Mialocq, Barat, Gilles, Hickel, and Lesigne
742
Whitten and Ilabinovitch.22 With this approximation the rate constant ratio may be cast conveniently into the form25,26
where the integral IS partitioned into three ranges of integration to tab e proper account of quantum restrictions and thc functional form of the quantum correction factor to the semiclassical energy density expression. The variable of integrstion k the reduced energy, x = Et/&?, where E t is the energy of the active modes of the activated complex in excess of its zero-point energy, E,T. In the absence of free internal rotations in the molecule and r t such rotations in the complex, the functions in eq A2.1 and limits of integration are given by GI(?)
=
+
St
Il'(ni t- l;n(.lAt/r(l r?/2)(EztYqxrt/2 0 5 x Cdx)
[X
G3(d= H(x)
A~ =
E {X
+ 1 - @t/(b,x f [X
b4Xba
+ bs)Int 0 5 x
+ I - k3t exp(-b,~"~)]"~1.0 5 x
i- (EL/Ezl)x
+
where the dagger (7) refers to the activated complex, and n, nf are defined as n = s - l (r=O) nf=s.i+ ~ r t The upper limit of integration of the first range, 0, is taken here for convenience as 0 < o,,,t/E,f, where wmlnt is the lowest vibrational frequency in the complex. Other quantities not previously defined are r, the gamma function; w L , vibrational frequencies in energy units (cm-l; UI,CQ?, symmetry numbers for adiabatic rotations; 2; , partition function for active (internal) rotations; Zvt7 quantum vibrational partition function; h, Planck's constant. The quantities b, are empirical constants relating to the quantum correction factor to the zero point ener-
