© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L08 Ch121a Atomic Level Simulations of Materials and Molecules William A. Goddard

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L08

Ch121a Atomic Level Simulations of Materials and Molecules

William A. Goddard III, [email protected] and Mary Ferkel Professor of Chemistry,

Materials Science, and Applied Physics, California Institute of Technology

BI 115Hours: 2:30-3:30 Monday and Wednesday

Lecture or Lab: Friday 2-3pm (+3-4pm)

Teaching Assistants Wei-Guang Liu, Fan Lu, Jose Mendoza, Andrea Kirkpatrick

Lecture 8, April 18, 2011Statistical Mech. Thermo, 2PT


Thermodynamics

Describe a system in terms of Hamiltonian H(p,q) where p is generalized momentum and q is generalized coordinate

Here Q, the Partition function, is a normalization constant

For a system in equilibrium, probability of a state with energy H(p,q) is

P(p,q) = exp[-H(p,q)/kBT]/Q

which is referred to as a Boltzmann distribution,

Q = S exp[-H(p,q)/kBT] summed over all states of the system


Thermodynamic functions can all be derived from Q

QkTT

QkTQkS

T

QkTE

VN

VN

lnAEnergy Free Helmholtz

)ln

(ln Entropy

)ln

( Energy

,

,2

VNVNV

TN

TV

T

QkT

T

QkTC

V

QkTp

N

QkT

,2

22

,

,

,

)ln

()ln

(2 Capacity Heat

)ln

( Pressure

)ln

( Potential Chemical


,2,1 8 2

22

xx

n nmL

nhx

Vh

mkTdneeeeTVq mL

nβh

nnn yyx

2/32

3

0

8

1

βε

1

βε

1

βε

trans )π2

()(),(2

22

ynynxn

The partition function for translation

Assume a cubic periodic box of side L

The QM Hamiltonian is

The QM eigenfunctions are just periodic functions for x, y, and z directions, sin(nxxp/L) etc

Leading to

2

22

2 xm

H

Thus the partition function for translation becomes


Thermodynamic functions for translation

VNVNV

TN

TV

T

QkT

T

QkTC

V

QkTp

N

QkT

,2

22

,

,

,

)ln

()ln

(2 Capacity Heat

)ln

( Pressure

)ln


Vh

MkT 2/32

)π2

(Q =

= (3/2) kT

2/5

2/3

2

π2ln e

N

V

h

MkT

=

= -kT

= (3/2) k

V

NkT

= k

Ideal gas

equipartition


2

2

2

2

sin

1

θθsin

θθsin

1

2 I

H

,2,1,0 )12( 2

)1( 2J

2

JJI

JJJ

2/12

22/1

2

22/1

2

21/2

0

2

)1(2

rot )π8

()π8

()π8

(σ

π)12(

σ

1)(

2

h

kTI

h

kTI

h

kTIdJeJTq CBAI

JβJ

The partition function for rotation

This leads to energy levels of

Thus the partition function becomes

This is 2 the Laplacian

I = moment of inertia


Thermodynamic functions for rotation (non linear)

VNVNV

TN

TV

T

QkT

T

QkTC

V

QkTp

N

QkT

,2

22

,

,

,

)ln

()ln

(2 Capacity Heat

)ln

( Pressure

)ln


Q =

= (3/2) kT

= 0

= -kT

= (3/2) k

= k

2/131/2

)(σ

π

CBA

T

2/132/31/2

)(σ

πln

CBA

Te

equipartition

equipartition


,2,1,0 )2

1( nnn

Summing over all normal modes leads to

The partition function for vibrations

An isolated harmonic oscillator with vibrational frequency ω

Has a spectrum of energies

Substituting into the Boltzmann expression leads to

1ωβ

ω/2β

1 0

β

1jj nvib e

eeq n

q =


Thermodynamic functions for vibration (harmonic oscillator)

Q =

= (3/2) kT

= 0

= -kT

= k

= k

63

1/

/2

1

n

jT

T

jv

jv

e

e

63

1/

1

/

2

n

jT

vv

jv

jj

e

T

T

63

1

/

/ )1ln(1

/n

j

T

T

v jv

jv

j ee

T

63

1

/)1ln(

2

n

j

Tv jvj eT

2/

/263

1 )1(

T

Tn

j

v

jv

jv

j

e

e

T


Q =

= - kT De

= 0

= -De - kT

= 0

= k

kTDe

ee /1

1ln e

1nuclear

β1

β1electronic

q

eeq ee De

De

Assuming the reference state has free atoms

Thermodynamic functions for electronic states

we will assume qelect=1


)/2βexp(-1

)/2βexp()βexp()( n

h

hq

n

QHO

Harmonic oscillator Partition function

)(ln)(ln0

HOqSdQ

Write partition function of the systemAs a continuous superposition of oscillators

)()(βln

β0

10

,

10 E

VN

WSdVT

QTVE

)()(ln

βln0,

1 SBVN

WSdkT

QQkS

)()(βlnβ0

10

10 AWSdVQVA

Thermodynamic properties

1)βexp(

β

2

β)(

h

hhW QE

)]βexp(1ln[1)βexp(

β)(

hh

hW QS

)/2βexp(

)βexp(1ln)(

h

hW QA

Zero point energyReference energy

)()(0,

vCB

VNv WSdk

T

EC

2

2

)]βexp(1[

)βexp()β()(

h

hhW QCv

hnn )2

1(

Weighting functions

Thermodynamic Properties for a Crystal

where


Where do we get the vibrational density of states DoS(n)?

