My final defense presentation!

1. Theory and applications of
fluctuating charge models
Jiahao Chen
Martínez Group
Dept. of Chemistry, Frederick Seitz Materials
Research Laboratory and the Beckman Institute
University of Illinois at Urbana-Champaign
Stanford Linear Accelerator Center
Dept. of Chemistry and Dept. of Photon Sciences
Stanford University

2. Acknowledgments
Committee
Prof. Nancy Makri
Prof. Duane Johnson
Prof. Dirk Hundertmark
Discussions
Prof. Susan Atlas (UNM)
Dr. Ben Levine (UPenn)
Prof. Todd J. Martínez Dr. Steve Valone (LANL)
Martínez Group and friends Prof. Troy van Voorhis (MIT)
$: DOE

3. “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.

4. Electronic structure and dynamics
ˆ What is the charge distribution?
HΨ = EΨ
direct density coarse-
semiempirical molecular continuum
ab initio
numerical functional grained
theories methods models (MM) electrostatics
quadrature theory models
numerical quadrature classical coarse-
finite element
ab initio molecular dynamics
path integrals molecular grained
methods
emiclassical dynamics dynamics dynamics
ˆ ˙ What does the system do?
HΨ = iΨ

5. Electronic structure and dynamics
ˆ What is the charge distribution?
HΨ = EΨ
direct density coarse-
semiempirical molecular continuum
ab initio
numerical functional grained
theories methods models (MM) electrostatics
quadrature theory models
numerical quadrature classical coarse-
finite element
ab initio molecular dynamics
path integrals molecular grained
methods
emiclassical dynamics dynamics dynamics
ˆ ˙ What does the system do?
HΨ = iΨ

6. Molecular models/force fields
Typical energy function
E = covalent bond effects
+
noncovalent interactions

7. Molecular models/force fields
Typical energy function
kb (rb − rb )2 + κa (θa − θa )2 +
E= ldn cos(nπ)
0
0
d∈dihedrals n
a∈angles
b∈bonds
bond stretch angle torsion dihedrals
+
-
12 6
σij σij
qi qj
+ 4 −
+
+ ij
rij rij rij
i<j∈atoms i<j∈atoms
dispersion
electrostatics
Usually fixed charges

8. Molecular models/force fields
Typical energy function
kb (rb − rb )2 + κa (θa − θa )2 +
E= ldn cos(nπ)
0
0
d∈dihedrals n
a∈angles
b∈bonds
bond stretch angle torsion dihedrals
+
-
12 6
σij σij
qi qj
+ 4 −
+
+ ij
rij rij rij
i<j∈atoms i<j∈atoms
dispersion
electrostatics
Usually fixed charges

9. Why care about polarization
and charge transfer?
They are important in
condensed phases, where
most chemistry and
biology happens

10. Polarization in chemistry
• Ex. 1: Stabilizes carbonium in lysozyme
carbonium
forms
sugar bond
cleaved
• Ex. 2: Hydrates chloride in water clusters
TIP4P/FQ OPLS/AA
non-polarizable
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), 11934 -11943.

11. Fluctuating charges
-0.3
charge transfer = 1.1e charge transfer = 0.2 e
-0.5 χ2 , η2
+0.8 charge transfer = 1.3 e
χ3 , η3
Response = change in atomic charges
Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.

12. Charge formation vs.
charge-charge interactions
Electronic Coulomb
energy of atom interactions
1
= Eat (qi ) +
E qi qj Jij
2
i i=j
12 1
∂2E
∂E
= + + ··· +
qi qi qi qj Jij
2 2
2
∂qi ∂qi
qi =0 qi =0
i i i=j
12 1
= qi χi + qi ηi + · · · + qi qj Jij
2 2
i i i=j
chemical
hardness
electronegativity
R. P. Iczkowski and J. L. Margrave J. Am. Chem. Soc. 83:(1961), 3547–3551

13. Electronegativity
IP + EA
χ=
2
R. S. Mulliken J. Chem. Phys 2:(1934), 782–793
Electronegativity: “Concept introduced by L. Pauling as
the power of an atom to attract electrons to itself.”
IUPAC Compendium of Chemical Terminology,
aka “The Gold Book”, goldbook.iupac.org

14. A quantitative definition
IP + EA
=
χ
2
E(N − 1) − E(N + 1)
=
2
∂E
∼
∂N
R. S. Mulliken J. Chem. Phys 2:(1934), 782–793
R. P. Iczkowski and J. L. Margrave J. Am. Chem. Soc. 83:(1961), 3547–3551
R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke J. Chem. Phys. 68:(1978), 3801–3807

15. Chemical hardness
= IP − EA
η
2
∂E
= 2
∂N
R. G. Parr, R. G. Pearson J. Am. Chem. Soc. 105:(1983), 7512–7516

16. QEq, a fluctuating-
charge model
1
E= qi χi + qi qj Jij
2
atomic screened
i ij Coulomb
electronegativities
“voltages” interactions
φ2 (r1 )φ2 (r2 )
i j
Jij = dr1 dr2
|r1 − r2 |
R3×2
ni −1 −ζi |r−Ri |
φi (r) = Ni |r − R| e
AK Rappé and WA Goddard III J. Phys. Chem. 95 (1991), 3358-3363.

