New aspects of the solvent mode in the photoinduced electron transfer in polar solution

Journal of Luminescence 40&41 (1988) 43 46 North-Holland, Amsterdam

43

NEW ASPECTS OF THE SOLVENT MODE IN THE PHOTOINDUCED ELECTRON TRANSFER IN POLAR SOLUTION Toshiaki KAKITAWI Department of Physics, Nagoya University Furo—cho, Chikusa—ku, Nagoya 464, Japan Asuming that the vibrational frequency of possibily coordinated solvent molecules around the charged reactant becomes considerably larger than that around the neutral reactant, we have derived formulas for the electron transfer rate as a function of the energy gap which differ greatly among the three kinds of electron transfer reactions; photoinduced charge separation, charge recombination and charge shift reactions. Those theoretical predictions are in good agreement with the experimental data so far available. By performing the Monte Carlo simulation study, it is confirmed that a considerable dielectric saturation of the coordinated solvent exists around the molecular ion and a substantial difference of the potential curvature is realized for the coordinated solvent mode around the charged molecule from that around the neutral molecule.

1. INTRODUCTION 1 of the

According to the Marcus theory electron transfer (ET), its rate becomes large as the free energy gap -AG increases, and reaches its maximum when —AG equals the total

around the neutral molecule. If we adopt this idea to the photoinduced charge separation reaction, a fortunate energy gap is readily obtained by means of the solvent reorientational fluctuation in the initial state,

reorientation energy Er of the solvent and

irrespective of the energy gap.

intramolecular vibration. The ET rate again decreases rapidly when -AG exceeds Er~ The

virtually leads to the disappearance of the inverted region.

This property

above relation is usually called the energy gap law. The energy gap region which is larger

2. THEORETICAL FORMULATION

than Er is called the inverted region. A systematic experimental study of the energy gap law for the photoinduced ET was

Based on the fermi’s Golden rule, the ET rate is expressed as

first done by Rehm and Weller.2 Although its reaction involves the diffusion process to form

W(AG)

an encounter complex, the experimental result

where H~fis the electron tunneling matrix

showed no evidence of the quenching rate of the fluorescer being decreased until as much as -AG~3eV, in contradiction to the energy gap

element between the initial and final electronic states, B~ the normalized Boltzmann function for the initial vibrational state u,

law of the current theory.

-

2T1 ~HifJ2~Bu~HuIv>I2~(EuEv+AE), (1) U v

~2 the Franck-Condon factor, AE the

Under such a situation, we propose that the solvent orientational motion around the charged molecule will be much restricted than that

energy gap, and Ev the vibrational energy, respectively. We consider the following three kinds of the

around the neutral molecule. In other word, the potential curvature for the solvent

vibrational modes; (i) intramolecular quantum mode (q-mode), (ii) coordinated solvent mode

reorientational motion around the charged molecule can be greatly different from that

(c-mode) and (iii) coordinated solvent mode (s-mode). Then, we obtain from eq.(l)

0022 2313/88 $03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

44

T. Kakitani

W(-AG)

/ New aspects of the solvent mode

JlWql)f25c2)5sE1~2)~

=

In the CSH reaction, (2)

2/kT] 1 ÷~ 4kT {exp[~(/(l+8)(8Ac~c2) - 8A~)

Sc(E

2)

where Wq(s

-

1) is the ET rate due to the q—mode,

____________

+

Wq(C1)

A exp[-S(2~+l)]I

=

(2S/~+l))

IPI

+

2exp[~{~(l+2~)Ac- (l+8)~2}/kT]

x

I0(13/(l+~)Ac(8Ac_E2)/kT)}

(3)

=

-

exp[~(/(l+8)(8Ac~E2) +

—l , [exp(4~i/kT)-l] p = ~/41<~>, 2/~2 ~ , A = 2~IH~fI

()

for for c>BA C

0

=

(7)

.

In the CSH reaction, In eq.(4), ~ and are the normal coordinate displacement and angular frequency of the q-mode.

I

is the modified Bessel function of

—

Sc(E 2)

the first kind.

Sc and

_______

~kT

-

exp[~E2~8(l+2~)Ac}/2kT]

are the thermally cosh(~/2(l+P)AclE2I /kT)

averaged Franck-Condon factor due to the c-mode and s-mode, respectively. Using the Hopfield method, we obtain

K

___________

~~‘2(l+6)A~2~/2kT

((l+28)Js2j/2kT).

2/4A

_______

=

x

1 exp[-(AE-c1-c2) 2VTrA5 kT

5kT],

(8)

In the above equations

(5)

where A5 is the total reorientation energy due to the s-mode. We define A*+B

~

A++B_ A+B

-i-

~

A+B

(CR reaction),

the neutral and charged reactants, respectively.

A+B

(CSH reaction).

Ac is the total reorientation energy due to the c-mode. In eq.(8), K is the modified Bessel

I0(8/(l+~)Ac(c2+8Ac)/kT)}

0

for £<_8A~ .

function of the second type. Numerical calculations show that ~ in eq.(6) becomes almost a step function while sc(E2) in eq.(7) becomes a sharp triangle for 8<
~-

-AG

for E>_IlAc —

(9)

,

where kn and kc are the potential curvatures for the motion of solvent molecules surrounding

2IkT] ~XP[~(/E~48A~- ~1(l+8)Ac1 + exp[_13(/c 2/kT] 2+~Ac + V(l+8)A) + 2exp[_8~a 2+(l+26)Ac}/kT] x

k n /(kc -k n )

A~+B+(CS reaction),

In the CS reaction,

=

=

(6)

-

AE—kTtn{(l+8)/~}

for

CS

AE+kTkn{(l+8)/5}

for

CR

AE

for CSH

,

.