Experimentally from Inelastic neutron scattering

Compare to phonon dispersion curves. Peak is for phonons with little dispersion

“Phonon Densities of States and Related ThermodynamicProperties of High Temperature Ceramics” C.-K Loong, J.European Ceramic Society, 1998

Can use to calculate thermodynamic properties


How can we get the vibrational DoS(n) from MD?

Consider mass transport( , )

0c t

t

r

j

Fick’s Law D c j

combine 2 ( , ) 0cD c t

t

r

Boundary condition: ( , ) ( )c t r r At t=0

2nd moment

D=diffusion coefficient

solution

=6Dt

2/ 2( , ) (2 ) exp

2d r

c t DtDt

r Gaussian

Ballistic r = v t


Time correlation

0 0

( ) ( )t td

t d ddt

rr v

Write r at time t as sum of displacement

1

1

21 1 2 2

0 0

1 2 2 10 0

1 2 2 10 0

1 2 1 20 0

10 0

0

( ) ( ) ( )

( ) ( )

2 ( ) ( )

2 (0) ( )

2 (0) ( )

2 2 (0) ( )

t t

t t

t

t

t t

t

r t d d

d d

d d

d d

d d

dDt t d

v v

v v

v v

v v

v v

v v

Take limit as t∞

3D=

6D=Green-Kubo Equation

From David A. [email protected]

get


Velocity autocorrelation function, C(t), VAC

( ) (0) ( )C t t v v

Zero slope

backscattering

Lennard-Jones 12-6 particles

2(0) /C v dkT m 3kT/m

3D=

Diffusion coefficient = area under VAC curve


Density of States function

Density of states, DoS(n)

where the atomic DoS is

and

Can also get DoS from Fourier transform of VAC

In terms of atomic vac

Wiener-Khintchine theorem


Density of States is Fourier transform of vac

Density of states, DoS(n)

DoS(n) = is number of states in dn around n

Summing over all DoS gives the total number of normal modes

since

We have


Evaluation of Thermodynamic Functions from DoS

Note that as n 0 get 0 in denominator

And also ln(0)

Reference energy

Weighting functions

Zero point energy


Vibrational Spectrum for a solid

0

5

10

15

20

25

30

35

0 50 100 150 [cm-1]

S(

) [c

m]

S jk()lim

1

2

jk(t) j

k(t t ')dt'e i2tdtlim

c jk (t)e i2tdt

Need DoS() 0 as 0 to avoid singularities

No problem for solids since Debye theory DoS 3 as 0


Density of states for a liquid

Since DoS(0) is proportional to the diffusion constant, have finite DoS(0)


Density of states of a gas


The velocity autocorrelation function of a hard sphere gas decays exponentially

) exp(3

) exp()0()( tm

kTtctc HSHS

where a is the Enskog friction constant related to the collisions between hard spheres.

Ng is the number of effective hard sphere particles in the system

222

0

01

3

1

4

12

)2cos() exp(3

4

)2cos()(

4)(

g

g

N

j k

kjj

HS

N

dtttkTNkT

dtttcmkT

Sg

222

0

01

3

1

4

12

)2cos() exp(3

4

)2cos()(

4)(

g

g

N

j k

kjj

HS

N

dtttkTNkT

dtttcmkT

Sg

Velocity autocorrelation for a hard sphere gas

222

0

01

3

1

4

12

)2cos() exp(3

4

)2cos()(

4)(

g

g

N

j k

kjj

HS

N

dtttkTNkT

dtttcmkT

Sg

222

0

01

3

1

4

12

)2cos() exp(3

4

)2cos()(

4)(

g

g

N

j k

kjj

HS

N

dtttkTNkT

dtttcmkT

Sg


The Gas phase component

fN

sS HS12

)0( 0

2

0

0

61

)()(

fN

s

sSS HSg

The Density of vibrational states for the gas component is completely determined with two parameters: s0 and f.

S ( )

Gas

exponentialdecay

Need to define exact value of “fluidicity” factor f that determines the conceptual partition of the whole system between solid and gas components. f must satisfy two limiting conditions: High temperature (low density) limit, the system behaves like hard spheres, therefore f =1 (no solid component).High density limit, system is a solid, we expect f=0 (no gas component)


Define f as proportional to the diffusivity, which automatically satisfies the high density and low density conditions.

