Slides for my talk at the APS (March 2007) national meeting.

1. Representing molecules as atomic‐scale electrical
circuits with fluctuating‐charge models
+ +
‐ ‐
q1 q2
Jiahao Chen
Department of Chemistry and Beckman Institute
University of Illinois at Urbana‐Champaign
APS Meeting P19.5, 2007‐03‐07
Acknowledgments Funding
•Todd Martínez •NSF DMR‐03 25939 ITR
•Martínez Group members •DOE DE‐FG02‐05ER46260

2. Polarization and charge transfer in
molecular mechanics (MM)
• Want to describe both polarization and charge
transfer with reasonable computational cost
• Common models to describe polarization:
– Charge‐on‐spring/Drude oscillator, e.g. Drude (1902)
– Point‐polarizable dipole, e.g. Vesely (1977)
– Chemical potential equilibration (CPE), a.k.a.
fluctuating‐charge: Rappé and Goddard (1991); Rick,
Stuart and Berne (1994)
• Only CPE models can account for both effects
P. Drude, The Theory of Optics, Longmans, Green and Co., New York (1902); F.J. Vesely,
J. Comp. Phys. 24 (1977), 361‐371; A. K. Rappé, W. A. Goddard, III, J. Phys. Chem. 95
(1991), 3358‐3363; S. W. Rick, S. J. Stuart, B. J. Berne, J. Chem. Phys. 101 (1994), 6141‐6156.

3. A simple DC circuit
DC source + capacitor
V C
‐
ground
0 V

4. A simple DC circuit
What is the charge q on C?
energy depleted energy gain
from DC source of capacitor
DC source + capacitor charge
V
‐
C q E = −qV + 1 C −1 q 2
2
∂E −1
= −V + C q = 0
ground ∂q
0 V ∴q =VC
This Hamiltonian approach works for molecules too:
fluctuating‐charge/electronegativity equilibration models

5. CPE models: The QEq model
QEq model for a diatomic molecule
source capacitance
term term
electronegativity X 1 2
+ +
χ1 χ2
E = qi χi + ηi qi
i
2
‐ ‐
Coulomb 1X Coulomb
interaction
+ qi qj Jij term
chemical 2
hardness η q J12 ∂E i6=j
1 1 η2 q2 =μ
∂qi
chemical
μ
potential
A. K. Rappé, W. A. Goddard III, J. Phys. Chem. 95 (1991), 3358‐3363.

6. QEq: wrong NaCl dissociation
1.0
q/e equilibrium geometry
0.9
0.8
+ +
0.7 ‐ ‐
0.6
0.5 QEq
0.4 QEq, R → ∞
0.3 + +
‐ J12 → 0 ‐
0.2
0.1
ab initio DMA0
CASSCF(8/5)/6‐31G*
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R/Å 8.0
DMA0 = distributed multipole analysis restricted to point charges only
CASSCF = complete active space self‐consistent field method

7. The QTPIE model: Motivation
X
1. Introduce charge transfer variables qi = pji
X 1 2 X j
EQEq = qi χi + ηi qi + qi qj Jij
i
2
i6=j
X X1 1X
= pji χi + ηi pji pki + pki plj Jij
ij
2 2
ijk ijkl
2. Introduce overlap integral: explicit notion of distance
X X1 1X
EQTPIE = pji χi Sij + ηi pji pki + pki plj Jij
ij
2 2
ijk ijkl
∂EQTPIE
=0
∂pji
J. Chen, T. J. Martínez, Chem. Phys. Lett., in press.

8. QTPIE: Correct NaCl asymptote
1.0
0.9
q/e equilibrium geometry
0.8
0.7
0.6
0.5 QEq
0.4
0.3
QTPIE
0.2
0.1
ab initio
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R/Å 8.0
QTPIE prediction improved over QEq without reoptimizing
parameters, but variation is still slower than ab initio

9. Water fragments correctly
• Asymmetric dissociation: correct asymptotics, charge
transfer on OH fragment retained
1.0
q/e equilibrium geometry
ab initio R
0.5
QEq
R/Å
QTPIE
0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
‐0.5
‐1.0

10. Water parameters transferable
1.0 • Parameters transferable across geometries
q/e 1.0
q/e
0.8
O H 0.8
0.6 O H
H 0.6
0.4
0.4
H
DMA
0.2 0.2 DMA
0.0 QEq 0.0 QEq
R/Å QTPIE R/Å QTPIE
‐0.2 0.5 1.5 2.5 3.5 4.5
QTPIE‐0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
QTPIE
‐0.4 DMA ‐0.4 DMA
‐0.6 ‐0.6
‐0.8 QEq ‐0.8 QEq
‐1.0 ‐1.0
1.0 1.0
q/e q/e
0.8 0.8
O H O H
0.6 0.6 H
0.4 H 0.4
0.2
DMA 0.2 DMA
0.0 QEq 0.0 QEq
R/Å QTPIE R/Å QTPIE
‐0.2 0.5 1.5 2.5 3.5 4.5 ‐0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
QTPIE
QTPIE
‐0.4 DMA ‐0.4
DMA
‐0.6 ‐0.6
‐0.8 QEq ‐0.8 QEq
‐1.0 ‐1.0

11. Dipole polarizability of phenol
• Response of dipole moment to external electric
field
• QTPIE: overestimates less than QEq
QEq QTPIE ab initio*
x 24.6244 13.0298 13.6758
y 20.3270 10.7566 12.3621
z 0.0000 0.0000 6.9981 (ų)
*ab initio method: MP2/aug‐cc‐pVDZ

12. Conclusions
• Fluctuating‐charge models are analogous to DC
electrical circuits
• QTPIE (our new charge model) predicts correct
dissociation behavior of atomic charges
• Explicit distance cutoff for electronegativities
improves qualitative behavior
Thank You

13. QEq v. ab initio charges
1.2
q/e
equilibrium geometry
1.0
0.8
QEq
0.6
Mulliken
ab initio
0.4 DMA
charges
Ideal dipole
0.2
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R/Å 8.0

14. QEq1, a fluctuating charge model
• Given geometry, find charge distribution
energy to charge atom Coulomb interaction
q1
q2
q3
• Minimization with fixed total charge q4 q5
defines Lagrange multiplier μ
1. A. K. Rappe, W. A. Goddard III, J. Phys. Chem. 95 (1991) 3358‐3363.

15. QTPIE: charge transfer with
polarization current equilibration
• Shift focus to charge transfer variables pji:
– Charge accounting: where it came from, where it’s
going p 12
p23
p34
p45
– Explicitly penalize long‐distance charge transfer