Ch121a Atomic Level Simulations of Materials
and Molecules
Room BI 115
Lecture: Monday, Wednesday Friday 2-3pm
Lecture 6 and 7, April 16 and 21, 2013
MD3: vibrations
Lecture 6 Presented by Jason Crowley
William A. Goddard III, wag@wag.caltech.edu
Charles and Mary Ferkel Professor of Chemistry,
Materials Science, and Applied Physics,
California Institute of Technology
TA’s Jason Crowley and Jialiu Wang
Ch120a-Goddard-L06
© copyright 2013-William A. Goddard III, all rights reserved
1
Homework and Research Project
First 5 weeks: The homework each week uses generally available
computer software implementing the basic methods on
applications aimed at exposing the students to understanding how
to use atomistic simulations to solve problems.
Each calculation requires making decisions on the specific
approaches and parameters relevant and how to analyze the
results.
Midterm: each student submits proposal for a project using the
methods of Ch121a to solve a research problem that can be
completed in the final 5 weeks.
The homework for the last 5 weeks is to turn in a one page report
on progress with the project
The final is a research report describing the calculations and
conclusions
Ch120a-Goddard-L06
© copyright 2013-William A. Goddard III, all rights reserved
2
Outline of today’s lecture
• Vibration of molecules
– Classical and quantum harmonic oscillators
– Internal vibrations and normal modes
– Rotations and selection rules
• Experimentally probing the vibrations
– Dipoles and polarizabilities
– IR and Raman spectra
– Selection rules
• Thermodynamics of molecules
– Definition of functions
– Relationship to normal modes
– Deviations from ideal classical behavior
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
3
Simple vibrations
• Starting with an atom inside a molecule at equilibrium,
we can expand its potential energy as a power series.
The second order term gives the local spring constant
• We conceptualize molecular vibrations as coupled
quantum mechanical harmonic oscillators (which have
constant differences between energy levels)
• Including Anharmonicity in the interactions, the energy
levels become closer with higher energy
• Some (but not all) of the vibrational modes of molecules
interact with or emit photons This provides a
spectroscopic fingerprint to characterize the molecule
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
4
Vibration in one dimension – Harmonic Oscillator
Consider a one dimensional spring with equilibrium length xe
which is fixed at one end with a mass M at the other.
If we extend the spring to some new distance x and let go, it will
oscillate with some frequency, w, which is related to the M and
spring constant k.
To determine the relation we solve Newton’s equation 
M (d2x/dt2) = F = -k (x-xe)
Assume x-x0=d = A cos(wt) then
E= ½ k d2
–Mw2 Acos(wt) = -k A cos(wt)
Hence –Mw2 = -k or w = Sqrt(k/M).
Stiffer force constant k  higher w and
higher M  lower w
Ch121a-Goddard-L07
No friction
© copyright 2012 William A. Goddard III, all rights reserved
5
Reduced Mass
Put M1 at R1 and M2 at R2
CM = Center of mass
Fix Rcm = (M1R1 + M2R2)/(M1+ M2) = 0
Relative coordinate R=(R2-R1)
M1
Then Pcm = (M1+ M2)*Vcm = 0
And P2 = - P1
Thus KE = ½ P12/M1 + ½ P22/M2 = ½ P12/m
Where 1/m = (1/M1 + 1/M2) or m = M1M2/(M1+ M2)
Is the reduced mass.
Get w = sqrt(k/m).
Thus we can treat the diatomic molecule as a simple
mass on a spring but with a reduced mass, m
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
M2
6
For molecules the energy is harmonic near equilibrium
but for large distortions the bond can break.
The simplest case is the Morse Potential:
V ( x) = hcDe (1  e
 ax 2
)
k 1/ 2
a=(
)
2hcDe
Exact solution
En = (n  1 / 2)w  ((n
v +1½)
/ 2)22 w e
a 2  Successive vibrational
w e =E = (n levels
2
are
closer
by
1
/
2
)

w

(
v

1
/
2
)
w e
n2 m
2
a
 are more complex; in general:
Realw
potentials
e =
2 2
2
m
2
E
=
(
n

1
/
2
)

w

(
v

1
/
2
w
(
n+1/½)
12/)23)wd
wd
 ....
En =n (n  1 / 2)w  ((n
v +1½)
/ 2) )w
(n(n
e 
e ....
e 
e 
(Philip
Morse a professor
at MIT, did not manufacture cigarettes)
Ch121a-Goddard-L07
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7
Vibration for a molecule with N particles
There are 3N degrees of freedom (dof) which we collect together
into the 3N vector, Rk where k=1,2..3N
The interactions then lead to 3N net forces,
Fk = -(∂E(Rnew)/∂Rk) all of which are zero at equilibrium, R0
Now consider that every particle is moved a small amount leading
to a 3N distortion vector, (dR)m = Rnew – R0
Expanding the force in a Taylor’s series leads to
Fk = -(∂E(Rnew)/∂Rk) = -(∂E/∂Rk)0 - Sm (∂2E/∂Rk∂Rm) (dR)m
Where we have neglected terms of order d2.
Writing the 2nd derivatives as a matrix (the Hessian)
Hkm = (∂2E/∂Rk∂Rm) and setting (∂E/∂Rk)0 = 0, we get
Newton’s equation
Fk = - Sm Hkm (dR)m = Mk (∂2Rk/∂t2)
To find the normal modes we write (dR)m = Am cos wt leading to
Mk(∂2Rk/∂t2) = Mk w2 (Ak cos wt) = Sm Hkm (Amcos wt)
2A - S H
Here
the coefficient of© cos
wt2012
must
be
{Mk wIII,
Ch121a-Goddard-L07
copyright
William
A. Goddard
allkrights reserved
m
km Am}=0
8
Solving for the Vibrational modes
The normal modes satisfy
{Mk w2 Ak - Sm Hkm Am}=0
To solve this we mass weight the coordinates as Bk = sqrt(Mk)Ak
leading to
Sqrt(Mk) w2 Bk - Sm Hkm [1/sqrt(Mm)]Bm}=0 leading to
Sm Gkm Bm = wk2 Bk
where Gkm = Hkm/sqrt(MkMm)
G is referred to as the reduced Hessian
For M degrees of freedom this has M eigenstates
Sm Gkm Bmp = dkp Bk (w2)p
where the M eigenvalues (w2)p are the squares of the vibrational
energies.
If the Hessian includes the 6 translation and rotation modes then
there will be 6 zero frequency modes, wp = 0
Ch121a-Goddard-L07
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9
Saddle points
If the point of interest were a saddle point rather than a
minimum, G would have one negative eigenvalue, (w2)p = - A2
where A is a positive number
This leads to an imaginary frequency, wp = iA ,
Saddle point
Ch121a-Goddard-L07
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10
For practical simulations
• We can obtain reasonably accurate vibrational modes from just
the classical harmonic oscillators, usually within a few %
• N atoms => 3N degrees of freedom
• However, there are 3 degrees for translation, n = 0
• 3 degrees for rotation for non-linear molecules, n = 0
• 2 degrees if linear
• The rest are vibrational modes
Ch121a-Goddard-L07
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11
Normal Modes of Vibration H2O
H2O
Sym. stretch
Bend
D2O
3657 cm-1
2671 cm-1
Ratio: 0.730
1595 cm-1
1178 cm-1
Ratio: 0.735
3756 cm-1
2788 cm-1
Isotope effect: n ~ sqrt(k/M): Ratio: 0.742
Antisym. stretch
Simple nD/nH ~ 1/sqrt(2) = 0.707:
More accurately, reduced masses
Most accurately
mOH = MHMO/(MH+MO)
MH=1.007825
mOD = MDMO/(MD+MO)
MD=2.0141
Ratio = sqrt[MD(MH+MO)/MH(MD+MO)]
MO=15.99492
~ sqrt(2*17/1*18)
= 0.728
Ch121a-Goddard-L07
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reserved= 0.728 12
The Infrared (IR) Spectrum
Characteristic vibrational modes
•EM energy absorbed by
interatomic bonds in organic
compounds
•frequencies between 4000
and 400 cm-1 (wavenumbers)
•Useful for resolving molecular
vibrations
13
http://webbook.nist.gov/chemistry/
Ch121a-Goddard-L07
13
© copyright 2012 William A. Goddard
III, all rights reserved
http://wwwchem.csustan.edu/Tutorials/INFRARED.HTM
Normal Modes of Vibration CH4
4 independent CH bonds  4 CH stretch
modes, by symmetry one is triply degenerate
6 possible angle terms HCH  5 HCH modes,
one doubly degenate, on triply deg.
Reason only 5 linearly independent HCH
3
2
3
1
Sym. stretch
A1
CH4
CD4
Anti. stretch
T2
Sym. bend
E
Sym. bend
T2
2917
cm-1
3019 cm-1
1534 cm-1
1306 cm-1
1178
cm-1
2259 cm-1
1092 cm-1
996 cm-1
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
14
Fitting force fields to Vibrational
frequencies and force constants
Hessian-Biased Force Fields from Combining
Theory and Experiment; S. Dasgupta and W.
A. Goddard III; J. Chem. Phys. 90, 7207 (1989)
MC: Morse bond stretch and cosine angle bend
MCX: include 1 center cross terms
CH sym str
CO stretch
CH2 scis
H2CO
CH2 rock
CH asym str
CH2 wag
4 atoms
 12-6=6
vibrations
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
15
1/ 4  y / 2

