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