fluctuating-charge models: theory and applications

Fluctuating charge models: applications and illustrations

Jiahao ChenMartínez Group

Dept. of Chemistry, Frederick Seitz Materials Research Lab. and the Beckman Institute

University of Illinois at Urbana-Champaign

Acknowledgments

Prof. Troy van VoorhisProf. Alán Aspuru-Guzik

Prof. Todd J. MartínezMartínez Group and friends

$: DOE

Discussions

Harvard/MIT visit

Prof. Susan Atlas (UNM)Dr. Ben Levine (UPenn)Dr. Steve Valone (LANL)

Prof. Troy van Voorhis (MIT)

“The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without

having to surrender the adequate representation of a single datum of

experience.”Albert Einstein, “On the Method of Theoretical Physics”, Phil. Sci. 1 (1934), 163-9.

What is electronegativity?

“Concept introduced by L. Pauling as the power of an atom to attract electrons to itself.”

“The quantity that measures the escaping tendency of electrons from a species in its ground state.” IUPAC Compendium of Chemical Terminology,

aka “The Gold Book”, goldbook.iupac.org

http://goldbook.iupac.org/G02704.html

http://goldbook.iupac.org/G02704.html

A quantitative definition

R. S. Mulliken J. Chem. Phys 2:(1934), 782–793

! =IP + EA

2

A quantitative definition

R. S. Mulliken J. Chem. Phys 2:(1934), 782–793

! =IP + EA

2

=E(N ! 1)! E(N + 1)

2

" "E

"N

Electronic structure and dynamics

Electronic structure and dynamicsWhat is the charge distribution?

What does the system do?


H! = i!

H! = E! What is the charge distribution?



less variablesmore variables

H! = i!

H! = E! What is the charge distribution?




H! = i!

H! = E!

directnumericalquadrature

ab initiotheories

semiempiricalmethods

density functional

theory

coarse-grained models

continuumelectrostatics

molecular models (MM)

classicalmoleculardynamics

finite element methods

coarse-grained

dynamics

numerical quadrature, e.g. real-time path

integral propagatorsab initio molecular dynamics

What is the charge distribution?




H! = i!

H! = E!

directnumericalquadrature

ab initiotheories

semiempiricalmethods

density functional

theory

coarse-grained models

continuumelectrostatics



finite element methods

coarse-grained

dynamics

numerical quadrature, e.g. real-time path

integral propagatorsab initio molecular dynamics

What is the charge distribution?




Molecular models/force fields

covalent bond effectsE =

+

Typical energy function

noncovalent interactions

bond stretch angle torsion dihedrals

electrostatics dispersion

+-

+

Typical energy function

!

i<j!atoms

!ij

"#"ij

rij

$12

!#

"ij

rij

$6%

E =!

b!bonds

kb(rb ! r0b )2 +

!

a!angles

!a("a ! "0a)2

!

d!dihedrals

!

n

ldn cos(n!)+

++!

i<j!atoms

qiqj

rij

Molecular models/force fields

Unique to condensed phases, where most

chemistry and biology happens

Why care about polarization and charge transfer?

Polarization in chemistry• Effect of local environment in liquid phases

• Ex. 1: Stabilizes carbonium in lysozyme

• Ex. 2: Hydrates chloride in water clusters

OPLS/AAnon-polarizable

force field

TIP4P/FQpolarizableforce field

1. A Warshel and M Levitt J. Mol. Biol. 103 (1976), 227-249. 2. SJ Stuart and BJ Berne J. Phys. Chem. 100 (1996), 11934 -11943.

3 models for polarization

Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.

Drude oscillatorsor charge-on-spring

or shell modelsQ

q !Q

kR

Response = change in RReview: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.

Ideal spring

Inducible dipoles

!1 !2

µinduced,1 µinduced,2

Response = change in induced dipoles


-0.3

Fluctuating charges

+0.8

-0.5

charge transfer = 1.1e charge transfer = 0.2 e

charge transfer = 1.3 e

Response = change in atomic charges

!2, "2

!3, "3


Better electrostatics

Model Polari-zation

Charge transfer

Cost

Pairwise fixed charges

Drude oscillator

Inducible dipoles

Fluctuating charges

Implicit, at best ❙

✓ ❙ ❙

✓ ❙ ❙ ❙ ❙ ❙ ❙

✓ ✓ ❙ ❙ ❙

!

i<j!atoms

qiqj

rij

QEq, a fluctuating-charge model

AK Rappé and WA Goddard III J. Phys. Chem. 95 (1991), 3358-3363.

atomicelectronegativities

“voltages”

screenedCoulomb

interactions

Jij =!

