Constructing a rigorous fluctuating-charge model for molecular mechanics
Jiahao ChenSeptember 19, 2006
Acknowledgments•Todd Martínez•Martínez Group members,esp. Ben Levine
FundingNSF DMR-03 25939 ITRDOE DE-FG02-05ER46260
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Molecular mechanics is useful
• Since atomic nuclei behave mostly classically, molecularmechanics (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.
water flow in aquaporins1 mechanical deformation in ceramics2
Molecular mechanics• Classical energy function with bonded and
nonbonded terms
• Nuclear motions propagated using classicalequations of motion
Molecular electrostatics
Van der Waals interactions
time 0• Ab initio molecular dynamics (MD)
nuclear forces from wavefunction
• MM/MDnuclear forces from fixed charge distribution
• MM/MD cannot describe chemical reactions
MM leaves out something
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time
specified
QEq1, a fluctuating charge model
• Given geometry, find charge distribution
• Minimization with fixed total chargedefines Lagrange multiplier μ
1. A. K. Rappe, W. A. Goddard III, J. Phys. Chem. 95 (1991) 3358-3363.
q5q4
q3
q2
q1
energy to charge atom Coulomb interaction
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
N, V, T
i
moleculeatom
Energy derivatives: chemical potential μ, hardness
Physical interpretation of QEq
• Three-point approximation for derivatives
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.
E
N
EA
IP
N0+1N0-1 N0
Mulliken1
Parr-Pearson2
Why QEq is bad
• Wrong asymptotic charges predicted
• No penalty for long-range charge transfer• Overestimates molecular electrostatic properties• Especially bad far from equilibrium
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0R / Å
q/e
QEq
equilibrium geometry
MullikenDMA
Ideal dipole
ab initiocharges
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
QTPIE: charge transfer withpolarization current equilibration
• Shift focus to charge transfer variables pji:– Charge accounting: where it came from, where it’s
going
– Explicitly penalize long-distance charge transfer
p12
p23
p34
p45
NaCl asymptote correct
• QTPIE prediction improved over QEq, even withoutreoptimized parameters
• Slope wrong: cannot capture nonadiabatic effects0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0R / Å
q/e
QEq
QTPIE
DMA
equilibrium geometry
Water fragments correctly
• Asymmetric dissociation: correct asymptotics, chargetransfer on OH fragment retained
-1.0
-0.5
0.0
0.5
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
R/Å
q/e equilibrium geometry
R
Water parameters transferable
• Parameters transferable across geometries
H
O H
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.5 2.5 3.5 4.5R/Å
q/e
QEq
QEq
QTPIE
QTPIEDMA
DMA
Water parameters transferable
• Parameters transferable across geometries
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.5 2.5 3.5 4.5R/Å
q/e
H
O H
QEq
QEq
QTPIE
QTPIEDMA
DMA
Water parameters transferable
• Parameters transferable across geometries
H
O H
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0R/Å
q/e
QEq
QEq
QTPIEQTPIEDMA
DMA
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0R/Å
q/e
H
O H
Water parameters transferable
• Parameters transferable across geometries
QEq
QEq
QTPIEQTPIEDMA
DMA
Water parameters transferable
• Parameters transferable across geometries
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0R/Å
q/e
H
O H
QEq
QEq
QTPIEQTPIEDMA
DMA
H
O H
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0R/Å
q/e
QEq
QEq
QTPIEQTPIEDMA
DMA
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.5 2.5 3.5 4.5R/Å
q/e
H
O H
QEq
QEq
QTPIEQTPIEDMA
DMA
H
O H
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.5 2.5 3.5 4.5R/Å
q/e
QEq
QEq
QTPIEQTPIEDMA
DMA
Dipole polarizability of phenol• Response of dipole moment to external electricfield
• QTPIE: overestimates less than QEq
• Out-of-plane component missing in QEq, QTPIE• MP2/STO-3G suggests this is largely because of
inflexible basis set
z
y
x
6.99810.85950.00000.0000
12.36217.048810.756620.3270
13.67588.424013.029824.6244
MP2/aug-cc-pVDZ
MP2/STO-3G
QTPIE/STOQEq/STO
(Å )
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
• Similar results for other ab initio methods, e.g.FCI/STO-3G, RHF/aug-cc-pVDZ…
3.1272.059EA(O)
1.0000.280EA(C)
-0.068-2.417EA(H)
14.56515.423IP(O)
9.60710.406IP(C)
13.58811.473IP(H)
ab initioOriginal(eV)
6.99810.00000.0000
12.362111.131610.7566
13.675813.428513.0298
ab initioNew QTPIEOld QTPIE
Eigenvalues of dipolepolarizability tensor/Å
Dealing with charged systems I• Constrained minimization with Lagrange
multipliers
– Problem 1: Cannot be enforced for diatomic molecule
– Problem 2: Generalizing to non-zero diagonal chargetransfer variables destroys asymptotic property
– Model has insufficient constraints at large bondlengths to guarantee integer charges
and
Dealing with charged systems II• Redefine atoms with formal charges
• Problem: must account for multiple references
IP0, EA0
IP+1, EA+1
IP0, EA0
IP0, EA0
IP0, EA0
IP0, EA0
IP+1, EA+1
IP0, EA0
IP0, EA0
+ …+- e-
++
E
NEA
IP
N0+1N0-1 N0
E
N
EA+1
IP+1
N0-1 N0N0-2
- e-
Test case - water : phenol : sodium -stack• Chemically “obvious”
localized charge• Reparameterization
appears to work well forQTPIE
• Need to figure outextension to generalsystems
0.86480.4798reparam.
0.18760.6177Lagrange
QTPIEQEqqNa/e
Mulliken/MP2/cc-pVDZ charge: 0.7394
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?
Conclusions
• Focus on charge transfer and including distance penaltyimproves description of atomic charges
NoNoExcited states
No, O(N4)Yes, O(N2)Practical scaling
Some evidenceYes*Coarse-graining picture
NoYes*Arbitrary total charge
Almost!NoCorrect molecular electrostatics
YesNoCorrect asymptotics
YesYesTransferable parameters
QTPIE (now)QEqFluctuating-charge model
EstablishedNew resultIn progressNeed ideas
*with caveats