Idea Transcript
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)