=
(
)
e
0


The QM Harmonic Oscillator
 1/ 4

y2 / 2
 1 = ( )  2 =ye( )1/ 4 2 ye  y / 2
1

 H=e
The Schrödinger equation
 oscillator
1/ 4 1
2
 y2 / 2
for harmonic

1
 2 = 2( )2  =((2 y )1/ 4 1)e (2 y 2  1)e  y / 2
2
 
1
2kx2
2

H = 
2
2mx1/ 4 12
3 1 / 4 1  y 2 /32

 3 = ( )1 13 =(2( y )  3 y )e(2 y  3 y )e  y / 2
energy
e n =e(nn= (n)
w3)w n = 0n,31=,20, 
,1,2, 
2 2
mw
wavefunctions
 =1/ 4  y / 2  = mw y =  x y =  x
 0 = ( ) e

0
2
2
2
2

 1/ 4
1 = ( )
2 ye  y / 2


1
 2 = ( )1/ 4
(2 y 2  1)e  y / 2

2
 1/ 4 1
3 = ( )
( 2 y 3  3 y )e  y / 2

3
Gaussian
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
reference
mhttp://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1
w
2
2
2
16
Raman and InfraRed spectroscopy
• IR
– Vibrations at same frequency as radiation
– To be observable, there must be a finite dipole derivative
– Thus homonuclear diatomic molecule (O2 , N2 ,etc.) does not
lead to IR absorption or emission.
• Raman spectroscopy is complementary to IR
spectroscopy.
– radiation at some frequency, n, is scattered by the molecule to
frequency, n’, shifted observed frequency shifts are related to
vibrational modes in the molecule
• IR and Raman have symmetry based selection rules
that specify active or inactive modes
Ch121a-Goddard-L07
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17
IR and Raman selection rules for vibrations
The electrical dipole moment is responsible for IR
m (t ) =   (r, t )rd 3r
The intensity is proportional to
dm/dR averaged over the
vibrational state
The polarizability is responsible for Raman
m (t ) =  (t )e (t )
where e is the external electric field
at frequency n
For both, we consider transition matrix elements of the form
 ' | m (t ) | 
 =  i ,n (Qi )
i
Ch121a-Goddard-L07
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18
IR selection rules, continued
• For IR, we expand dipole moment
m = m 0  (
i
m
) 0 Qi  ....
Qi
We see that the transition elements are
m
(
) 0  ni ' | Qi | ni 
Qi
The dipole changes during the vibration
Can show that n can only change 1 level at a
time
Ch121a-Goddard-L07
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19
Raman selection rules
• For Raman, we expand polarizability

 = 0  (
) 0 Qi  ....
Qi
i
substitute the dipole expression for the induced dipole
 
( ( ) 0  e)=0(t)e n(ti )' |Q
nii| ni 
nii ' || Q
Qi Qi
Same rules except now it’s the polarizability that has to
change
For both Raman and IR, our expansion of the dipole
and alpha shows higher order effects possible
Ch121a-Goddard-L07
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20
Translation and Rotation Modes
•center of mass translation
Dx= Dx Dy=0
Dz=0 E is a constant dE/dx = 0 d2E/dx2 = 0
Dx=0
Dy=Dy
Dz=0 Thus the eigenmode l=0
Dx=0
Dy=0
Dz=Dz
•center of mass rotation (nonlinear molecules)
E is a constant dE/dx = 0
Dx=0
Dy=-cDqx Dz=bDqx
Dx= cDqy
Dy=0
Dz=-aDqy d2E/dx2 = 0
Thus the eigenmode l=0
Dx= -bDqz Dy=aDqx
Dz=0
•linear molecules have only 2 rotational degrees of freedom
•The translational and rotational degrees of freedom can be
removed beforehand by using internal coordinates or by
transforming to a new coordinate system in which these 6 modes
are separated out
21
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
21
Classical Rotations
• The moment of inertia about an axis q is defined as
I qq =  mk xk2 (q)
xk(q) is the perpendicular distance to the axis q
k
Can also define a moment of inertia tensor where (just replace the mass
density with point masses and the integral with a summation.
Diagonalization of this matrix gives the principle moments of inertia!
r m
=
m
k
R0
k
k
k
the rotational energy has the form
Erot
k
2
J
1
q
=  I qqwq2 (q ) = 
2 q
q 2 I qq
J q = I qqwq
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
22
Quantum Rotations
The rotational Hamiltonian has no
associated potential energy
J y2
J x2
J z2
H=


2 I xx 2 I yy 2 I zz
For symmetric rotors, two of the
moments of inertia are equivalent,
combine:
J2
1
1
H=
(

)J z
2I 
2I 2I 
Eigenfunctions are spherical
harmonic functions YJ,K or Zlm with
eigenvalues
J ( J  1) 2
1
1
Erot ( J , K J , M J ) =
(

) K 2 2
2I 
2I 2I 
J = 0,1,2,...
K J = J , J  1,..., J
M
M J =Ch121a-Goddard-L07
J , J  1,..., J
© copyright 2012 William A. Goddard III, all rights reserved
23
Transition rules for rotations
• For rotations
– Wavefunctions are spherical harmonics
– Project the dipole and polarizability due to rotation
• It can be shown that for IR
– Delta J changes by +/- 1
– Delta MJ changes by 0 or +/-1
– Delta K does not change
• For Raman
– Delta J could be 1 or 2
– Delta K = 0
– But for K=0, delta J cannot be +/- 1
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
24
Raman scattering
• Phonons are the normal modes of lattice vibrations
(thermal + zero point energy)
• When a photon absorbs/emits a single phonon,
momentum and energy conservation  the photon
gains/loses the energy and the crystal momentum of the
phonon.
– q ~ q` => K = 0
– The process is called anti-Stokes for absorption and
Stokes for emission.
– Alternatively, one could look at the process as a
Doppler shift in the incident photon caused by a first
order Bragg reflection off the phonon with group
velocity v = (ω/ k)*k
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
25
Raman selection rules
• For Raman, we expand polarizability