Definition of fluidicity, f

); ,(

),(

0HSHS TD

TDf

D, the self-diffusivity of the system is the hard sphere diffusivity determined in the zero pressure limit (the Chapman-Enskog result)

2/120 )(

1

8

3);,(

m

kTTD

HS

HSHS

To determine f, need now only determine sHS from the MD


Determining the hard sphere diameter sHS . Require diffusivity of the gas component (at temperature T and density fr) to agree with prediction from Enskog theory (gives the best transport properties

for dense hard sphere fluids)

1)(

4);,(),( 0

fyz

fyfTDfTD HSHSHS

mfN

kTs

m

kTdttcfTD HSHS

12) (

3

1),( 0

0

The Enskog theory predicts the deviation of diffusivity for a dense hard sphere fluid from its zero pressure limit as

where z is the compressibility, which can be obtained from the accurate Carnahan-Starling equation of state for hard spheres

3

32

)1(

1)(

y

yyyyz

y is the hard sphere packing fraction. For a given value of f, these equations

can be solved for y, and thus sHS . However, f is also a function of sHS , must

solve simultaneously for both f and sHS .

The diffusivity of the gas component is determined from the VAC


S ( )

Finite density of states at =0Proportional to diffusion coefficient Harmonic Approximation Entropy= ∞

•Also strong anharmonicity at low frequencies

Liquid

S ( )

Gasexponentialdecay

S ( )

SolidDebye crystalS(v) ~v2New Model

2 phase theory (2PT )Liquid Solid + Gas

•Two-Phase Thermodynamics Model (2PT)• Decompose liquid S(v) to a gas and a solid contribution• S(0) attributed to gas phase diffusion• Gas component contains anharmonic effects• Solid component contains quantum effects

S ( )

solid-likegas-like

For crystals get Free Energies from phonon density of states. Cannot do this for Liquids since S(0)≠0

The two-phase model for calculating thermodynamic properties of liquids from molecular dynamics: Validation for the phase diagram of Lennard-Jones fluids; Lin, Blanco, Goddard; JCP, 119:11792(2003)


Describe the diffusional gas-like component as a hard sphere fluid. The velocity autocorrelation function of a hard sphere gas decays exponentially

The 2PT Model: Describe the vibrational density of states of a liquid as a superposition of a diffusional gas-like phase and a

vibrational non-diffusional phase

) exp(3

) exp()0()( tm

kTtctc HSHS

Ng = f N is the number of effective hard sphere particles in the system

f = fractional hard sphere component in overall system. Measures “fluidicity” of the system (depends on both temperature and density).

222

0

01

3

1

4

12

)2cos() exp(3

4

)2cos()(

4)(

g

g

N

j k

kjj

HS

N

dtttkTNkT

dtttcmkT

Sg

222

0

01

3

1

4

12

)2cos() exp(3

4

)2cos()(

4)(

g

g

N

j k

kjj

HS

N

dtttkTNkT

dtttcmkT

Sg

)()()()(00

gP

gHOP

s WSdWSdP

Property =


0

200

400

600

800

1000

1200

0 5 10 [cm-1]

S(

) [c

m]

0

5

10

15

20

25

30

0 50 100 150 [cm-1]

S(

) [c

m]

0

5

10

15

20

25

30

35

0 50 100 150 [cm-1]

S(

) [c

m]

• Examples

LJ gas liquid FCC solid

solid-likegas-likegas-like

solid-like

solid-likegas-like


Solid phase system, gas part negligible


T - diagram for Lennard Jones Fluid

0.6

1.0

1.4

1.8

0.0 0.4 0.8 1.2*

T*

Solid

LiquidGas

Supercritical Fluid

●metastable●unstable

Validate Accuracy of Predicted Free Energy from MD

For Lennard-Jones Fluid have essentially exact free energies for full phase diagram from extensive Monte Carlo Calculations (by others)

Include: solid, liquid, gas,

supercritical, metastable and

unstable


Free Energy

•Accurate for gas, liquid, and crystal•Accurate in metastable and unstable regimes

-30

-25

-20

-15

-10

-5

0

5

0 0.2 0.4 0.6 0.8 1 1.2 *

G *

T*=1.8T*=1.4T*=1.1T*=0.92PT(Q)2PT(C)

Free energy 2PT model

Entropy and Free Energy from MD nearly exact

Lines exactCircles: MD-2PT

UM

U

M

SL

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2 *

S *

T*=1.8T*=1.4T*=1.1T*=0.9

2PT(Q)2PT(C)

Entropy 2PT model

• Accurate for gas, liquid, and crystal• Accurate in metastable regime• Quantum Effects most important for crystals (~1.5%)



Pressures and Energies from MD agree with exact EOSFor all phases

Pressure and Energy from MD nearly exact

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 *

E *

T*=1.8T*=1.4T*=1.1T*=0.9MD2PT(Q)

