Chemistry Reference
In-Depth Information
Fig. 6.3
Ring closure of
cis
-butadiene to
cyclo
-butene. In
C
2
v
symmetry there is a symmetry
mismatch between the
a
2
and
a
1
occupied orbitals. Vibronic orbital coupling requires a concerted
mechanism, based on a conrotatory ring closure, which conserves only the
C
2
axis
The summation over
λ
∈
Λ
thus covers the entire product space of
Γ
×
Σ
.Com-
bining the sum rule of Eq. (
6.83
) with the total expression for the PJT, one finds:
1
2
K
Γ
+
Q
2
Γγ
2
|
Λ
Γ
Σ
|
E(Q)
=
E
0
+
(6.85)
E
0
−
E
Λ
γ
Γ
Λ
Hence, when the ground state is non-degenerate, the first-order dependence of the
energy on symmetry-lowering displacement vanishes, and the second-order term
contains two contributions: the diagonal harmonic force constant, which is always
positive, and the bilinear relaxation term, which is always negative. If the excited
states are close in energy to the ground state, and if the vibronic coupling is strong,
the relaxation term may be dominant, and a second-order symmetry-breaking effect
will result. This is known as the pseudo-JT effect. There are two main applications
of this effect: in geometry optimization, and in reaction dynamics.
In reaction dynamics the PJT may be responsible for stereoselectivity, because of
the selection rules for vibronic coupling matrix elements. Via these relaxation matrix
elements the Wigner-Eckart theorem is at the basis of the Woodward-Hoffmann
rules [
12
]. We shall not discuss these rules in general, but consider some simple
illustrations, related to electrocyclic reactions.
6
Take as a simple example the ring
closure of
cis
-butadiene, as illustrated in Fig.
6.3
. The relevant occupied orbitals are
6
Basedon[
13
].
