Calculation of entropy from MolecularDynamics: First Principles Thermodynamics

Mario Blanco*, Tod Pascal*, Shiang-Tai Lin#, and W. A. Goddard III

Beckman Institute *Caltech Pasadena, California, USA

# National Taiwan University, Taipei, Taiwan

Outline

• Motivation– Free Energy: Enthalpy and Entropy Components

• First Principles Thermodynamics– Thermodynamic Integration

– Umbrella Sampling

– Umbrella Integration

• 2PT Model– Lennard-Jonesium

• Water Results– Precision and Accuracy

• Other Common Solvents

• Conclusions

Material

ScienceChemistry

QUANTUM

MECHANICS

Molecular Dynamics

Force Fields

Hierarchical First Principles Simulations

G = H - T S < 0

Meso-scale

Modeling

Equilibrium

& Rate

Constants

Molecular

Self-Assembly

Catalysis

Biochemistry

Organelle

Modeling

Genetic

Engineering

Pharmaceuticals

Specialty

Chemicals

& CatalystsMetal

Alloys

Ceramics

Polymers

Receptor

Modeling

C1 Chemistry

Electronic

& Optical

Materials

Cancer

Research

Fossil Energy

Fuel CellsNanotechnology

Atoms

Electrons

Molecules

Materials

Design

Femptoseconds

Angstroms

Microseconds

Microns

Years

Yards

Picoseconds

Nanometers

Seconds

Inches

H = E

F=ma

•

=

=2

Multi-scale

© W.A. Goddard III, M. Blanco, 1998

Entropy

The internal energy U might be thought of as the energyrequired to create a system in the absence of changes intemperature or volume. But if the system is created in anenvironment of temperature T, then some of the energycan be obtained by spontaneous heat transfer from theenvironment to the system -> - TS

S more fundamental than E

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/seclaw.html#c4

Continuous Dielectric Models: Poisson

Equation

•Poisson Eq.: Interaction between Solute and Continuum Solvent

[ ] )(4)()( rrrrrr

=rr

rr

rr

rr

rr

rr

at density charge :)(

at potential ticelectrosta :)(

position at constant dielectric :)(

law sCoulomb' '

)'(')( 1 if

rr

rdrr ==

==S

screensolute

V

solutesolvent

ele

SirrrdrrrdE )()()()( 23*/rrrrrr

•Energy of Interaction

====

=

= V

ele

Si

ele

Si

ele

SirrrdEEdG )()(

2

1)1(

2

1)( 3*/

*

/

1

0

*

/

rrr

•Electrostatic Solvation Free Energy

•Apparent Surface Charges

nin

r=

4

1

1

2

3

4

5

6

1

2

3

n

Linear response

Estimation of F

An indirect method which is very similar to the way in which free

energies are obtained in real experiments leads to Free energy

differences, not absolute values

MD is used to obtain derivatives of the free energy such as pressure

or energy:

Integrating these derivatives between two well definedthermodynamic states leads to a change in free energy F

Thermodynamic IntegrationThe reaction is divided into windows with a specific value i

assigned to each window.

with an additional term correcting for incomplete momentum

sampling, the so-called metric-tensor correction

Review: Kastner & Thiel, J. Chem. Phys. 123, 144104 (2005)

Thermodynamic Integration

Review: Kastner & Thiel, J. Chem. Phys. 123, 144104 (2005)

Umbrella Sampling

Review: Kastner & Thiel, J. Chem. Phys. 123, 144104 (2005)

Umbrella Sampling

Review: Kastner & Thiel, J. Chem. Phys. 123, 144104 (2005)

Umbrella Integration

Review: Kastner & Thiel, J. Chem. Phys. 123, 144104 (2005)

Results

Results

Water properties

Results

Timings: only 8.4 CPU years!

Precision and Accuracy

Any new thermodynamic model to predict Free Energies comes

Once every 10 years. It definitely needs validation!

a) Precision: How reproducible are the results

b) Accuracy: How well results compare to experiment

Precision: Model & MD Integration parameters

Accuracy: Model, MD integration &Force Field parameters

In an effort to validate the 2PT model we worked on a further tuning

Levitt’s F3C water model, commonly used in our group, to leave

Out issues regarding FF parameters.

Primary validation focus: Entropy predictions in a about one CPU hour!

Molecular Thermodynamics

S jk ( ) = lim

1

2 jk (t) j

k (t + t ')dt'e i2 tdt = lim c jk (t)e i2 tdt

Lin, S.-T.; Blanco, M.; Goddard-III, W. A. J Chem Phys 2003, 119(22), 11792-11806.

Molecular Thermodynamics

S jk ( ) = lim

1

2 jk (t) j

k (t + t ')dt'e i2 tdt = lim c jk (t)e i2 tdt

+=+=0

E

1

0

V,N

1

0 )(W)(SdVT

QlnTVE

=+=0

s

V,N

1 )(W)(SdkT

QlnQlnkS

1)hexp(

h

2

h)(W

Q

E +=

[ ])hexp(1ln1)hexp(

h)(W

Q

S =

Molecular Thermodynamics

Helmholtz Free energies determined using a Quantum and a Classically corrected versions of the 2PT method. The curves are the exact results from the equations of state for Lennard-Jones liquids.

Lin, S.-T.; Blanco, M.; Goddard-III, W. A. J Chem Phys 2003, 119(22), 11792-11806.

