce T
en
sio
n (
)
[m
N/m
]
Bulk valueSPC/E Molecular Dynamics
Classical DFT
Exclus
ion
Cohesion
Hard Sphere radius R [A]
SPC/E data: K. Ding et al Phys Rev Lett 59 1698 (1987)R. Sundararaman and T.A. Arias, arXiv :1302.0026
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Nonlinear dielectric response
0
25
50
75
100
0 0.5 1 1.5 2 2.5
ε(E0)
E0 [V/Å]
Molecular dynamics (SPC/E)Molecular dynamics (BJH)
Classical DFT
MD data: I.-C. Yeh and M. Berkowitz, J. Chem. Phys. 110, 7935 (1999)R. Sundararaman and T.A. Arias, arXiv :1302.0026
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Solvation energy of molecules
-8
-6
-4
-2
0
2
wate
r
meth
anol
eth
anol
pro
panol
aceto
ne
dim
eth
yle
ther
eth
ane
meth
ane
pro
pane
∆G
so
lv [kcal/m
ol]
RMS error: 1.2 kcal/mol
ExperimentJoint density functional
Single fit parameter: dispersion prefactor ‘s6’
K. L.-Weaver, R. Sundararaman and T.A. Arias, (under preparation)
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Software for joint density functional theory
Direct free energy minimization for stable convergence
Plane-wave electronic DFT in:
VacuumPolarizable continuaClassical density functional fluids
Grand canonical calculations for electrochemistry
DFT++ algebraic formulation: S. Ismail-Beigi andT.A. Arias, Comp. Phys. Comm. 128, 1 (2000)
Readable, efficient C++11 code
Supports nVidia GPUs using CUDATM
Intel MIC support under developement
R.Sundararaman, D. Gunceler, K. L.-Weaver and T.A. Arias, ‘JDFTx’,
available from http://jdftx.sourceforge.net (2012)
Nonlocal polarizable continuum models from joint density functional theory
Joint density functional theory (JDFT) overview
Software for joint density functional theory
Direct free energy minimization for stable convergence
Plane-wave electronic DFT in:
VacuumPolarizable continuaClassical density functional fluids
Grand canonical calculations for electrochemistry
DFT++ algebraic formulation: S. Ismail-Beigi andT.A. Arias, Comp. Phys. Comm. 128, 1 (2000)
Readable, efficient C++11 code
Supports nVidia GPUs using CUDATM
Intel MIC support under developement
R.Sundararaman, D. Gunceler, K. L.-Weaver and T.A. Arias, ‘JDFTx’,
available from http://jdftx.sourceforge.net (2012)
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Why construct simplified models?
Generality: JDFT requires an accurate classical densityfunctional approximation for each solvent of interest
Cost: JDFT requires 3-10x computational effortcompared to vacuum electronic DFT
Intuition: A simpler model would, by construction,disentangle physical effects and provide insight into theresults of the more complex model
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Why construct simplified models?
Generality: JDFT requires an accurate classical densityfunctional approximation for each solvent of interest
Cost: JDFT requires 3-10x computational effortcompared to vacuum electronic DFT
Intuition: A simpler model would, by construction,disentangle physical effects and provide insight into theresults of the more complex model
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Why construct simplified models?
Generality: JDFT requires an accurate classical densityfunctional approximation for each solvent of interest
Cost: JDFT requires 3-10x computational effortcompared to vacuum electronic DFT
Intuition: A simpler model would, by construction,disentangle physical effects and provide insight into theresults of the more complex model
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Simplified models
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Simplified models
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Simplified models
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Simplified models
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Simplified models
Given 4 parameters ...
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Simplified models
Given 4 parameters ...
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Point dipoles at the molecular scale?
Drastic approximation: fitting cavity size to solvationenergies compensates empirically
Fit parameters are solvent dependent ⇒ needexperimental solvation energy data for each solvent ofinterest
Uncontrolled extrapolation when applying to a solutedrastically different from fit set
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Point dipoles at the molecular scale?
Drastic approximation: fitting cavity size to solvationenergies compensates empirically
Fit parameters are solvent dependent ⇒ needexperimental solvation energy data for each solvent ofinterest
Uncontrolled extrapolation when applying to a solutedrastically different from fit set
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Point dipoles at the molecular scale?
