5.4 Conical Intersections
155
CNNC angle, which acts as a very efficient funnel. In fact, the S 1 lifetime is well
below 1 ps in the gas phase, especially if the cis isomer is excited. At CNNC = 0
◦
or 180
◦ (i.e., either for the trans or for the cis isomer) the CNN angles have larger
equilibrium values for S 1 than for S 0 . For example, the trans isomer has CNN = 115
◦
for S 0 and CNN = 129
◦ for S 1 . So, opening the CNN angles the S 0 PES rises more
steeply with respect to S 1 , leading to a sloped conical intersection, which actually
belongs to the same crossing seam as the peaked one referred above. Clearly, after
S 0 → S 1 excitation the vibrational coordinate CNN gets excited (see Sect. 3.7), and
the sloped conical intersection may be reached, giving rise to “early” decay to S 0
(i.e., at transoid or cisoid geometries), which in turn leads to a decrease of the photoisomerization quantum yield. This phenomenon is more important for the trans
than for the cis isomer, due to the fact that the torsion of the CNNC dihedral after
excitation is much faster for the latter. So, in azobenzene the cis → trans photoisomerization quantum yield Φ cis→trans is close to 0.6, while Φ trans→cis is considerably
lower (about 0.3 after n → π
∗ excitation).
In monoalkenes the S 0 and S 1 PES are still quite separated at 90
◦ of torsion around
the double bond. However, as discussed in Sect. 2.6.4, the two states get closer by
pyramidalization of one of the two carbon atoms. In ethylene, this leads to a conical
intersection, which is evidently very easily accessed from the Franck–Condon point.
As a consequence, the S 1 lifetime in ethylene is very short (∼10
2 fs).
In acetone a crossing is found between the S 1 and T 1 n → π
∗ states by stretching
the C-O bond and keeping the C 2v geometry of the ground state minimum. Taking
into account the spin–orbit coupling, such a crossing is actually a symmetry-allowed
conical intersection. In fact, as for the spatial part S 1 and T 1 both belong to the same
A 2 irrep, but considering also the symmetry of the spin part, the three components of
the triplet actually belong to the B 1 , B 2 , and A 1 irreps (while the singlet retains A 2
symmetry). The S 1 /T 1 spin–orbit coupling is therefore zero at C 2v geometry, and the
degeneracy is removed along nonsymmetric coordinates. In general, the true crossings of singlet and triplet PESs become avoided crossings or conical intersections
when the spin–orbit coupling is introduced in the Hamiltonian. In the approximation
of neglecting the coupling among the triplet degenerate states, one can identify a
linear combination of them that interacts with the crossing singlet, and two noninteracting orthogonal combinations. So, in this approximation, only one triplet PES gives
place to avoided crossings or conical intersections with the singlet one, while the two
other degenerate PESs cross the singlet one without constraints. Some consequences
for nonadiabatic dynamics are examined in Ref. [9].
5.4.2 Branching Plane (Real Hamiltonian)
The plane spanned by the two vectors q and h defined in Eq. (5.36) is called branching
plane. As far as the first-order approximation is valid, only a displacement from
Q x in the branching plane is able to produce a value different from zero for ΔH
and/or H 12 , so removing the degeneracy. On the contrary, the degeneracy is kept
155
CNNC angle, which acts as a very efficient funnel. In fact, the S 1 lifetime is well
below 1 ps in the gas phase, especially if the cis isomer is excited. At CNNC = 0
◦
or 180
◦ (i.e., either for the trans or for the cis isomer) the CNN angles have larger
equilibrium values for S 1 than for S 0 . For example, the trans isomer has CNN = 115
◦
for S 0 and CNN = 129
◦ for S 1 . So, opening the CNN angles the S 0 PES rises more
steeply with respect to S 1 , leading to a sloped conical intersection, which actually
belongs to the same crossing seam as the peaked one referred above. Clearly, after
S 0 → S 1 excitation the vibrational coordinate CNN gets excited (see Sect. 3.7), and
the sloped conical intersection may be reached, giving rise to “early” decay to S 0
(i.e., at transoid or cisoid geometries), which in turn leads to a decrease of the photoisomerization quantum yield. This phenomenon is more important for the trans
than for the cis isomer, due to the fact that the torsion of the CNNC dihedral after
excitation is much faster for the latter. So, in azobenzene the cis → trans photoisomerization quantum yield Φ cis→trans is close to 0.6, while Φ trans→cis is considerably
lower (about 0.3 after n → π
∗ excitation).
In monoalkenes the S 0 and S 1 PES are still quite separated at 90
◦ of torsion around
the double bond. However, as discussed in Sect. 2.6.4, the two states get closer by
pyramidalization of one of the two carbon atoms. In ethylene, this leads to a conical
intersection, which is evidently very easily accessed from the Franck–Condon point.
As a consequence, the S 1 lifetime in ethylene is very short (∼10
2 fs).
In acetone a crossing is found between the S 1 and T 1 n → π
∗ states by stretching
the C-O bond and keeping the C 2v geometry of the ground state minimum. Taking
into account the spin–orbit coupling, such a crossing is actually a symmetry-allowed
conical intersection. In fact, as for the spatial part S 1 and T 1 both belong to the same
A 2 irrep, but considering also the symmetry of the spin part, the three components of
the triplet actually belong to the B 1 , B 2 , and A 1 irreps (while the singlet retains A 2
symmetry). The S 1 /T 1 spin–orbit coupling is therefore zero at C 2v geometry, and the
degeneracy is removed along nonsymmetric coordinates. In general, the true crossings of singlet and triplet PESs become avoided crossings or conical intersections
when the spin–orbit coupling is introduced in the Hamiltonian. In the approximation
of neglecting the coupling among the triplet degenerate states, one can identify a
linear combination of them that interacts with the crossing singlet, and two noninteracting orthogonal combinations. So, in this approximation, only one triplet PES gives
place to avoided crossings or conical intersections with the singlet one, while the two
other degenerate PESs cross the singlet one without constraints. Some consequences
for nonadiabatic dynamics are examined in Ref. [9].
5.4.2 Branching Plane (Real Hamiltonian)
The plane spanned by the two vectors q and h defined in Eq. (5.36) is called branching
plane. As far as the first-order approximation is valid, only a displacement from
Q x in the branching plane is able to produce a value different from zero for ΔH
and/or H 12 , so removing the degeneracy. On the contrary, the degeneracy is kept