erin e. dahlke department of chemistry university of minnesota vlab tutorial may 25, 2006

Computational Challenges in the Simulation of Water & Ice: the

Motivation for an Improved Description of Water-Water Interactions

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r (Å)

g(r)

g(r)O-Og(r)O-Hg(r)H-H

ROH = 0.9572Å

HOH = 104.52º

Erin E. Dahlke

Department of Chemistry

University of Minnesota

VLab Tutorial

May 25, 2006

ROH = 0.9572

HOH = 104.52

= 1.86D

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r (Å)

g(r)

g(r)O-O

g(r)O-H

g(r)H-H

ISIS Disordered Materials Group Neutron Database http://www.isis.rl.ac.uk/disordered/database/DBMain.htm

ROH = 0.9572

HOH = 104.52

= 1.86D

Umemoto, K.; Wentzcovich, R. M. Phys. Rev. B 2004, 69, 180103.

ROH = 0.9572

HOH = 104.52

= 1.86D

QuickTime™ and aTIFF (Uncompressed) decompressor

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are needed to see this picture.QuickTime™ and a

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Uranus Neptune

Triton

Titan

Ganymede Callisto



http://www.solarviews.com/eng/uranus.htm

HOH = 104.52°= 2.18 D

= 0.6492 kJ/mol = 3.153 Å

ROH = 0.9572 Å

0.520.52

-1.04

TIP4P

Jorgensen, W. L.; Chandrasekhar, J.; Madura, J., D.; Impey, R. W.; Klein, M. L. J. Chem. Phys.1983, 79, 926.

Analytic potentials for water are generally parameterized to get a specific physical property right (i.e., vapor pressure, density, structural properties)

+ Fast+ Accurate*- Non-transferable

Variations of TIP4P1. TIP4P-FQ2. TIP4P-POL23. TIP4P-EW4. TIP4P-ice5. TIP4P-m

€

υ(r)⇒ −h2∇2

2m+ υ (r)

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥Ψ(r) = EΨ(r)⇒ Ψ(r)⇒ observable properties

€

n(r)⇒ Ψ(r)⇒ υ (r)⇒ observable properties

Wave Function Theory:

Density Functional Theory:

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HΨ(r) = EΨ(r)

€

H = T +V

Quantum Mechanics

Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864.

Kuo, I.–F. W.; Mundy, C. J.; Eggimann, B. L.; McGrath, M. J.; Siepmann, J. I.; Chen, B.; Vieceli, J.; Tobias, D. J. J. Phys. Chem. B 2006, 110, 3738.McGrath, M. J.; Siepmann, J. I.; Kuo, I.–F. W.; Mundy, C. J.; VandeVondele, J.; Hutter, J.; Mohamed, F.; Krack, M. J. Phys. Chem. A 2006, 110, 640.Umemoto, K.; Wentzcovich, R. M. Phys. Rev. B 2004, 69, 180103

a

(Å)

c

(Å)

O-H

(Å)

O…H

(Å)

O-O

(Å)

O…O

(Å)

V

(Å3)

Calc 4.727 6.817 0.986 1.929 2.914 2.77 76.16

Expt 4.656 6.775 0.968 1.911 2.879 2.743 73.43

Ice VIII (T=10K P=24GPa)

Perdew, J. P.; Schmidt, K.; Density Functional Theory and Its Application to Materials, Doren,V., Alsenoy, C. V., Geerlings, P. Eds.; American Institute of Physics: New York 2001

LSDA

GGA

meta GGA

hybrid meta GGAhybrid GGA

fully non-local

‘Earth’ Hartree Theory

‘Heaven’Chemical Accuracy

QuantumChemistry

MaterialsScience

LSDA

GGA

meta GGA

hybrid meta GGAhybrid GGA

fully non-local



Perdew, J. P.; Schmidt, K.; Density Functional Theory and Its Application to Materials, Doren,V., Alsenoy, C. V., Geerlings, P. Eds.; American Institute of Physics: New York 2001

High-level wave function theory

Density functional theory

Coming up with a test set…

Literature clusters

QuickTime™ and aTIFF (LZW) decompressor


Solid-phase simulations

Calculate AccurateBinding Energies

Compare 25 DFTMethods to Accurate

Energies

Liquid/vaporsimulations

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and

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0 0.5 1 1.5 2 2.5 3 3.5

SVWN5

mPWLYP

PBE

BLYP

PBELYP

HCTH

OLYP

TPSS

TPSSLYP

BB95

B98

B97-1

X3LYP

B3LYP

MPW1K

mPW1PW91

PBE1PBE

MPW3LYP

B97-2

PW6B95

MPWB1K

MPW1B95

PBE1KCIS

PWB6K

B1B95

Mean Unisgned Error per Molecule (kcal/mol)

LSDA

GGA

meta GGA

hybridGGA

hybridmetaGGA

light dimermedium trimerdark all

Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 15677

All DFT calculations use the MG3S (6-311+G(2df,2p)) basis set.