Drastic approximation: fitting cavity size to solvationenergies compensates empirically
Fit parameters are solvent dependent ⇒ needexperimental solvation energy data for each solvent ofinterest
Uncontrolled extrapolation when applying to a solutedrastically different from fit set
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Spherically-averaged liquid susceptibility ansatz (SaLSA)
ΦJDFT[n, {Nα}] = FHKM[n]︸ ︷︷ ︸Electrons
+ ΦCDFT[{Nα}]︸ ︷︷ ︸Liquid
+ ∆Φ[n, {Nα}]︸ ︷︷ ︸Coupling
Assumptions:
Cavity density determined from overlap of solute and solvent electron densities
Solvent molecules oriented randomly in the absence of electrostatics
Consequences:
Repulsion ← Kinetic-exchange-correlation terms in coupling
Dispersion ← Pair potential vdW terms in coupling
Cavitation ← Exclusion of classical density functional fluid
Electrostatics ← Linear rotational and polarization response
R. Sundararaman, K. L.-Weaver, K. Schwarz and T.A. Arias, (under preparation)
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Spherically-averaged liquid susceptibility ansatz (SaLSA)
ΦJDFT[n, {Nα}] = FHKM[n]︸ ︷︷ ︸Electrons
+ ΦCDFT[{Nα}]︸ ︷︷ ︸Liquid
+ ∆Φ[n, {Nα}]︸ ︷︷ ︸Coupling
Assumptions:
Cavity density determined from overlap of solute and solvent electron densities
Solvent molecules oriented randomly in the absence of electrostatics
Consequences:
Repulsion ← Kinetic-exchange-correlation terms in coupling
Dispersion ← Pair potential vdW terms in coupling
Cavitation ← Exclusion of classical density functional fluid
Electrostatics ← Linear rotational and polarization response
R. Sundararaman, K. L.-Weaver, K. Schwarz and T.A. Arias, (under preparation)
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Nonlocal electrostatic response
Solvent molecule with charge density ρmol(~r) and expand electronic andvibrational susceptibility in eigenbasis (obtain from DFT/DFPT)
χmol(~r ,~r′) = −
∑i
Xiρi (~r)ρi (~r′)
Decompose each ρ into angular momentum channels
=
︸ ︷︷ ︸l=0
+
︸ ︷︷ ︸l=1
+
︸ ︷︷ ︸l=2
+ · · ·
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Nonlocal electrostatic response
Solvent molecule with charge density ρmol(~r) and expand electronic andvibrational susceptibility in eigenbasis (obtain from DFT/DFPT)
χmol(~r ,~r′) = −
∑i
Xiρi (~r)ρi (~r′)
Decompose each ρ into angular momentum channels
=
︸ ︷︷ ︸l=0
+
︸ ︷︷ ︸l=1
+
︸ ︷︷ ︸l=2
+ · · ·
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Nonlocal electrostatic response
Susceptibility of fluid of such molecules distributed as N(~r) is exactly
χ(~G , ~G ′) = −N(~G − ~G ′)∑l
Pl(G · G ′)4π
∑m
(1T ρ
lmmol(G )ρlm∗mol(G
′)+∑
i Xi ρlmi (G )ρlm∗i (G ′)
)
Non-local generalization of Poisson (or linearized Poisson-Boltzmann) equation(−∇
2
4π+ χ
)φ = ρ
At l = 1 with rotations alone, computational costs of SaLSA and PCM areidentical!
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Nonlocal electrostatic response
Susceptibility of fluid of such molecules distributed as N(~r) is exactly
χ(~G , ~G ′) = −N(~G − ~G ′)∑l
Pl(G · G ′)4π
∑m
(1T ρ
lmmol(G )ρlm∗mol(G
′)+∑
i Xi ρlmi (G )ρlm∗i (G ′)
)
Non-local generalization of Poisson (or linearized Poisson-Boltzmann) equation(−∇
2
4π+ χ
)φ = ρ
At l = 1 with rotations alone, computational costs of SaLSA and PCM areidentical!
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Nonlocal electrostatic response
Susceptibility of fluid of such molecules distributed as N(~r) is exactly
χ(~G , ~G ′) = −N(~G − ~G ′)∑l
Pl(G · G ′)4π
∑m
(1T ρ
lmmol(G )ρlm∗mol(G
′)+∑
i Xi ρlmi (G )ρlm∗i (G ′)
)
Non-local generalization of Poisson (or linearized Poisson-Boltzmann) equation(−∇
2
4π+ χ
)φ = ρ
At l = 1 with rotations alone, computational costs of SaLSA and PCM areidentical!
