1. Resolving the dissociation
catastrophe in fluctuating-
charge models
Jiahao Chen
MIT
UIUC
Thanks to:
Susan R. Atlas, UNM
Benjamin G. Levine, Temple
Todd J. Martínez, Stanford
Steven M. Valone, LANL
Troy van Voorhis, MIT University of
Chicago
2010-06-29

2. Polarization is key to describing
condensed phase chemistry
Ex. 1: Stabilizes carbonium in
lysozyme
carbonium
forms
sugar bond
cleaved
TIP4P/FQ OPLS/AA
polarizable non-polarizable
force field force 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),

3. Outline
The concept of fluctuating charges
Principle of electronegativity equalization
Circuit analogy
The dissociation catastrophe
Fixing the dissociation catastrophe
Charge transfer variables
Charge transfer topologies
Applications
Water
Metal clusters

4. The idea of fluctuating
charges
A
Atoms in isolation have no charge*
*In general, ions in isolation have integer charges

5. The idea of fluctuating
charges
+q -q
A B
coupling, J
Atoms in isolation have no charge
but atoms nearby interact by
inducing charges on each other

6. The idea of fluctuating
charges
+q
A
To determine the charge, let
electronegativity vary with charge
χeff (q) = χ + ηq
∂χeff
effective (chemical) hardness η = ∂q q=0
electronegativity electronegativity

7. The idea of fluctuating
charges
+q -q
A B
coupling, J
To determine the charge, let
electronegativity vary with charge
and allow for interactions with
other charges
χeff,A (qA , qB ) = χA + ηA qA + JAB qB
χeff,B (qB , qA ) = χB + ηB qB + JBA qA

8. The idea of fluctuating
charges
+q -q
A B
Assume that the electronegativities
on all atoms become equal
χeff,A = χeff,B
Principle of electronegativity
equalization (Sanderson, 1951)
χeff,A (qA , qB ) = χA + ηA qA + JAB qB
χeff,B (qB , qA ) = χB + ηB qB + JBA qA

9. The idea of fluctuating
charges Modern examples:
Electronegativity equalization model (EEM)
+q -q Mortier, Shankar and Ghosh, 1985/6
A B Charge equilibration model (QEq)
Rappé and Goddard, 1991
Fluctuating-charge model ( uc-q)
e charge distribution Rick, Stuart and Berne, 1996
minimizes the energy
plus many more variants since
1 2 1
E= qi χi + qi ηi + Jij
i
2 2
i=j
J 1 q −χ
P min E(q1 , . . . qN ) =⇒ =
i qi =Q 1T
0 −X Q
Lagrange multiplier = global electronegativity

10. Analogies to electrical
circuits
screened
electro- chemical Coulomb
molecule negativity hardness interaction
electrical electric (inverse) Coulomb
circuits potential capacitance interaction
More electropositive
χ
- Voltage +
η 1
1
χ
η 2
2 0V
More electronegative

11. The dissociation catastrophe
+q -q
A B
We immediately see a problem.
e solution for two atoms is
χA − χB
qA =
ηA + ηB − 2JAB
and so at in nite separation
χA − χB
qA = =0
ηA + ηB
Wrong long-range behavior!

12. The dissociation catastrophe
+q -q
A B
R
1.0
q/e
0.8
0.6
QEq
0.4
0.2
ab initio
0.0 R/Å
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Long-range CT = metallic

13. Outline
The concept of fluctuating charges
Principle of electronegativity equalization
Circuit analogy
The dissociation catastrophe
Fixing the dissociation catastrophe
Charge transfer variables
Charge transfer topologies
Applications
Water
Metal clusters

14. Fixing the dissociation
catastrophe
One solution is to constrain certain atoms to be
unable to transfer charge.
A B
Another is to modify the electronegativity difference
to be distance dependent
q q
A B A B
(χA − χB ) (χA − χB ) SAB
Morales and Martínez, 2001, 2004
Chen and Martínez, 2007
Long-range charge transfer goes smoothly to 0 at
dissociation

15. Fixing the dissociation
catastrophe
How to apply this distance dependent
electronegativity to many atoms? p C pBC
AC
q pAB
A B A B
pCD
pAD
D
Introduce charge transfer variables
qi = pj→i
j
that account for the amount of charge transfer
between each pair of atoms Chelli, Procacci, Righini, and
Califano, 1999
E= pj→i (χi − χj ) Sij + pj→i pl→k Jik
ji ijkl
Chen and Martínez, 2007

16. Attenuation fixes long-range
CT
1.0
q/e
Na Cl
R
0.8
χ1 − χ2
q=
J11 + J22 − J12
0.6
QEq
0.4 (χ1 − χ2 )S12
q=
J11 + J22 − J12
QTPIE
0.2
ab initio
0.0 R/Å
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

