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)