Models for computing partialcharges

Jiahao ChenMartínez Group Meeting

September 27, 2005

Outline

• An atom-site charge model: QEq– Results for amino acids– NaCl dissociation– Reparameterization study

• A minimal bond-space model– Study of NaCl.6H2O dissociation

• Quantum mechanical analogs– Derivative discontinuities

Molecular charge distributions

• Molecules as clusters of point charges• Electrostatics in the classical limit• Useful for molecular modeling

Point charge models

• Key atomic parameters:– Electronegativity– Hardness

• Mulliken definitions– Ionization potential– Electron affinity

• Sanderson electronegativity equilibration

Iczkowsky, R. P.; Margrave, J. L., J. Am. Chem. Soc. 83, 1961, 3547-3553.

QEq: Rappé and Goddard, 1991

• Parameters: Mulliken electronegativitiesand hardnesses

internal energy Coulombinteraction

Rappé, A. K.; Goddard, W. A. III, J. Phys. Chem. 95, 1991, 3358-3363.

QEq (continued)• Screened Coulomb interaction: two-

electron integrals over ns-ms STOs

• Sanderson electronegativity equalizationprinciple

• Linear system of simultaneous equations

QEq: Electrical interpretation• Molecules as classical

circuits

(Ideal)Wire

Bond

Capacitor+ Resistor

Atom

QEqCircuitelement

QEq on equilibrium geometries

• Compare QEq results with ab initiocalculations for ground state geometries

• Molecules: 20 naturally occurring aminoacids

• Ab initio method:– MP2 geometry optimization– DMA0 (distributed multipole analysis)

charges: 0th order = monopoles

-1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

QEq v. DMA0 on MP2/6-31G*

S

C

CHx

NH2

N, NH

OHO

QEq v. DMA0 on MP2/cc-pVDZ

ab initio

QEq

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

S

CCHx

NH2

N, NH

OHO

QEq v. DMA0 on MP2: Results

• Only singly bonded atoms have goodagreement (Δq<0.1)– Deviations: 1° > 2° > aromatic > 3°– N termini– Hydrocarbons– Carboxyls, imines…

• Higher correlation between QEq andDMA0 on MP2/cc-pVDZ

Does QEq neglect polarizability?

• 6-31G v. 6-31G* on Cys: very similar

-1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

QEq on Diatomics• Compare QEq results with experimental

results for diatomics• Molecule: NaCl (g)• Dipole moments from experimental

literature• Given bond length, can QEq predict the

dipole moment ?• QEq parameters derived from fit to

experimental dipole moments

0 2 4 6 8 10 12 14 16 18 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

QEq results: NaCl dissociation

qN a_R ! 1 __ __ :___6_ _

Â_R_

qNa _R _

qNa _R _ R eq_

Too slow!

Not zero!

QEq: What is Missing?

• No HOMO-LUMO band gap!– All bonding is completely metallic

• Wrong asymptotic limit of quantumstatistical mechanics– Have: No Fermi gap => T ∞ limit– Need: Ground state only => Want T 0 limit!

• No notion of bond length and bond order– All atoms are pairwise “σ”–bonded together!

• No out-of-plane polarizability

QEq: Parameterization

• Can reparameterizing QEq improve itsaccuracy?

• Molecules: 94 diatomics• Benchmark: experimental (and high-

precision computational) dipole moments• Partial charges from ideal dipole model

• χ² goodness-of-fit minimization

+q -qr

Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules, VanNostrand Reinhold, 1978, New York, NY.

NaClNaBr

NaI

KCl

KBr

KI

RbCl

RbBr

RbI

CsCl

CsBr

CsI

LiF

LiCl

LiBrLiI

NaF

KFRbF

CsF

CO

CF

NO

NS

OH

SiO

PNSO

SH

ClF

ClO

BrCl

BrF

IBr

ICl

HI

HBr

HCl

HF

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

QEq: Original parameters

Expt.

QEq

NaClNaBrNaI

KCl

KBr

KI

RbCl

RbBr

RbI

CsCl

CsBr

CsI

LiF

LiClLiBrLiI

NaF

KFRbF

CsF

CO

CF

NO

NSOH

SiO

PN

SO

SHClF

ClO

BrCl

BrF

IBr

ICl

HI

HBr

HClHF

NaClNaBrNaI

KCl

KBr

KI

RbCl

RbBr

RbI

CsCl

CsBr

CsI

LiF

LiClLiBrLiI

NaF

KFRbF

CsF

CO

CF

NO

NSOH

SiO

PN

SO

SHClF

ClO

BrCl

BrF

IBr

ICl

HI

HBr

HClHF

NaClNaBrNaI

KCl

KBr

KI

RbCl

RbBr

RbI

CsCl

CsBr

CsI

LiF

LiClLiBrLiI

NaF

KFRbF

CsF

CO

CF

NO

NSOH

SiO

PN

SO

SHClF

ClO

BrCl

BrF

IBr

ICl

HI

HBr

HClHF

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

QEq: Optimized parameters

Expt.

