Slides for my prelim exam in September 2006.

1. Constructing a rigorous fluctuating-
charge model for molecular mechanics
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Funding Acknowledgments
NSF DMR-03 25939 ITR •Todd Martínez
DOE DE-FG02-05ER46260 •Martínez Group members,
esp. Ben Levine
Jiahao Chen
September 19, 2006

2. Molecular mechanics is useful
water flow in aquaporins1 mechanical deformation in ceramics2
• Since atomic nuclei behave mostly classically, molecular
mechanics (MM) is a useful method for doing dynamics
• In MM, classical electrostatic effects are important,
including polarization
1. E. Tajkhorshid et. al., Science 296 (2002), 525-530.
2. P. S. Branicio, R. K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 96 (2006), art. no. 065502.

3. Molecular mechanics
• Classical energy function with bonded and
nonbonded terms
Van der Waals interactions
Molecular electrostatics
• Nuclear motions propagated using classical
equations of motion

4. MM leaves out something
time 0 time
• Ab initio molecular dynamics (MD)
nuclear forces from wavefunction
• MM/MD
nuclear forces from fixed charge distribution
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specified
• MM/MD cannot describe chemical reactions

5. 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.

6. Physical interpretation of QEq
• In equilibrium:
– each atom i has the same chemical potential μ
– μ uniquely determines the atomic charges qi
• Atoms interpreted as subsystems in equilibrium
molecule
i atom
N, V, T
Energy derivatives: chemical potential μ, hardness

7. Physical interpretation of QEq
• Three-point approximation for derivatives
Mulliken1
E Parr-Pearson2
IP
EA
N
N0-1 N0 N0+1
1. R. S. Mulliken, J. Chem. Phys. 2 (1934) 782-793.
2. R. G. Parr, R. G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512-7516.

8. Why QEq is bad
• Wrong asymptotic charges predicted
1.2
q/e
equilibrium geometry
1.0
0.8
0.6 QEq
Mulliken
0.4 ab initio
DMA
charges
0.2 Ideal dipole
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R/Å
8.0
• No penalty for long-range charge transfer
• Overestimates molecular electrostatic properties
• Especially bad far from equilibrium

9. New charge model: Desiderata
• Transferable parameters
– Generic, application-independent
– No atom typing
• Accurate
– Able to describe polarization and charge transfer
– Correct asymptotic charge distributions
– Predicts electrostatic properties accurately
• Flexible
– Able to handle arbitrary total charge
– Able to describe electronic excited states
• Rigorous
– Well-defined coarse-graining picture from conventional
electronic structure methods
• Practical to compute
– O(N ) or better
– Faster than conventional electronic structure methods

10. 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

11. NaCl asymptote correct
• QTPIE prediction improved over QEq, even without
reoptimized parameters
1.2
q/e
1.0 equilibrium geometry
0.8
0.6
QEq
0.4
QTPIE
0.2
DMA
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R / Å8.0
• Slope wrong: cannot capture nonadiabatic effects

12. Water fragments correctly
• Asymmetric dissociation: correct asymptotics, charge
transfer on OH fragment retained
1.0
q/e equilibrium geometry
R
0.5
R/Å
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

13. Water parameters transferable
• Parameters transferable across geometries
1.0
q/e
0.8
O H
0.6
H
0.4 DMA
0.2
QEq
0.0 QTPIE
R/Å
QTPIE
-0.20.5 1.5 2.5 3.5 4.5
DMA
-0.4
-0.6
QEq
-0.8
-1.0

14. Water parameters transferable
• Parameters transferable across geometries
1.0
q/e
0.8
O H
0.6
0.4 DMA
H
0.2
QEq
0.0 QTPIE
R/Å
-0.20.5 1.5 2.5 3.5 4.5 QTPIE
DMA
-0.4
-0.6
QEq
-0.8
-1.0

15. Water parameters transferable
• Parameters transferable across geometries
1.0
q/e
0.8
O H
0.6
0.4 H
DMA
0.2
QEq
0.0 QTPIE
R / Å QTPIE
-0.20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
DMA
-0.4
-0.6
QEq
-0.8
-1.0