R3!2

!2i (r1)!2

j (r2)|r1 ! r2| dr1dr2

!i(r) = Ni |r !R|ni!1 e!!i|r!Ri|

E =!

i

qi!i +12

!

ij

qiqjJij

Principle of electronegativity equalization

Minimize energy

subject to charge constraint!

i

qi = Q

Using the method of Lagrange multipliers, reduces to solving the linear equation

(electronic) chemical potential

!J 11T 0

" !qµ

"=

!!!0

"

E =!

i

qi!i +12

!

ij

qiqjJij

Physical interpretationIn equilibrium:

oeach atom i has the same chemical potential µo µ uniquely determines the atomic charges qi

Atoms are subsystems in equilibrium

N, V, T

ΩΩi

moleculeatom

Energy derivatives: chemical potential µ, hardness η

QEq has wrong asymptotics

q =!1 ! !2

J11 + J22 ! J12

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

R/Å

q/e

QEq

ab initio

Na ClR

asymptote ~ 0.43 ≠ 0

Problems with QEqFractional charge distributions predicted

for dissociated systems

Wrong direction of intermolecular charge transfer predicted in some systems

No out-of-plane dipole polarizability

Overestimates in-plane dipole polarizability

Fluctuating-charge models map molecules onto electrical circuits

screenedCoulomb

interactionchemicalhardness

electro-negativitymolecule

screenedCoulombchemicalelectro-


screenedCoulomb


electro-negativitymoleculeelectric

potential(inverse)

capacitanceelectricalcircuits

Coulombinteraction



screenedCoulomb


electro-negativity

More electropositive

More electronegative0 V

χ2

χ1η

1

η2

- V

olta

ge +

moleculeelectric

potential(inverse)

capacitanceelectricalcircuits

Coulombinteraction


QEq has wrong asymptotics

q =!1 ! !2

J11 + J22 ! J12

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

R/Å

q/e

QEq

ab initio

Na ClR

asymptote ~ 0.43 ≠ 0

J12 → 0+

-

+

-

+

-

+

-

In fluctuating-charge models like QEq, all

molecules are metallic

η2

QTPIE, our new charge modelCharge-transfer with polarization current

equilibrationVoltage attenuates with increasing distance

J. Chen and T. J. Martínez, Chem. Phys. Lett. 438 (2007) 315-320.

voltage

distance

η2

η2

QTPIE, our new charge modelCharge-transfer with polarization current

equilibrationVoltage attenuates with increasing distance

J. Chen and T. J. Martínez, Chem. Phys. Lett. 438 (2007) 315-320.

voltage

distance

η2

η2

Making QTPIE (Step 1)

J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.

To make the proposed change, first change variables

E =!

i

qi!i +12

!

ij

qiqjJij

=!

ij

pji!i +12

!

ijkl

pkipljJij

qi =!

j

pji

Charge transfer variables quantify how much charge went from one atom to another, and are indexed over pairs

p12

p23

p34

p45

Still QEq!Same model,new representation

Making QTPIE (Step 2)

J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.

atomic electronegativities become bond electronegativities

Sij =!

R3!i(r)!j(r)dr

EQEq =!

ij

pji!i +12

!

ijkl

pkipljJij

EQTPIE =!

ij

pji!ikijSij +12

!

ijkl

pkipljJij

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

R/Å

q/e

QEq

QTPIE

ab initio

QTPIE has correct limit

q =(!1 ! !2)S12

J11 + J22 ! J12

q =!1 ! !2

J11 + J22 ! J12

Na ClR

Reverting to charge variables

p12p13

p34

p14

p23

p24

q1

q2 q3

q4qi =

!

j

pji

?

Adjacency matrix of an oriented complete graph with 4 vertices

!

""#

q1

q2

q3

q4

$

%%& =

!

""#

!1 !1 !1 0 0 01 0 0 !1 !1 00 1 0 1 0 !10 0 1 0 1 1

$

%%&

!

""""""#

p12

p13

p14

p23

p24

p34

$

%%%%%%&


p12p13

p34

p14

p23

p24

q1

q2 q3

q4qi =

!

j

pji

?

Inverse transformation is determined by pseudoinverse of adjacency matrix

!

""""""#

p12

p13

p14

p23

p24

p34

$

%%%%%%&=

!

""#

!1 !1 !1 0 0 01 0 0 !1 !1 00 1 0 1 0 !10 0 1 0 1 1

$

%%&

+ !

""#

q1

q2

q3

q4

$

%%&

=14

!

""""""#

!1 1 0 0!1 0 1 0!1 0 0 10 !1 1 00 !1 0 10 0 !1 1

$

%%%%%%&

!

""#

q1

q2

q3

q4

$

%%&


p12p13

p34

p14

p23

p24

q1

q2 q3

q4qi =

!

j

pji

Inverse transformation is determined by pseudoinverse of adjacency matrix

!

""""""#

p12

p13

p14

p23

p24

p34

$

%%%%%%&=

!

""#

!1 !1 !1 0 0 01 0 0 !1 !1 00 1 0 1 0 !10 0 1 0 1 1

$

%%&

+ !

""#

q1

q2

q3

q4

$

%%&

=14

!

