The functions f ij (R, H, r) are the amplitudes of the probability of scattering from
channel i to channel j. The transition probability and the CINAT cross-section are
proportional to f
2
ij .
The wave functions of the diabatic representation are not eigenfunctions of
b
H el r; R
ð Þ. The electronic states in the diabatic representation are connected by
off-diagonal matrix elements hw
d
i j b
V el jw
d
j i (potential binding), i.e., a non-adiabatic
transition between them occurs as it does in the case of weak interaction (Fig. 3.16).
The set of the b
H el r; R
ð Þ eigenfunctions corresponds to the adiabatic representation. In this case, there is no potential binding, and a nonadiabatic transition
between these states occurs as it does in the case of the strong interaction (see
Sect. 3.6, Fig. 3.17 and Sect. 4.6.1.1). The wave functions of the adiabatic and
diabatic representations are related by a linear transformation mixing the diabatic
functions of the same symmetry (see below).
Now, let us consider the results obtained in [36] on the calculation of processes
in the CN + He system (see [37], also). What happens when a He atom collides
with the CN radical (Fig. 5.10)? If the collinear collisions, He–CN, or CN–He,
occur, then the symmetry type does not change compared to the CN symmetry type,
i.e., remains C ∞v , if non-collinear one, it is lifted to C s point group. In this symmetry group, the state R
+ corresponds to A
0 state and the degenerate П state splits
into two, A
0
þ A
00 states. The mixing of the R
+
, П states in the C ∞v point group, and
A
0
; A
00 in C s one is allowed only taking into account the electron-rotational or
vibronic interactions.
Let R is the distance from the atom to the center of mass of the molecule, r is the
internuclear distance of the molecule, and the z axis coincides with CN internuclear
axis (Fig. 5.10). Then the totally-symmetric component of the state П, A
0 , corresponds to П x , and non-totally symmetric A
00 to П y . The resulting adiabatic wave
functions of symmetry A
0 are linear combinations of the diabatic wave functions of
the R
+
, and П states (see above):
w 1 ðR; r; hÞ ¼ cos v R
þ
j iÀ sin v P x
j i
ð5:5:21Þ
w 2 ðR; r; hÞ ¼ sin vjR
þ
À cos v P x
j i
ð5:5:22Þ
where v is the mixing angle, a coefficient depending on the geometry of the
complex, i.e., bond length C-N, distance R, and angle of attack (see below). Here
|R
+ > , |P x > are diabatic wave functions obeying the relation
R
þ
h j
@
@q
P x
j i ¼ 0
ð5:5:23Þ
for all three internal coordinates of the complex (see Sect. 4.6.1.1).
The matrix elements of the electronic Hamiltonian b
H el , by definition being
potential surfaces, can be represented as:
5.5 Collision-Induced Nonadiabatic Transitions
179
channel i to channel j. The transition probability and the CINAT cross-section are
proportional to f
2
ij .
The wave functions of the diabatic representation are not eigenfunctions of
b
H el r; R
ð Þ. The electronic states in the diabatic representation are connected by
off-diagonal matrix elements hw
d
i j b
V el jw
d
j i (potential binding), i.e., a non-adiabatic
transition between them occurs as it does in the case of weak interaction (Fig. 3.16).
The set of the b
H el r; R
ð Þ eigenfunctions corresponds to the adiabatic representation. In this case, there is no potential binding, and a nonadiabatic transition
between these states occurs as it does in the case of the strong interaction (see
Sect. 3.6, Fig. 3.17 and Sect. 4.6.1.1). The wave functions of the adiabatic and
diabatic representations are related by a linear transformation mixing the diabatic
functions of the same symmetry (see below).
Now, let us consider the results obtained in [36] on the calculation of processes
in the CN + He system (see [37], also). What happens when a He atom collides
with the CN radical (Fig. 5.10)? If the collinear collisions, He–CN, or CN–He,
occur, then the symmetry type does not change compared to the CN symmetry type,
i.e., remains C ∞v , if non-collinear one, it is lifted to C s point group. In this symmetry group, the state R
+ corresponds to A
0 state and the degenerate П state splits
into two, A
0
þ A
00 states. The mixing of the R
+
, П states in the C ∞v point group, and
A
0
; A
00 in C s one is allowed only taking into account the electron-rotational or
vibronic interactions.
Let R is the distance from the atom to the center of mass of the molecule, r is the
internuclear distance of the molecule, and the z axis coincides with CN internuclear
axis (Fig. 5.10). Then the totally-symmetric component of the state П, A
0 , corresponds to П x , and non-totally symmetric A
00 to П y . The resulting adiabatic wave
functions of symmetry A
0 are linear combinations of the diabatic wave functions of
the R
+
, and П states (see above):
w 1 ðR; r; hÞ ¼ cos v R
þ
j iÀ sin v P x
j i
ð5:5:21Þ
w 2 ðR; r; hÞ ¼ sin vjR
þ
À cos v P x
j i
ð5:5:22Þ
where v is the mixing angle, a coefficient depending on the geometry of the
complex, i.e., bond length C-N, distance R, and angle of attack (see below). Here
|R
+ > , |P x > are diabatic wave functions obeying the relation
R
þ
h j
@
@q
P x
j i ¼ 0
ð5:5:23Þ
for all three internal coordinates of the complex (see Sect. 4.6.1.1).
The matrix elements of the electronic Hamiltonian b
H el , by definition being
potential surfaces, can be represented as:
5.5 Collision-Induced Nonadiabatic Transitions
179