 = 0  (
) 0 Qi  ....
Qi
i
substitute the dipole expression for the induced dipole
 
( ( ) 0  e)=0(t)e n(ti )' |Q
nii| ni 
nii ' || Q
Qi Qi
Same rules except now it’s the polarizability that has to
change
For both Raman and IR, our expansion of the dipole
and alpha shows higher order effects possible
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
26
Another simple way of looking at Raman
Take our earlier expression for the time dependent dipole and
expose it to an ideal monochromatic light (electric field)
m (t ) =  (t )e (t ) = 2 (t )e 0 cos(wt )


1
2


m (t ) = 2  0  D cos(wintt )e 0 cos(wt )
1
m (t ) = 2e 0 cos(wt )  De 0 cos(wt  wintt )  cos(wt  wintt )
2
We get the Stokes lines when we add the frequency and the antiStokes when we substract
The peak of the incident light is called the Rayleigh line
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
27
Skip The Sorption lineshape - 1


•The external EM field is monochromatic E (t ) = E0 ε cos(ωt )
  n n 
  N Ni i i
Total
Dipole
Dipole
μ =μ
=
μ i μ i Molecular
Molecular
Dipole
Dipoleμ i μ
=i 
=
rj qrjj q
•Dipole moment of the system Total


•Interaction between the field and the molecules  (t ) = μ  E (t )
i =1 i =1
j =1 j =1
•Probability for a transition from the state i to the state f (the
Golden Rule)
πE
Pi  f (ω) = 0
2
2
  2
f | ε  μ | i [δ(ω fi  ω)  δ(ω fi  ω)]
ω fi = ω f  ω f
•Rate of energy loss from the radiation to the system
 E rad (ω) =  ρ i ω fi Pi  f (ω)
i
f
•The flux of the incident radiation
c: speed of light
cn 2
S=
E0
n: index of refraction of the medium
28
8π
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
28
Skip The Sorption lineshape - II
 E rad (ω)
α(ω) =
S
Beer-Lambert law Log(P/P0)=bc
•Define absorption linshape I(w) as
•Absorption cross section (w)
I (ω) =
3cnα(ω)
  2
=
3
ρ
f
|
ε
 μ | i δ(ω fi  ω)

i
2
βω
4π ω(1  e )
i
f
•It is more convenient to express I(w) in the time domain
1  iωt
I(w) is just the Fourier transform of the
δ(ω) =
e
dt
2π 
autocorrelation function of the dipole moment
3
 
I (ω) =
ρi i | ε  μ | f

2π i f
 
f | ε μ | i



e
i[
E f  Ei

-ω]t
dt
ensemble average
=
1  

iωt
29
=
μ
(
0
)

μ
(
t
)
e
dt



© copyright 2012 William A. Goddard III, all rights reserved
2πCh121a-Goddard-L07
29
Non idealities and surprising behavior
• Anharmonicity – bonds do
eventually dissociate
• Coriolis forces
– Interaction between vibration
and rotation
• Inversion doubling
• Identical atoms on rotation
– need to obey the Pauli
Principle
N
E V
N

=




– Total wavefunction symmetric
for Boson and antisymmetric YJ , M (q   ,  ) = (1) J YJ , M (q ,  )
for Fermion
Ch121a-Goddard-L07
J
© copyright 2012 William A. Goddard III, all rights reserved
J
30
Electromagnetic Spectrum
How does a Molecule response to an
oscillating external electric field (of
frequency w)? Absorption of
radiation via exciting to a higher
energy state ħw ~ (Ef - Ei)
Ch121a-Goddard-L07
© copyright 2012 William A. Goddard III, all rights reserved
Figure taken from Streitwiser & Heathcock, Introduction to
Organic Chemistry, Chapter 14, 1976
31
31
Using the vibrational modes: thermodynamics
In QM and MM the Energy at minima = motionless state at 0K
BUT, experiments are made at finite T, hence corrections are
required to allow for rotational, translational and vibrational
motion.
The internal energy of the system:
U(T)=Urot(T)+Utran(T)+Uvib(T)+Uvib(0)
From equipartition theorem: Urot(T) = (3/2)KBT , Utran(T) = (3/2)KBT per
molecule (except Urot(T)=KBT for linear molecules)
BUT, vibrational energy levels are often only partially excited at room T, thus
Uvib(T) requires knowledge of vibrational frequencies
Uvib(T) = vibrational enthalpy @ T - vibrational enthalpy @ 0K
Vibrational frequencies can be
N mo d

used to calculate entropies and
hn i
hn i
i


U
T
=

vib
free energies, or to compare
 2 exp hn K T 1  n i =
i
B
2

with results of spectroscopic
i=1 
experiments
32-AJB
The vibrational frequencies  i) of the normal modes (Nmod) calculated from the
eigenvalues  i) of the ©
force-constant
of Hessian
of second derivatives32
Ch121a-Goddard-L07
copyright 2012 equivalent
William A. Goddard
III, allmatrix
rights reserved
) 


Thermodynamics
Describe a system in terms of Hamiltonian H(p,q) where p is
generalized momentum and q is generalized coordinate
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,
Here Q, the Partition function, is a normalization constant
Q = S exp[-H(p,q)/kBT] summed over all states of the system
Ch121a-Goddard-L08
© copyright 2012 William A. Goddard III, all rights reserved
33
Thermodynamic functions can all be derived from Q
 ln Q
Energy
E = kT (
) N ,V
T
 ln Q
Entropy
S = k ln Q  kT (
) N ,V
T
Helmholtz Free Energy A =  kT ln Q
2
Chemical Potential
Pressure
Heat Capacity
Ch121a-Goddard-L08
 ln Q
)V ,T
N
 ln Q
p = kT (
) N ,T
V
2
 ln Q

ln Q
CV = 2kT (
) N ,V  kT 2 (
) N ,V
2
T
T
m = kT (
© copyright 2012 William A. Goddard III, all rights reserved
34
The partition function for translation
Assume a cubic periodic box of side L
2  2
The QM Hamiltonian is H = 
2m x 2
The QM eigenfunctions are just periodic functions for x, y, and z
directions, sin(nxxp/L) etc
Leading to
en
2
x
h 2 nx
=
8mL2
nx = 1,2, 
Thus the partition function for translation becomes

qtrans (V , T ) =  e
n x =1
Ch121a-Goddard-L08
βεn x

e
n y =1
βεn y

e
n y =1
βεn y

= ( e
0
βh2 n 2

8 mL2
2πmkT 3 / 2
dn) = (
) V
2
h
3
© copyright 2012 William A. Goddard III, all rights reserved
35
Thermodynamic functions for translation
2 πMkT 3 / 2
Q= (
) V
2
h
 ln Q
 ln
Q

ln
Q
2
equipartition
E= =
kT( ( E) N=),VNkT
 ln )QNkT
E
kT
Energy
,V ( =
2 (3/2)
,
V
Energy
TT E = kT (TT ) N ,V
3/ 2


2
π
MkT
V


5
/
2

ln
Q
(ln Q )
Q
= klnln
e 


)Q
S
=
k
ln
Q

kT
ln
2
S
=
k
ln
Q

kT
(
)
Entropy
S
=
k
ln
Q

kT
(
N
,
V
Entropy
ST
=Tk lnN ,VQ  kT (T N ,)VN ,h
 N

V
T
3/ 2
Free
Energy
A
=

kT
ln
Q

2πMkT  V 
ee
Energy
A
=

kT
ln
Q
Helmholtz
Free
Energy
A
=

kT
ln
Q

Helmholtz Free Energy A =  kT ln Q = -kT ln 
e

2
2 2

Chemical Potential
Pressure
Heat Capacity
Ch121a-Goddard-L08
h

N 
 ln Q
)V ,T
N
 ln Q
Ideal gas
p = kT (
) N ,T = NkT
V
V
2
 ln Q

ln Q
CV = 2kT (
) N ,V  kT 2 (
) N ,V = (3/2) k
2
T
T
m = kT (
© copyright 2012 William A. Goddard III, all rights reserved
36
The partition function for rotation
2
H=
2I
 1  

