4.5 Chaos in the HOCl Molecular System
125
Fig. 4.18 Contour plot of the
2D HOCl potential energy
surface (for the case when the
HO bond is in its ground
state) (plot based on Lin et al.
2015)
H =
p 2
R
2μ 1
+
p 2
θ
2μ 2 r 2
o
+
p 2
θ
2μ 1 R 2 + D e V (R, θ) = E,
(4.24)
where V (R, θ) is the potential energy of the interaction between the atoms in
the HOCl molecule, and E is the total energy of the system. The experimentally
obtained potential energy V (R, θ) (Weiss et al. 2000), is shown in Fig. 4.18.
We can now assign numbers to this model of HOCl. The HOCl molecule
dissociates into a free Cl atom and a bound HO molecule at energy D e =
20,312.3 cm −1 = 2.518 eV. The reduced masses have values μ 1 =
m Cl m d
M
=
595
52 u
and μ 2 =
m H m O
m d
=
16
17 u (u the atomic mass unit). It is useful to write the above
quantities in terms of dimensionless units (d.u.). We parametrize all energies in
units of D e , lengths in units of the Bohr radius a B = 5.2917×10 −11 m, and angular
momenta in terms of Planck’s constant ¯
h = 1.05457×10 −34 J·s. The length of the
H − O bond, in the ground state, is r o = 1.85 (in multiples of Bohr radii). Then
H = D e H , E = D e E , R = a B R , r = a B r , p R = ¯
h
a B
p
R , p r = ¯
h
a B
p
r , p θ = ¯
hp
θ ,
p β = ¯
hp
β , and time t = ¯
h
D e
t , where primed quantities are dimensionless. If we
now drop the primes on dimensionless quantities, the (dimensionless) Hamiltonian
takes the form
H =
p 2
R
2 2
R
+
p 2
θ
2 2
θ
+
p 2
θ
2 2
R R 2
+ V (R, θ)
(4.25)