Calculation of entropy from Molecular Dynamics: First ... - Krell Institute [PDF]

Motivation. – Free Energy: Enthalpy and Entropy Components. • First Principles Thermodynamics. – Thermodynamic Int

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Calculation of entropy from Molecular Dynamics: 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

Multi-scale

Hierarchical First Principles Simulations  G = H - T  S < 0

Years Yards

Cancer Research Genetic Engineering Seconds Inches

 2 =  

Fossil Energy Fuel Cells

Nanotechnology

C1 Chemistry Organelle Modeling Receptor Modeling

Pharmaceuticals

Polymers Electronic & Optical Ceramics Materials Specialty Chemicals Metal & Catalysts Alloys

Materials

Catalysis Microseconds Microns



 = 

Biochemistry Molecular Self-Assembly

Picoseconds Nanometers

Material Science

Chemistry Equilibrium & Rate Constants

Design

Meso-scale Modeling Molecules

F=ma Molecular Dynamics Force Fields

Femptoseconds Angstroms

H = E 

QUANTUM MECHANICS

Atoms Electrons © W.A. Goddard III, M. Blanco, 1998

Entropy

S more fundamental than E The internal energy U might be thought of as the energy required to create a system in the absence of changes in temperature or volume. But if the system is created in an environment of temperature T, then some of the energy can be obtained by spontaneous heat transfer from the environment to the system -> - TS http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/seclaw.html#c4

Continuous Dielectric Models: Poisson Equation •Poisson Eq.: Interaction between Solute and Continuum Solvent r r  (r ) : dielectric constant at position r r r  (r ) : electrostatic potential at r r r  (r ) : charge density at r

r r r    [ (r ) (r )]= 4 (r )

if  = 1   (r ) =  dr '

•Apparent Surface Charges =

 (r ' ) Coulomb' s law r  r'

r  1 in  n 4

2 n

•Energy of Interaction r r r r 3r 2r Ei*/ele = d r  ( r )  ( r ) = d r  ( r )  ( r S  solvent solute  solute screen ) V

S

•Electrostatic Solvation Free Energy  =1 *ele i/S

G

=

*ele d   E i / S ( ) = 

 =0

r r r 1 *ele 1 Ei / S ( = 1) =  d 3 r  (r )  (r ) 2 2V

Linear response

1 2

5

3 1 3

4 6

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 defined thermodynamic states leads to a change in free energy F

Thermodynamic Integration The 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

1   2

S kj ( ) = lim



 





   kj (t) kj (t + t')dt'e i2t dt = lim  c kj (t)e i2t dt  

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

Molecular Thermodynamics 1   2

S kj ( ) = lim











   kj (t) kj (t + t')dt'e i2t dt = lim  c kj (t)e i2t dt  

   ln Q 1 E = V0 + T = V0 +  0 dS (  )WE (  )  T N ,V 1

   ln Q S = k ln Q +  = k  dS (  )Ws (  ) 0  T N ,V 1

 h  h W ( ) = + 2 exp( h )  1 Q E

WSQ (  ) =

 h  ln[1  exp(   h )] exp( h )  1

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.

other QH1 QH2 QH3 LMP2

H-Charge 0.4014 0.39 0.3846 0.36433

Hvap (cal/cc) -618.35 -541.07 -521.46 -406.49

rms cal/cc ++++-

16.97 12.58 7.02 10.21

density (g/cc) 0.99 0.97 0.97 0.93

rms 0.02 0.02 0.01 0.02

Calculation of Interfacial Tension Kirkwood-buff theory       =  dz[PN ( z )  PT ( z )]

    1 PN ( z ) =  ( z )k BT  Vs 1 PT ( z ) =  ( z )k BT  Vs

zij2 du (rij )

r

( i , j ) ij



(i , j )

rij

xij2 + yij2 du (rij ) 2rij

rij

       ( z) =

n( z ) Vs

Vs = Lx Ly z

z

y x

Comparison of Calculated and Predicted Surface Tensions 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

Dielectric Constant Kirkwood-Frohlich Equation

F3C H-opt Model: Electrostatic Sensitivity Q(H)

Hvap (cal/cc)

a,b Exp. F3C QHOpt

other QH1 QH2 QH3 LMP2

a) b)

0.41 0.39697

H-Charge 0.4014 0.39 0.3846 0.36433

rms cal/cc

(300K) Dielectrms Constant

density (g/cc)

(Dyn/cm) Surface Tension rms

-582.53 +-

0.0001

0.997

0

79.5

0.01

71.55

0.01

-689.71 +-580.68 +-

6.82 7.3

1.02 0.98

0.01 0.01

104 80.6

1.5 1.5

70.94 69.21

2.25 2.25

Hvap (cal/cc) -618.35 -541.07 -521.46 -406.49

rms cal/cc ++++-

16.97 12.58 7.02 10.21

density (g/cc) 0.99 0.97 0.97 0.93

rms 0.02 0.02 0.01 0.02

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

Quantum vs Classical Entropy MD Simulation VAC time

Joules/K*mol Gas Solid Total 30.4 38.1 68.6 30.9 37.6 68.6 31.0 37.6 68.6 31.3 37.3 68.6 31.2 37.5 68.7 31.1 37.6 68.7 30.6 38.3 68.9

Entropy with Quantum Correction

10 12 14 16 18 20 22

Classical Entropy

10 12 14 16 18 20 22

30.4 30.9 31.0 31.3 31.2 31.1 30.6

-1.8 -2.3 -2.3 -2.5 -2.3 -2.2 -1.4

28.6 28.7 28.7 28.7 28.8 28.9 29.3

Entropy with Flory Huggins Correction Undistinguishable Molecules

10 12 14 16 18 20 22

42.6 43.4 43.5 44.0 43.8 43.7 42.9

38.1 37.6 37.6 37.3 37.5 37.6 38.3

80.8 81.1 81.1 81.3 81.3 81.3 81.2

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

Velocity Auto-Correlation Function F3C/HQopt water C(t)

time(ps)

Water Power Spectrum (DoS) 25 ps, 1fs steps ()

 (cm-1) 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 .

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*K gas

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 Density

1.1

DMSO

benzene

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

S experimental 174.5

188.7

Joules/K*mol

185.7 174.3

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)

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