1 2 
 sin θ   2

2
θ  sin   
 sin θ θ 
This leads to energy levels of
This is 2 the
Laplacian
I = moment of inertia
J ( J  1) 2
eJ =
w J = (2 J  1) 2 J = 0,1,2,
2I
Thus the partition function becomes
βJ ( J 1) 2

2
2I
1 
qrot (T ) =  (2 J  1) e
σ 0
π1/2 8π2 I AkT 1/ 2 8π 2 I B kT 1/ 2 8π 2 I C kT 1/ 2
dJ =
(
) (
) (
)
2
2
2
σ
h
h
h
h2
rotational temperatu re  A = 2
8π I A k
ω
hv
vibrationa
l temperatu re
v =2012 William
= A. Goddard III, all rights reserved
Ch121a-Goddard-L08
© copyright
37
Thermodynamic functions for rotation (non linear)
π1/2
T3
1/ 2
(
)
Q=
σ  A B C
 ln
Q
2 ln
Q
2
2  ln Q
E
=
kT
(
)
 ln )QNkT
equipartition
E
=
kT
(
)
Energy
E
=
kT
N
,V ( =
2 (3/2)
N
,
V
,
V
Energy
TT E = kT (TT ) N ,V
 ln
Q
 π1/2e3 / 2
T3

ln
Q

ln
Q
1/ 2 
S
=
k
ln
Q

kT
(
)
ln)Q
(
) 
 ln
S
=
k
ln
Q

kT
(
)
Entropy
S
=
k
ln
Q

N ,kT
V (= k

N
,
V
N
,
V
Entropy
ST
=Tk ln Q  kT (T  ) Nσ,V  A B C 
T
1/2
3
Free
Energy
A
=

kT
ln
Q
ee
Energy
A
=

kT
ln
Q
Helmholtz
Free
Energy
A
=

kT
ln
Q

π
T
1/ 2 
Helmholtz Free Energy A =  kT ln Q = -kT ln 
(
) 
 σ
Chemical Potential
Pressure
Heat Capacity
Ch121a-Goddard-L08
 A B C

 ln Q
)V ,T
N
 ln Q
p = kT (
) N ,T = 0
equipartition
V
2
 ln Q

ln Q
CV = 2kT (
) N ,V  kT 2 (
) N ,V = (3/2) k
2
T
T
m = kT (
© copyright 2012 William A. Goddard III, all rights reserved
38
The partition function for vibrations
An isolated harmonic oscillator with vibrational frequency ω
Has a spectrum of energies
1
e n = (n  )w
2
n = 0,1,2, 
h2
rotational temperatu re  A = 2
Substituting into the Boltzmann expression leads to 8π I A k
 βω/2

e

n
q==
βω
 j =1 1  e
ω hv
vibrationa l temperatu re v =
=
k
k
Summing over all normal modes leads to
  βe n   e βω/2
=   e  = 
βω
j =1  n =0
 j =1 1  e

qvib
Ch121a-Goddard-L08
© copyright 2012 William A. Goddard III, all rights reserved
39
Thermodynamic functions for vibration
(harmonic oscillator)
  v j /2T
3n 6
e
Q=
 1 e
j =1
  v j /T
 v /T 
 v

ln
Q

ln
Q
2 2  ln Q
2


2

=
(3/2)
kT

Energy E =E kT
E
=
kT
(
)
(
)

/
T

= kT (
N ,)
V N ,V
lnTQ
ln QNN ,,VV j =1  2T e

1
2
2 

T

E
T =E kT
Energy
= kT( (
Energy
) N ,)VN ,V
TT  ln Q3n 6  v /T
 ln Q
ln
Q
  /T 

kT
(
)
Entropy S =S k=lnk Q
S
=
k
ln
Q

kT
(
)
 ln( 1  e
) 
ln Q  kT (
N ,)
V N ,V
N,,V
V  /T
N

=
k

ln
Q

ln
Q

kTQ
Entropy
=ln
ln QkT
 kT
Entropy
S =SkT
( (T j)=1N,)VeN ,V  1

TT 3n 6 
ee
Energy
=
kT
ln Q
Helmholtz
A =  kT ln Q
Free
EnergyAFree
A =Energy
 kT
ln Q
 v
  /T 
 lnAQ=
ln
 ln( 1  e
) 
HelmholtzFree
Energy
A) =kT
 kT
ln) Q = -kT  
Helmholtz
Energy
lnQQ
Potential Potential
mFree
= kT ( m = kT
Chemical
(
2T
V ,T
V ,T
j =1
3n 6
j
j
vj
vj
j
vj
vj
j

N
N
 ln Q
 ln Q
p = kT ( p = kT
) N ,T(
Pressure
) N ,T = 0
V
V
3 n  6 
 ln Q
 ln Q2  2 ln Q 2  2 ln
Q vj
)
acity
) N ,V(  kT )(N ,V  2kT) N(,=V k 
Heat Capacity CV = 2kT (CV = 2kT
2  N ,V
T
T
T
Tj =1  T
Ch121a-Goddard-L08

 v j /T

 e /T
 (e v j  1) 2

40
© copyright 2012 William A. Goddard III, all rights reserved
2
qelectronic = we1e
βDe
qnuclear = 1
Q=
we1e
De / kT
   we1e
βDe
Thermodynamic functions for
electronic states
we will assume qelect=1
Assuming the reference
 ln Q
2  ln Q
ln
Q
2
2
= - kT) ND,Ve
Energy E =E kT
(
= kT( (T E) N=,)VkT
N ,V 2 
lnTQ
ln QN ,V state has free atoms
2
E
T =E kT
Energy
= kT( (
Energy
) N ,)VN ,V
TT  ln Q
 ln Q
ln
Q
( =(k ln )QN ,
Entropy S =S k=lnk Q
)QNNw
V kT ( = k ln
ln QkT
 SkT
)
V
,,V
N
,
V

ln

ln
Q

T

T
kTQ
Entropy
ln QkT
 kT
Entropy
S =S k=ln
( (
) N ,)VNe,1V
TT
ee
Energy
=
kT
ln Q
Helmholtz
A =  kT ln Q
Free
EnergyAFree
A =Energy
 kT
ln Q
D
e
 lnAQ=
 kT
ln
QQ
Helmholtz
Free
Energy
A
=

ln
Q
Helmholtz
Free
Energy

kT
ln

 ln we1
=
-De
kT
Potential Potential
m = kT ( m = kT
)V ,T(
Chemical
)V ,T
2
kT
N
N
 ln Q
 ln Q
p = kT ( p = kT
) N ,T(
Pressure
) N ,T = 0
V
V
 ln Q
 ln Q2  2 ln Q 2  2 ln Q
acity
) N ,V(  kT )(N ,V  2kT) N(,V = 20 ) N ,V
Heat Capacity CV = 2kT (CV = 2kT
T
T
T
T
Ch121a-Goddard-L08
© copyright 2012 William A. Goddard III, all rights reserved
41
Thermodynamic Properties for a Crystal

Write partition function of the system
ln Q =  dS ( ) ln q HO ( )
0
As a continuous superposition of oscillators
Harmonic oscillator Partition function
Q
qHO
( ) =  exp( βe n ) =
n
exp( βh/2)
where e n = (n  1 )h
1 - exp( βh/2)
2
Thermodynamic properties
Weighting functions
  ln Q 
βh
βh
1
E = V0  Tβ 1 
WEQ ( ) =

 = V0  β  dS ( )WE ( )
2
exp( βh )  1
 T  N ,V
0
Reference energy
Zero point energy

  ln Q 
βh
S = k ln Q  β 1 
= k B  dS ( )WS ( )

WSQ ( ) =
 ln[ 1  exp( βh )]
 T  N ,V
exp(
β
h

)

