Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

12.4 Radiationless processes in photoexcited molecules 252
With some rewriting 53 ,weobtain
± = 1 ³ ´
(0)
1 + (0)
2 ± 1 r ³
(0)
1 + (0)
´2
(0)
2 − 4 1 (0)
2 +4| 12 | 2 (12.58)
2
2
y
± = 1 2
³ ´
(0)
1 + (0)
2 ± 1 r ³
(0)
1 − (0)
´2
2 +4|12 | 2 (12.59)
2
I Shorthand form: Using the abbreviations
± (R) = ± (12.60)
1 (R) = (0)
1 (12.61)
2 (R) = (0)
2 (12.62)
2 (R) =| 12 | 2 (12.63)
Σ (R) = 1 2 ( 1 (R)+ 2 (R)) (12.64)
∆ (R) = 1 2 ( 1 (R) − 2 (R)) (12.65)
we can rewrite the solution for the eigenvalues + and − in a compact form as
± (R) =Σ (R) ±
q
(∆ (R)) 2 +( (R)) 2 (12.66)
12.4.4 Radiationless processes in polyatomic molecules: Conical intersections
I “Traditional” view on the cis-trans isomerization of C 2 H 4 :
• Reaction coordinate can be approximated by torsion angle (0 ≤ ≤ 180 ◦ ).
• Reaction proceeds via a perpendicular transition state ( =90 ◦ ), in which the C
atoms are connected by a single bond.
• Correlation diagram shows that the orbital of the trans-isomer transforms into
the ∗ orbital of the cis-isomer and vice versa.
• Crossing of the two diabatic states corresponds to an avoided crossing of the
adiabatic states.
53
³
(0)
1 + (0)
´2 ³
(0)
2 − 4 1 (0) 2 =
(0)
1
´2
+
³
³
=
(0)
2
(0)
1
´2
+2
(0)
´2
+
³
1 (0) 2 − 4 (0)
1 (0) 2
(0)
2
´2
− 2
(0)
1 (0) 2 =
³
(0)
1 − (0)
´2
2