Electron Transfer, Optical Spectroscopy, and Solvation in Polar 

Liquids

MIT, November 17, 2004

Outline●Linear response approximation●Polarization correlations ●Critical tests●Formalism●Applications

Linear Response Approximation

Linear Response/MC

y=4/9m2

m is the dipole moment    is the density  =1/kT

u0 s=−∫E0 r ⋅P r d r=−∑ jm j⋅E0 j

y p=4/9m ' 2

m' is the condensed­phasedipole moment

Solvent Correlations

Solvent correlations/MC

Longitudinal/Transverse Correlations

P=PL

PT , P=∫ P r eik⋅r d r

PL=k k⋅P , PT= P−PL

⟨ P k P −k⟩= 3 y4

[SL k JLST k JT ]

JL=k k , JT=1−k k

Longitudinal/Transverse Structure Factors

Longitudinal – short­rangedTransverse ­ long­ranged

S k =⟨k−k⟩

Longitudinal/Transverse Dynamics

Longitudinal/transverse relaxation times:

⟨ PL ,T k , t PL ,T −k , 0⟩=⟨∣PL ,T k∣2⟩e−t /L ,T k

L ,T k =D k −1 SL ,T k , D k =2 DRk2 DT

k=0 structure factors:

SL 0=3 y−11−1 , ST 0=3 y−1−1

k=0 relaxation times:

L 0/T 0=∞/

Both static and dynamic propertiesof “L” and “T” are distinctly different

Goal

⟨ Pδ r Pδ 0 ⟩∝ 1r

e−r /Λ

Theory input:

SL(k) and ST(k) for an arbitrarydielectric 

Solute of arbitrary shape (atomicresolution) and arbitrary chargedistribution

Theory output:

Solvation/reorganization free energy in the linear responseapproximation

hst=−solvsolvel

hst∝[ −121

−∞−12∞1 ]

Optical spectroscopy:

S t =E t −E ∞E 0 −E ∞

Solvation dynamics:

Equilibrium solvation/electron transfer:

Gact=Gs

2

4s

s=hst /2=1/∞−1/ gGs=solv final −solv initial

Experiment

µsolv saturates with increasing εs

µsolv is made by both L and T polarization

Reorganization energy is about twice smaller than µsolv 

Properties of the reorganization energy are largely defined by ε∞ in strongly polar solvents

D A

Qualitative results

s=∞−1−−1 g g /∞

Saturation limit

Dependence on high-frequency diel. conts.

s∝[ −121

−∞−12 ∞1 ]Lippert­Mataga equation:

Response function:

Reorganization energy:

Formalism

Integral equation:

Generating functional:

step function projecting inside the solute

Felderhof-Li-Kardar-Chandler

Response  function:

Solution

Properties of ' '

Dipole solvation (exact result)

Transverse part of the solvation free energy disappears in polar solvents! 

Dipole solvation (results)

Mean-Field Solution

Calculation Method

(bpy)2Ru2+(bpy’)­(pro)4­O­Co3+(NH3)5

2.78Resp. Func

2.64MD

λs, eVMethod

TIP3P water

ET through a polypeptide

=V molecules/V liquid

S2=12⟨3cos2−1⟩

ET in Nematics

Nematics

ET Rates in Nematics

director ≃10−3 s

DA≃10−9 s

Solvation Dynamics

Laplace transform of the emission energy:

Continuum solvation dynamics is fundamentally faster than microscopic solvation dynamics

Coumarin­153dynamics come in throughε(s)

E s=E0−2 s−1∫E0⋅s⋅E0 d k ' d k ' '

T= 92 K

Solvent=2­methylTHF

Solute:

Solvation Dynamics: Low T

Solvation in quadrupolar solvents

Quadrupolar structure factors in site­site benzene (from MD simulations)

Charge transfer in quadrupolar solvents

D

A

e­

solvent =

Reorganization energy:

Difference in solvation free energy:

quarupoles = ­ 0.24 eV

induction = ­ 0.32 eV

dispersion = 0.2­0.4 eV

0.20 eV