Performance of Molecular Polarization Methods Marco Masia

Performance of Molecular Polarization Methods

Marco Masia

Performance of Molecular Polarization Methods - BCN: april 2005

Overview

Nonpolarizable ModelsAlgorithms Incorporating Polarizability

Fluctuacting Charges (FQ) Point Dipoles (PD) Shell Models (SH)

Comparison Among Methods: Case of a Positive Point Charge Case of Cations Damping Methods

Conclusions


A: repulsive short range term

B: attractive term depending on the (dipole-dipole) London dispersion

Nonpolarizable Models

ij

jiqq

LJ

qqLJ

r

qqrU

rB

BArr

rU

UUU

)(

4)(

6

2

612

qO

qH

qH


Nonpolarizable Models

Drawback: no dynamical response to the fluctuations of the electric fields is considered!

We need to implement polarizability in an explicit way!

O

H

H

E


Algorithms Incorporating Polarizability

Several methods have been developed for the last 30 years.

0

),...,,(0

2100

i

n

xU

EUU

xxxUU

Minimization of the energy respect to some parameter


Fluctuacting Charges (FQ)

Charges are allowed to fluctuate according to the electronic properties of the molecule as atomic electronegativity and atomic hardness.

jj

i

q

qU

0

0

E=0 E=E(r)

q1

q2

q3

q1+dq1

q2+dq2

q3 +dq3

dq1 + dq2 + dq3 = 0


Point Dipoles (PD)

E=0 E=E(r)

q1

q2

q3

q1

q2

q3

Atomic polarizabilities i are assigned to some molecular site

The electric field induces the formation of a point dipole i


Point Dipoles (PD)

iii

ijij

ijijijij

ij

ijij

jiijjijji

E

rr

rrTT

r

rT

TTqE

35

3

3The calculation is repeated iteratively till convergence.

Charge and

Dipole Field Tensors


Molecular Polarizability

3336

33323332

3)(

)(sin16)(sin82)(sin32)(sin32)(sin16

)2(

),,,,,,(

00

00

00

2

dddd

ZYXzyxf

HOH

HOHOHOHHO

OHzz

iiiati

Tr

zz

yy

xx

mol

mol

Dependence of the molecular polarizability tensor from the atomic polarizabilities


Shell Model (SH)

The point dipole is mapped to a system of two point charges linked by a spring.

i

Sii

iSii

qk

rq

2,

,


Comparison Among Methods

Water:• Low polarizability (1.47 Å3)• Anisotropic

Carbon Tetrachloride:• High polarizability (10.5 Å3)• Isotropic


Case of a Positive Charge Close to Water

Five configurations were considered:

1 2 3 4

-400

-200

0

200

400

C2v

-face

trans

top

cis

po

tent

ial e

nerg

y (k

J m

ol-1

)

distance (+)-O (Å)

C2v

-back


Case of a Positive Point Charge Close to Water

2 3 4 5

2

3

4

5 G03 PDM-H2O PD2-H2O SH-H2O

distance (+)-O (Å)

dipo

le m

omen

t (D

ebye

)

Similar results were obtained for all the other configurations considered

O

HH M

dOH

dOM


Case of a Positive Point Charge Close to Water

What about the performance with double point charges?

2 3 4 5 6 72

4

6

8

PDM-H2O SH-H2O G03

distance (++)-O (Å)

dipo

le m

omen

t (D

ebye

)


Case of a Positive Point ChargeClose to Carbon Tetrachloride

2 3 4 5 6 7

-300

-200

-100

0

100

face

edge

corner

po

tent

ial e

nerg

y (k

J m

ol-1

)

distance (+)-C (Å)

Three configurations were considered:



3 4 5 6 7

2

4

6

8 G03 FQ PD PD-C PD opt SH opt

distance (+)-C (Å)

dipo

le m

omen

t (D

ebye

)

C

Cl

ClCl

Cl



2 3 4 5 6 70

4

8

12

16

20 PD opt SH opt G03

distance (++)-C (Å)

dipo

le m

omen

t (D

ebye

)


Case of a Positive Point Charge

PD and SH models can be reparametrized to reproduce the polarizability tensor of the molecule & the dipole moment induced by a point charge;

Also at short distances there is no need to use damping functions;

High electric fields cause the linear models to fail due to hyperpolarizability effects;


Case of Cations

2 3 4 5-200

-150

-100

-50

0

distance ion-O (Å)

Pot

enti

al E

nerg

y (k

J m

ol-1

)

3 4 5-200

-150

-100

-50

0

(+); Li+; Na+; K+

distance ion-C (Å)

Potential energy: importance of electron repulsion


Case of Cations

2 3 4 5

2.0

2.5

3.0

3.5

4.0

4.5 (+)

Li+

Na+

K+

distance ion-O (Å)

dipo

le m

omen

t (D

ebye

)

2 3 4 52

3

4

5

2 3 4 52

3

4

5

6

7

8

PDM-H2O; PD2-H2O; SH-H2O; G03

dip

ole

mom

ent (

Deb

ye)

distance ion-O (Å)

Li+ Ca2+

distance ion-O (Å)


Case of Cations

3 4 5

2

3

4

5

6

3 4 5

4

6

8

10

12

PD opt; SH opt; PD; G03

Li+

distance ion-C (Å)

dip

ole

mom

ent (

Deb

ye)

Mg2+

distance ion-C (Å)


Damping Functions

Thole (1981): for intramolecular interactions the molecular polarizability diverges at short distances

r

=(r,)

Many functional forms for the charge density have been proposed.

The most used are the exponential and the linear forms.


Damping Functions

35

3

3

ijij

ijijijij

ij

ijij

jiijjijji

rr

rrTT

r

rT

TTqE


Damping Functions

),,,(

)(3

)(

;)(

3553

31

wrf

rf

r

rrfTT

r

rfT

jiij

ij

ij

ij

ijijijijij

ij

ijijij


Damping Functions

2 3 4 52.0

2.5

3.0

3.5

4.0

4.5

2 3 4 52

3

4

5

6

7

PDM; PDM-exp; PDM-lin; G03

distance ion-O (Å)

Na+

dipo

le m

omen

t (D

ebye

)

Ca2+

distance ion-O (Å)


Conclusions and Future Work

Dimers with cations show a different behaviour from the case of positive point charges;

In the case of cations the use of damping functions for the electrostatic interactions is needed;

The Thole linear and exponential models have been applied to intermolecular interactions and reparametrized for the interactions cation-water and cation-CCl4.

Study the performance of the same methods with

anions (high polarizabilities!)


Bibliography

Review Rev. in Comput. Chem. 18, 89 (2002).

Methods FQ: J. Chem. Phys. 101, 6141 (1994) PD: J. Am. Chem. Soc. 94, 2952 (1972) SH: The Theory of Optics (Longmans, N. Y., 1902) Damping: Chem. Phys. 59, 341 (1981)

Results J. Chem. Phys. 121, 7362 (2004) Comp. Phys. Commun. In press Manuscript in preparation


Aknowledgements

Rossend ReyMichael Probst

EUMinisterio EspañolRegione Sardegna

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Performance of Molecular Polarization Methods Marco Masia