Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Classical density functional theory for water

David Roundy Sahak Petrosyan Tomas A. Arias

Oregon State University Cornell University

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Outline

1 Introduction to Density Functional Theory

2 Properties of fluids

3 Classical DFT for water

4 ResultsLiquid-vapor interfaceHydration of hard spheres

5 Future work

6 Where are we going?

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Motivation for Density Functional Theory

Old-fashioned variational principle

The ground state energy of a system of N electrons in an externalpotential V (r) can be written as

Eo = minΨ(r1,r2,...,rN)

{〈Ψ|Ho |Ψ〉+

∫n(r)V (r)dr

}where Ho is a universal hamiltonian independent of V (r), and n(r)is the single-particle density

n(r) =

∫∫∫dr2dr3 · · · drN |Ψ(r , r2, ..., rN)|2

i.e. the probability of finding any electron at position r .

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Density Functional Theory

Hohenberg-Kohn theorem

The ground state energy of a system of N electrons in an externalpotential V (r) can be written as

Eo = minn(r)

{F [n(r)] +

∫n(r)V (r)dr

}where F [n(r)] is a universal functional of the electron density n(r),independent of V (r).

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Kohn-Sham Density Functional Theory

Kohn-Sham approach

The ground state energy of a system of N electrons in an externalpotential V (r) can be written as

FKS [n(r)] = min{φi (r)}

{∑i

〈φi |T |φi 〉+

1

2

∫∫n(r)n(r ′)

|r − r ′|drdr ′ + Fxc [n(r)]

}where F [n(r)] is a universal functional of the electron density n(r),independent of V (r).

Kinetic energy of non-interacting electrons

Hartree energy

“Exchange-correlation” energy

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Kohn-Sham Density Functional Theory

Kohn-Sham approach

The ground state energy of a system of N electrons in an externalpotential V (r) can be written as

FKS [n(r)] = min{φi (r)}

{∑i

〈φi |T |φi 〉+

1

2

∫∫n(r)n(r ′)

|r − r ′|drdr ′ + Fxc [n(r)]

}where F [n(r)] is a universal functional of the electron density n(r),independent of V (r).

Kinetic energy of non-interacting electrons

Hartree energy

“Exchange-correlation” energy

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Kohn-Sham Density Functional Theory

Kohn-Sham approach

The ground state energy of a system of N electrons in an externalpotential V (r) can be written as

FKS [n(r)] = min{φi (r)}

{∑i

〈φi |T |φi 〉+

1

2

∫∫n(r)n(r ′)

|r − r ′|drdr ′ + Fxc [n(r)]

}where F [n(r)] is a universal functional of the electron density n(r),independent of V (r).

Kinetic energy of non-interacting electrons

Hartree energy

“Exchange-correlation” energy

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Kohn-Sham Density Functional Theory

Kohn-Sham approach

The ground state energy of a system of N electrons in an externalpotential V (r) can be written as

FKS [n(r)] = min{φi (r)}

{∑i

〈φi |T |φi 〉+

1

2

∫∫n(r)n(r ′)

|r − r ′|drdr ′ + Fxc [n(r)]

}where F [n(r)] is a universal functional of the electron density n(r),independent of V (r).

Kinetic energy of non-interacting electrons

Hartree energy

“Exchange-correlation” energy

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Finite-temperature Density Functional Theory

Free Energy

A = −kBT log

(∑i

〈i |H |i〉

)

Mermin Theorem

A(T ) = minn(r)

{F [n(r),T ] +

∫Vext(r)n(r) d3r

}

An extension of Hohenberg-Kohn theorem to non-zerotemperatures.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

“Classical” Density-Functional Theory

Mermin Theorem

A(T ) = minn(r)

{F [n(r),T ] +

∫Vext(r)n(r) d3r

}

Applies to a density of any sort of particle in anexternal potential: we’ll consider the density ofwater molecules.

In principle, exact: no “classical” approximation.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Why do we care about water?

Most interesting chemistry happens in thepresence of water:

Solvation

Catalysis

Corrosion

Hydrophobic interactions

Protein folding

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Why bother with classical density functional theory?

Making ab initio predictions involving liquid water is a pain!

Requires large cell to simulateaperiodic liquid

Each water molecule adds eightelectrons to the problem (which isgenerally O(N3)

Need free energy not just the groundstate energy

Requires molecular dynamicscalculations to generate a thermalensemble of water molecule positions

Simpler models often lead to betterunderstanding!

