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Ionic and Molecular Liquids:Long-Range Forces and Rigid-Body Constraints
Charusita Chakravarty
Indian Institute of Technology Delhi
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Electrostatic Interactions
• Ionic liquids: NaCl, SiO2, NH4Cl• mobile charge carriers which are
atomic or molecular entities • Simple ionic melts: model ions as point
charges with Coulombic interactions.
Short-range repulsions control ionic radii.
• Molecular Liquids• Electronic charge distributions show
significant deviations from spherical symmetry
• Can be modeled by: (a) multipole moment expansions or (b) arrays of partial charges
water
benzene
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Multipole Expansion for Electrostatic Potential
rquadrupola
1cos3
2
dipolar cos
Coulombic 4
3
2
2
r
r
r
Q
Represent the electronic charge distribution of a molecule by a set of multipoles:
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Range of Electrostatic Interactions
Type Range Energy (kJ/mol) Comment
Ion-Ion 1/r 250
Ion-dipole 1/r2 15
Dipole-Dipole 1/r3 2 Static dipoles (solid phase)
Dipole-quadrupole 1/r4 Fixed Orientation / Linear
Quadrupole-quadrupole 1/r5 Fixed Orientation / Linear
Long-range interactions: Tail correction will diverge for 1/rn interactions with n greater than or equal to 3; therefore minimum image convention cannot be applied
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Partial Charge Distributions of Some Typical Molecules
109.47
H
O
H
0.95
72 Å
(+0.52e)
(+0.52e)
2(-
1.04e)
1.0 Å
109.47
OO
H H
2(-0.8472e)
(+0.4238e)
(+0.4238e)
TIP4P Water SPC/E Water
Dipole + Quadrupole
Quadrupole
Molecular Nitrogen
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Array of Point ChargesCoulombic contribution to the potential energy for an array
of N charges that form a charge-neutral system:
Electrostatic potential
• Particle i interacts with
all other charges and their mirror images but not with itself
• Gaussian units
• Cannot apply minimum image convention because
sum converges very slowly
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Electrostatic Potential
ChargeDistribution
Poisson Equation
Potential
Energy and
Forces
)(r )(r
)rU(rU
and )(
)(4)(2 rr
Linear differential equation:
ii
ii
r
r
)(:potential ticElectrosta
)(:ondistributi Charge
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Ewald Summation for Point Charges
Co
Point ChargeDistribution:
Converges slowly
)(rP
)(-
)()(
r
rr
G
GP
Screened charge distribution:
Converges fast in real space
Gaussian compensating charge distribution: can be analytically
evaluated in real space
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Ewald Summation
• Screening a point charge to convert the long-range Coulombic interaction into a short-range interaction
• Evaluating the real-space contribution due to the screened charges
• The Poisson equation in reciprocal space for the compensating screened charge distribution
• Evaluating the reciprocal space contribution
• Self-interaction correction
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Screening a point charge
)(4
)(1
:ondistributi
chargeGaussian todue potential ticElectrosta
)exp()/()(
:originat centred chargeGaussian Single
/)( and )()(
:originat centred chargepoint Single
2
2
22/3
rr
r
r
rqr
rqrrqr
SGSG
iSG
iSPiSP
)(4)(2 rr
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Electrostatic potential due to a Gaussian charge distribution
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)(erfc)/()(erf)/()/(
:density chargeGaussian ngcompensati aby
screened chargepoint of potential ticElectrosta
range-short rrqrrqrq iii
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Real Space Contribution
ijij
jSR
jij
rerfcr
qr
rqr
)(
:at charge todue at n interactio ticElectrosta
j
jSRjreal rqU )(2
1
:rat potential ticelectrosta with ther positionsat
located q charges ofarray an ofenergy n Interactio
jj
j
i ij
ijij
jireal rerfc
r
qqU
2
1
The value of must be chosen so that the range of interaction of thescreened charges is small enough that a real space cutoff of rc < L/2can be used and the minimum image convention can be applied
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Poisson Equation in Reciprocal Space
)(42 rr
)(~4)(~2 kkk
)(~
of FT kfkdX
df )(~
of FT kfkdX
fd n
n
n
Fourier series representation of a function in a cubic box with edge length Land volume V under periodic boundary conditions:
),,(2 where)exp()(~
)/1()( zyxl
lllkrkikfVrf
)exp()()( rkirfrdkfV
Fourier coefficients
Poisson equation in real space
Poisson equation in reciprocal space
Reminder: Fourier transforms of derivatives
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)()(~ rrS
2
4)(~
kkg
i
iiP rrqr )()(~
i
iiP rkqk )exp()(
i
ii rkqk
)exp(4
2
)()()(
~kkgk PP
Unit Point Charge at origin:
Array of point charges
Array of Gaussian charges
i
iiG rrqr )()( )()( rrrdr P
)()(~)(~)(~
kkrkgk P
Green’sFunction Fourier transform
of smearing Function
FT of Point charge Array
Convolution of Point Charge distribution and smearing function
)(~
)(~
)(~
then )()()( :Reminder 321321 xfxfxfxfxfxf
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Fourier Part of Ewald Sum
N
j njjG Lnrrqr
1
223 exp)(
N
ii krk
kk
1
22
4expexp4
)(
Corresponding Electrostatic potential in reciprocal space
Electrostatic potential in real space can be obtained using:
0
)exp()(1
)(k
rkkV
r
Array of N Gaussian point charges with periodic images:
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Fourier Part of Ewald Sum (contd.)