17. Principle of electronegativity
equalization
1
Minimize energy E= qi χi + qi qj Jij
2
i ij
qi = Q
subject to charge constraint
i
Using the method of Lagrange multipliers, reduces to
solving the linear equation
J 1 q −χ
=
0
0
1 T
µ
(electronic) chemical potential

18. Physical interpretation
In equilibrium:
o each atom i has the same chemical potential µ
o µ uniquely determines the atomic charges qi
Atoms are subsystems in equilibrium
molecule
Ω
Ωi atom
N, V, T
Energy derivatives: chemical potential µ, hardness η

19. QEq, a fluctuating-
charge model
1
E= qi χi + qi qj Jij
2
atomic screened
i ij Coulomb
electronegativities
“voltages” interactions
φ2 (r1 )φ2 (r2 )
i j
Jij = dr1 dr2
|r1 − r2 |
R3×2
ni −1 −ζi |r−Ri |
φi (r) = Ni |r − R| e
AK Rappé and WA Goddard III J. Phys. Chem. 95 (1991), 3358-3363.

20. QEq has wrong asymptotics
1.0
q/e
Na Cl
R
0.8
χ1 − χ2
q=
J11 + J22 − J12
0.6
QEq
asymptote ~ 0.43 ≠ 0
0.4
0.2
ab initio
R/Å
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

21. Fluctuating-charge models map
molecules onto electrical circuits
screened
electro- chemical Coulomb
molecule negativity hardness interaction

22. Fluctuating-charge models map
molecules onto electrical circuits
screened
electro- chemical Coulomb
molecule negativity hardness interaction
electric (inverse) Coulomb
electrical
potential capacitance interaction
circuits

23. Fluctuating-charge models map
molecules onto electrical circuits
screened
electro- chemical Coulomb
molecule negativity hardness interaction
electric (inverse) Coulomb
electrical
potential capacitance interaction
circuits
More electropositive
χ
- Voltage +
η 1
1
χ
η 2
0V
2
More electronegative

24. QEq has wrong asymptotics
1.0
q/e
Na Cl
R
0.8 +
+
χ1 − χ2
-
-
q=
J11 + J22 − J12
0.6
QEq
asymptote ~ 0.43 ≠ 0
0.4
+
+
J12 → 0 -
-
0.2
ab initio
R/Å
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

25. In fluctuating-charge
models like QEq, all
molecules are metallic

26. Problems due to metallicity
Fractional charge distributions predicted
for dissociated systems
Overestimates charge transfer for
stretched / reactive geometries
In practice, existing models must
introduce ad hoc cutoffs on charge flows
Polarizabilities are not size-extensive

27. QTPIE, our new charge model
Charge-transfer with polarization current
equilibration
Voltage attenuates with increasing distance
voltage
η
2
distance
J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.

28. QTPIE, our new charge model
Charge-transfer with polarization current
equilibration
Voltage attenuates with increasing distance
voltage
η
2
η
2
distance
J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.

29. Making QTPIE (Step 1)
To make the proposed change, first change variables
qi = pji
p12
j
Charge transfer variables quantify how much
charge went from one atom to another, and are p23
indexed over pairs
1 p34
E= qi χi + qi qj Jij p45
2 ij
i
Still QEq! 1
Same model, = pji χi + pki plj Jij
2
new representation ij ijkl
J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.

30. Making QTPIE (Step 2)
atomic electronegativities become bond electronegativities
1
= pji χi +
QEq
E pki plj Jij
2
ij ijkl
1
= pji χi kij Sij +
QT P IE
E pki plj Jij
2
ij ijkl
Sij = φi (r)φj (r)dr
R3
J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.

31. QTPIE has correct limit
1.0
q/e
Na Cl
R
0.8
χ1 − χ2
q=
J11 + J22 − J12
0.6
QEq
(χ1 − χ2 )S12
0.4 q=
J11 + J22 − J12
QTPIE
0.2
ab initio
R/Å
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

32. However...
1
= qi χi +
QEq
E qi qj Jij
2
i ij
1
= pji χi kij Sij +
QT P IE
E pki plj Jij
2
ij ijkl
N times as many variables as before - costly!
Equations are rank deficient - need SVD

33. Origin of rank deficiency
Charge transfer variables are massively
redundant due to Kirchhoff’s voltage law
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 atomic charges

34. 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

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

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

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

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

39. Atom-space QTPIE vs QEq
1
= qi χi +
QEq
E qi qj Jij
2
i ij
1
= qi χi +
¯
QT P IE
E qi qj Jij
2
i 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 −
N N N
j j j

40. 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,
e.g. Bernal-Fowler, SPC, TIPnP...
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.

41. Polarization in water chains
• Use parameters from gas phase data to
model chains of waters
• Compare QTPIE with:
QEq and reparameterized QEq
๏
ˆ
Ab initio DF-LMP2/aug-cc-pVTZ
๏ HΨ = 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), 5933-5947.

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

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

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

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

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

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

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

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

50. 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

51. Summary
• Polarization and charge transfer are important
effects usually neglected in classical MD
• Our new charge model corrects deficiencies in
existing fluctuating-charge models at similar
computational cost
• We obtain quantitative polarization and qualitative
charge transfer trends in linear water chains