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Slides for prospective postdoctoral interview at the Aspuru-Guzik group at Harvard University, 2008-10-09.
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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
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)
“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.
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
A quantitative definition
R. S. Mulliken J. Chem. Phys 2:(1934), 782–793
! =IP + EA
2
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
Electronic structure and dynamics
Electronic structure and dynamicsWhat is the charge distribution?
What does the system do?
Electronic structure and dynamics
H! = i!
H! = E! What is the charge distribution?
What does the system do?
Electronic structure and dynamics
less variablesmore variables
H! = i!
H! = E! What is the charge distribution?
What does the system do?
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?
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
Molecular models/force fields
covalent bond effectsE =
+
Typical energy function
noncovalent interactions
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
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
Unique to condensed phases, where most
chemistry and biology happens
Why care about polarization and charge transfer?
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.
3 models for polarization
Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.
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
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.
-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.
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
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
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
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
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
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
Fluctuating-charge models map molecules onto electrical circuits
screenedCoulomb
interactionchemicalhardness
electro-negativitymolecule
screenedCoulombchemicalelectro-
Fluctuating-charge models map molecules onto electrical circuits
screenedCoulomb
interactionchemicalhardness
electro-negativitymoleculeelectric
potential(inverse)
capacitanceelectricalcircuits
Coulombinteraction
screenedCoulombchemicalelectro-
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-
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+
-
+
-
+
-
+
-
In fluctuating-charge models like QEq, all
molecules are metallic
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
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
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
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
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
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
$
%%%%%%&
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
$
%%&
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
Reverting to charge variables
Charge transfer variables have massive linear redundancy due to Kirchhoff’s voltage law
p12
p23
p31
p12 + p13 + p31 = 0
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
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.
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!
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
$6
)
+EQTPIE
Flexible SPC/E, but with QTPIE electrostaticsFit to gas-phase data, and test transition to bulk in 1 dim.
!"!ij
rij
#12
!"
!ij
rij
#6$
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
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
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
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
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
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