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

12.4 Radiationless processes in photoexcited molecules 246
• The zero-order electronic wave functions el
(r; R) describe the electrons and
el
their distribution
¯¯ (r; R)¯¯2 for the case of fixed nuclei in the electronic state
|i withtheelectronicenergy el
.
• The el
(r; R) depend on R only as a parameter, not as a variable (we do not
differentiate or integrate with respect to the in the zero-order SE).
• The el
(r; R) form a complete orthonormal system.
• The eigenvalues (0) (R) of the electronic SE are what we have come to know as
the molecular potential energy functions (or hypersurfaces) (R) of the molecule
with all nuclei at rest.
I The perturbation Ĥ 0 :
• The perturbation ˆ 0 is the nuclear kinetic energy operator ˆ .
c) Solution of the complete SE
• To solve the complete SE
³
ˆ 0 + ˆ 0 − ´
(r R) =0 (12.25)
we use the ansatz
(r R) = X (R) · el (r; R) (12.26)
with the expansion coefficients = (R) depending only on R but not on r.
• Inserting the ansatz 12.26 into the complete SE,
³
ˆ 0 + ˆ 0 − ´ X
(R) el
(r; R) =0 (12.27)
and doing some algebraic manipulation, we arrive at a
• system of coupled equations for the electronic wavefunctions el (r; R) for the
electronic state el
(r; R) and the nuclear wavefunctions (R):
and
ˆ 0 el
(r; R) =
(0)
(R) el
(r; R) (12.28)
ˆ 0 (R)+ X (R) (R) = ¡ − (0)
(R) ¢ (R) (12.29)
with the coupling coefficients