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Eventual goal: Joint Density Functional Theory

Treat water as a continuum with classical density functional theory,solute with Kohn-Sham quantum mechanical density functionaltheory, and approximate the interaction between the two systems.

A = minne(r),nw (r)

{FKS [ne(r)] + Fclassical [nw (r)]

∫+ U[ne(r), nw (r)] +

∫ne(r)Ve(r) + nw (r)Vw (r)dr

}

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

A classical density functional theory for water

Treat water as a continuum with classical density functional theory,solute with Kohn-Sham quantum mechanical density functionaltheory, and approximate the interaction between the two systems.

A = minne(r),nw (r)

{FKS [ne(r)] + Fclassical [nw (r)]

∫+ U[ne(r), nw (r)] +

∫ne(r)Ve(r) + nw (r)Vw (r)dr

}

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Exact properties and experimentally known properties

Exact properties

Relation between contact density and pressure on a hard wall

Hydration energy of small hard spheres

Ideal gas law describes low-density limit

Experimentally accessible properties

Free energy of a homogeneous fluid

Surface tension

etc...

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Pressure on a hard wall

Hard wall

A hard wall is a surface with infinite potential on one side, andzero potential on the other.

The pressure on a hard wall is proportional to the contact density:

p = nckBT

x

n

nc

Exact ideal gas solution

Ideal gas free energy satisfies this property:

Fid [n] = kBT

∫n(r)log(n(r)) dr

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Free energy of a homogeneous fluid

The free energy of a homogeneous fluid may be determined atmost densities from its equation of state.

n

free

en

erg

y

nll

p

k Tvapor

B

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Surface tension

The energy of a macroscopic liquid is proportional to its totalsurface area. The proportionality constant is called the surfacetension.

E = Surface Tension · Area

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Classical Density-Functional Theory for Liquids

Universal functional

F [n(~r)] = Fid [n(~r)] + Fex [n(~r)]

Free energy of ideal gas

Fid [n(~r)] = kT

∫n(~r) log(n(~r)) d~r

Excess free energy

Fex is the usual “unknown functional” that crops up in any sort ofdensity-functional theory.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Free energy of a homogeneous aqueous system

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

f id(n

) +

f exc

(n)

(10-6

Har

tree

/boh

r3 )

n/nl

f (n) = fid(n) + fex(n)

Written as ideal gas free energy plus a polynomial.

Fit to liquid density, vapor pressure, bulk modulus and itsderivative.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Free energy of a homogeneous aqueous system

How about

Fex [n(r)] =

∫fex(n(r))dr

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

f id(n

) +

f exc

(n)

(10-6

Har

tree

/boh

r3 )

n/nl

Problems:

it would get the wrong contact density at a hard wall

hydration energy of small hard spheres would be wrong

this would predict zero surface tension

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Requirement for retaining the exact properties

The excess free energy functional Fex must be a “smooth” (i.e.purely non-local) functional of n(r)!

Several exact properties hold for any smooth Fex

Relationship between contact density and the pressure on ahard wall.

Hydration energy for hard spheres in small sphere limit.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Simplest approach for a non-local functional

Weighted density approximation

Fex [n(r)] =

∫fex(n(r))dr

where

n(r) =

∫W (r − r ′)n(r ′)dr ′

= W ◦ n

Constraints∫W (r)dr = 1, to reproduce homogeneous limit

W (r) must be smooth, so that the resulting functional ispurely non-local

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Properties acheived by simplest WDA

Simplest weighted density approximation:

Fex [n(r)] =∫

fex(n(r))dr

Exact properties

Relation between contact density and pressure on a hard wall

Hydration energy of small hard spheres

Ideal gas law describes low-density limit

Experimentally accessible properties

Free energy of a homogeneous fluid

Surface tension

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

More complicated functional in practice...

Fex [n(r)] =

∫fex(n(r))dr +

∫∫χ(r − r ′)n(r)n(r ′)drdr ′

where

n(r) = n(r ′)W (r − r ′)dr ′ = W ◦ n

This allows us to also reproduce the response function extractedfrom the direct correlation function, which I’ve skipped over forbrevity.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

A liquid-vapor interface

In order to calculate the surface tension,we must consider a simple planarliquid-vapor interface.

The surface tension is just the energy perunit area of a liquid-vapor interface.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Density profile for a liquid-vapor interface

0

0.001

0.002

0.003

0.004

0.005

0.006

-10 -5 0 5 10

wat

er d

ensi

ty (

1/bo

hr3 )

position (bohr)

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Hydration of hard spheres in water

A model for hydrophobic solvation.

We compute an effective surface tension bydividing the energy required to place thehard sphere in the water by its surface area.

