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

12.4 Radiationless processes in photoexcited molecules 250
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Figure 12.6: Conical intersection of potential energy surfaces of a polyatomic molecule.
Conical Intersections of Potential Surfaces in Polyatomic Molecules
V E W V V V V
1 2 1 2 1 2
0 E
W
with: ;
W V2
E
2 2
V
S 2
Conical Intersections
Diatomic Molecules: s = 1
2
V ( R) ( R) ( R) W( R)
S 1
S 0
Q 1
Q 2
Polyatomic Molecules: s = 3N-6
V ( Q , Q ) ( Q , Q )
1 2 1 2
( Q , Q ) W( Q , Q )
1 2 1 2
2
Physical Chemistry III: Chemical Kinetics • © F. Temps, IPC Kiel 257
12.4.3 Quantum mechanical treatment of a coupled 2×2 system
We can easily verify the “avoided crossing” of two coupled potential energy curves as
follows:
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Schrödinger equation:
( − ) |i =0 (12.36)
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Hamilton-Operator:
= (0) + (1) (12.37)
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Ansatz:
|i = 1 | 1 i + 2 | 2 i (12.38)
with orthonormal basis vectors that are eigenvectors of (0) , i.e.,
½
­ ® 1
| = =
0
for =
for 6=
(12.39)
and the zero-order energies
(0) | 1 i = (0)
1 | 1 i (12.40)
(0) | 2 i = (0)
2 | 2 i (12.41)