We formulate a general theory of the diffusion-influenced kinetics of irreversible bimolecular reactions occurring in the low concentration limit. Starting from the classical Liouville equation for the reactants and explicit solvent molecules, a formally exact expression for the bimolecular reaction rate coefficient is derived; the structures of reactant molecules and the sink functions may be arbitrarily complicated. The present theoretical formulation shows clearly how the well-known Noyes and Wilemski–Fixman rate theories are related and can be improved in a systematic manner. The general properties of the rate coefficient such as the long-time behavior and the upper and the lower bounds are analyzed. When the reaction can occur at a range of distance, the non-Markovianity of repeated encounter events between a reactant pair becomes significant and either the Noyes theory or the Wilemski–Fixman theory fails. The present theory provides a practical method for calculating the rate expression for such reactions, which improves significantly on the Wilemski–Fixman theory.

1.
S. A.
Rice
,
Diffusion-Limited Reactions
,
Comprehensive Chemical Kinetics
, Vol.
25
(
Elsevier
,
New York
,
1985
).
2.
M.
Smoluchowski
,
Ann. Phys.
48
,
1103
(
1915
);
M.
Smoluchowski
,
Z. Phys. Chem. (Leipzig)
92
,
129
(
1917
).
3.
F. C.
Collins
and
G. E.
Kimball
,
J. Colloid Sci.
4
,
425
(
1949
).
4.
G.
Wilemski
and
M.
Fixman
,
J. Chem. Phys.
58
,
4009
(
1973
).
5.
S.
Lee
and
M.
Karplus
,
J. Chem. Phys.
86
,
1883
(
1987
).
6.
S.
Lee
and
M.
Karplus
,
J. Chem. Phys.
86
,
1904
(
1987
).
7.
S. H.
Northrup
and
J. T.
Hynes
,
J. Chem. Phys.
68
,
3203
(
1978
).
8.
R. I.
Cukier
,
R.
Kapral
,
J. R.
Mehaffey
, and
K. J.
Shin
,
J. Chem. Phys.
72
,
1830
(
1980
);
R. I.
Cukier
,
R.
Kapral
,
J. R.
Mehaffey
, and
K. J.
Shin
,
J. Chem. Phys.
72
,
1844
(
1980
).
9.
R. M.
Noyes
,
J. Chem. Phys.
22
,
1349
(
1954
).
10.
A. A.
Kipriyanov
and
A. B.
Doktorov
,
Chem. Phys. Lett.
246
,
359
(
1995
);
A. A.
Kipriyanov
and
A. B.
Doktorov
,
Physica A
230
,
75
(
1996
).
11.
H. -X.
Zhou
and
A.
Szabo
,
Biophys. J.
71
,
2440
(
1996
).
12.
R.
Zwanzig
,
Nonequilibrium Statistical Mechanics
(
Oxford University Press
,
New York
,
2001
).
13.
Rigorously speaking, the Hamiltonian may vary with reaction progress, but the effects of this variation can be neglected in the low reactant concentration limit.
14.
J. -H.
Kim
and
S.
Lee
,
J. Phys. Chem. B
112
,
577
(
2008
).
15.
J.
Klafter
,
A.
Blumen
, and
M. F.
Shlesinger
,
Phys. Rev. A
35
,
3081
(
1987
).
16.
G. A.
Baker
, Jr.
and
P.
Graves-Morris
,
Padé Approximants
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
1996
).
18.
P.
Wynn
,
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
5
,
805
(
1968
).
19.
B.
Simon
,
Int. J. Quantum Chem.
21
,
3
(
1982
).
20.
U.
Gösele
,
M.
Hauser
,
U. K. A.
Klein
, and
R.
Frey
,
Chem. Phys. Lett.
34
,
519
(
1975
);
U. K. A.
Klein
,
R.
Frey
,
M.
Hauser
, and
U.
Gösele
,
Chem. Phys. Lett.
41
,
139
(
1976
);
U.
Gösele
,
Physica B
85
,
317
(
1977
).
21.
H.
Sumi
,
J. Phys. Chem.
100
,
4831
(
1996
).
22.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions
(
Dover
,
New York
,
1972
).
23.
A. B.
Doktorov
and
A. I.
Burshtein
,
Sov. Phys. JETP
41
,
671
(
1976
);
E. A.
Kotomin
and
A. B.
Doktorov
,
Phys. Status Solidi B
114
,
287
(
1982
).
24.
M.
Yokota
and
O.
Tanimoto
,
J. Phys. Soc. Jpn.
22
,
779
(
1967
).
25.
26.
J.
Lee
,
S.
Yang
,
J.
Kim
, and
S.
Lee
,
J. Chem. Phys.
120
,
7564
(
2004
).
27.
H.
Van Beijeren
,
W.
Dong
, and
L.
Bocquet
,
J. Chem. Phys.
114
,
6265
(
2001
).
28.
J. T.
Hynes
and
J. M.
Deutch
, in
Physical Chemistry
, edited by
D.
Henderson
(
Academic
,
New York
,
1976
), Vol.
XIB
, Chap. 8.
29.
C. W.
Gardiner
,
Handbook of Stochastic Methods
, 2nd ed. (
Springer-Verlag
,
Berlin
,
1985
).
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