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Fluctuating charge models: applications and illustrations
Jiahao ChenMartínez Group
Dept. of Chemistry, Frederick Seitz Materials Research Lab. and the Beckman Institute
University of Illinois at Urbana-Champaign
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Acknowledgments
Prof. Troy van VoorhisProf. Alán Aspuru-Guzik
Prof. Todd J. MartínezMartínez Group and friends
$: DOE
Discussions
Harvard/MIT visit
Prof. Susan Atlas (UNM)Dr. Ben Levine (UPenn)Dr. Steve Valone (LANL)
Prof. Troy van Voorhis (MIT)
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“The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without
having to surrender the adequate representation of a single datum of
experience.”Albert Einstein, “On the Method of Theoretical Physics”, Phil. Sci. 1 (1934), 163-9.
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What is electronegativity?
“Concept introduced by L. Pauling as the power of an atom to attract electrons to itself.”
“The quantity that measures the escaping tendency of electrons from a species in its ground state.” IUPAC Compendium of Chemical Terminology,
aka “The Gold Book”, goldbook.iupac.org
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A quantitative definition
R. S. Mulliken J. Chem. Phys 2:(1934), 782–793
! =IP + EA
2
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A quantitative definition
R. S. Mulliken J. Chem. Phys 2:(1934), 782–793
! =IP + EA
2
=E(N ! 1)! E(N + 1)
2
" "E
"N
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Electronic structure and dynamics
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Electronic structure and dynamicsWhat is the charge distribution?
What does the system do?
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Electronic structure and dynamics
H! = i!
H! = E! What is the charge distribution?
What does the system do?
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Electronic structure and dynamics
less variablesmore variables
H! = i!
H! = E! What is the charge distribution?
What does the system do?
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Electronic structure and dynamics
less variablesmore variables
H! = i!
H! = E!
directnumericalquadrature
ab initiotheories
semiempiricalmethods
density functional
theory
coarse-grained models
continuumelectrostatics
molecular models (MM)
classicalmoleculardynamics
finite element methods
coarse-grained
dynamics
numerical quadrature, e.g. real-time path
integral propagatorsab initio molecular dynamics
What is the charge distribution?
What does the system do?
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Electronic structure and dynamics
less variablesmore variables
H! = i!
H! = E!
directnumericalquadrature
ab initiotheories
semiempiricalmethods
density functional
theory
coarse-grained models
continuumelectrostatics
molecular models (MM)
classicalmoleculardynamics
finite element methods
coarse-grained
dynamics
numerical quadrature, e.g. real-time path
integral propagatorsab initio molecular dynamics
What is the charge distribution?
What does the system do?
molecular models (MM)
classicalmoleculardynamics
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Molecular models/force fields
covalent bond effectsE =
+
Typical energy function
noncovalent interactions
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bond stretch angle torsion dihedrals
electrostatics dispersion
+-
+
Typical energy function
!
i<j!atoms
!ij
"#"ij
rij
$12
!#
"ij
rij
$6%
E =!
b!bonds
kb(rb ! r0b )2 +
!
a!angles
!a("a ! "0a)2
!
d!dihedrals
!
n
ldn cos(n!)+
++!
i<j!atoms
qiqj
rij
Molecular models/force fields
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bond stretch angle torsion dihedrals
electrostatics dispersion
+-
+
Typical energy function
!
i<j!atoms
!ij
"#"ij
rij
$12
!#
"ij
rij
$6%
E =!
b!bonds
kb(rb ! r0b )2 +
!
a!angles
!a("a ! "0a)2
!
d!dihedrals
!
n
ldn cos(n!)+
++!
i<j!atoms
qiqj
rij
Molecular models/force fields
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Unique to condensed phases, where most
chemistry and biology happens
Why care about polarization and charge transfer?