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Typical cavities and bound charge (in liquid water)
JDFT(density
functional)
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Typical cavities and bound charge (in liquid water)
PCM(dielectric)
JDFT(density
functional)
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Typical cavities and bound charge (in liquid water)
PCM(dielectric)
SaLSA(nonlocalresponse)
JDFT(density
functional)
Nonlocal polarizable continuum models from joint density functional theory
Nonlocal polarizable continuum models from JDFT
Solvation energies
-10
-8
-6
-4
-2
0
2
aceta
mid
e
acetic a
cid
phenol
wate
r
meth
anol
eth
anol
isopro
panol
pyridin
e
meth
yla
min
e
tetr
ahydro
fura
n
anis
ole
benzene
tolu
ene
meth
ane
eth
ane
pro
pane
∆G
solv
[kcal/m
ol]
RMS error: 1.3 kcal/mol
ExperimentSaLSA
Single fit parameter: dispersion prefactor ‘s6’
Nonlocal polarizable continuum models from joint density functional theory
Underpotential deposition
Under-potential deposition: Cu/Pt(111) [Cl−]
Voltammetry
Nonlocal polarizable continuum models from joint density functional theory
Underpotential deposition
Under-potential deposition: Cu/Pt(111) [Cl−]
← ∆VUPD = 0.29 V→
Based on ex-situ LEED from N.M. Markovik et al, Surf. Sci. 335, 91 (1995)and in-situ EXAFS from Y. Soldo et al, Electrochimica Acta 47, 3081 (2002)
Nonlocal polarizable continuum models from joint density functional theory
Underpotential deposition
Explicit Pt and Cu, implicit solvent
-2
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
Φ -
ΦP
t(bulk
) a
t 1
mM
Cu
2+ [
eV
]
V against Standard Hydrogen Electrode [V]
(Pt)(Pt)Cu
(Pt)Cu2(Pt)Cu1/4
ExperimentBulk Monolayer Partial
Qualitative agreement, need explicit anions
SHE calibration: K. Letchworth-Weaver et al., Phys. Rev. B 86, 075140 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Underpotential deposition
Explicit Pt, Cu and Cl, implicit solvent
-2
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
Φ -
ΦP
t(bulk
) a
t 1
mM
Cu
2+ [
eV
]
V against Standard Hydrogen Electrode [V]
(Pt)(Pt)Cu
(Pt)Cu2(Pt)Cu1/4
ExperimentBulk Monolayer Partial
Partial monolayer susceptible to additional adsorbants (protons?)
Nonlocal polarizable continuum models from joint density functional theory
Outlook
Outlook
JDFT: efficient description of electronicsystems in liquids
Scope to develop more accurate functionals forliquids and coupling
Obtain efficient and minimally empiricalcontiuum models as limit of JDFT
Explore complex electrochemical systems byspecifying atomic positions only of relevantsubsystems
Nonlocal polarizable continuum models from joint density functional theory
Outlook
Thank you for your attention!
Acknowledgements:
This work was supported as a part of the Energy Materials Center at Cornell(EMC2), an Energy Frontier Research Center funded by the U.S. Departmentof Energy, Office of Basic Energy Sciences under Award No. DE-SC0001086.
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Iso-density-product cavity determination
Correlate overlap of atom electrondensities with vdW radii
Fit nc to minimize discrepancybetween R1 + R2 and R12 defined by∫
d~rn1(~r)n2(~r + R12ez) = nc
Best fit nc = 1.2× 10−3 insensitive toelectronic structure method.
Analogously determine cavity fromsolute and spherically-averagedsolvent molecule electron denisties:
s(~r) =1
2erfc log
∫d~r ′n(~r ′)nmol(|~r ′ −~r |)
nc
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
R12
[Å]
R1 + R2 [Å]
HFGGALDA
HF (pseudized)GGA (pseudized)LDA (pseudized)
Perfect agreement
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Weighted density cavitation energy
Measure neighbour occupancy with s = δ(r−Rvdw)4πR2
vdw∗ s(~r)
Constrain weighted density approximation to limits:
Φcav[s] = NbulkT
∫s(1− s) (s + γ(1− s) + Cs(1− s))
where γ ≡ lnpvap
NbulkT− 1
C ≡ σbulk
RvdwNbulkT+
1 + γ
6
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Cavity model vs classical DFT
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Size consistency
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Density functional theory
Ground state energy of many-electron system from ground state density n(~r) alone
E = minψ〈ψ|(H + V )|ψ〉 = min
n
[FHK[n] +
∫V · n
]
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Density functional theory
Helmholtz free energy of many-electron system from equilibrium density n(~r) alone
A = minρ
Tr(H + V )ρ = minn
[FHKM[n] +
∫V · n
]
N.D. Mermin, Phys. Rev. 137, A1441 (1965)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Density functional theory
Helmholtz free energy of many-electron system from equilibrium density n(~r) alone
A = minρ
Tr(H + V )ρ = minn
[FHKM[n] +
∫V · n
]
Grand free energy of many-body nuclear system from equilibrium densities{Nα(~r)} alone
Φ = minP
Tr(H + ~V )P = min{Nα}
[ΦCDFT[{Nα}] +
∑α
∫Vα · Nα
]
‘Classical’ DFT is fully quantum, not even Born-Oppenheimer approximation!
