WOLFRAM|DEMONSTRATIONS PROJECT

Non-Crossing Rule for Energy Curves in Diatomic Molecules

​
D
e
R
e
V
12
Let
E
1
(R)
and
E
2
(R)
be energy curves for two different electronic states of a diatomic molecule, both computed within the Born–Oppenheimer approximation. If the two states belong to different symmetry species, say
Σ
and
Π
,
u
and
g
, or singlet and triplet, there is no restriction on whether the curves can cross. If, however, the two states have the same symmetry, a non-crossing rule applies. Close approach of the two curves results in mutual repulsion, known as an anticrossing. For near degeneracy of
E
1
(R)
and
E
2
(R)
, a perturbation
V
12
(R)
, representing higher-order contributions in the Born–Oppenheimer approximation, becomes significant, giving mixed states that do not cross.
In this Demonstration, the lower energy state,
E
1
(R)
, is drawn in blue. It is assumed to be a bonding state, with dissociation energy
D
e
and equilibrium internuclear distance
R
e
, which can both be varied with sliders. The upper energy state,
E
2
(R)
, drawn in red, is assumed to be a repulsive state. The mixing parameter
V
12
can also be varied. In certain cases, the upper state can develop a minimum as a result of the
V
12
interaction. The dashed curves in the graphic pertain when
V
12
(R)=0
.