QEq

-1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

QEq: New parameters on aa’s

-1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Worse than before!

QEq Reparamet.: Conclusions

• Optimization procedure is insufficient toimprove parameter quality beyond thestandard values.

• Lack of sufficient data, esp. for radicalsand ions.

• Published parameters likely to be optimal,despite physical difficulty in interpretatione.g. EA(H) <0

Outline

• An atom-site charge model: QEq– Results for amino acids– NaCl dissociation– Reparameterization study

• A minimal bond-space model– Study of NaCl.6H2O dissociation

Electronegativity, revisited• Many definitions and scales

– Pauling, Mulliken– Different dimensionalities!

• Intrinsic chemical potential for electrons• Substantial empirical evidence for

variations depending on context, e.g., C-Cv. C=C

• Electronegativity a characteristic of bonds,rather than atoms?

Charge-transfer model

• “Derivation”– Replace electronegativity by distance-

dependant electronegativity– Replace charges by charge-transfer variables

– Impose detailed balance

• Sum over CTs are deviations fromreference charge, not actual charge per se

nQEq: Formulation

• Linear system of simultaneous equations

Computation

• Cast system into matrix problem• Degenerate system of equations

– Singular value decomposition– Generalized Moore-Penrose inverse

(psuedoinverse)

O+2δ

H+δ H+δ

O

H H

η η

Theoretical Results• Singular values/zero eigenvalues correspond to

closed loops of circulation– Faraday’s Law– Linear responses

• N-1 nonzero eigenvalues/singular values– N-1 linearly independent flow variables– Minimum spanning tree for N nodes has N-1 edges

0 2 4 6 8 10 12 14 16 18 20

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Results: NaCl

Correct asymptotic limit!

0 2 4 6 8 10 12 14 16 18 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14 16 18 20

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Results: H2O

Solvation of salt in H2O 6-mer• 6-mer known to be

smallest clusterneeded to fullysolvate NaCl

• Sudden limit ofdissociationdynamics: no solventreorganization

R(Na-Cl)/Å

q/e

0 2 4 6 8 10 12 14 16 18 20

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Results: NaCl.6H2O

nonvanishing residue

0 2 4 6 8 10 12 14 16 18 20

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Representations in Bond-space

• How to describe molecule in bond space?– Bonds : Adjacency matrices– Atoms and Bond lengths: Metrized graphs

• How to solve for electrostatic equilibrium?– Topological/geometric properties– Cutoffs for Coulomb interactions (optional)

Numerical Issues

• For large systems, algorithm does not findcorrect dissociation limits

• Large residual found• Low condition number• What’s going on?

Future work• Look at adiabatic limit of NaCl.6H2O

dissociation– Need ab initio equilibrium geometries

• Computation of molecular properties– Dipole moments– Polarizabilities– pKa?

• More efficient algorithm for solving model– Graph/network flow algorithms?

Outline

• An atom-site charge model: QEq– Results for amino acids– NaCl dissociation– Reparameterization study

• A minimal bond-space model– Study of NaCl.6H2O dissociation

• Quantum mechanical analogs– Derivative discontinuities

Quantum Analogues?

• Quantum analogue of partial charges?– Spin-statistics theorem– Anyons

• QEq analog: Heisenberg spin magnet

Janak’s Theorem

• Kohn-Sham one-particle orbital energiesdictate change in total energy

• Implies discontinuities as a function ofparticle number at integers:

Janak, J. F.; Phys. Rev. B, 18, 1978, 7165-7168.

Origin of discontinuity

• Which term in universal functionalcontributes the most?– Coulomb exchange– Kinetic: Pauli exclusion principle– Unsolved question!

Future work

• Notion of generating density matricescompatible with a given Hamiltonian

Derivative discontinuities andionization potentials

• Implementingdiscontinuities improveestimates of ionizationpotentials

• “Double knee” feature inlaser-induced ionizationof helium atoms

• Model discontinuity incorrelation potentialneeded to obtain correctlimit

Lein, M.; Kümmel, S. Phys. Rev. Lett., 94, 2005, 143003.

Acknowledgments