1
0

1
A = V0  β ln Q = V0  β
1

 dS ( )WA ( )
WAQ ( ) = ln
0

 E 
Cv =   = k B  dS ( )WCv ( )
 T  N ,V
0
Ch121a-Goddard-L08
1  exp( βh )
exp( βh/2)
(βh ) 2 exp( βh )
W ( ) =
[1  exp( βh )]242
Q
Cv
© copyright 2012 William A. Goddard III, all rights reserved
Where do we get the vibrational density of states
DoS(n)?
Experimentally from Inelastic
neutron scattering
Can use to calculate
thermodynamic properties
Compare to phonon dispersion
curves. Peak is for phonons
with little dispersion
“Phonon Densities of States and
Related Thermodynamic
Properties of High Temperature
Ceramics” C.-K Loong,
J.European Ceramic Society, 1998
Ch121a-Goddard-L08
© copyright 2012 William A. Goddard III, all rights reserved
43
Jason stopped on April 16, 2014
Ch121a-Goddard-L08
© copyright 2012 William A. Goddard III, all rights reserved
44
TOD A. PASCAL MSC, CALTECH
Tod A Pascal
Ravi Abrol
William A Goddard III
MSC 2013 Research Conference
TOD A. PASCAL MSC, CALTECH


Overview of the 2PT method for calculating
accurate entropies and free energies
Application to the solvation thermodynamics
of Amino Acid sidechains
TOD A. PASCAL MSC, CALTECH
Rudolf Clausius originator of the
concept of "entropy".
“Any method involving the notion of entropy, the very
existence of which depends on the second law of
thermodynamics, will doubtless seem to many farfetched, and may repel beginners as obscure and
difficult of comprehension.
”
--Willard Gibbs, Graphical Methods in the
Thermodynamics of Fluids (1873)
DGt = Dt  DSt
TOD A. PASCAL MSC, CALTECH
•
Test Particle Methods (insertion or deletion)
• Good for low density systems
• Perturbation Methods
• Thermodynamic integration, Thermodynamic perturbation
• Applicable to most problems
• Require long simulations to maintain “reversibility”
•
Nonequilibrium Methods (Jarzinski’s equality)
• Obtaining differential equilibrium properties from irreversible processes
• Require multiple samplings to ensure good statistics
•
Normal mode Methods
• Good for gas and solids
• Fast
• Not applicable for liquids
References: Frenkel, D.; Smit, B. Understanding Molecular Simulation from Algorithms to Applications. Academic press: Ed., New
McQuarrie, A. A. Statistical Mechanics. Harper & Row: Ed., New York, 1976.
Jarzynski, C. Nonequilibrium Equality for Free Energy Differences. Phys. Rev. Lett. 1997, 78, 2690.
Remaining issue: experimental energies are free
energies, need to calculate entropy
General approach to predict Entropy, S, and Free Energy
Free Energy, F = U – TS = − kBT ln Q(N,V,T)
J. G. Kirkwood. Statistical
Kirkwood thermodynamic integration
mechanics of fluid mixtures, J.
Chem. Phys., 3:300-313,1935
Enormous computational cost required for complete sampling of
the thermally relevant configurations of the system often makes
this impractical for realistic systems. Additional complexities,
choice of the appropriate approximation formalism or somewhat 49
ad-hoc parameterization of the “reaction coordinate”
Solvation free energies amino acid side chains
Pande and Shirts (JCP 122 134508 (2005) Thermodynamic
integration leads to accurate differential free energies
50
Solvation free energies amino acid side chains
Pande and Shirts (JCP 122 134508 (2005) Thermodynamic
integration leads to accurate differential free energies
But costs 8.4 CPU-years on 2.8 GHz processor
51
Starting point: Get Density of states from MD (Velocity
autocorrelation) Partition Function entropy
Velocity autocorrelation function
DoS(n) is the vibrational density of States
DoS(n)
Calculate entropy from DoS(n)
zero
zero
zero
Problem: as n  0 get DoS ∞ unless DoS(0) = 0
52
Problem with Liquids: S(0)≠0
S ( )
Finite density of states at n =0
Proportional to diffusion coefficie
DoS(n)
where D is the diffusion
coefficient
n
N=number of particles

M = mass
Also strong anharmonicity at low frequencies zero
zero
zero
53
Two-Phase Thermodynamics Model (2PT)
•Decompose liquid DoS(n) to a gas and a solid contribution
•DoS(n) total = DoS(n) gas + DoS(n) solid
•S(0) attributed to gas phase diffusion, fit to hard sphere theory
•Gas component contains small n anharmonic effects
•Solid component contains quantum effects
Total
Gas
solid-like
gas-like
S ( )

=

PropertyP==  dS ( )W
s
0
exponential
decay
S ( )

HO
P

( )   dS ( )W ( )
g
g
P
Solid
Debye crystal
S(v) ~v2
S ( )

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;
54
JCP, 119:11792(2003) wag536
TOD A. PASCAL MSC, CALTECH

The basic idea
S ( ) = S gas ( )  S solid ( )
 The DoS

 Thermodynamics


P =  dS ( )W ( )   dS s ( )WPHO ( )
g
g
P
0
The gas component
0
 VAC for hard sphere gas
c HS (t ) =
3kT
exp(  t )
m
 (T ,  HS ,  HS ) : friction coefficien t
 DoS for hard sphere gas
S HS ( ) =
12 N 
=
2
2 2
  4 
gas
N gas = fN
s0
 s  
1  0 
 6 fN 
2
s0 = S
HS
( 0) =
12 N gas

Two unknowns (a and Ngas) or (s0 and f)
TOD A. PASCAL MSC, CALTECH

so (DoS of the gas component at v=0)
 completely remove S(0) of the fluid


s0 = S (0), S solid (0) = 0
f (gas component fraction)
 T -> ∞ or ρ -> 0: f -> 1 (all gas)
 ρ -> ∞: f-> 0 (all solid)
f =
D(T ,  )
D0HS (T ,  ;  HS )
D(T ,  ) =
D0HS (T ,  ; HS ) =
3 1
kT 1/ 2
(
)
2
HS
8 
m
kTs(0)
12mN
(Chapman - Enskog)
one unknown σHS
TOD A. PASCAL MSC, CALTECH



σHS (hard sphere radius for describing the gas
molecules)
gas component diffusivity should agree with
statistical mechanical predictions at the same
T and ρ
gas component diffusivity from MD
simulation
kTs
D HS (T , f ) =

0
12mfN
HS diffusivity from the Enskog theory
D HS (T , f ) = D0HS (T , f ; HS )
y=

6
4 fy
z ( fy)  1
 HS Hard sphere packing
3
fraction
1 y  y2  y3
z( y) =
(1  y ) 3
Compressibility
TOD A. PASCAL MSC, CALTECH
A universal equation for f

2D9 / 2 f 15 / 2  6D3 f 5  D3 / 2 y 7 / 2  6D3 / 2 f 5 / 2  2 f  2 = 0
normalized diffusivit y : D(T ,  , m, s0 ) =
2s0 kT 1/ 2 1/ 3 6 2 / 3
(
)  ( )
9N m

Graphical representation

1.0
f or f y
0.8
f
fy
0.6
0.4
0.2
0.0
1.E-05
1.E-03
1.E-01
1.E+01
D (normalized diffusivity)
1.E+03
TOD A. PASCAL MSC, CALTECH
Run a MD simulation
(trajectory information saved)
Calculate VAC
Calculate DoS (FFT of VAC)
Apply HO
approximation
To S()
1PT
thermodynamic
predictions
Calculate S(0) and D
Solve for f
Determine Sgas(), Ssolid()
Apply HO statistics
To Ssolid()
Apply HS statistics to
Sgas()
2PT
thermodynamic
predictions
TOD A. PASCAL MSC, CALTECH
• Intermolecular potential
Lennard-Jones Potential
 