€

Exc = X100Ex

HF + (1− X100

)ExLDA + bΔEx

GGA + EcLSDA + Y

100ΔEc

GGA

€

Exc = Ex + EcLSDA + Y

100ΔEc

GGA

How should we parameterize our new method?The general form for a hybrid density functional method is:

What if instead…

mPWLYPPBEPBELYPTPSSLYP

Optimize Y mPWLYP1WPBE1WPBELYP1WTPSSLYP1W

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SVWN5

mPWLYP

PBE

BLYP

PBELYP

HCTH

OLYP

PBE1W

MPWLYP1W

TPSSLYP1W

PBELYP1W

TPSS

TPSSLYP

BB95

B98

B97-1

X3LYP

B3LYP

MPW1K

mPW1PW91

PBE1PBE

MPW3LYP

B97-2

PW6B95

MPWB1K

MPW1B95

PBE1KCIS

PWB6K

B1B95

Mean Unsigned Error per Molecule (kcal/mol)

LSDA

GGA

meta GGA

hybridGGA

hybridmetaGGA

newmethods

light dimermedium trimerdark all

Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 15677

All DFT calculations use the MG3S (6-311+G(2df,2p)) basis set.

Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 15677All DFT calculations use the MG3S (6-311+G(2df,2p)) basis set.

LSDA

GGA

meta GGA

hybrid-meta GGAhybrid GGA

fully non-local



aMUEPM denotes mean unsigned error per molecule

Dimers Trimers AllPW6B95 0.06 0.06 0.06MPWB1K 0.06 0.08 0.07B98 0.05 0.11 0.06B3LYP 0.13 0.25 0.15TPSS 0.10 0.21 0.12TPSSLYP 0.21 0.44 0.26PBE1W 0.05 0.07 0.05MPWLYP1W 0.08 0.07 0.08TPSSLYP1W 0.09 0.07 0.09PBELYP1W 0.13 0.05 0.11mPWLYP 0.09 0.20 0.12PBE 0.22 0.27 0.23BLYP 0.41 0.59 0.45PBELYP 0.40 0.78 0.48SVWN5 1.99 3.22 2.26

MUEPMa (kcal/mol)

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0.80

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1.40

1.60

1.80

6-31+G(d,p) 6-31+G(d,2p)6-311+G(2d,2p)6-311+G(2df,2p)

aug-cc-pVDZ aug-cc-pVTZ

Mean Unsigned Error (kcal/mol)

BLYP

B3LYP

PBE1W

PBE

Csonka, G. I.; Ruzsinsky, A.; Perdew, J. P. J. Phys. Chem. B, 2006, 109, 21475.Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 15677Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B In Press.

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6-31+G(d,p) 6-31+G(d,2p)6-311+G(2d,2p)6-311+G(2df,2p)

aug-cc-pVDZ aug-cc-pVTZ


BLYP

MUEPM(kcal/mol)

Dimers Trimers All

PBE 0.22 0.27 0.23

BLYP 0.41 0.59 0.45

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mPWLYP

PBE B98B97-1X3LYPB3LYP

PW6B95MPWB1KMPW1B95

PBE1W

MPWLYP1WTPSSLYP1WPBELYP1W


light = AE6dark = BH6

Is getting the energies right enough??

What about other things like geometries or polarizabilities?

R(O-H) Θ( - - )H O H

LSDA 0.9701 104.9605PBE 0.9690 104.2068

1PBE W 0.9686 104.2883BLYP 0.9706 104.5230

1PBE PBE 0.9575 104.90513B LYP 0.9604 105.1265

–Hartree Fock 0.9399 106.2709

ROH = 0.9572

HOH = 104.52

Fitting of the functional to get the best bond length possible gives reallybad energies for the clusters.