Total Energy

Density-2

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 *

P *

T*=1.8

T*=1.4

T*=1.1

T*=0.9

MD

Pressure

Density



T - diagram for Lennard Jones Fluid

0.6

1.0

1.4

1.8

0.0 0.4 0.8 1.2*

T*

Solid

LiquidGas

Supercritical Fluid

●metastable●unstable

Validate Accuracy of Predicted Free Energy from MD

For Lennard-Jones Fluid have essentially exact free energies for full phase diagram from extensive Monte Carlo Calculations (by others)

Include: solid, liquid, gas,

supercritical, metastable and

unstable


6.5

7.5

8.5

9.5

10.5

11.5

12.5

13.5

14.5

15.5

100 1000 10000 100000 1000000

MD steps

S*

2PT(Q)

2PT(C)

MBWR EOS

gas (r*=0.05 T*=1.8)

liquid (r*=0.85 T*=0.9)

• For gas, the entropy

converges to within

0.2% with 2500 MD

steps (20 ps)

• For liquid, the entropy

converges to within

1.5% with 2500 MD

steps (20 ps).


0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.00 0.40 0.80 1.20

T*

Simulation conditions

solid

supercritical fluid

•Initial amorphous structure is used in the cooling process

•The fluid remains amorphous in simulation even down to T*=0.8 (supercooled)

•The predicted entropy for the fluid and supercooled fluid agree well with EOS for LJ fluids

•Initial fcc crystal is used in the heating process

•The crystal appears stable in simulation even up to T*=1.8 (superheated)

•The predicted entropies for the crystal and superheated crystal agree well with EOS for LJ solids

3

4

5

6

7

8

0.80 1.20 1.60 2.00T*

S*liquid (EOS)

solid (EOS)

heating

cooling

classical

Ent

rop

y

solidmetatstable

unstablesupercritical

fluid

starting withfcc crystal


Diffusion Coefficient from S(0)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 0.2 0.4 0.6 0.8 1 1.2 *

D *

T*=1.8

T*=1.4

T*=1.1

T*=0.9

DoS

)0(12

SmN

kTD

Diffusion Coefficient From MD (2PT)



Application to Water


Velocity Auto-Correlation Function

F3C/HQopt water

time(ps)

VAC[tot]

-500000

0

500000

1000000

1500000

2000000

0 0.25 0.5 0.75 1


log (cm-1)

0

2

4

6

8

10

12

14

16

1 10 100 1000 10000

S_hs(v)[cm]

S_s(v)[cm]

Stot(v)[cm]

0

2

4

6

8

10

12

14

16

0 500 1000 1500 2000 2500 3000

S_hs(v)[cm]

S_s(v)[cm]

Stot(v)[cm]

Total power spectrum (Fourier transform of velocity

autocorrelation function

)

2PT decomposition for H2O (300K)

25 ps, 1fs steps

) total

vibration

diffusional

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.

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.


• Theory: 69.6 +/- 0.2 J/K*mol• Experimental Entropy: 69.9 J/K*mol (NIST)

Statistics collected over 20ps of MD , no additional costStatistics collected over 20ps of MD , no additional cost


Plot of Sq (J mol-1 molecule-1) vs timestep

0

50

100

150

200

250

300

350

1 10 100 1000 10000

trajectory read timestep (²t

qu

an

tum

en

trop

y (

J/m

ol)

Dependence of Accuracy on time step used in velocity autocorrelation analysis of the dynamics

Need to keep

velocities at every 9fs or

more frequently

MD was at 1 fs time step


Statistics: Precision across frequency of sampling


ResultsWater properties


Liquid Entropy of Water (300 K)

68.5068.55

68.6068.65

68.7068.75

68.8068.8568.9068.95

0 5 10 15 20 25

MD Time (ps)

En

tro

py (

Jou

les/K

*mo

l)

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

Precision: Across total length of MD simulation


Applications to several liquids

Thermodynamics of liquids: Absolute molar entropies and heat capacities of common solvents from 2PT Molecular Dynamics; Tod A Pascal ,Shiang-Tai Lin and William A Goddard III, Phys Chem Chem Phys 13 (1): 169-181 (2011)


2PT liquid Chloroform, CHCl3


Benzene S and Cv convergence

Converged at 20 ps


Standard Absolute Molar Entropy


Specific Heat Capacity


Accuracy in predicted S from various FF


Accuracy in predicted Cp from various FF


Components of S from 2PT

0.050.240.0749.790.079.53--TIP4P-Ew91

0.070.290.1453.050.0312.03--SPC/E90

0.060.250.2550.500.0611.540.000.04F3C89

Waterd

0.61.561.320.160.200.1218.750.1116.540.0411.21TFE

3.002.750.230.260.1119.770.0717.450.0911.45toluene

4.633.920.310.330.1520.050.0717.900.069.01THF

1.21.731.520.100.210.0316.320.0413.590.0113.29NMA

2.23.683.390.200.320.0616.030.0411.380.011.71methanol

4.853.550.300.350.1219.870.1216.760.043.40furan

0.390.330.100.160.0814.660.0812.030.026.94ethylene glycol

1.821.540.150.200.0915.950.0613.160.014.51ethanol

1.090.630.130.160.1016.310.0813.930.068.87DMSO

1.821.690.200.200.1517.970.0916.430.058.431,4 dioxane

3.763.220.300.330.1621.120.1519.200.025.65chloroform

3.773.450.290.300.0919.830.0616.630.064.74benzene

7.937.250.300.400.1419.200.0813.860.020.93acetonitrile

5.094.390.290.340.0519.350.0416.840.0411.08acetone

1.181.030.120.160.1015.480.0813.380.066.28acetic acid

±avg±avg±avg

ExpcGKbMSDafrotftransStransSrotSvib

D x10-5(cm2/s)fluidicityfactor

Standard Molar Entropy S0(cal/mol/K)