V (r ) = 4e ( )12  ( ) 6 
r 
 r
V(r)
r=
0
-e
r
T -  diagram for Lennard Jones Fluid
• Phase diagram
●stable
– critical point
1.8
Tc* = 1.316  0.006
T  0.69
●unstable
1.4
c* = 0.304  0.006
– triple point
Gas
T*
Liquid
1.0
*
tp
(T*=kT/e *=3)
●metastable
Supercritical Fluid
Solid
0.6
0.0
0.4
*
0.8
1.2
TOD A. PASCAL MSC, CALTECH
Velocity Autocorrelation
Density of States

2
S ( υ) =
lim  C (t )e i 2 πυt dt
kT   
C(t ) = v(0)  v(t )
1600
100
800
C (t)
90
=
80
=
70
DoS S(cm)
1200
=
=
=
400
0
gas
liquid
solid
60
=
=
=
=
=
50
40
30
gas
liquid
solid
-400
-800
0
0.5
1
1.5
time (ps)
2
2.5
20
10
0
3
0
20
40
60
frequency v(cm-1)
80
100
TOD A. PASCAL MSC, CALTECH
• Examples
liquid
LJ gas
1200
30
1000
25
FCC solid
35
30
25
600
gas-like
solid-like
400
0
0
5
solid-like
gas-like
15
10
5
 [cm-1]
20
10
200
0
solid-like
gas-like
15
S ( ) [cm]
20
S ( ) [cm]
S ( ) [cm]
800
10
5
0
0
50
100
 [cm-1]
150
0
50
100
 [cm-1]
150
TOD A. PASCAL MSC, CALTECH
Total Energy
Pressure
3
18
T*=1.8
16
T*=1.4
14
T*=1.1
1
12
T*=0.9
0
MD
10
P*
-1
E*
8
-2
6
-3
4
-4
2
-5
0
-6
-2
-7
0
0.2
0.4
0.6
*
0.8
T*=1.8
T*=1.4
T*=1.1
T*=0.9
MD
2PT(Q)
2
1
1.2
0
0.2
0.4
0.6
*
Pressures and MD Energies agree with EOS values
Quantum Effect (ZPE) most significant for crystals (~2%)
0.8
1
1.2
TOD A. PASCAL MSC, CALTECH
2PT model
1PT
20
20
T*=1.8
T*=1.4
T*=1.1
T*=0.9
2PT(Q)
2PT(C)
T*=1.8
18
gas
16
T*=1.4
18
T*=1.1
16
T*=0.9
14
S*
14
1PT(Q)
S*
12
crystal
10
8
12
10
8
6
6
liquid
4
4
0
0.2
0.4
0.6
*
0.8
1
• Overestimate entropy for low
density gases
• Underestimate entropy for liquids
• Accurate for crystals
1.2
0
0.2
0.4
0.6
*
0.8
1
1.2
• Accurate for gas, liquid, and crystal
• Accurate in metastable regime
• Quantum Effects most important for
crystals (~1.5%)
TOD A. PASCAL MSC, CALTECH
1PT
2PT model
5
5
liquid
0
0
-5
G*
-5
-10
crystal
G*
-15
-10
-15
T*=1.8
-20
T*=1.4
T*=1.1
-20
-25
T*=0.9
1PT(Q)
-25
gas
-30
T*=1.8
T*=1.4
T*=1.1
T*=0.9
2PT(Q)
2PT(C)
-30
0
0.2
0.4
0.6
*
0.8
1
• Underestimate free energy for
low density gases
• overestimate entropy for liquids
• Accurate for crystals
1.2
0
0.2
0.4
0.6
*
0.8
1
1.2
• Accurate for gas, liquid, and crystal
• Accurate in metastable regime
TOD A. PASCAL MSC, CALTECH

P =  dS ( )W
s
1200

HO
P
( )   dS g ( )WPg ( )
0
HS fy = 0.036
0
HS fy = 0.309
4
gas
1000
WS( )
800
S( ) [cm]
5
600
3
2
400
QHO
1
200
CHO
0
0
0
2
4
6
 [cm-1]
8
10
0
50
100
150
 [cm ]
-1
30
liquid
25
• 1PT overestimates Wsgas for gas for
modes < 5 cm-1
S( ) [cm]
20
• 1PT underestimates Wsgas for liquid for
modes between 5 and 100 cm-1
15
10
5
• 2PT properly corrects these errors
0
0
50
100
 [cm-1]
150
TOD A. PASCAL MSC, CALTECH
15.5
14.5
• For gas, the entropy
13.5
2PT(Q)
2PT(C)
12.5
converges to within 0.2% with
MBWR EOS
11.5
S*
gas (*=0.05 T*=1.8)
10.5
• For liquid, the entropy
liquid (*=0.85 T*=0.9)
9.5
2500 MD steps (20 ps)
8.5
converges to within 1.5% with
7.5
2500 MD steps (20 ps).
6.5
100
1000
10000
MD steps
100000
1000000
TOD A. PASCAL MSC, CALTECH
• 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
metatstable supercritical
solid
unstable
fluid
Simulation conditions
2.0
supercritical
fluid
1.8
1.6
1.4
T*
1.2
1.0
0.8
solid
0.6
0.00
0.40
0.80
8
1.20
starting with
amorphous liquid

• 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
Entropy
7
6
S*
liquid (EOS)
5
solid (EOS)
4
starting with
fcc crystal
heating
cooling
classical
3
0.80
1.20
1.60
T*
2.00
TOD A. PASCAL MSC, CALTECH
16
S_hs(v)[cm]
S_s(v)[cm]
Stot(v)[cm]
14
12
16
S_hs(v)[cm]
S_s(v)[cm]
Stot(v)[cm]
14
12
10
10
8
w)
8
6
4
6
2
0
1
4
10
100
1000
10000
2
0
0
500
1000
1500
w (cm-1)
2000
2500
3000
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.
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.
TOD A. PASCAL MSC, CALTECH


