Calculation of entropy from Molecular Dynamics: First Principles Thermodynamics Mario Blanco, Tod Pascal, Shiang-Tai Lin#, and W. A.

Size: px

Start display at page:

Download "Calculation of entropy from Molecular Dynamics: First Principles Thermodynamics Mario Blanco*, Tod Pascal*, Shiang-Tai Lin#, and W. A."

Frederick Stewart
6 years ago
Views:

1 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

2 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 Years Yards Seconds Inches Microseconds Microns Picoseconds Nanometers Hierarchical First

Modeling Biochemistry Molecular Self-Assembly Fossil Energy Fuel Cells Receptor Modeling C 1 Chemistry

Ceramics Specialty Chemicals & Catalysts Material Science Meso-scale Modeling Electronic & Optical Materials

3 Multi-scale Years Yards Seconds Inches Microseconds Microns Picoseconds Nanometers Hierarchical First Principles Simulations G = H - T S < 0 2 = = F=ma Genetic Engineering Pharmaceuticals Cancer Research Organelle Modeling Biochemistry Molecular Self-Assembly Fossil Energy Fuel Cells Receptor Modeling C 1 Chemistry Catalysis Chemistry Equilibrium & Rate Constants Molecular Dynamics Force Fields Nanotechnology Polymers Ceramics Specialty Chemicals & Catalysts Material Science Meso-scale Modeling Electronic & Optical Materials Metal Alloys Design Materials Molecules Femptoseconds Angstroms H = E QUANTUM MECHANICS W.A. Goddard III, M. Blanco, 1998 Atoms Electrons

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.

4 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

5 Continuous Dielectric Models: Poisson Equation Poisson Eq.: Interaction between Solute and Continuum Solvent r r [ ( r ) ( r )] = 4( r ) Apparent Surface Charges 1 = 4 Energy of Interaction 3 = d rsolvent ( r ) solute ( r ) = r r ( ) : r ( ) : r ( ) : r dielectric constant at position r electrostatic potential at r charge density at ( r') if = 1 ( r) = dr' r r' * ele r r r 2r r r E d ( ) ( ) i / S = 1 V in r n Electrostatic Solvation Free Energy S solute screen * ele * ele 1 * ele 1 3r r r G i / S = dei / ( ) = Ei ( = 1) = d ( ) ( ) S / 2 S 2 = 0 V Linear response n Coulomb's law

is used to obtain derivatives of the free energy such as pressure or energy: Integrating

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

7 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, (2005)

8 Thermodynamic Integration Review: Kastner & Thiel, J. Chem. Phys. 123, (2005)

9 Umbrella Sampling Review: Kastner & Thiel, J. Chem. Phys. 123, (2005)

10 Umbrella Sampling Review: Kastner & Thiel, J. Chem. Phys. 123, (2005)

11 Umbrella Integration Review: Kastner & Thiel, J. Chem. Phys. 123, (2005)

12 Results

13 Results Water properties

14 Results Timings: only 8.4 CPU years!

15 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!

16 Molecular Thermodynamics S k 1 j () = lim 2 j k (t) j k (t + t')dt'e i2t dt = lim c j k (t)e i2t dt Lin, S.-T.; Blanco, M.; Goddard-III, W. A. J Chem Phys 2003, 119(22),

17 Molecular Thermodynamics S k 1 j () = lim 2 j k (t) j k (t + t')dt'e i2t dt = lim c j k (t)e i2t dt E = V lnq 1 1 = T V0 ds( )W E( ) T 0 N,V S = k lnq W Q E + 1 ( ) = lnq T h + 2 N V, = k 0 h exp( h ) 1 ds( )W ( ) s W Q ( ) = S h exp( h ) 1 ln 1 [ exp( h )]

18 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),

21 Hvap rms density other H-Charge (cal/cc) cal/cc (g/cc) rms QH QH QH LMP

23 Calculation of Interfacial Tension Kirkwood-buff theory [ P ( z) P ( z ] = dz ) N T P N ( z) = ( z) k B T 1 V s ( i, j) z r 2 ij ij du( r ) r ij ij P ( z) = ( z) k T T B 1 V s x 2 ij + ( i, j) 2rij y 2 ij du( r ) r ij ij ( z) = n( z) V s V s = L L z x y z y x

24 Comparison of Calculated and Predicted Surface Tensions Liquid Experimental (dynes/cm) Calculated (dynes/cm) Liquid Argon (57K) Water (298K) Cyclohexane (298K) Decane (298K)

25 Dielectric Constant Kirkwood-Frohlich Equation

26 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 F3C QHOpt Hvap rms density other H-Charge (cal/cc) cal/cc (g/cc) rms QH QH QH LMP a) Dielectric Constant CRC Handbook (interpolated between C) b) Surf Tension CRC Handbook (interpolated between C) Cohesive energy NIST Values: Hf = (gas-liquid) Kcal/mol => cal/cc with density=0.997 g/cc at K

27 Quantum vs Classical Entropy MD Simulation Joules/K*mol VAC time Gas Solid Total Entropy with Quantum Correction Classical Entropy Entropy with Flory Huggins Correction Undistinguishable Molecules Experimental Entropy: 69.9 J/K*mol (NIST)

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

29 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.

30 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.

31 Statistics: Precision across frequency of sampling

32 Statistics

33 Statistics: Precision across Independent Simulations

34 Precision: Across total length of MD simulation Experimental Entropy: 69.9 J/K*mol (NIST)

35 Accuracy of 2PT Model (FF dependent) J/mol*K gas solid total Sc Sq Sexp 69.9 % error +/ % (0.2 Joule/mole*K)

36 Non-protic Solvents Dichloromethane DMSO benzene Density Exp S_classic S_quantum S experimental Joules/K*mol

37 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

38 Acknowledgments Bill: providing support basic research Dow Corning NSF NIRT Shiang-Tai Lin Dr. Mario Blanco DOE CSGF Entire Krell Staff (Dr. Edelson, Rachel)

The role of specific cations and water entropy on the stability of branched DNA motif structures

The role of specific cations and water entropy on the stability of branched DNA motif structures Tod A Pascal, William A Goddard III, Prabal K Maiti and Nagarajan Vaidehi 1. Figures Supporting Information