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Polarization in chemistry• Effect of local environment in liquid phases
• Ex. 1: Stabilizes carbonium in lysozyme
• Ex. 2: Hydrates chloride in water clusters
OPLS/AAnon-polarizable
force field
TIP4P/FQpolarizableforce field
1. A Warshel and M Levitt J. Mol. Biol. 103 (1976), 227-249. 2. SJ Stuart and BJ Berne J. Phys. Chem. 100 (1996), 11934 -11943.
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3 models for polarization
Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.
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Drude oscillatorsor charge-on-spring
or shell modelsQ
q !Q
kR
Response = change in RReview: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.
Ideal spring
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Inducible dipoles
!1 !2
µinduced,1 µinduced,2
Response = change in induced dipoles
Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.
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-0.3
Fluctuating charges
+0.8
-0.5
charge transfer = 1.1e charge transfer = 0.2 e
charge transfer = 1.3 e
Response = change in atomic charges
!2, "2
!3, "3
Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.
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Better electrostatics
Model Polari-zation
Charge transfer
Cost
Pairwise fixed charges
Drude oscillator
Inducible dipoles
Fluctuating charges
Implicit, at best ❙
✓ ❙ ❙
✓ ❙ ❙ ❙ ❙ ❙ ❙
✓ ✓ ❙ ❙ ❙
!
i<j!atoms
qiqj
rij
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QEq, a fluctuating-charge model
AK Rappé and WA Goddard III J. Phys. Chem. 95 (1991), 3358-3363.
atomicelectronegativities
“voltages”
screenedCoulomb
interactions
Jij =!
R3!2
!2i (r1)!2
j (r2)|r1 ! r2| dr1dr2
!i(r) = Ni |r !R|ni!1 e!!i|r!Ri|
E =!
i
qi!i +12
!
ij
qiqjJij
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Principle of electronegativity equalization
Minimize energy
subject to charge constraint!
i
qi = Q
Using the method of Lagrange multipliers, reduces to solving the linear equation
(electronic) chemical potential
!J 11T 0
" !qµ
"=
!!!0
"
E =!
i
qi!i +12
!
ij
qiqjJij
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Physical interpretationIn equilibrium:
oeach atom i has the same chemical potential µo µ uniquely determines the atomic charges qi
Atoms are subsystems in equilibrium
N, V, T
ΩΩi
moleculeatom
Energy derivatives: chemical potential µ, hardness η
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QEq has wrong asymptotics
q =!1 ! !2
J11 + J22 ! J12
0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
R/Å
q/e
QEq
ab initio
Na ClR
asymptote ~ 0.43 ≠ 0
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Problems with QEqFractional charge distributions predicted
for dissociated systems
Wrong direction of intermolecular charge transfer predicted in some systems
No out-of-plane dipole polarizability
Overestimates in-plane dipole polarizability
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Fluctuating-charge models map molecules onto electrical circuits
screenedCoulomb
interactionchemicalhardness
electro-negativitymolecule
screenedCoulombchemicalelectro-
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Fluctuating-charge models map molecules onto electrical circuits
screenedCoulomb
interactionchemicalhardness
electro-negativitymoleculeelectric
potential(inverse)
capacitanceelectricalcircuits
Coulombinteraction
screenedCoulombchemicalelectro-
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Fluctuating-charge models map molecules onto electrical circuits
screenedCoulomb
interactionchemicalhardness
electro-negativity
More electropositive
More electronegative0 V
χ2
χ1η
1
η2
- V
olta
ge +
moleculeelectric
potential(inverse)
capacitanceelectricalcircuits
Coulombinteraction
screenedCoulombchemicalelectro-
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QEq has wrong asymptotics
q =!1 ! !2
J11 + J22 ! J12
0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
R/Å
q/e
QEq
ab initio
Na ClR
asymptote ~ 0.43 ≠ 0
J12 → 0+
-
+
-
+
-
+
-
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In fluctuating-charge models like QEq, all
molecules are metallic
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η2
QTPIE, our new charge modelCharge-transfer with polarization current
equilibrationVoltage attenuates with increasing distance
J. Chen and T. J. Martínez, Chem. Phys. Lett. 438 (2007) 315-320.
voltage
distance
η2
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η2
QTPIE, our new charge modelCharge-transfer with polarization current
equilibrationVoltage attenuates with increasing distance
J. Chen and T. J. Martínez, Chem. Phys. Lett. 438 (2007) 315-320.
voltage
distance
η2
η2
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Making QTPIE (Step 1)
J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.