Theory: 69.6 +/- 0.2 J/K*mol
Experimental Entropy: 69.9 J/K*mol (NIST)
Statistics collected over 20ps of dynamics , no
additional computation cost
TOD A. PASCAL MSC, CALTECH
temp (K)
ZPE_(kJ/mol/SimBox)=
Vo__(kJ/mol/SimBox)=
Eq__(kJ/mol/SimBox)=
Ec__(kJ/mol/SimBox)=
Sq_(J/mol_K/SimBox)=
Sc_(J/mol_K/SimBox)=
Aq__(kJ/mol/SimBox)=
Ac__(kJ/mol/SimBox)=
Cvq(J/mol_K/SimBox)=
Cvc(J/mol_K/SimBox)=
S(0)(cm/mol/SimBox)=
fluidicity___factor=
constant__________D=
POPC
303.40
3101.97
-19542.51
-16270.76
-18529.66
1211.99
-1275.28
-16638.48
-18142.74
1019.57
3338.36
0.05
0.19
0.10
Bulk
298.72
63.28
-436.24
-365.83
-414.84
68.08
23.47
-386.17
-421.85
37.84
71.62
0.02
0.26
0.19
Shell 1
298.39
65.21
-436.71
-364.49
-414.84
60.07
14.07
-382.42
-419.04
38.16
73.29
0.00
0.12
0.04
Shell 2
294.27
61.52
-436.07
-367.32
-414.84
74.77
31.00
-389.33
-423.97
38.81
72.12
0.01
0.22
0.13
Shell 3
298.28
63.32
-436.29
-365.85
-414.84
69.14
24.42
-386.47
-422.13
37.94
71.89
0.01
0.23
0.16
Exp
-378.23
69.90
37.27
0.13
TOD A. PASCAL MSC, CALTECH
Fourier Transform VAC
to get Density of
Vibrational States
Get Enthalpy and Free
Energy using quantum
100
partition
function
90
gas =
Do 25 picosec MD,
Extract Velocity
Autocorrelation (VAC)
Function
1600
=
=
=
=
800
C (t)
70
DoS S(cm)
1200
liquid
solid
80
=
400
60
=
=
=
=
50
40
30
20
0
10
gas
liquid
solid
-400
-800
0
0.5
1
1.5
time (ps)
2
2.5
0
0
20
40
60
80
frequency v(cm-1)
3
Works well for solids
100
TOD A. PASCAL MSC, CALTECH
16
S_hs(v)[cm]
S_s(v)[cm]
Stot(v)[cm]
14
12
16
S_hs(v)[cm]
S_s(v)[cm]
Stot(v)[cm]
14
12
10
10
8
w)
8
6
4
6
2
0
1
4
10
100
1000
10000
2
0
0
500
1000
1500
2000
w (cm-1)
2500
3000
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.
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.
• Theory: 69.6 +/- 0.2 J/K*mol
• Experimental Entropy: 69.9 J/K*mol (NIST)
Statistics collected over 20ps of MD , no additional cost
74
Absolute Entropies of common solvents
solvent
Expa AMBER 03
Dreiding
GAFF
OPLS AA/L
37.76
43.49 ± 1.04
32.58 ± 0.24
acetic acid
47.90 40.92 ± 0.18 40.99 ± 0.20 41.01 ± 0.29 43.58 ± 0.44
acetone
35.76 33.78 ± 0.24 30.20 ± 0.35 30.63 ± 0.52 31.17 ± 0.20
acetonitrile
41.41 38.20 ± 0.43 36.61 ± 0.48 36.63 ± 0.89 38.65 ± 0.39
benzene
46.99 36.91 ± 0.36 39.32 ± 0.12 35.92 ± 0.43 40.15 ± 0.32
1,4 dioxane
45.12 36.94 ± 0.28 36.40 ± 0.20 35.23 ± 0.33 37.11 ± 0.17
DMSO
38.21 30.90 ± 0.47 33.29 ± 0.44 27.90 ± 0.20 30.90 ± 0.22
ethanol
ethylene glycol 39.89 28.53 ± 0.09 29.44 ± 0.22 26.66 ± 0.30 31.43 ± 0.23
42.22 35.45 ± 0.10 35.65 ± 0.56 34.60 ± 0.43 36.96 ± 0.25
furan
30.40 25.21 ± 0.39 26.38 ± 0.31 23.57 ± 0.34 25.57 ± 0.12
methanol
48.71 38.61 ± 0.36 35.76 ± 0.24 34.96 ± 0.22 43.53 ± 0.45
THF
52.81 45.56 ± 0.35 42.44 ± 0.25 41.85 ± 0.27 45.40 ± 0.34
toluene
2.09
2.27
2.62
1.47
M.A.D.b
-7.13
-6.43
-9.13
-5.85 molar
Avg. Error predicted
Thermodynamics
of
liquids: standard
Accuracy
entropies
capacities of 6.08
common
7.63
7.84 and heat9.59
R.M.S. Error
entropy
only limited
2
solvents
molecular dynamics;
0.82
0.85 from 2PT 0.81
0.92
R
by accuracy of force
Pascal; Lin; Goddard; Phys. Chem. Chem.
Best
estimate*
75
field
Phys., 13: 169-181 (2011) wag897
TOD A. PASCAL MSC, CALTECH
water model
TIP4PEw
Strn(2P 50.590 53.050 49.870 55.590 49.790
T)
.25
.14
.14
.15
.07
Srot(2P 11.540 12.030 10.410 12.900 9.530.
T)
.06
.03
.04
.04
07
Svib(2P 0.040.
T)a
00
ftrn(2PT 0.250. 0.290. 0.230. 0.340. 0.240.
)b
01
01
01
00
01
frot(2PT 0.060. 0.070. 0.050. 0.080. 0.050.
)b
00
00
00
00
00
S(2PT) 62.180 65.090 60.280 68.490 59.320
.30
.13
.16
.14
.12
S(1PT) 53.820 56.240 52.280 59.370 51.390
.13
.13
.18
.17
.09
c
S(CT)
70.10
66.60
72.70
66.30
S(NN)d
73.51
66.91
80.19
65.46
d
S(FD)
65.10
64.48
70.86
S(FEP)
68.20
63.36
72.58
63.62
d
F3C
S(expt)
SPC
SPC/E
TIP3P
69.95
e
Within 3% of FEP method, while being 3
orders of magnitude more efficient
Lin, Maiti and Goddard, JPC-B, 2011
TOD A. PASCAL MSC, CALTECH
TOD A. PASCAL MSC, CALTECH
Amino-acid
Alanine
Arginine
abbreviation
ala
arn
Asparagine
Aspartate
Cysteine
Glutamine
Glutamine
Histidine
ash
asn
cys
gln
glh
hie
Histidine
hid
Isoleucine
Leucine
Lysine
Methionine
ile
leu
lyn
met
Phenylalanin
e
Serine
Threonine
Tryptophan
phe
Tryosine
Valine
tyr
val
ser
thr
trp
analogue
Methane
NPropylguanidi
ne
Acetamide
Acetic Acid
Methanethiol
Propionamide
Propionic Acid
4methylimidaz
ole
4methylimidaz
ole
1-butane
isobutane
N-butylamine
Methyl Ethyl
Sulfide
Toluene
Methanol
Ethanol
3methylindole
4-cresol
Propane
pKaa
13.65
3.86
4.24
6.00
6.00
10.79

ss
TOD A. PASCAL MSC, CALTECH

ss
TOD A. PASCAL MSC, CALTECH
TOD A. PASCAL MSC, CALTECH

Convergence from ~ 2ns of MD: time required for
system equilibration
TOD A. PASCAL MSC, CALTECH


94% correlation with experiment
Solvation entropy and enthalpy underestimated
TOD A. PASCAL MSC, CALTECH


94% agreement with Free Energy Perturbation
3 orders of magnitude more efficient!
2PT has 98% correlation with results of Shirts and Pande
2PT has 88% correlation with experiments – measure of
accuracy of forcefield
But Total Simulation time: 36.8 CPU hrs doing each 10 times
84
Factor of 2000 improvement
TOD A. PASCAL MSC, CALTECH



After equilibration, can obtain converged
thermodynamics of solvation from as little as
10ps of MD dynamics
Both solvation enthalpy and entropy of
common forcefields are significantly
underestimated
Water structure around
“hydrophobic”/hydrophilic small molecules
have dramatically different enthalpic
signatures
TOD A. PASCAL MSC, CALTECH
Tod A Pascal
William A Goddard III
MSC 2013 Research Conference
TOD A. PASCAL MSC, CALTECH

Surface tension of water:
 What is it
 Why is it important

Methods of Calculating Surface tension
 Kirkwood-Buff Theory
 Evaluation of Surface Free Energy
 Other schemes



Comparison of KB theory and 2PT method
Temperature dependence and energy profiles
Molecular orientations at the air-water
interface
TOD A. PASCAL MSC, CALTECH
“contractive tendency of the surface
of a liquid that allows it to resist an
external force”
Hydrophobicity
Liquid
Temperature Surface
°C
tension, γ
20
27.60
Acetic acid
Acetic acid
(40.1%) +
30
Water
Acetic acid
(10.0%) +
30
Water
Acetone
20
Diethyl ether 20
Ethanol
20
Ethanol
(40%) +
25
Water
Ethanol
(11.1%) +
25
Water
Glycerol
20
n-Hexane
20
Hydrochloric
acid 17.7M aq
20
ueous
solution
Isopropanol 20
Liquid helium
-273
II
Liquid
-196
nitrogen
Mercury
15
Methanol
20
n-Octane
20
Sodium
chloride 6.0
20
M aqueous
solution
40.68
54.56
23.70
17.00
22.27
29.63
46.03
63.00
18.40
65.95
21.70
[19]0.37
8.85
487.00
22.60
21.80
82.55
Terrestrial Life (Insects)
TOD A. PASCAL MSC, CALTECH