Hvap rms density

other H-Charge (cal/cc) cal/cc (g/cc) rms

QH1 0.4014 -618.35 +- 16.97 0.99 0.02

QH2 0.39 -541.07 +- 12.58 0.97 0.02

QH3 0.3846 -521.46 +- 7.02 0.97 0.01

LMP2 0.36433 -406.49 +- 10.21 0.93 0.02

Calculation of Interfacial Tension

sV

znz

)()( =

ij

ij

ji ij

ij

s

BNr

rdu

r

z

VTkzzP

)(1)()(

),(

2

=

ij

ij

ji ij

ijij

s

BTr

rdu

r

yx

VTkzzP

)(

2

1)()(

),(

22+

=

[ ]= )()( zPzPdzTN

zLLV yxs =

Kirkwood-buff theory

x

y

z

Liquid Experimental (dynes/cm) Calculated (dynes/cm)

Liquid Argon (57K) 14.5 15.5

Water (298K) 72 69.5

Cyclohexane (298K) 23 33

Decane (298K) 23.4 16.6

Comparison of Calculated and Predicted Surface Tensions

Dielectric Constant

Kirkwood-Frohlich Equation

a) Dielectric Constant CRC Handbook (interpolated between 20-30 C)

b) Surf Tension CRC Handbook (interpolated between 20-30 C)

Cohesive energy NIST Values: Hf = 10.5172 (gas-liquid) Kcal/mol =>582.5359 cal/cc

with density=0.997 g/cc at 298.15K

http://webbook.nist.gov/cgi/cbook.cgi?ID=C7732185&Units=CAL&Mask=1#Thermo-Gas

F3C H-opt Model: Electrostatic Sensitivity

(300K) (Dyn/cm)

Q(H) Hvap rms density Dielectrms Surface

(cal/cc) cal/cc (g/cc) Constant Tension rms

a,b Exp. -582.53 +- 0.0001 0.997 0 79.5 0.01 71.55 0.01

F3C 0.41 -689.71 +- 6.82 1.02 0.01 104 1.5 70.94 2.25

QHOpt 0.39697 -580.68 +- 7.3 0.98 0.01 80.6 1.5 69.21 2.25

Hvap rms density

other H-Charge (cal/cc) cal/cc (g/cc) rms

QH1 0.4014 -618.35 +- 16.97 0.99 0.02

QH2 0.39 -541.07 +- 12.58 0.97 0.02

QH3 0.3846 -521.46 +- 7.02 0.97 0.01

LMP2 0.36433 -406.49 +- 10.21 0.93 0.02

Quantum vs Classical Entropy

MD

Simulation Joules/K*mol

VAC time Gas Solid Total

Entropy with 10 30.4 38.1 68.6

Quantum Correction 12 30.9 37.6 68.6

14 31.0 37.6 68.6

16 31.3 37.3 68.6

18 31.2 37.5 68.7

20 31.1 37.6 68.7

22 30.6 38.3 68.9

Classical Entropy 10 30.4 -1.8 28.6

12 30.9 -2.3 28.7

14 31.0 -2.3 28.7

16 31.3 -2.5 28.7

18 31.2 -2.3 28.8

20 31.1 -2.2 28.9

22 30.6 -1.4 29.3

Entropy with 10 42.6 38.1 80.8

Flory Huggins Correction 12 43.4 37.6 81.1

Undistinguishable Molecules 14 43.5 37.6 81.1

16 44.0 37.3 81.3

18 43.8 37.5 81.3

20 43.7 37.6 81.3

22 42.9 38.3 81.2

Experimental Entropy: 69.9 J/K*mol (NIST)

Velocity Auto-Correlation Function

F3C/HQopt water

time(ps)

C(t)

Water Power Spectrum (DoS)

25 ps, 1fs steps

Power spectrum for water at 300 K. The power spectrum is decomposed into a gas (diffusive)

and a solid (fixed) spectra and their contributions added to yield the free energy of the liquid

state .

(cm-1)

( )

Water Power Spectrum (DoS)

Log (w)

Power spectrum for water at 300 K. The power spectrum is decomposed into a gas (diffusive)

and a solid (fixed) spectra and their contributions added to yield the free energy of the liquid

state .

Statistics: Precision across

frequency of sampling

Statistics

Statistics: Precision across

Independent Simulations

Precision: Across total length of

MD simulation

Experimental Entropy: 69.9 J/K*mol (NIST)

Accuracy of 2PT Model

(FF dependent)

J/mol*Kgas solid total

Sc 30.6 -1.4 29.3

Sq 30.6 38.3 69.6

Sexp 69.9

% error +/- 0. 0.4% (0.2 Joule/mole*K)

Non-protic Solvents

Dichloromethane DMSO benzene

Density 1.1 0.92824 0.80126

Exp 1.326 1.1004 8.7381

S_classic 95.2757 9.116 -16.189

S_quantum 162.5 181.9 185.7

S experimental 174.5 188.7 174.3

Joules/K*mol

Conclusions

• New first principles thermodynamics

model: 2PT

• Provided good potential results are

within 0.4% experimental entropy water

• Errors of 7% for other solvents

• Results in 1-2 CPU hours

• Full Statistical analysis in progress

Acknowledgments

• Bill: providing support basic research

• Dow Corning

• NSF NIRT

• Shiang-Tai Lin

• Dr. Mario Blanco

• DOE CSGF

• Entire Krell Staff (Dr. Edelson, Rachel)