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

12.4 Radiationless processes in photoexcited molecules 251
I Solution of the Schrödinger equation: Insertion of the ansatz (Eq. 12.38) into the
SE and multiplication from the left by either h 1 | or by h 2 | gives the two equations:
1 h 1 | ( − ) | 1 i + 2 h 1 | ( − ) | 2 i =0 (12.42)
1 h 2 | ( − ) | 1 i + 2 h 2 | ( − ) | 2 i =0 (12.43)
This is a system of coupled linear equations (“secular equations”):
with the “matrix elements” (integrals over all r resp. R):
Specifically, we have
1 ( 11 − )+ 2 12 =0 (12.44)
1 21 + 2 ( 22 − ) =0 (12.45)
= h | | i (12.46)
= h | (0) + (1) | i (12.47)
= h | (0) | i + h | (1) | i (12.48)
11 = h 1 | (0) | 1 i = (0)
1 (12.49)
22 = h 2 | (0) | 2 i = (0)
2 (12.50)
and
12 = h 1 | (1) | 2 i = h 2 | (1) | 1 i = 21 (12.51)
Therefore, we obtain the secular equations as follows:
I
Secular equations:
³ ´
1 (0)
1 − + 2 12 =0 (12.52)
³ ´
1 21 + 2 (0)
2 − =0 (12.53)
I Secular determinant: A non-trivial solution for the secular equations requires that
(0)
1 − 12
¯ 21 (0)
=0 (12.54)
2 − ¯
I
Eigenvalues:
0=
³ ´³
(0)
1 −
= (0)
1 (0)
2 −
´
(0)
2 − − | 12 | 2 (12.55)
³ ´
(0)
1 + (0)
2 + 2 − | 12 | 2 (12.56)
with
because (1) is Hermitian.
| 12 | 2 = 12 21 (12.57)