Computational Chemistry

Our calculations predict room temperature halflives for cyclopropylidene of about
10 #^9 –10#^10 s. Attempts to generate cyclopropylidene at 77 K gave allene [ 225 ]. We
can calculate the halflife at 77 K, instructing Gaussian 03 to use this temperature for
thermochemistry. Using CBS-QB3, the resultingDG{is 25.1 kJ mol#^1 (very little
change from the 298 K value of 23.8 kJ mol#^1 ), and with this andT¼77 K,
Eq. (5.197) giveskr¼1.49' 10 #^5 and a halflife of 4.7' 104 s, ca. 13 h.
Cyclopropylidene ought to be observable at 77 K.
From Eq. (5.197) and the fact that for a unimolecular reactiont 1 = 2 ¼ln 2=krit
follows that

logt 1 = 2 ¼logðln 2Þ

h

kBT



þ

DGz

RT

(loge ð 5 : 201 Þ

At 298 K (about room temperature) this becomes

logt 1 = 2 ¼ 0 : 175 DGz# 13 : 0 ð 5 : 202 Þ

whereDG{is in kJ mol#^1. Eq (5.202) shows that forDG{¼0 kJ mol#^1 ,t 1 = 2 is
about# 10 #^13 s; this is as expected, since the period of a molecular vibration
is about 10#^13 # 10 #^14 s and with no barrier a species should survive for only
about one vibrational motion (that along the reaction coordinate, corresponding to
the imaginary frequency) as it passes through the saddle region (e.g. Fig.5.29).
Figure5.31, a graph of Eq. (5.202), can be used to estimate halflives at room

log t1/2 (t in s)

0 50 100 150

10

20

30

40

50

–10

–20

∆G , kJ mol–1

Fig. 5.31 Graph of log t1/2¼0.175DG{#13.0. If this equation for the halflife of a unimolecular
reaction were strictly true, then the threshold value ofDG{for ready observability at room
temperature would be about 85 kJ mol#^1 , corresponding to t1/2¼75 s. Actually, a rough rule of
thumb is that the threshold barrier for observability at room temperature is about 100 kJ mol#^1

5.5 Applications of the Ab initio Method 329