R

R

R

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Hydration of hard spheres in water

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9

Sur

face

tens

ion

(mJ/

m2 )

R (Angstrom)

CDFTmolecular dynamics

From the slope of this curve, we can determine the pressure, andthus the contact density, which we can compare with thecomputed density.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Density profile for hard spheres

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14

n/nl

r (bohr)

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Finished and Future Work

Finished work

Have found several exact and experimental constraints we canplace on a classical density functional.

Developed a classical density functional for water.

Future work

Improved functional?

Introduce orientational degrees of freedom

Couple with electrostatic fields

Couple with quantum-mechanically described electronicsystems.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Prettier functional

Fex [n(r)] =

∫fex(n(r))dr

where

n = W ◦ n + |ξ ◦ n|2

Constraints∫W (r)dr = 1 and

∫ξ(r)dr = 0, in order to reproduce

homogeneous limit.

W (r) and ξ(r) smooth so that the functional is purelynon-local;

Given W , ξ is constrained by the direct correlation function.

Choose W to be a gaussian?

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Orientational degrees of freedom

As a first attempt, include polarization in the functional:

F [n(~r), ~P(~r)] = Fold [n(~r)] + Fpol [n(~r), ~P(~r)]

where

~P(~r) ≡ electrostatic polarization at ~r

A naive expansion would involve something like

Fpol [n(~r), ~P(~r)] =

∫d~r

1

2χn(~r)

∣∣∣~P(~r)∣∣∣2

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Electrostatic interactions

Electrostatic energy

U =

∫d~r

∫d~r ′

ρtot(~r)ρtot(~r′)

|~r −~r ′|where

ρtot(~r) = ρf (~r) + ρb(~r)

ρf is just the free charge density (i.e. that which generates theexternal fields

What is ρb? We read in Jackson

ρb = −~∇ · ~P

but this is only valid on macroscopic lengthscales!

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Electrostatic interactions at atomic length scales

Electrostatic energy

U =

∫d~r

∫d~r ′

ρtot(~r)ρtot(~r′)

|~r −~r ′|where

ρtot(~r) = ρf (~r) + ρb(~r)

In reality, bound charge is smooth on atomic length scales!

We resolve this dilemma (as usual) with a convolution:

ρb = −W ◦ ~∇ · ~P

where W (~r) is an appropriately chosen weighting function.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Coupling with quantum systems

When can we hope to represent the interaction of an electronicsystem treated quantum mechanically with a continuum waterdensity?

When the interaction between the electronic system and the wateris not chemical—which is the common case.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Coupling with quantum systems

When can we hope to represent the interaction of an electronicsystem treated quantum mechanically with a continuum waterdensity?

When the interaction between the electronic system and the wateris not chemical—which is the common case.

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Coupling with quantum systems

Attractive interactions

Electrostaticlarge in polar or charged systems

Hydrogen-bondinglarge in polar systems with first-row elements

Van der Waalssmall and hard to compute

Repulsive interactions

Pauli repulsionvery short length-scale

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Coupling with quantum systems

Attractive interactions

Electrostaticlarge in polar or charged systems

Hydrogen-bondinglarge in polar systems with first-row elements

Van der Waalssmall and hard to compute

Repulsive interactions

Pauli repulsionvery short length-scale

Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?

Pauli repulsion

The “Pauli” repulsion is simply an effect of the Pauli exclusionprinciple.

We will try representing the Pauli repulsion using a“pseudopotential”, which approximates this effect.

Non-local pseudopotentials are known to be able to accuratelyrepresent the Pauli repulsion of valence electrons with coreelectrons in an atom.

Fpauli [ne , nw ] =

∫dr

∫dr ′Vp(r − r ′)ne(r)nw (r ′)

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And then what?

Verify accuracy

Compute the solvation energy of water in wateri.e. its latent heat of vaporization

Compute the energy required to pull a proton from a watermolecule

i.e. why does pure water have a pH of seven?

Compute the solubility of common ionsi.e. why does baking soda dissolve more easily than chalk?

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And then what?

Make predictions

Predict surface tension changes caused by surfactantsi.e. which soap makes the best bubbles?

Predict the elastic properties of a lipid bilayeri.e. how intrinsically strong is a cell membrane?

Predict the energy of interaction of water with a solid surfacei.e. what’s the angle of the meniscus in a tin cup?

Predict the properties of biologically active moleculesi.e. why does that new medicine actually work?

Predict the properties of systems under hydrostatic pressurei.e. how does this enzyme work at the bottom of the ocean?