Interaction-induced effects in dipole and polarizability...
Transcript of Interaction-induced effects in dipole and polarizability...
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
∞
′ ′′= = +∫