Kirkwood-Buff Theory (or is it Tolman Theory)?
• Interfacial tension
 =  dzPN ( z )  PT ( z )
• Stress profile
1
PN ( z ) =  ( z )k BT 
Vs
1
PT ( z ) =  ( z )k BT 
Vs
zij2 du (rij )
r
( i , j ) ij

(i , j )
n( z )
Vs
Richard C.
Tolman
xij2  yij2 du (rij )
2rij
• Density profile
 ( z) =
rij
Vs = Lx Ly Dz
rij
z
y
x
John Gamble Kirkwoo
TOD A. PASCAL MSC, CALTECH





75x75x100 cell
6900 F3C water
molecules (20Å
thick layer)
500ps NVT
equilibration
dynamics with
LAMMPS
Stress per atom
dumped every
10fs for 100ps
59.9 dynes/cm3
obtained
(Experimental:
70 dynes/cm3)
TOD A. PASCAL MSC, CALTECH

Direct evaluation of surface free energysurface
𝜕𝐺
𝛾=
𝜕𝐴
𝑁,𝑇,𝑃
𝐺𝑠𝑢𝑟𝑓𝑎𝑐𝑒 − 𝐺𝑏𝑢𝑙𝑘
=
𝜕𝐴
𝑁,𝑇,𝑃
𝐺 = 𝐻 − 𝑃𝑉
= 𝑈 − 𝑇𝑆 − 𝑃𝑉

Requires evaluation of the surface entropy
 Requires extensive simulation time for
convergence
 Can be approximated from potential of mean
force calculations – large uncertainties
bul
k
TOD A. PASCAL MSC, CALTECH
Q
qHO
( ) =  exp( βe n ) =
n
Liquid
S ( )

New Model
2 phase theory
(2PT )
Liquid  Solid +
Gas
solidS ( )
like
gas-like
exp( βh/2)
1 - exp( βh/2)
Finite density of states at n =0
Proportional to diffusion
coefficient
Harmonic
Approximation
 at
•Also strong
anharmonicity
Entropy=
∞
low frequencies
Solid
Gas
S ( )
exponential
decay

Debye crystal
S(v) ~v2
S ( )

•Two-Phase Thermodynamics
Model (2PT)
• Decompose liquid S(v) to a gas and a solid
contribution

• S(0) attributed to gas phase diffusion
The two-phase model for•calculating
thermodynamic
properties
of liquids from
Gas component
contains
anharmonic
92
molecular dynamics: Validation for the phase diagram of Lennard-Jones fluids;
TOD A. PASCAL MSC, CALTECH
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Excellent agreement
Systematic underestimation of experimental value: Deficiency of forcefield (SPC-Ew) used
TOD A. PASCAL MSC, CALTECH
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The decrease in surface tension with increasing
temperature is an entropically driven process
TOD A. PASCAL MSC, CALTECH
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Both rotational and translational entropy increases with increasing temperature
TOD A. PASCAL MSC, CALTECH
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Hydrogen bonding in supercooled water is different from ambient
water
 Subsurface water molecules are enthalpically stabilized
 Effect is reduced with increasing temperature
TOD A. PASCAL MSC, CALTECH
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Surface tension effects propagate into subsurface
Implications for propensity of ions at the interface
TOD A. PASCAL MSC, CALTECH
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Can evaluate the surface tension of liquids
from direct evaluation of the surface energy
Reduction in surface tension with
temperature is entropically driven
Sub-surface water molecules are
preferentially stabilized enthalpically,
especially for super-cooled water
TOD A. PASCAL MSC, CALTECH
Collaborators:
Yousung Jung (KAIST – Korea)
William A. Goddard III (Caltech)
TOD A. PASCAL MSC, CALTECH
"The antipathy of the paraffin chain for water is, however,
frequently misunderstood. There is no question of actual
repulsion between individual water molecules and paraffin
chains, nor is there any very strong attraction of paraffin chains
for one another. There is, however, a very strong attraction of
water molecules for one another in comparison with which the
DGt = Dtattractions
 DSt
paraffin-paraffin or paraffin-water
are slight." - G. S.
Hartley 1936
 Driving force in formation of
 Membranes
 Micelles
 Globular proteins

 Nonpolar
groups
(alkane
chains)
are
hydroph
obic
 Polar
Hydrophobic DNA bases stack so
as to exclude water molecules
 What are the
microscopic
thermodynamic
forces involved?
Enthalpy and/or
Entropy?
 How does structure
TOD A. PASCAL MSC, CALTECH
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Synthetic analogue of biological aquaporins
TOD A. PASCAL MSC, CALTECH
TOD A. PASCAL MSC, CALTECH
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Holt, Park et al, Science
(2006)
sub-2nm vertically aligned CNTs,
microfabricated into membranes
Flux estimated: 10-40 water/nm^2/ns (1000-10,000 times faster)
Slip length 1.4 micro-m, breakdown of continuum Hagen-Poiseuille
theory
TOD A. PASCAL MSC, CALTECH
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Aluru, Nano Lett
(2008)
Atomic smoothness?
Depletion layer (hydrophobicity) & dangling OH bonds near the
interface?
TOD A. PASCAL MSC, CALTECH
Enthalpically, water-water H-bonds are broken upon
creating a surface (unfavorable)
 Entropically, going into a confined space reduces entropy
(unfavorable)

∆𝐺 = ∆𝐻 − 𝑇∆𝑆
Spontaneous filling of
∆𝐺 > 0‼
CNT with water appears
to be against textbook
concept!
Nonetheless,
experimentally water
spontaneously wets the
internal
CNT origin?
What pores
is the of
physical
TOD A. PASCAL MSC, CALTECH
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SPC-E water
model
QMFF-Cx
forcefield for
graphite
Carbon –
water
interactions
obtained
from QM
  12   6 
E LJ 126 = 4e      
 r   r  

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εO-C =
0.65
kJ/mol
50ns
εH-CMD
=
simulation
0.29
LAMMPS
simulation
σO-C =
engine
3.166 Å
TOD A. PASCAL MSC, CALTECH
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Single file waters insiide (6,6) CNT located in center of CNT
Ice – like waters inside (8,8) and (9,9) CNTs absorbed on walls
TOD A. PASCAL MSC, CALTECH
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Decreased enthalpy is observed due to confinement and shows excellent
correlation to the average number of HB per water molecule
Enthalpy of ice-like (8,8) and (9,9) waters more favorable than bulk
Enthalpy of waters in all other CNTs less favorable than bulk; worse in case of
single-file (6,6)
TOD A. PASCAL MSC, CALTECH
Entropic trends exactly opposite to enthalpy
Free rotations contributes 60% (6,6) - 20%
(11,11 and onwards) to entropic gain
 Translational entropy (due to reduced density
near hydrophobic interface) responsible for rest
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TOD A. PASCAL MSC, CALTECH
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water molecules inside the CNTs have lower free energies
than bulk water
entropy dominates for tube diameters less than 1.0 nm
(gas phase), the enthalpy dominates for tubes between 1.1
and 1.2 nm (ice phase), and both energies compensate for
tubes larger than 1.4 nm (liquid phase)
TOD A. PASCAL MSC, CALTECH
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Water inside (6,6) and (7,7) resemble as gas – increase rotational entropy
Water inside (8,8) and (9,9) resemble ice/water – decrease rotational entropy
Water inside larger CNTs resemble bulk water – same rotational entropy
TOD A. PASCAL MSC, CALTECH
M3
B
m
W
StillingerWeber 3-body
qH:
+0.423
SP
8 eCE with same interactions with CNT as
LJ liquid
water (M3B) has unfavorable free energies
 Thermodynamics recovered by including 3body H-bond (mW)

TOD A. PASCAL MSC, CALTECH

Favorable local chemical potential
inside CNT
 Lower free energy due to lower
enthalpy for ice-like CNTs but higher
entropy for all others
 Loss of hydrogen bonding inside tube
overcome by increased entropy due to
confinement