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

12.4 Radiationless processes in photoexcited molecules 247
(R) =
Z
∗ (r; R) ˆ 0 (r; R) r
r
−~ 2 Z
r
∗ (r; R) X
1
(r; R) r
(12.30)
• We interprete these Eqs. as follows:
(1)
(0) (R) can be interpreted as the zero-order potential functions of the th
zero-order electronic state that is given by the solution of the electronic SE
(Eq. 12.24) for fixed nuclei R.
(2) Without the (R), Eq. 12.29 describes the kinetic energy of the nuclei
in the potential (0) (R). 52 Eq. 12.29 is thus simply the vibrational SE for
the electronic potential
(0) (R).
(3) The coefficients (R) defined by Eq. 12.30 are coupling terms that connect
(i.e., mix) the electronic states and . They describe how the
different electronic states and are coupled by the nuclear motion (the
two terms on the RHS resulting from the nuclear kinetic energy operator
ˆ ;seeEq.12.30).
d) Born-Oppenheimer approximation
The Born-Oppenheimer (BO) approximation completely neglects the coupling coefficients
. The complete SE is therefore reduced to two uncoupled equations, which
we denote as electronic SE
ˆ 0 el
(r) =
(0)
(R) el
(r) (12.31)
and nuclear (vibrational) SE
h i
ˆ +
(0) (R) = (R) (12.32)
which describe, respectively, the electronic wavefunction el
for fixed nuclear coordinates
R in the electronic state el
and the set of nuclear wavefunctions (R) for the energy
state of the nuclei in the electronic state el
.
e) Breakdown of the Born-Oppenheimer approximnation
When two (or more) electronic states el
(R), el
(R) become nearly isoenergetic or,
at a crossing, even degenerate, the coupling matrix element (R) may no longer be
neglected, because through the (R), thenuclear motion leads to a mixing between
the different electronic BO-states.
52 We used to denote these potential energy functions (≡ potential energy hypersurfaces) as (R).