oregon state university cornell universityroundyd/talks/research...introduction to dft properties of...
TRANSCRIPT
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
Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?
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
Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?
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?
Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?
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 ′)
Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?
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?
Introduction to DFT Properties of fluids Classical DFT for water Results Future work Where are we going?
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?