To make the proposed change, first change variables
E =!
i
qi!i +12
!
ij
qiqjJij
=!
ij
pji!i +12
!
ijkl
pkipljJij
qi =!
j
pji
Charge transfer variables quantify how much charge went from one atom to another, and are indexed over pairs
p12
p23
p34
p45
Still QEq!Same model,new representation
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Making QTPIE (Step 2)
J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.
atomic electronegativities become bond electronegativities
Sij =!
R3!i(r)!j(r)dr
EQEq =!
ij
pji!i +12
!
ijkl
pkipljJij
EQTPIE =!
ij
pji!ikijSij +12
!
ijkl
pkipljJij
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0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
R/Å
q/e
QEq
QTPIE
ab initio
QTPIE has correct limit
q =(!1 ! !2)S12
J11 + J22 ! J12
q =!1 ! !2
J11 + J22 ! J12
Na ClR
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Reverting to charge variables
p12p13
p34
p14
p23
p24
q1
q2 q3
q4qi =
!
j
pji
?
Adjacency matrix of an oriented complete graph with 4 vertices
!
""#
q1
q2
q3
q4
$
%%& =
!
""#
!1 !1 !1 0 0 01 0 0 !1 !1 00 1 0 1 0 !10 0 1 0 1 1
$
%%&
!
""""""#
p12
p13
p14
p23
p24
p34
$
%%%%%%&
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Reverting to charge variables
p12p13
p34
p14
p23
p24
q1
q2 q3
q4qi =
!
j
pji
?
Inverse transformation is determined by pseudoinverse of adjacency matrix
!
""""""#
p12
p13
p14
p23
p24
p34
$
%%%%%%&=
!
""#
!1 !1 !1 0 0 01 0 0 !1 !1 00 1 0 1 0 !10 0 1 0 1 1
$
%%&
+ !
""#
q1
q2
q3
q4
$
%%&
=14
!
""""""#
!1 1 0 0!1 0 1 0!1 0 0 10 !1 1 00 !1 0 10 0 !1 1
$
%%%%%%&
!
""#
q1
q2
q3
q4
$
%%&
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Reverting to charge variables
p12p13
p34
p14
p23
p24
q1
q2 q3
q4qi =
!
j
pji
Inverse transformation is determined by pseudoinverse of adjacency matrix
!
""""""#
p12
p13
p14
p23
p24
p34
$
%%%%%%&=
!
""#
!1 !1 !1 0 0 01 0 0 !1 !1 00 1 0 1 0 !10 0 1 0 1 1
$
%%&
+ !
""#
q1
q2
q3
q4
$
%%&
=14
!
""""""#
!1 1 0 0!1 0 1 0!1 0 0 10 !1 1 00 !1 0 10 0 !1 1
$
%%%%%%&
!
""#
q1
q2
q3
q4
$
%%&
pji =qi ! qj
N
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Reverting to charge variables
Charge transfer variables have massive linear redundancy due to Kirchhoff’s voltage law
p12
p23
p31
p12 + p13 + p31 = 0
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Execution times
0.01
0.1
1
10
100
1000
104
10 100 1000 104 105
TImes to solve the QTPIE model
Bond-space SVDBond-space COFAtom-space iterative solverAtom-space direct solver
Sol
utio
n tim
e (s
)
Number of atoms
N1.81N6.20
N
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Cooperative polarization in water
• Dipole moment of water increases from 1.854 Debye1 in gas phase to 2.95±0.20 Debye2 at r.t.p. (liquid phase)
• Polarization enhances dipole moments
• Missing in models with implicit or no polarization
!"+
1. D R Lide, CRC Handbook of Chemistry and Physics, 73rd ed., 1992.2. AV Gubskaya and PG Kusalik J. Chem. Phys. 117 (2002) 5290-5302.
