* is the path for which
f
,I'
fll exp({3U)ds = minimum .
(10)
Using the terminology proposed in Ref. 16 we call the integral in Eq. (10) the "generalized resistance" and the minimum condition of Eq. (10) the minimum resistance principle. 17
B. Euler-lagrange equations for the minimum resistance principle The minimum resistance principle [Eq. (10)] can be written in the form 61 == 6
Jo eau Sf
'ftt ds
== 0 ,
(11)
which can be used to obtain Euler-Lagrange equations for the minimum resistance path. Here, sf is the total arclength of the path; this length is in general different for all the paths being varied over. In order to eliminate the variation of sf we consider s to be a function of a new parameter t which varies between the fixed limits of 0 and 1. Using the dot to represent differentiation with respect to t, and explicitly representing the tangent in the coordinates x"' one obtains from Eq. (11):
The Euler-Lagrange equations corresponding to Eq. (12) are
a~K [eau~ ;;x"Xvf"v/(~ x~r2J
.
Eq. (16) becomes
~ (eau~) _~e6uJ [L:(X,,)2]1I2 =0 . [ds V ds ax~ "
(18)
Equation (18) can be reduced to
d2~~ + ~ d{3U _ a{3U == 0 ds
ds
ds
ax~
(19)
or, in vector form (20)
Kn+t(VBU' t)-Vau==o,
where K is the curvature of the path, and nand tare the principal unit vectors normal and tangent to the path, respectively. Equation (20) describes the minimum resistance path (MRP), which we designate the optimum reaction coordinate. The latter is often identified instead with the steepest descent path (SDP), which is given by the equation (21)
-V/3U/jV8Uj==t.
Comparison of Eqs. (20) and (21) shows that they describe paths which COincide if and only if the path is a straight line (i. e., K == 0). This can be the case when the potential surface is symmetrical. In general, the MRP differs from the SDP and may not go through the expected saddle points. 12 The deviation between these paths will typically be larger for higher temperatures or flatter potential surfaces. Another interesting case for which one can obtain a simple differential equation for the MRP is that of nonuniform isotropic friction, defined by the following relation:
The Euler-Lagrange equations nOw have the form (13)
Note that j ". is in general anisotropic and a function of the coordinates. While Eq. (13) is very general, it is cumbersome for analysis. Therefore we consider two special cases for purposes of illustration. The first case is that of uniform isotropic friction, which is defined by the following relation: j"v==f6"v==const.
S
(22)
0= :t[a~. (e6U~ ~X"Xvf"v/(~xiY'2)] -
(17c)
d'
"
(14)
In this case, Eq. (13) reduces to
:t [a~l (e8Uj(x)(~x~y/2)]
=
a~l [eBUj(X)(~x~rl
(23)
If the friction at same arbitrary reference point Xo is taken to be j(x o) and the function W(x) is defined through the relation W(x) == U(x) + kB T lnf(x)/j(x o) ,
(24)
the Euler-Lagrange equations become
:t{a~~ [e8W(Z)(~>~r2]}== a~~ ~BW(S)(~:X~ f2]
(25)
Equation (25) is formally identical to Eq. (15), with Proceeding as before, one finds that the equation for the MRP has the form W(x) in place of U(x).
Kn=VBW-t(t· VBW) .
or (16)
(17a) (17b)
(26)
Thus, for the case of nonuniform isotropic friction, the system moves on an effective potential W(x) that depends on both the true potential U(x) and the friction along the pathj(x).
III. DISCUSSION Equation (10) provides a variational expression for the optimum reaction coordinate of a diffusional process.
J. Chern. Phys., Vol. 79, No. 11, 1 December 1983
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Berkowitz, Morgan, McCammon, and Northrup: Diffusion-controlled reactions
Physically, the best single path is seen to be one that balances small frictional reSistance and small energy cost in terms of the potential of mean force_ If there are large changes in the potential of mean force as a function of configuration, only the direction of passage through the saddle point between minima on the surface may be importantll ; more generally, the global character of the friction and potential determine the nature of the preferred motion on the surface. The actual rate constant for a diffusion-controlled reaction will, of course, involve contributions from all possible paths between reactant and product states and not just from the MRP; this is an area of continuing investigation. For many systems, it is expected that the rate will be dominated by paths similar to the MRP. When this is the case, the MRP will be useful in choosing the optimum dividing surface between reactants and products for calculation of rate constants by the Brownian dynamics trajectory method. 13 The MRP may also be useful in constructing the best one-dimensional approximation to the system for a Kramers-type analysis of the reaction dynamics. 4 In general, the Kramers rate constant will be inversely proportional to the integral in Eq. (10) or, using notation Similar to that of Eq. (24),
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NIH (Houston) and from the Research Corporation and the Donors of the Petroleum Research Fund as administered by the American Chemical SOCiety (Tennessee Tech). JAM is a Sloan Fellow and the recipient of NIH Research Career Development and Dreyfus TeacherScholar Awards.
10. F. Calef and J. M. Deutch, Annu. Rev. Phys. Chem. (in press). 2M. v. Smoluchowski, Phys. Z. 17, 557 (1916). 3p. Debye, Trans. Electrochem. Soc. 82, 265 (1943). 'H. A. Kramers, Physica 7, 284 (1940). 5T. F. Schatzki, J. Polym. Sci. 57, 496 (1962); Polym. Prepr. 6, 646 (1965). 6E. Helfand, J. Chem. Phys. 54, 465 (1971). fE. Helfand, Z. R. Wasserman, and T. A. Weber, J. Chem. Phys. 70, 2016 (1979). So. C. Knauss and G. T. Evans, J. Chem. Phys. 72, 1504 (1980).
SM. R. Pear, S. H. Northrup, and J. A. McCammon, J. Chem. Phys. 73, 4703 (1980). 10M. R. Pear, S. H. Northrup, J. A. McCammon, M. Karplus, and R. Levy, Biopolymers 20, 629 (1981). ltG. van der Zwan and J. T. Hynes, J. Chem. Phys. 77, 1295 (1982).
k- 1CX !o(so>1 exp[{3W(s)]ds,