17. Origin of rank deficiency
Charge transfer variables are
massively redundant due to
p12
p31
p23
p12 + p13 + p31 = 0
only N-1 of these variables are linearly
independent!
Therefore, charge transfer variables contain
exactly the same amount of information as

18. Reverting to atomic charges
qi = pji q1
p12 j
p31
p23 q2 q3
?
Topological analysis of the relationship between
charges and charge transfer variables allows the
reverse transformation to be derived as
qi − qj
pji =
N

19. Reverting to charge variables
qi = pji q4
p14
j q1
p24 p34
p12
p13
q2 q3
p23 ?  
    p12
q1 −1 −1 −1 0 0 0  p13 
 
 q2   1 0 0 −1 −1 0  p14 
 =  
 q3   0 1 0 1 0 −1  
 p23 

q4 0 0 1 0 1 1  p24 
Adjacency matrix of an p34
oriented complete graph
with 4 vertices

20. Reverting to charge variables
qi = pji q4
p14
j q1
p24 p34
p12
p13
q2 q3
p
 23 
?
p12  +  
 p13  −1 −1 −1 0 0 0 q1
 
 p14   1 0 0 −1 −1 0   q2 
  =    
 p23   0 1 0 1 0 −1   q3 
 
 p24  0 0 1 0 1 1 q4
p34
 
−1 1 0 0  
 −1 0 1 0  q1
 
1
 −1 0 0 1 
 q2 

=
4
 0 −1 1 0 
 q3 
 0 −1 0 1  q4
0 0 −1 1
Inverse transformation is determined by
pseudoinverse of adjacency matrix

21. Reverting to charge variables
qi = pji q4
p14
j q1
p24 p34
p12
p13
qi − qj q2 q3
p
 23 
pji =
p12  N +  
 p13  −1 −1 −1 0 0 0 q1
 
 p14   1 0 0 −1 −1 0   q2 
  =    
 p23   0 1 0 1 0 −1   q3 
 
 p24  0 0 1 0 1 1 q4
p34
 
−1 1 0 0  
 −1 0 1 0  q1
 
1
 −1 0 0 1 
 q2 

=
4
 0 −1 1 0 
 q3 
 0 −1 0 1  q4
0 0 −1 1
Inverse transformation is determined by
pseudoinverse of adjacency matrix

22. Execution times
TImes to solve the QTPIE model
4
10
N6.20
1000
N1.81
100
Solution time (s)
10
1
0.1
Bond-space SVD
Bond-space COF
Atom-space iterative solver
Atom-space direct solver
0.01
4 5
10 100 1000 10 10
N
Number of atoms

23. Atom-space QTPIE vs QEq
1
E QEq
= qi χi + qi qj Jij
i
2 ij
1
E QT P IE
= qi χi +
¯ qi qj Jij
i
2 ij
A charge model with bond
electronegativities is equivalent to one with
renormalized atomic electronegativities
kij Sij (χi − χj ) kij Sij kij Sij χj
χ=
¯ = χi −
j
N j
N j
N

24. Cooperative polarization in
water
+ −→
• Dipole moment of water increases from
1.854 Debye 1 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, e.g. Bernal-Fowler, SPC,
1. D R Lide, CRC Handbook of Chemistry and Physics,
73rd ed., 1992.

25. Outline
The concept of fluctuating charges
Principle of electronegativity equalization
Circuit analogy
The dissociation catastrophe
Fixing the dissociation catastrophe
Charge transfer variables
Charge transfer topologies
Applications
Water
Metal clusters

26. Polarization in water chains
• Use parameters from gas phase
data 1 to model chains of waters
• Compare QTPIE with:
๏ QEq and reparameterized QEq
๏ ˆ
Ab initio DF-LMP2/aug-cc-pVTZHΨ = EΨ
๏ AMOEBA2, an inducible dipole model
1. WF Murphy J. Chem. Phys. 67 (1977), 5877-5882.
2. P Ren and JW Ponder J. Phys. Chem. B 107 (2003),

27. The flexible SPC model
E = kO–H RO–H − 0
RO–H
2 bond
stretch
O–H Urey-
2
+ UB
kH—H RH—H − 0
RH—H Bradley
H—H 1,3 term
angle
2
+ κ∠HOH θ∠HOH − 0
θ∠HOH
∠HOH
torsion
12 6
σO—H σO—H
+ 4 O—H −
RO—H RO—H
O—H,nonbonded
qi qj dispersion
+
Rij
electrostati
ij,nonbonded
cs Dang and Pettitt,

28. Our new water model
2
E = kO–H RO–H − 0
RO–H bond stretch
O–H
Urey-
2
+ UB
kH—H RH—H − 0
RH—H Bradley 1,3
H—H term
2
+ κ∠HOH θ∠HOH − 0
θ∠HOH angle torsion
∠HOH
12 6
σO—H σO—H
+ 4 O—H −
RO—H RO—H
O—H,nonbonded
qi qj dispersion
+ EQTPIE
Rij
ij,nonbonded
electrostatics
LX Dang and BM Pettitt J. Phys. Chem. 91 (1987) 3349-3354.