There’s no simple fix to this problem. Best geometry possible with this optimization procedure:R(O-H) = 0.9675 Å Θ(H-O-H) = 104.5008

One way to get a feeling for whether a method is getting the polarizability right is to look at the many-body effects in the structure.

€

V (r1,r2,...,rN ) = V2(ri ,rj )i< j

N∑ + V3(ri ,rj ,rk ) + ...+ Vn (ri ,rj ,rk ,...,rn )

i< j<k,...,n

N∑

i< j<k

N∑

€

V (r1,r2,r3,r4 ,r5) = V2(ri ,rj )i< j

5

∑ + V3(ri ,rj ,rk ) + V4 (ri ,rj ,rk ,rl ) +V5i< j<k<l

5

∑i< j<k

5

∑

€

V4 (rA,rB,rC,rD) =V (rA,rB,rC,rD) − V3(ri ,rj ,rk )i< j<k

4

∑ − V2(ri ,rj )i< j

4

∑

€

V3(rA,rB,rC ) =V (rA,rB,rC ) − V2(ri ,rj )i< j

3

∑

€

=V (rA,rB,rC ) −V2(rA,rB) −V2(rA,rC ) −V2(rB,rC )

Gas phase optimized

Monte Carlo simulation of bulk water

Monte Carlo simulation of bulk water

Gas phase optimized

MD simulation of ice VIII

What do we hope to learn?

1. Relative magnitudes of many-body terms in small clusters.

2. Differences in many-body terms between gas-phase structures and those taken from simulation.

3. Performance of common density functionals in the prediction of many-body effects.

4. Performance of density functionals is the prediction of binding energies for larger water clusters.

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V3 V4 V5

Magnitude (kcal/mol)

2.01

5.86

-0.460.14

0.53

-0.24 0.01

Average absolute magnitudeMax valueMinimum value

Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B in press

Relative magnitudes of many-body terms

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V3 V4

Average Absolute Magnitude (kcal/mol)

2.01

2.74

1.18

0.140.31

0.07

All structuresGas phaseSimulation

PBE1WPBEBLYPB3LYP

New functional parameterized specifically for water

Most commonly used GGAs in simulation

Most commonly used hybrid GGA in chemistry, recently used in water simulation


Gas-phase versus simulation

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PBE1W/MG3SPBE/MG3S BLYP/MG3S B3LYP/MG3S

PBE/aug-cc-pVTZBLYP/6-31G+(d,p)B3LYP/6-31+G(d,2p)

PBE1W/6-311+G(2d,2p)


V2 (13.46)V3 (2.01)V4 & V5 (0.12)All (6.13)


Performance of density functionals for many-body terms

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PBE1W/MG3SPBE/MG3S BLYP/MG3S B3LYP/MG3S

PBE1W/6-311+G(2d,2p)


Mean Unsigned Error in Binding Energies (kcal/mol)

Trimers

Tet & Pent

All


Performance of density functionals for binding energies

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PBE1W/MG3SPBE/MG3SBLYP/MG3SB3LYP/MG3S

PBE1W/6-311+G(2d,2p)


Mean Unsigned Error in Binding Energies (kcal/mol)

Dimers

Trimers

Tet & Pent

All


Performance of density functionals for binding energies - large data set

Conclusions• Different density functional methods give vastly different results for different functionals.

• PBE1W shows improved performance over other GGA methods for small water clusters-and is competitive with hybrid and hybrid-meta methods.

• Selection of basis set is crucial to performance.

• All GGAs have shortcomings at predicting many-body effects.

Future Work

• Use PBE1W in the simulation of liquid water.

• Examine the use of PBE1W for structural properties and larger water clusters

Method 1 2 3 4 5 6 7 8 9 10

CCSD(T) 0 1 2 1 2 3 2 3 1 2PBE 0 1 2 1 1 3 2 3 1 2PBE1PBE 0 1 2 1 1 3 2 3 1 2B3LYP 0 1 2 1 1 3 2 3 1 2BHandHLYP 0 1 2 1 2 3 2 3 1 2

Anderson, J. A.; Tchumper, G. S. J. Phys. Chem. A 2006, published on the web 05/12/06

Future Work

Acknowledgments



Don TruhlarNate SchultzYan Zhao

Ilja Siepmann, Matt McGrath Renata Wentzcovitch, Koichiro Umemoto



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erin e. dahlke department of chemistry university of minnesota vlab tutorial may 25, 2006

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