(10)

3. COMPARISON WITH EXPERIMENTAL DATA The reaction scheme for the photoinduced CS

/

T. Kakitan,

reaction is written as A*+B

k

12 k21

10~

~ _____ k32

_____

k30

45

New aspects of the solvent mode

(11)

______________________________________

IOu 2

Under the condition k30>>k32, the quenching rate kq is expressed by kq

=

kl2 l+k21/W

(12) IO~

The rate constants k12 and k21 in acetonitrile 10M~s’ are estimated as 2xl0 and experimentally 2.3xl010s~, respectively. The calculated values of k with use of eqs.(2), (6) and (12) q are compared with the experiment by Rehm and Weller in Figure 1.

The theoretical curve for

8=0.3 0.1 fits very well the experimental data.

~9=0.030.1

10

0

0.5

1.0

1.5

2.0

2.5

3.0

3 5

—4G(ev)

FIGURE 2 The CR rate as a function of -AG. The curves denote theory: 1, A=3xl0’2s~, 8=0.3; 2, A= 4x1 0 12s~, 8=0.3. In both cases Ac=l.OeV, A 5= 0.9eV. The open circle shows the experimental data by Mataga et al.

:::~~----~~.

10~o

~ io~

F

10~ j~l~0.3 10~

—0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

—4G(ev)

FIGURE 1 The quenching rate due to the photoinduced CS reaction. The curves denote theory: 12s~,Ac=l.OeV, A A=5xlO 5=Oev. The most extensive experimental study for

!.~Jo —1.5

0

0.5

1.0

1.5

2.0

2.5

3.0

—~G(eV) FIGURE 3

The CSH rate as a function of -AG.

The curve

the CR reaction between uncombined donor and acceptor molecules beenbell-shaped done by Mataga al.3 Figure 2 showshasthat energyet

denote theory: 1, A=1.4xl0’°s~, 8=0.1, Ac=l~O Ac= 10H, 8=0.3, 1.0eV, eV, A A5=0.3eV; 2, A=3xl0 10s~, 8=~,A

gap law is realized for the CR reaction in both

5=0.3eV; 3, A=3xl0 5=0.9eV, Am~nta1 =0eV. data The by open circle denotes the experiMiller et al.

thoretical and experimental results. A thorough experimental study for the CSH reaction has been performed for the linked 4 donor and acceptor et al. From Figure 3, it ismolecules seen thatby a Miller slow decrease of W in the inverted region is theoretically well reproduced using rather a small value of 8.

~.

MONTE CARLO SIMULATION So far, we have intuitively introduced an

assumption that the vibrational frequency of the c-mode differs considerably between the initial and final states. But, this is not so obvious fact because the vibrational frequency

46

T. Kakitan,

/ New aspects of the solvent mode

of the solvent mode does not change if we adopt

Ag

=

_kTtnf(P,,)

(13)

the dielectric continuum model of solvent without dielectric saturation. Considerable amount of dielectric saturation is necessary

where ~ is the frequency that the polarization happens to be between P and P +ISP in the r r r lO~MC steps. The curvature of the free energy surface considerably increases when z increases,

for the frequency change of the solvent mode accompanying the ET reaction. Under these situations, we have conducted the Monte Carlo simulation using the following model system.

supporting our initial idea of the frequency change of the c-mode.

Solute and solvent molecules are assumed to be sphere with a hard core radius a.

A solute

0

molecule has a charge ze and a solvent molecule has a permanent dipole moment i. One solute

—

molecule and N solvent molecules are confined

~

in a large vessel. At the surface of the vessel, solute molecules are buried by fixing the orientation of dipole moment randomly. In

—l

IONCHARGE

.~

ION CHARGE

0

._.

000

0 57

—l

the actual calculation, the number of movable solvent molecues are 380, and the Monte Carlo step is lO~,with ~i=l.2D, a=2.2A. The solvent molecules which contact with the charged reac— tant are 5 6 and mostly locate in 4.4A 4.9A from the center.

The average polarization ~r

ION CHARGE

1

0

//

—

2

—V

in the radial direction per solvent molecule as a function of z is shown in Figure 4. The increase rate of P r is small when z increases, suggesting that a considerable dielectric

saturation occurs at z-l.0. Figure 5 is the calculated free energy change Ag defined by

100

0

IONCUARGE

200

—1

2 ~

06

0 4 —02

0

02

04

06

0.8

.0

Pr p

1.0

0.8

FIGURE 5 Free energy surfaces as a function of the

0.6

for some values of z. of the coordinated solvent averaged polarization

0.4

REFERENCES 1. R.A. Marcus, J. Chem. Phys. 24 (1956) 966.

a.

______________________________________

a

l;O

2.0

2. D. Rehm and A. Weller, Israel J. Chem. 8 (1970) 259. 3. N. t1ataga, Pure Appl. Chern. 56 (1984) 1255; N. Mataga, V. Kanda and T. Okada, J. Phys. Chem. 90 (1986) 3880.

FIGURE 4 Averaged polarization of the coordinated solvent as a function of z.

4. J.R. Miller, L.T. Calcaterra and G.L. Closs, J. Am. Chem. Soc. 106 (1984) 3047.