Electrostatic potential in real space
)4/exp()](exp[4
)exp()(1
)(
2
0k 12
0
krrkik
q
rkkV
r
j
N
j
j
k
0
22
2
2
0k 1,2
4exp)(4
2
1
)4/exp()](exp[4
)2/1(
)()2/1(
k
ji
N
ji
ij
iirec
kkkV
krrkiVk
rqU
Reciprocal Space contribution to potential energy
System is embedded in a medium with an inifinite dielectric constant
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Correction for Self-Interaction
rerfr
qr i
Gauss )(
r
qr i
Gauss
2)0(
N
iiselfiSelf rqU
1
)(2
1
N
iiq
1
221
Must remove potential energy contribution due to a continuous Gaussian cloudof charge qi and a point charge qi
located at the centre of the cloud.
Electrostatic potential due to Gaussian centred at origin
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Coulombic Interaction expressed as an Ewald Sum
Reciprocal space
Self-Interaction
Real Space
0
22
24exp
4
2
1
kCoul kk
kVU
N
iiq
1
22
1
N
ji ij
ijji
r
rerfcqq 2
1
Important: For molecules, the self-interaction correction must be modified because partial charges on the same molecule will not interact
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Accuracy of Ewald Summation
• Convergence parameters:
– Width of Gaussian in real space, – Real space cut-off, rc
– Cutoff in Fourier or reciprocal space, kc=2/Lnc
/
/
sLn
sr
c
c
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Calculating Ewald Sums for NaCl
Na+
Cl-
Liquid Crystal
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Hands-on Exercise:Calculating the Madelung Constant for NaCl
The electrostatic energy of a structure of 2N ions of charge +/- q is
where is the Madelung constant and rnn is the distance between the nearest neighbours.
ijnn r
qN
r
qNU
22
Structure Sodium Chloride (NaCl) 1.747565
Cesium Chloride (CsCl) 1.762675
Zinc blende (cubic ZnS) 1.6381
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Rotational and Vibrational Modes of Water
Symmetric Stretch3657cm-1
http
://chsfp
c5.ch
em
.ncsu
.ed
u/~
fran
zen
/CH
79
5N
/lectu
re/X
IV/X
IV.h
tml
Rotational
Constants (cm-1)
A 40.1
B 20.9
C 13.4
Bend1595cm-1
Asymmetric Strech3756cm-1
http
://w
ww
1.ls
bu.
ac.
uk/w
ater
/vib
rat.
