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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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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,
,
Performance of Molecular Polarization Methods - BCN: april 2005
Comparison Among Methods
Water:• Low polarizability (1.47 Å3)• Anisotropic
Carbon Tetrachloride:• High polarizability (10.5 Å3)• Isotropic
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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
)
Performance of Molecular Polarization Methods - BCN: april 2005
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:
Performance of Molecular Polarization Methods - BCN: april 2005
Case of a Positive Point ChargeClose to Carbon Tetrachloride
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
Performance of Molecular Polarization Methods - BCN: april 2005
Case of a Positive Point ChargeClose to Carbon Tetrachloride
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
)
Performance of Molecular Polarization Methods - BCN: april 2005
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;
Performance of Molecular Polarization Methods - BCN: april 2005
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
Performance of Molecular Polarization Methods - BCN: april 2005
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)
Performance of Molecular Polarization Methods - BCN: april 2005
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)
Performance of Molecular Polarization Methods - BCN: april 2005
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.
Performance of Molecular Polarization Methods - BCN: april 2005
Damping Functions
35
3
3
ijij
ijijijij
ij
ijij
jiijjijji
rr
rrTT
r
rT
TTqE
Performance of Molecular Polarization Methods - BCN: april 2005
Damping Functions
),,,(
)(3
)(
;)(
3553
31
wrf
rf
r
rrfTT
r
rfT
jiij
ij
ij
ij
ijijijijij
ij
ijijij
Performance of Molecular Polarization Methods - BCN: april 2005
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)
Performance of Molecular Polarization Methods - BCN: april 2005
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!)
Performance of Molecular Polarization Methods - BCN: april 2005
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
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