16. Water parameters transferable
• Parameters transferable across geometries
1.0
q/e
0.8
O H
0.6 H
0.4
DMA
0.2
QEq
0.0 QTPIE
R / Å QTPIE
-0.20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
DMA
-0.4
-0.6
QEq
-0.8
-1.0

17. 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
0.6
H
0.4
0.4 H
DMA
0.2 0.2 DMA
0.0 QEq QEq
R/Å QTPIE 0.0 R / Å QTPIE
-0.20.5 1.5 2.5 3.5 4.5
QTPIE-0.20.5 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 R / Å QTPIE
QTPIE
-0.20.5 1.5 2.5 3.5 4.5 -0.20.5 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

18. Dipole polarizability of phenol
• Response of dipole moment to external electric
field
• QTPIE: overestimates less than QEq
QEq/STO QTPIE/STO MP2/STO- MP2/aug-cc-
3G pVDZ
x 24.6244 13.0298 8.4240 13.6758
y 20.3270 10.7566 7.0488 12.3621
z 0.0000 0.0000 0.8595 6.9981 (Å )
• Out-of-plane component missing in QEq, QTPIE
• MP2/STO-3G suggests this is largely because of
inflexible basis set

19. QTPIE = coarse-grained ab initio?
• Reparameterizing with ab initio (MP2/aug-cc-
pVDZ) IPs and EAs improves agreement of in-
plane polarizabilities at same level of theory
(eV) Original ab initio Eigenvalues of dipole
IP(H) 11.473 13.588 polarizability tensor/Å
IP(C) 10.406 9.607 Old QTPIE New QTPIE ab initio
IP(O) 15.423 14.565 13.0298 13.4285 13.6758
EA(H) -2.417 -0.068 10.7566 11.1316 12.3621
EA(C) 0.280 1.000 0.0000 0.0000 6.9981
EA(O) 2.059 3.127
• Similar results for other ab initio methods, e.g.
FCI/STO-3G, RHF/aug-cc-pVDZ…

20. Dealing with charged systems I
• Constrained minimization with Lagrange
multipliers
– Problem 1: Cannot be enforced for diatomic molecule
and
– Problem 2: Generalizing to non-zero diagonal charge
transfer variables destroys asymptotic property
– Model has insufficient constraints at large bond
lengths to guarantee integer charges

21. Dealing with charged systems II
• Redefine atoms with formal charges
E E
IP+1
IP - e-
EA+1
EA
N N
N0-1 N0 N0+1 N0-2 N0-1 N0
• Problem: must account for multiple references
IP0, EA0 IP+1, EA+1 IP0, EA0
- e- +
+ +…
IP0, EA0 IP0, EA0 + IP0, EA0
IP0, EA0 IP0, EA0 IP+1, EA+1

22. Test case - water : phenol : sodium -stack
• Chemically “obvious”
localized charge
• Reparameterization
appears to work well for
QTPIE
• Need to figure out
extension to general
systems
qNa/e QEq QTPIE
Lagrange 0.6177 0.1876
reparam. 0.4798 0.8648
Mulliken/MP2/cc-pVDZ charge: 0.7394

23. Outlook
• QTPIE is a promising new charge model
– Implement scalable solution algorithm
– Interface with MD code
– Chemical applications, e.g. enzyme-substrate
docking, electrochemistry
• Many open theoretical questions, e.g.:
– How to account for out-of-plane polarizabilities?
– When does a molecule stop being a molecule?
– What is the quantum-mechanical analogue of charge
transfer variables?
– How to deal with excited states?

24. Conclusions
• Focus on charge transfer and including distance penalty
improves description of atomic charges
Fluctuating-charge model QEq QTPIE (now)
Transferable parameters Yes Yes
Correct asymptotics No Yes
Correct molecular electrostatics No Almost!
Established
Arbitrary total charge Yes* No
New result
Coarse-graining picture Yes* Some evidence In progress
Practical scaling Yes, O(N2) No, O(N4) Need ideas
Excited states No No
*with caveats