0.050.240.0749.790.079.53--TIP4P-Ew91

0.070.290.1453.050.0312.03--SPC/E90

0.060.250.2550.500.0611.540.000.04F3C89

Waterd

0.61.561.320.160.200.1218.750.1116.540.0411.21TFE

3.002.750.230.260.1119.770.0717.450.0911.45toluene

4.633.920.310.330.1520.050.0717.900.069.01THF

1.21.731.520.100.210.0316.320.0413.590.0113.29NMA

2.23.683.390.200.320.0616.030.0411.380.011.71methanol

4.853.550.300.350.1219.870.1216.760.043.40furan

0.390.330.100.160.0814.660.0812.030.026.94ethylene glycol

1.821.540.150.200.0915.950.0613.160.014.51ethanol

1.090.630.130.160.1016.310.0813.930.068.87DMSO

1.821.690.200.200.1517.970.0916.430.058.431,4 dioxane

3.763.220.300.330.1621.120.1519.200.025.65chloroform

3.773.450.290.300.0919.830.0616.630.064.74benzene

7.937.250.300.400.1419.200.0813.860.020.93acetonitrile

5.094.390.290.340.0519.350.0416.840.0411.08acetone

1.181.030.120.160.1015.480.0813.380.066.28acetic acid

±avg±avg±avg





Components of S from 2PT

0.050.240.0749.790.079.53--TIP4P-Ew91

0.070.290.1453.050.0312.03--SPC/E90

0.060.250.2550.500.0611.540.000.04F3C89

Waterd

0.61.561.320.160.200.1218.750.1116.540.0411.21TFE

3.002.750.230.260.1119.770.0717.450.0911.45toluene

4.633.920.310.330.1520.050.0717.900.069.01THF

1.21.731.520.100.210.0316.320.0413.590.0113.29NMA

2.23.683.390.200.320.0616.030.0411.380.011.71methanol

4.853.550.300.350.1219.870.1216.760.043.40furan

0.390.330.100.160.0814.660.0812.030.026.94ethylene glycol

1.821.540.150.200.0915.950.0613.160.014.51ethanol

1.090.630.130.160.1016.310.0813.930.068.87DMSO

1.821.690.200.200.1517.970.0916.430.058.431,4 dioxane

3.763.220.300.330.1621.120.1519.200.025.65chloroform

3.773.450.290.300.0919.830.0616.630.064.74benzene

7.937.250.300.400.1419.200.0813.860.020.93acetonitrile

5.094.390.290.340.0519.350.0416.840.0411.08acetone

1.181.030.120.160.1015.480.0813.380.066.28acetic acid

±avg±avg±avg




0.050.240.0749.790.079.53--TIP4P-Ew91

0.070.290.1453.050.0312.03--SPC/E90

0.060.250.2550.500.0611.540.000.04F3C89

Waterd

0.61.561.320.160.200.1218.750.1116.540.0411.21TFE

3.002.750.230.260.1119.770.0717.450.0911.45toluene

4.633.920.310.330.1520.050.0717.900.069.01THF

1.21.731.520.100.210.0316.320.0413.590.0113.29NMA

2.23.683.390.200.320.0616.030.0411.380.011.71methanol

4.853.550.300.350.1219.870.1216.760.043.40furan

0.390.330.100.160.0814.660.0812.030.026.94ethylene glycol

1.821.540.150.200.0915.950.0613.160.014.51ethanol

1.090.630.130.160.1016.310.0813.930.068.87DMSO

1.821.690.200.200.1517.970.0916.430.058.431,4 dioxane

3.763.220.300.330.1621.120.1519.200.025.65chloroform

3.773.450.290.300.0919.830.0616.630.064.74benzene

7.937.250.300.400.1419.200.0813.860.020.93acetonitrile

5.094.390.290.340.0519.350.0416.840.0411.08acetone

1.181.030.120.160.1015.480.0813.380.066.28acetic acid

±avg±avg±avg




0.050.240.0749.790.079.53--TIP4P-Ew91

0.070.290.1453.050.0312.03--SPC/E90

0.060.250.2550.500.0611.540.000.04F3C89

Waterd

0.61.561.320.160.200.1218.750.1116.540.0411.21TFE

3.002.750.230.260.1119.770.0717.450.0911.45toluene

4.633.920.310.330.1520.050.0717.900.069.01THF

1.21.731.520.100.210.0316.320.0413.590.0113.29NMA

2.23.683.390.200.320.0616.030.0411.380.011.71methanol

4.853.550.300.350.1219.870.1216.760.043.40furan

0.390.330.100.160.0814.660.0812.030.026.94ethylene glycol

1.821.540.150.200.0915.950.0613.160.014.51ethanol

1.090.630.130.160.1016.310.0813.930.068.87DMSO

1.821.690.200.200.1517.970.0916.430.058.431,4 dioxane

3.763.220.300.330.1621.120.1519.200.025.65chloroform

