Once we know the potential U n (R), we can take into account the dynamics of
nuclei describing the rotational and vibrational quantum states [4]. For the background state of molecular hydrogen isotopologues, we can introduce the rotational
energy as U rot (R) ¼ E rot K(K + 1), where E rot ¼ ħ
2
=2 e
MR
2 , e
M is the reduced mass of
the nuclei and K is the quantum number (K ¼ 0, 1, 2, . . .) of the total angular
momentum of the molecule. Then, using the potential U K (R) ¼ U n (R) + U rot (R) and
expanding U K (R) near the minimum at R ¼ R min we can write
U K R
ð Þ ¼ U K R min
ð
Þþ
e
M
2
ω
2
v δR
ð Þ
2 ,
ð2:4Þ
where δR ¼ R À R min and ω
2
v ¼ e
M
À1 d
2 U K R
ð Þ=dR
2
R¼R min
. The quantum mechanical motion of a particle in a quadratic potential gives the following expression for
the energy of the vibrational quantum states: E v ¼ ħω v (v + 1/2), where v is the
vibrational quantum number (v ¼ 0, 1, 2, . . .). As a result, the energy terms of the
diatomic molecule include three components: electronic, rotational, and vibrational,
which gives
U tot R
ð Þ ¼ U n R
ð Þ þ E rot K K þ 1
ð
Þþħω v v þ 1=2
ð
Þ :
ð2:5Þ
The expression (2.5), where the contributions of vibrational and rotational states to
the total energy are additive, is only valid for relatively low K and v values (e.g. see
[4]), and for the higher ones their contributions are mixed. Therefore, the vibrational
and rotational states are often called ro-vibrational states. Because the function
Fig. 2.3 Molecular
hydrogen terms.
(Reproduced with
permission from [16],
© Springer 2012)
18
2 Atomic Physics Relevant to Fusion Plasmas