v 1 ðR; t; t
0
Þ
v 2 ðR; t; t
0
Þ
!
¼ e
ÀiE F ðt
0 Þ= h / 1 ðR; t; t
0
Þ
/ 2 ðR; t; t
0
Þ
!
ð9Þ
E F is the Floquet quasi-energy. The functions / k ðR; t; t
0
Þ (k ¼ 1; 2) are periodic in
time. They can be expanded in Fourier series
/ k ðR; t; t
0
Þ ¼
X þ1
n¼À1
e
inxðt
0 Þt
u
n
k ðR; t
0
Þ:
ð10Þ
The functions u
n
k ðR; t
0
Þ are the solutions of the coupled differential equations [16]
T N þ V 1;2 ðRÞ þ n hxðt
0
Þ À E F ðt
0
Þ
Â
à u
n
1;2 ðR; t
0
Þ
À 1=2E 0 ðt
0
ÞlðRÞ u
nÀ1
2;1 ðR; t
0
Þ þ u
nþ1
2;1 ðR; t
0
Þ
h
i
¼ 0
ð11Þ
If the field amplitude is small enough to avoid going beyond 2 channels (or onephoton absorption) there results the two coupled equations
T N þ V g ðRÞ þ hxðt
0
Þ À E F ðt
0
Þ
Â
Ã
u g ðR; t
0
Þ À 1=2E 0 ðt
0
ÞlðRÞu u ðR; t
0
Þ ¼ 0
ð12Þ
and
0
1
2
3
4
0
1 0
2 0
3 0
-3000
-2000
-1000
0
Time (fs)
Barrier top (cm )
-1
(a)
(b)
R
Γ
-1
(t)
(cm )
Fig. 3 Panel a The instantaneous rate as a function of time. Panel b The instantaneous position of
the top of the barrier as a function of time. The dashed horizontal line shows the position of the
unperturbed vibrational level with quantum number t ¼ 12, at À2;673 cm
À1 . Only when the
barrier top is below this position is there a non-zero rate. The rate maxima are strictly correlated
with the barrier top minima
140
R. Lefebvre