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Polarization in water chains• Use parameters from single water molecule
to model chains of waters
• Compare QEq and QTPIE with:
๏ Gas phase experimental data1
๏ Ab initio DF-LMP2/aug-cc-pVDZ
๏ AMOEBA2, an inducible dipole model
๏ TIP3P, a common implicit polarization model
1. WF Murphy J. Chem. Phys. 67 (1977), 5877-5882.2. P Ren and JW Ponder J. Phys. Chem. B 107 (2003), 5933-5947.
H! = E!
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Our new water model
EMM =!
i!bonds
ki (Ri !Reqi )2
+!
i!1,3-bonded
kUBi
"RUB
i !RUB,eqi
#2
+!
i!angles
!i ("i ! "eqi )2
+!
ij
4#ij
$%$ij
rij
&12
!%
$ij
rij
&6'
+EQTPIE
E =!
b!bonds
kb(rb ! r0b )2 +
!
b!1,3int.
kUBb (rUB
b ! rUB,0b )2
+!
a!angles
!a("a ! "0a)2
+!
i<j!atoms
4#ij
"#$ij
rij
$1
2!#
$ij
rij
$6
)
+EQTPIE
Flexible SPC/E, but with QTPIE electrostaticsFit to gas-phase data, and test transition to bulk in 1 dim.
!"!ij
rij
#12
!"
!ij
rij
#6$
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Dipole moment per water
0 5 10 15 20 25
Number of molecules
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
Dip
ole
mom
ent/
mol. (
Debye)
DF-LMP2/aug-cc-PVTZ
QTPIE
QEq
AMOEBA
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Polarizability per water
0 5 10 15 20 25
Number of molecules
1.0
1.5
2.0
2.5
3.0
3.5Tra
nve
rse p
ola
riza
bili
ty/m
ol.
(Å!)
DF-LMP2/aug-cc-PVTZ
QTPIE
QEq
AMOEBA
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Polarizability per water
0 5 10 15 20 25
Number of molecules
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5Longit
udin
al p
ola
riza
bili
ty (
Å!)
DF-LMP2/aug-cc-PVTZ
QTPIE
QEq
AMOEBA
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Polarizability per water
0 5 10 15 20 25
Number of molecules
.0
.5
1.0
1.5
Out-
of-
pla
ne
pola
riza
bili
ty (
Å!)
DF-LMP2/aug-cc-PVTZ
QTPIE QEq
AMOEBA
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Charge transfer in 15 waters
1 3 5 7 9 11 13 15Index of water molecule
-.05
-.03
-.01
.01
.03
.05
Ch
arg
e o
n w
ate
r m
ole
cu
le
QEq
QTPIE
Mulliken
1 3 5 7 9 11 13 15Index of water molecule
-.05
-.03
-.01
.01
.03
.05
Ch
arg
e o
n w
ate
r m
ole
cu
le
QEq
QTPIE
Mulliken
1 3 5 7 9 11 13 15Index of water molecule
-.05
-.03
-.01
.01
.03
.05
Ch
arg
e o
n w
ate
r m
ole
cu
le
QEq
QTPIE
Mulliken
1 3 5 7 9 11 13 15Index of water molecule
-.05
-.03
-.01
.01
.03
.05
Ch
arg
e o
n w
ate
r m
ole
cu
le
QEq
QTPIE
Mulliken
1 3 5 7 9 11 13 15Index of water molecule
-.05
-.03
-.01
.01
.03
.05C
ha
rge
on
wa
ter
mo
lec
ule
QEq
QTPIE
Mulliken
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Summary
• Polarization and charge transfer are important effects usually neglected in classical MD
• Our new charge model corrects deficiencies in existing fluctuating-charge model at similar computational cost
• We obtain quantitative polarization and qualitative charge transfer trends in linear water chains