""""""#

!1 1 0 0!1 0 1 0!1 0 0 10 !1 1 00 !1 0 10 0 !1 1

$

%%%%%%&

!

""#

q1

q2

q3

q4

$

%%&

pji =qi ! qj

N


Charge transfer variables have massive linear redundancy due to Kirchhoff’s voltage law

p12

p23

p31

p12 + p13 + p31 = 0

Execution times

0.01

0.1

1

10

100

1000

104

10 100 1000 104 105

TImes to solve the QTPIE model

Bond-space SVDBond-space COFAtom-space iterative solverAtom-space direct solver

Sol

utio

n tim

e (s

)

Number of atoms

N1.81N6.20

N

Cooperative polarization in water

• Dipole moment of water increases from 1.854 Debye1 in gas phase to 2.95±0.20 Debye2 at r.t.p. (liquid phase)

• Polarization enhances dipole moments

• Missing in models with implicit or no polarization

!"+

1. D R Lide, CRC Handbook of Chemistry and Physics, 73rd ed., 1992.2. AV Gubskaya and PG Kusalik J. Chem. Phys. 117 (2002) 5290-5302.

Polarization in water chains• Use parameters from single water molecule

to model chains of waters

• Compare QEq and QTPIE with:

๏ Gas phase experimental data1

๏ Ab initio DF-LMP2/aug-cc-pVDZ

๏ AMOEBA2, an inducible dipole model

๏ TIP3P, a common implicit polarization model

1. WF Murphy J. Chem. Phys. 67 (1977), 5877-5882.2. P Ren and JW Ponder J. Phys. Chem. B 107 (2003), 5933-5947.

H! = E!

Our new water model

EMM =!

i!bonds

ki (Ri !Reqi )2

+!

i!1,3-bonded

kUBi

"RUB

i !RUB,eqi

#2

+!

i!angles

!i ("i ! "eqi )2

+!

ij

4#ij

$%$ij

rij

&12

!%

$ij

rij

&6'

+EQTPIE

E =!

b!bonds

kb(rb ! r0b )2 +

!

b!1,3int.

kUBb (rUB

b ! rUB,0b )2

+!

a!angles

!a("a ! "0a)2

+!

i<j!atoms

4#ij

"#$ij

rij

$1

2!#

$ij

rij

$6

)

+EQTPIE

Flexible SPC/E, but with QTPIE electrostaticsFit to gas-phase data, and test transition to bulk in 1 dim.

!"!ij

rij

#12

!"

!ij

rij

#6$

Dipole moment per water

0 5 10 15 20 25

Number of molecules

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

Dip

ole

mom

ent/

mol. (

Debye)

DF-LMP2/aug-cc-PVTZ

QTPIE

QEq

AMOEBA

Polarizability per water

0 5 10 15 20 25

Number of molecules

1.0

1.5

2.0

2.5

3.0

3.5Tra

nve

rse p

ola

riza

bili

ty/m

ol.

(Å!)

DF-LMP2/aug-cc-PVTZ

QTPIE

QEq

AMOEBA


0 5 10 15 20 25

Number of molecules

.0

.5

1.0

1.5

2.0

2.5

3.0

3.5Longit

udin

al p

ola

riza

bili

ty (

Å!)

DF-LMP2/aug-cc-PVTZ

QTPIE

QEq

AMOEBA


0 5 10 15 20 25

Number of molecules

.0

.5

1.0

1.5

Out-

of-

pla

ne

pola

riza

bili

ty (

Å!)

DF-LMP2/aug-cc-PVTZ

QTPIE QEq

AMOEBA

Charge transfer in 15 waters

1 3 5 7 9 11 13 15Index of water molecule

-.05

-.03

-.01

.01

.03

.05

Ch

arg

e o

n w

ate

r m

ole

cu

le

QEq

QTPIE

Mulliken


-.05

-.03

-.01

.01

.03

.05

Ch

arg

e o

n w

ate

r m

ole

cu

le

QEq

QTPIE

Mulliken


-.05

-.03

-.01

.01

.03

.05

Ch

arg

e o

n w

ate

r m

ole

cu

le

QEq

QTPIE

Mulliken


-.05

-.03

-.01

.01

.03

.05

Ch

arg

e o

n w

ate

r m

ole

cu

le

QEq

QTPIE

Mulliken


-.05

-.03

-.01

.01

.03

.05C

ha

rge

on

wa

ter

mo

lec

ule

QEq

QTPIE

Mulliken

Summary

• Polarization and charge transfer are important effects usually neglected in classical MD

• Our new charge model corrects deficiencies in existing fluctuating-charge model at similar computational cost

• We obtain quantitative polarization and qualitative charge transfer trends in linear water chains

fluctuating-charge models: theory and applications

Education

fluctuating charge models

e charge transfer

drude oscillatorsor

wf van gunsteren comput

enelectronic structure

dynamicselectronic structure

echarge transfer

troy van voorhisprof