htm
Intermolecular
vibrations (cm-1)
Librations 800
OO stretch 200
OOO bend 60
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Molecules: Multiple Time-scales
• Bonded interactions are much stronger than non-bonded interactions
• Intramolecular vibrations have frequencies that are typically an order magnitude greater than those of intermolecular vibrations
• MD/MC: time step will be dictated by fastest vibrational mode
• Fast, intramolecular vibrational modes do not explore much of configuration space- rapid, essentially harmonic, small amplitude motion about equilibrium geometry
• Require efficient sampling of orientational and intermolecular vibrational motions
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Simulation Methods for Molecules
• Freeze out all or some intramolecular modes:– Serve to define vibrationally averaged molecular
structure and are completely decoupled from intermolecular vibrations, librations or rotations
– Rigid-Body Rotations:• Characterize the mass distribution of the molecule by its
moment of inertia tensor• MC: Use orientational moves• MD: Propagate rigid-body equations of motion• Will not work if there are low-frequency vibrational modes
– Apply Geometrical Constraints• MD: SHAKE• MD: RATTLE
• Multiple Time-Scale Algorithms
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Rotations of Rigid-Bodies
H2
N2
CH4SF6
NH3 lCl5
H2O
Ix Iy Iz
Space-fixed (SF) and Body-fixed (BF)axes (Goldstein, Classical Mechanics)
Moments of Inertia of Molecules:Ia < Ib < Ic
Linear: Spherical polar angles: Non-linear:Euler angles: (Atkins, Physical Chemistry)
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Monte Carlo Orientational Moves for Linear Molecules
Orientation of a linear molecule is specified by a unit vector u . To change it by a small amount:
1. Generate a unit vector v with a random orientation. See algorithm to generate random vector on unit sphere
2. Multiply vector v with a scale factor g, which determines acceptance probability of trial orientational move
3. Create a sum vector: t = u + gv
4. Normalise t to obtain trial orientation
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Euler Angles (
http://stackoverflow.com/questionshttp://mathworld.wolfram.com/EulerAngles.html
Euler’s rotation theorem: Any rotation of a rigid-body may bedescribed by a set of three angles
•Rotation, A: Initial orientation of body-fixed axes (X,Y,Z) to final orientation (X’’,Y’’’, Z’)
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Euler angles ((contd.)• Rotate the X-, Y-, and Z-axes about the Z-axis by
resulting in the X'-, Y'-, and Z-axes.
• Rotate the X'-, Y'-, and Z-axes about the X'-axis by resulting in the X'-, Y''-, and Z'-axes.
• Rotate the X'-, Y''-, and Z'-axes about the Z'-axis by resulting in the X''-, Y'''-, and Z'-axes.
Rotation A = BCD, therefore new coordinates are:
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Monte Carlo Orientational Moves for Non-linear Molecules
• Specify the orientation of a rigid body by a quaternion of unit norm Q that may be thought of as a unit vector in four-dimensional space.
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Applying Geometrical Constraints
)(2
1),( 2 xUxmxxL
q
L
q
L
t )()( qUqKL
Lagrangian Equations of Motion Kinetic Energy Potential Energy
Cartesian coordinates
x
Uxm
x
L
x
L
t
Geometrical Constraints
00
0)(
qt
dqq
d
q
ii i
Define constraint equations and require that system moves tangential to the constraint plane.
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Introduce a new Lagrangian that contains all the constraints:
q
L
q
L
t
''
qLL'
ii
iiii
GF
Uqm
The equations of motion of the constrained system are:
Constraint force acting along coordinate qi
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Bond Length Constraint for Diatomic Molecule
022
21 drr
)(or )(2 211211
rr Grrr
)(or )(2 212212
rr Grrr
m1 m2 Bond constraint
Constraint forces:• lie along bond direction• are equal in magnitude • opposite in direction• do no work
Verlet algorithm:
)()('
)()()()(2)(
2
22
tGm
tttr
tGm
ttF
m
tttrtrttr
ii
i
ii
ii
iii
New positionwith constraint
Unconstrainedposition
Constraint forceson atom i
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)()(
))()()(()(')(
))()()(()(')(
22
21
212
22
212
11
dttrttr
trtrmtttrttr
trtrmtttrttr
i
i
Diatomic molecule (contd)
Three unknowns
In the case of a diatomic molecule, can obtain quadratic equation in . However, cannot be conveniently generalisedfor larger molecules with more constraints.
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Multiple Constraints
Verlet algorithm for N atoms in the presence of l constraints:
Constraint force actingon i-th atom due to k-thconstraint
Taylor expansion of the constraints with respect to unconstrained positions.Impose the requirement that after one timestep, the constraints must be satisfied.
positions nedunconstrai
using )( ttki Substitute from Taylorexpansion above
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Multiple Constraints (contd.)
Solution by Matrix Inversion
Need to find the unknown Lagrange multipliers:
Since the Taylor expansion was truncated at first-orderiterative scheme will be required to obtain convergence.
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SHAKE Algorithm
• Iterative scheme is applied to each constraint in turn:
RATTLE: Similar approach within a velocity Verlet scheme
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References
• D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications
• A. R. Leach, Molecular Modelling: Principles and Applications
• D. C. Rapaport, The Art of Molecular Dynamics Simulation (Details of how to implement algorithms for molecular systems)
• M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (SHAKE, RATTLE, Ewald subroutines)