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Free energy functionals for real liquids
Employ exact free energy of non-interacting classical fluid (analogous toKohn-Sham) in terms of orientation probability density pω
ΦCDFT[{Nα}]→ T
∫pω(ln pω − 1) Non-interacting free energy
+
R. Sundararaman and T.A. Arias, arXiv :1302.0026R. Sundararaman, K. L.-Weaver and T.A. Arias, J. Chem. Phys 137, 044107 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Free energy functionals for real liquids
Employ exact free energy of non-interacting classical fluid (analogous toKohn-Sham) in terms of orientation probability density pω
ΦCDFT[{Nα}]→ T
∫pω(ln pω − 1) Non-interacting free energy
+ ΦHS[N] Hard-sphere entropy corrections
+
Fundamental measure theory: Y. Rosenfeld, Phys. Rev. Lett. 63, 980 (1989)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Free energy functionals for real liquids
Employ exact free energy of non-interacting classical fluid (analogous toKohn-Sham) in terms of orientation probability density pω
ΦCDFT[{Nα}]→ T
∫pω(ln pω − 1) Non-interacting free energy
+ ΦHS[N] Hard-sphere entropy corrections
+
∫NαZα ·
1
2|~r −~r ′|· NβZβ Mean-field Coulomb interactions
+
R. Sundararaman and T.A. Arias, arXiv :1302.0026R. Sundararaman, K. L.-Weaver and T.A. Arias, J. Chem. Phys 137, 044107 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Free energy functionals for real liquids
Employ exact free energy of non-interacting classical fluid (analogous toKohn-Sham) in terms of orientation probability density pω
ΦCDFT[{Nα}]→ T
∫pω(ln pω − 1) Non-interacting free energy
+ ΦHS[N] Hard-sphere entropy corrections
+
∫NαZα ·
1
2|~r −~r ′|· NβZβ Mean-field Coulomb interactions
+
∫NAex(w ∗ N) Weighted density excess functional
+
R. Sundararaman and T.A. Arias, arXiv :1302.0026R. Sundararaman, K. L.-Weaver and T.A. Arias, J. Chem. Phys 137, 044107 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Free energy functionals for real liquids
Employ exact free energy of non-interacting classical fluid (analogous toKohn-Sham) in terms of orientation probability density pω
ΦCDFT[{Nα}]→ T
∫pω(ln pω − 1) Non-interacting free energy
+ ΦHS[N] Hard-sphere entropy corrections
+
∫NαZα ·
1
2|~r −~r ′|· NβZβ Mean-field Coulomb interactions
+
∫NAex(w ∗ N) Weighted density excess functional
+ α
∫P2 Local polarization correlations
R. Sundararaman and T.A. Arias, arXiv :1302.0026R. Sundararaman, K. L.-Weaver and T.A. Arias, J. Chem. Phys 137, 044107 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Radial distribution around hard spheres
4A
r
0
0.5
1
1.5
2
2 4 6 8 10 12
NO
/Nbulk
r [A]
SPC/E Molecular DynamicsClassical DFT
SPC/E data: K. Ding et al Phys Rev Lett 59 1698 (1987)R. Sundararaman and T.A. Arias, arXiv :1302.0026
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Voltage reference calibration
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
3.6 3.8 4 4.2 4.4 4.6 4.8 5
Pote
ntial of zero
charg
e a
gain
st S
HE
[V
]
DFT µe [V]
Au(100)
Au(110)
Au(111)
Ag(100)Ag(110)
Ag(111)
Cu(100)
Cu(110)
Cu(111)
Cu(100)[NaF]
Cu(110)[NaF]Cu(111)
[NaF]
Best fit ⇒ µSHE = (−4.44± 0.02) V
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Cu2+ chemical potential calibration
-2
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
Φ a
t 1
mM
Cu
2+ [
eV
]
V against Standard Hydrogen Electrode [V]
5 layers Cu7 layers Cu9 layers Cu
ExperimentBulk
Relevant free energy: Φ(µe) = A(Ne)− µeNe − µCu2+NCu − µCl−NCl−
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Vacuum DFT predictions
-2
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
Φ -
ΦP
t(bulk
) a
t 1
mM
Cu
2+ [
eV
]
V against Standard Hydrogen Electrode [V]
(Pt)(Pt)Cu
(Pt)Cu2(Pt)Cu1/4
ExperimentBulk Monolayer Partial
Linearized Φ(µe) = A(N0e )− µeN
0e assumes fixed Ne
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Neutral PCM predictions
-2
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
Φ -
ΦP
t(bulk
) a
t 1
mM
Cu
2+ [
eV
]
V against Standard Hydrogen Electrode [V]
(Pt)(Pt)Cu
(Pt)Cu2(Pt)Cu1/4
ExperimentBulk Monolayer Partial
Linearized Φ(µe) = A(N0e )− µeN
0e assumes fixed Ne
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Polarizable continuum models
Threshold electron density at nc to define a cavity in the fluid (described by shapefunction s(~r) = 0 in the cavity and 1 in the fluid)
Approximate effect of fluid by electrostatic response of nonuniform dielectric withε(~r) = 1 + (εb − 1)s(~r)
Add empirical corrections for subdominant contributions (dispersion, cavityformation etc.)