3.773.450.290.300.0919.830.0616.630.064.74benzene

7.937.250.300.400.1419.200.0813.860.020.93acetonitrile

5.094.390.290.340.0519.350.0416.840.0411.08acetone

1.181.030.120.160.1015.480.0813.380.066.28acetic acid

±avg±avg±avg




0.050.240.0749.790.079.53--TIP4P-Ew91

0.070.290.1453.050.0312.03--SPC/E90

0.060.250.2550.500.0611.540.000.04F3C89

Waterd

0.61.561.320.160.200.1218.750.1116.540.0411.21TFE

3.002.750.230.260.1119.770.0717.450.0911.45toluene

4.633.920.310.330.1520.050.0717.900.069.01THF

1.21.731.520.100.210.0316.320.0413.590.0113.29NMA

2.23.683.390.200.320.0616.030.0411.380.011.71methanol

4.853.550.300.350.1219.870.1216.760.043.40furan

0.390.330.100.160.0814.660.0812.030.026.94ethylene glycol

1.821.540.150.200.0915.950.0613.160.014.51ethanol

1.090.630.130.160.1016.310.0813.930.068.87DMSO

1.821.690.200.200.1517.970.0916.430.058.431,4 dioxane

3.763.220.300.330.1621.120.1519.200.025.65chloroform

3.773.450.290.300.0919.830.0616.630.064.74benzene

7.937.250.300.400.1419.200.0813.860.020.93acetonitrile

5.094.390.290.340.0519.350.0416.840.0411.08acetone

1.181.030.120.160.1015.480.0813.380.066.28acetic acid

±avg±avg±avg





Conclusions

• New first principles thermodynamics model: 2PT

• Provides good results within 0.4% experimental entropy water

• Errors of 7% for other solvents

• Results in 1-2 CPU hours

• Full Statistical analysis in progress


Alternative approaches to Estimation of F

Common indirect method 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



Umbrella Sampling - 1


In umbrella sampling, a restraint (a ξ dependent bias) is applied in each window along the path from state a to state b.

For example

Calculate the distribution P(ξ ) along the reaction coordinate from a to b

Calculate the dependence of the free energy on ξ (the potential of mean force

where b = 1/kBT and kB is the Boltzmann Constant.


Umbrella Sampling - 2


The constant Fi is not known. It is determined by combining unbiased free energies Ai

u of the different windows

The unbiased free energy is


Results

Timings: only 8.4 CPU years on 2.8 GHz CPU


Put in 2PT results from Tod


Equilibrium Stat. Mech.Partition FunctionsW(N,V,E), Q(NVT),(T,V,), (N,T,P)

Thermodynamic PropertiesT,P,V,E,H,S,A,G, ,Cp,Cv

Non-Equilibrium Stat. Mech.Time Correlation Functions<A(0)A(t)>

Transport PropertiesD,Dynamic ProcessIR, Dielectric RelaxationThermodynamic Properties???

Time correlation functions are important for studying dynamical properties of a system


'

')'()'(Alim)();()0;()()0()(

dt

dttAtttAAdtAAtC

Dynamical function (velocity, momentum, ect)

Phase space coordinates (e.g. position and momentum)

Equilibrium distributionEnsemble average

Definition of Time Correlation Function


Properties of Autocorrelation Function

)()( tCtC

)()()()0()( tAAtAAtC1. starting time is arbitrary

2. symmetric in time space

3. short time value ~ <A2>

4. long time value ~ <A>2)(tC

t

<A2>

<A>2

Can you prove that C(0)C(t) for any t?


Diffusivity

)()0(3

1

0

tvvdtD

Shear viscosity

)()0(1

0

tdtVkT

xzxz

N

j

N

ji

xj

xiji

zj

xjj

xz FFzzvvm1

))((2

1

Thermal Conductivity

)()0(1

02

tqqdtVkT

N

j

N

jiijjj rVvm

dt

dq

1

2)(

2

1

2

1

Absorption lineshape (Rotational-vibrational spectra)

QttidtI )(μ)0(μ)ωexp(

2

1)ω(

Some commonly used time correlation functions


Computing dielectric constant

The dielectric constant ε

where

and finally

Need to add slides showing results bulk water and also

as function of distance from POPC and water


What can we learn from VAC?

