22
B. Sutcliffe and R.G. Woolley
of the internal Hamiltonian, ˆ
H , would actually be those that would have been obtained from (1.10) after separation of the centre-of-mass term, by letting the nuclear
masses increase without limit. