Interaction-induced effects in dipole and polarizability relaxation

Branka M. LadanyiDepartment of ChemistryColorado State University

Dipole relaxation

Contribution to the dipole density TCF from permanent and induced dipoles:

M I= +M M M

2

( ,0) ( , )( , ) ; , ; 1; 2A AA L T

A

tk t A L T

Nν ν

µ ν⋅ −

Ψ = = = =M k M k

( , ) ( , ) ( , ) ( , )MM MI IIA A A Ak t k t k t k tΨ = Ψ + Ψ + Ψ

Dipole TCFs for longitudinal and transverse components

Acetonitrile (CH3CN)*

*D. M. F. Edwards and P. A. Madden, Mol. Phys. 51, 1163 (1984).

Results for the transverse component for low k (recall, smallest k for cubic box = k1 = 2π/L)All contributions are of comparable magnitude and decay on similar time scales.Can separate contributions according to their relaxation properties.

[ ]2

( ) 1 ( ) ( ); ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

R M I MA A A A A A A

M I MA A A A

G k G k

G k

= + ∆ = −

= ⋅ −

M k M k M k M k M k

M k M k M k

( , ) ( , ) ( , ) ( , )RR RA A A Ak t k t k t k t∆ ∆∆Ψ = Ψ + Ψ + Ψ

This gives the dipole density TCF in the form:

Edwards-Madden notation:

( ) ( )R OA A⇒M k M k

We see that at low k, most of the transverse induced dipole projects along the molecular dipole. To a good approximation:

[ ]2

( , ) ( , )

1 ( ) ( , )

RRT T

MMT T

k t k t

G k k t

Ψ ≅ Ψ

≅ + Ψ

Methanol example*

*Results from M. S. Skaf, T. Fonseca, and B. M. Ladanyi, J. Chem. Phys. 98, 8929 (1993).

In this case too, all contributions are of comparable magnitude and decay on similar time scales.

Methanol: transverse dipole TCF in terms of projected variables

Interpretation in terms of local field factors

0 0

( ) 2 ( ) 2lim [1 ( )] ; lim [1 ( )]3 3 ( )T Lk k

G k G kε εε→ →

∞ + ∞ ++ = + =∞

Kirkwood-Fröhlich theory would predict:

These agree well with low-k simulation data for both acetonitrile and methanol.

Induced dipole contributions –

frequency domain

This methanol example illustrates a significant contribution to higher-frequency dielectric relaxation.

Collective polarizability anisotropy relaxation

Recall, M I= +Π Π Π

Polarizability anisotropy – any off-diagonal component or the traceless part of a diagonal component. We will choose the xz component.

2

(0) ( )( )

/15xz xz

xz

tt

NγΠ Π

Ψ =

γ is the molecular polarizability anisotropy:

( ) ( ) ( )2 2 22 111 22 11 33 22 332γ α α α α α α = − + − + −

( ) ( ) ( ) ( )MM MI IIxz xz xz xzt t t tΨ = Ψ + Ψ + Ψ

In most cases there is a considerable amount of molecular-induced cross correlation. A dynamical separation can be implemented by using a projection scheme analogous to what we used in the case of dipole density relaxation:

( )2wher

(1

e

) ;R M I Mxz xz xz xz xz xz xz

I M Mxz xz xz xz

G G

G

Π = + Π ∆Π = Π − Π

= Π Π Π

This is illustrated using our MD results for acetonitrile and methanol.*

(Next two slides)

*Results from B. M. Ladanyi and Y. Q. Liang, J. Chem. Phys. 103, 6325 (1995).

Acetonitrile example

Methanol example

Note that in both cases Gxz < 0.

More about interaction-induced polarizability

First order center-center model:1

NI

i ij ji j i= ≠

= ⋅ ⋅∑∑ TΠ α α

All-orders center-center model: 0i i i ij jj i≠

= ⋅ + ⋅ ⋅∑m E T mα α

This is a set of coupled linear equations - can be solved by matrix inversion.

Site-site model (first order):

Can use, for example, the Thole model* to get site polarizabilitytensors and modified dipole tensors

, , ,1

NI

i a ia jb j bi j i a i b j= ≠ ∈ ∈

= ⋅ ⋅∑∑∑∑a τ aΠ

,i aa ,ia jbτ

*B. T. Thole, Chem. Phys. 59, 341 (1981).

First vs. all-order interaction-induced polarizability*

0 1 2 3 4time (ps)

-1

-0.5

0

0.5

1

Ψxz

(t)

CC1CCA

acetonitrile

MI

MM

II0i i i ij j

j i≠

= ⋅ + ⋅ ⋅∑m E T mα α

A set of coupled linear equations - can be solved by matrix inversion.

*M.D. Elola and B.M. Ladanyi

First order vs all-orders (within the center-center model – for notational simplicity)Dipole induced in molecule i

Center-center vs site-site

Center-center vs site-site

Optical Kerr effectNuclear response (there is also an electronic response due to molecular second hyperpolarizability. That response is essentially instantaneous within experimental time-resolution):

1( ) ( ) ( )n xzB

R t t tk T

θ∝ − Ψ

Heaviside step func n( o) titθ =

Simulation data: M.D. Elola and B.M. Ladanyi;Expt.: S. Park and N.F. Scherer.

Kerr spectral density

0

( ) ( ) ( ) ( )i tnR t e dt iωχ ω χ ω χ ω

∞

′ ′′= = +∫