1) C(t=0) ~ kinetic energy (temperature) of the system

K.E.2)0()0()0()0(1

2

11

3

1

N

jj

N

jj

N

j

kj

kj vmvvmcmC

2) Integration over time (Area underneath): Self-Diffusion Coefficient

The diffusion equation

),(),( 2 trGD

t

trG

Initial condition )()0,( rrG

20

0

4 |)(|6

1),(4

6

1rtr

tdrtrGr

tD

mean square displacement

)0()(3

1

0

vvdD

3) Dynamical behaviors of the system


Free energy profile over volume at 277K

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Volume (A^3)

A(k

J/m

ol/

den

dro

n)

AA15

Critical pressure: 0.033GPa

a. Condense phase

b. Isolated Micelle phase

Study of Percec Dendrimer Crystals with 2PT

References: Li, Y., S.-T. Lin, and W. A. Goddard, accepted in J. Am. Chem. Soc. (2003), accepted.

OH

O

O

O

O

RO

RO OR

RO

RO

RO

RO

RO

OR

R=C12H25

Cub

m3ImTet mnmP /42


Thermodynamics of Na Three way junction at 285K

Free energy

-TS entropy

enthalpy


sV

znz

)()(

ij

ij

ji ij

ij

sBN r

rdu

r

z

VTkzzP

)(1)()(

),(

2

ij

ij

ji ij

ijij

sBT r

rdu

r

yx

VTkzzP

)(

2

1)()(

),(

22

• Density profile

• Stress profile

• Interfacial tension )()( zPzPdz TN

zLLV yxs

Kirkwood-buff theory

xy

z

Calculation of Interfacial Tension


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


Quench anneal cycle for PET polymer

0.4 rexp 0.8 rexp1.2 rexp ~1.0 rexp


Entropy, heat capacity and free energy over the phase diagram using 2PT molecular dynamics


Issues with current approaches•CPMD simulations (32 molecules) too small to describe phase behavior•Rigid empirical models give inaccurate super-critical behavior•Flexible empirical model not fit to thermodynamic properties

Issues with current approaches•CPMD simulations (32 molecules) too small to describe phase behavior•Rigid empirical models give inaccurate super-critical behavior•Flexible empirical model not fit to thermodynamic properties

Physical PropertiesMolar mass 44.010 g/molAppearance colorless, odorless gas

Density1.562 g/mL (solid/1 atm/195K)0.770 g/mL (liquid/56 atm293K)1.977 g/L (gas /1 atm/273K)849.6 g/L (supercritical/150 atm/305K)

Melting point 194.7 KBoiling point 216.6 K (at 5.185 bar)Solubility in water 1.45 g/L at 300K/1 barAcidity (pKa) 6.35, 10.33Viscosity 0.07 cP/195KDipole moment 0

Needs accurate forcefields that accounts for physical and thermodynamic properties

Needs accurate forcefields that accounts for physical and thermodynamic properties


εC σC εO σO α qC r01MSM 0.058 2.785 0.165 3.01 ¯ 0.596 1.1602EPM2 0.056 2.757 0.160 3.03 ¯ 0.651 1.1493EPM3 0.056 2.800 0.160 3.03 - 0.652 1.1624TraPPE 0.054 2.800 0.157 3.05 ¯ 0.700 1.1605Errington* 0.058 2.753 0.165 3.03 14 0.647 1.1436Zhang 0.057 2.792 0.164 3.00 ¯ 0.589 1.1637COMPASS* 0.136 3.420 0.134 2.94 - 0.800 1.160• Errington uses Exponential-6 for VDW• COMPASS uses Bond-Bond stretch term to match vib.

frequencies• Models optimized to reproduce experimental physical properties

• Errington uses Exponential-6 for VDW• COMPASS uses Bond-Bond stretch term to match vib.

frequencies• Models optimized to reproduce experimental physical properties

Tc(K) ρc (g/cm3)Pc (MPa)304.9 0.4642 7.17303.2 0.4664 7.07304.0 0.4679 7.39309.1 0.462 7.2302.5 0.4728 7.31

304.0 0.467 7.23316.1 0.4621 6.92

Exp 304.1 0.4676 7.377

How well do they reproduce experimental thermodynamics?How well do they reproduce experimental thermodynamics?

Critical PropertiesCritical Properties


• COMPASS has reasonable description of liquid, poor description of Sc-Co2 at low pressures• EPM3 more accurate for both liquid and Sc-C02

• COMPASS has reasonable description of liquid, poor description of Sc-Co2 at low pressures• EPM3 more accurate for both liquid and Sc-C02

LiquidLiquid Super CriticalSuper Critical


• COMPASS has large errors at high pressure liquid phase• EPM3 superior for both liquid and Sc-CO2

• COMPASS has large errors at high pressure liquid phase• EPM3 superior for both liquid and Sc-CO2