ΦPCM[n] = FHKM[n]︸ ︷︷ ︸Electrons
+1
2
∫ρel
((K−1 − χ
)−1
− K
)ρel︸ ︷︷ ︸
Dielectric response
+ σeff
∫∇s︸ ︷︷ ︸
Cavity tension
A. Andreussi, I. Dabo and N. Marzari, J. Chem. Phys. 136, 064102 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Polarizable continuum models
Threshold electron density at nc to define a cavity in the fluid (described by shapefunction s(~r) = 0 in the cavity and 1 in the fluid)
Approximate effect of fluid by electrostatic response of nonuniform dielectric withε(~r) = 1 + (εb − 1)s(~r)
Add empirical corrections for subdominant contributions (dispersion, cavityformation etc.)
ΦPCM[n] = FHKM[n]︸ ︷︷ ︸Electrons
+1
2
∫ρel
((K−1 − χ
)−1
− K
)ρel︸ ︷︷ ︸
Dielectric response
+ σeff
∫∇s︸ ︷︷ ︸
Cavity tension
A. Andreussi, I. Dabo and N. Marzari, J. Chem. Phys. 136, 064102 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Polarizable continuum models
Threshold electron density at nc to define a cavity in the fluid (described by shapefunction s(~r) = 0 in the cavity and 1 in the fluid)
Approximate effect of fluid by electrostatic response of nonuniform dielectric withε(~r) = 1 + (εb − 1)s(~r)
Add empirical corrections for subdominant contributions (dispersion, cavityformation etc.)
ΦPCM[n] = FHKM[n]︸ ︷︷ ︸Electrons
+1
2
∫ρel
((K−1 − χ
)−1
− K
)ρel︸ ︷︷ ︸
Dielectric response
+ σeff
∫∇s︸ ︷︷ ︸
Cavity tension
A. Andreussi, I. Dabo and N. Marzari, J. Chem. Phys. 136, 064102 (2012)
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Parameter fit: nc and σeff to solvation energies
-10
-8
-6
-4
-2
0
2
aceta
mid
e
wate
r
acetic a
cid
meth
anol
aceto
ne
eth
anol
isopro
panol
meth
yla
min
e
eth
yla
min
e
meth
aneth
iol
tetr
ahydro
fura
n
benzene
die
thyl eth
er
meth
ane
eth
ane
pro
pane
∆G
solv
[kcal/m
ol]
RMS error: 1.1 kcal/mol
ExperimentPCM
Nonlocal polarizable continuum models from joint density functional theory
Backup slides
Nonlinear PCM free energy functional
Electrostatic contribution to the free energy is
Φε =
∫TNmols(~r)
[ε2
(f (ε)− α
2f 2(ε) +
X
2(1− αf (ε))2
)− log
sinhε
ε
]+
∫ρlq(ρel + ρlq/2)
|~r −~r ′|
where
ρε(~r) ≡ −~∇ · [pmolNmols(~r)~ε (f (ε) + X (1− αf (ε)))]
f (ε) ≡ (ε coth ε− 1)/ε2
X =T (ε∞ − 1)
4πNmolp2mol
α = 3− 4πNmolp2mol
T (εb − ε∞)
D. Gunceler, K. L.-Weaver, R. Sundararaman et al., arXiv :1301.6189