nonlocal polarizable continuum models from joint density functional

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


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)



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



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



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)



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:






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)






∫|∇φa|2

2+

∫n(~r)n(~r ′)

2|~r −~r ′|+ EXC[n].






∆Φ[n, {Nα}] ≈∫

ρelρlq



−∫

C6iαNα(~r)


Systematically improvable- can replace KS DFT with QMC:

K. Schwarz et al., Phys. Rev. B 85, 201102(R) (2012)






∫|∇φa|2

2+

∫n(~r)n(~r ′)

2|~r −~r ′|+ EXC[n].






∆Φ[n, {Nα}] ≈∫

ρelρlq



−∫

C6iαNα(~r)


R. Sundararaman and T.A. Arias, arXiv :1302.0026






∫|∇φa|2

2+

∫n(~r)n(~r ′)

2|~r −~r ′|+ EXC[n].






∆Φ[n, {Nα}] ≈∫

ρelρlq



−∫

C6iαNα(~r)


K. L.-Weaver, R. Sundararaman and T.A. Arias, (under preparation)



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



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



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)



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)

http://jdftx.sourceforge.net



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



Simplified models



Simplified models

Given 4 parameters ...



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



Spherically-averaged liquid susceptibility ansatz (SaLSA)




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 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 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!



Typical cavities and bound charge (in liquid water)

JDFT(density

functional)




PCM(dielectric)

JDFT(density

functional)




PCM(dielectric)

SaLSA(nonlocalresponse)

JDFT(density

functional)



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]


ExperimentSaLSA

Single fit parameter: dispersion prefactor ‘s6’


Underpotential deposition

Under-potential deposition: Cu/Pt(111) [Cl−]

Voltammetry



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)



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)



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

]


(Pt)(Pt)Cu

(Pt)Cu2(Pt)Cu1/4


Partial monolayer susceptible to additional adsorbants (protons?)


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


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.


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


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


Backup slides

Cavity model vs classical DFT


Backup slides

Size consistency


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)


Backup slides


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)


Backup slides


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!


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)


Backup slides



ΦCDFT[{Nα}]→ T


+ ΦHS[N] Hard-sphere entropy corrections

+

Fundamental measure theory: Y. Rosenfeld, Phys. Rev. Lett. 63, 980 (1989)


Backup slides



ΦCDFT[{Nα}]→ T



+

∫NαZα ·

1

2|~r −~r ′|· NβZβ Mean-field Coulomb interactions

+



Backup slides



ΦCDFT[{Nα}]→ T



+

∫NαZα ·

1


+

∫NAex(w ∗ N) Weighted density excess functional

+



Backup slides



ΦCDFT[{Nα}]→ T



+

∫NαZα ·

1


+

∫NAex(w ∗ N) Weighted density excess functional

+ α

∫P2 Local polarization correlations



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


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


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

]


5 layers Cu7 layers Cu9 layers Cu

ExperimentBulk

Relevant free energy: Φ(µe) = A(Ne)− µeNe − µCu2+NCu − µCl−NCl−


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

]


(Pt)(Pt)Cu

(Pt)Cu2(Pt)Cu1/4


Linearized Φ(µe) = A(N0e )− µeN

0e assumes fixed Ne


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

]


(Pt)(Pt)Cu

(Pt)Cu2(Pt)Cu1/4


Linearized Φ(µe) = A(N0e )− µeN

0e assumes fixed Ne


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)


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]


ExperimentPCM


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

nonlocal polarizable continuum models from joint density functional

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