LiquidLiquid Super CriticalSuper Critical


Crystal structure of dry ice•Pa3 (#205) space group

•a0: 5.624 (2) Å

•C-O distance: 1.155 (1) Å

Free Energy is dominated by entropyFree Energy is dominated by entropy


• Entropy dominated by diffusion (50 % - solid, 55% liquid, 66% super critical

• Melting of Co2 corresponding to dramatic increase in diffusional entropy• Small increase in rotational entropy: C02 not a free rotor in liquid phase• Monotonic increase in vibrational entropy

from solid-> liquid ->super critical


Conclusion

• The recent EMP3 forcefield is accurate at describing the physical and thermodynamics of liquid and Sc-Co2– Correctly predicts critical properties– Melting temperature is 20K too high

• COMPASS forcefield is not as accurate

• Free Energy is dominated by entropy

• Diffusional entropy accounts for 50 – 66% of total entropy


Stopped Lecture 8


Predict 4 %wt H2 at 273K and 100 bar

Cheap as water

Tod Pascal and wag

Tod Pascal

New candidate advanced material for H2 storage


New candidate advanced material for H2 storage

Predict 4 %wt H2 at 273K and 100 bar

Cheap as water

Tod Pascal and wag

Ice I has large hexagonal channels

As T is increased to melting at 273K, individual water molecules leave the framework and rush through the channels

These channels also accommodate H2

4.4 wt% at 273K and 100 bar


Movie showing H2O molecules going into channels just before melting

entropy

• The first step of melting is Evaporation of water molecules one by one into hexagonal channels

• Entropy increases, enthalpy remains almost constant

• As more H2O evaporates into channels, Stresses build on framework

• at the melting temperature the whole framework collapses Rapidly


Cerius2 Sorption Module– 1,000,000 steps of GCMC– Pressure from 0.1 – 250,000 Kpa (0.001 – 2500 bar)– Temperatures: 77K,150K,273K,300K (still running)

4.5 wt% storage at 250K and 100bar (0.5% for amorphous phase)

Amorphous iceAmorphous ice

Hexagonal Ice 1hHexagonal Ice 1h

4% difference in storage

BE: -20.25 kcal/mol

300K273K250K


How would the H2/Ice system work?

273K 275K

Melting is very sharp

The melting process is very sharp, rapidly decreasing the stored H2 from 4.4 wt% in ice at 273K to 0.5 wt% in the liquid H2O at 275K

GCMC calculations based on a very accurate QM force field. Expect loading curve to be correct

Potential problems. Ice structure at surface may impeded H2O loading, requiring higher loading pressure.

Time scale for forming ice from the liquid may be slow, could require a see or high H2 pressure


Homework

• Use the F3C force field to estimate the entropy of liquid water at room temperature

• Estimate the change of entropy of liquid water from just above the freezing to just below the boiling point. Compare to experiment (NIST Webbook)


Resources

• You can find 5 equilibrated water boxes in /ul/mario/water/F3C/0.39697Hq/ced. Choose any of the bgf files.

• In this directory you will also find the force field to be used with the script to create the input file for LAMMPS– Force Field: F3C_H+-ewald.par

• Use Tod’s link /ul/tpascal/scripts/createLammpsInput.pl to convert the bgf to the LAMMPS input file

• Use the serial lammps code to get the velocities of all atoms– ~tpascal/programs/bin/lmp_serial < in.myrun

• Run 25 ps saving the trajectory each 4 fs for determining the velocity autocorrelation function

• /ul/tpascal/vac/vac_linux


• TIP4P water model (w/ SHAKE constraints)• Initial Minimization

– 100 steps SD with solute restrained by 500 kcal/mol harmonic spring

– 500 steps CG (harmonic restraints removed)• System heated from 0K -> 330 K

– Langevin thermostat – NVE ensemble– Ramped by 30K increments over 10 ps– 2.0 fs timestep

• 2 ns NPT dynamics for correct density– Nose-Hoover thermostat – Elevated temperature (330 K) for accessing more

energy states – Anderson barostat (1 atm)– “Best” snapshot: snapshot with closest volume to

average volume during last 200 ps• 2ns ps NVT dynamics at 300K (STP)• Nose-Hoover thermostat (300 K) for 100 ps NVT

dynamics– Velocities and coordinates dumped every 4 fs– 1.0 fs timestep– SHAKE constraints removed

• 2PT Analysis for system thermodynamics– 5 consecutive 20 ps window analysis of system

during NVT for thermodynamics averaging– System partitioned/group thermodynamics computed

• Protein• Chitobiose• Membrane• 1st shell water molecules (within 3.6Å of protein

surface)• Bulk waters/ions

2ns NVT Dynamics (300K)

100 ps NVT Dynamics

2PT Analysis of 10 segments

Minimization: Solute Fixed

Minimization: Solute Movable

NVE Dynamics: Heat System

Equilibration Procedure

select best snapshot

2ns NPT Dynamics

MD Simulation Flowchart for POPC and graphene


H2 complexes binds into the ice structure

Experimental evidence for H2 storage in Ice

No gases

Added H2

Added He

Ice Ih

16 C increase in melting temperature of Ice with

2.2 kbar H2

Documents

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L08 Ch121a Atomic Level Simulations of Materials and Molecules William A. Goddard