29. Our new water model
reparameterized
E = kO–H RO–H − 0
RO–H
2 to ab initio (DF-
O–H LMP2/aug-cc-
+ UB 0
kH—H RH—H − RH—H
2
pVTZ) energies,
H—H dipoles and
+ 0
κ∠HOH θ∠HOH − θ∠HOH
2
polarizabilities
∠HOH of sampled
σO—H monomer and
12
σO—H
6
+ 4 O—H −
RO—H dimer
RO—H
O—H,nonbonded
qi qj geometries
+ EQTPIE
Rij
ij,nonbonded

30. Parameterization
1 230 monomers sampled by systematic
variation of coords.
890 dimers sampled from flexible SPC at 30 000
K
Step 2: Fit non-electrostatic parameters with ab
Step 1: Fit electrostatics to dipoles andflexible work
initio energies New QEq QTPIE
Parameter/eV QEq
Parameter
SPC
This
H 4.528 3.678 4.528 LJ radius of 3.1656 1.7055
electronegativit
H hardness 13.89 18.448 11.774 OH/Å
y LJ well depth/ 0.1554 0.2798
O 8.741 9.591 7.651 kcm
bond stretch 527.2 226
electronegativit
O hardness 13.364 17.448 13.364
y eq. bond 1 1.118
length /Å
angle stretch 37.95 40.81
eq. angle/deg. 109.47 111.48
UB stretch 39.9 54.32
UB eq. length/Å 1.633 1.518

31. Dipole moment per water
2.6
Dipole moment per molecule (Debye)
DF-LMP2/aug-cc-PVTZ
2.5 AMOEBA
2.4 QTPIE
2.3 QEq (reparameterized)
2.2
2.1
2.0
1.9
QEq
1.8
0 5 10 15 20 25
Number of molecules

32. Polarizability per water
Longitudinal polarizability per molecule (Å!)
5.0
QEq
4.0 QEq (reparameterized)
3.0
2.0 AMOEBA
DF-LMP2/aug-cc-PVTZ QTPIE
1.0
.0
0 5 10 15 20 25
Number of molecules

33. Polarizability per water
Transverse polarizability per molecule (Å!)
3.5
3.0
QEq
2.5
2.0
QTPIE
1.5 AMOEBA
QEq (reparameterized) DF-LMP2/aug-cc-PVTZ
1.0
0 5 10 15 20 25
Number of molecules

34. Polarizability per water
Out of plane polarizability per molecule (Å!)
1.5
DF-LMP2/aug-cc-PVTZ
1.0 AMOEBA
.5
QTPIE, QEq (reparameterized) and QEq
.0
-.5
0 5 10 15 20 25
Number of molecules

35. Charge transfer in 15 waters
.20
.10
Molecular charge
.00
QEq
-.10
QEq (reparameterized)
QTPIE
DMA Charges
-.20
-.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Index of water molecule

36. Interaction energies in water
clusters
Interaction energy per molecule (kcal/mol)
0
DF-LMP2/aug-cc-pVTZ
-50
QTPIE
AMOEBA
-100
flexible SPC
-150
2 3 4 5 6 7 8 9 10 11 12
Number of water molecules

37. Polarizability of sodium clusters

38. Polarizability of sodium clusters
√
3
3
α= N rs + δ
rs (Å) δ (Å)
QEq 0.987(10)
QTPIE 2.0896(23
Knight )
0.4070(13 4.2573(45)
0.631(73)
Rayane )
2.320(38) 0.55(13)
Tikhonov 2.339(71) 0.71(14)
Liang 2.298(47) 0.763(17)
bulk 2.2370(43
2.12 -
)

39. Freidel oscillation in polarized
Na55(Ih)
QEq QTPIE

40. Freidel oscillations in Na309(Ih)
QEq QTPIE

41. Conclusions
Long-range charge transfer can
be attenuated smoothly at
similar computational cost to
non-attenuated models
A three-site water model can
correctly describe in-plane
polarizability scaling
quantitatively, and charge
transfer behavior qualitatively
Data from sodium clusters
suggest need to develop
models that interpolate
smoothly between metallic and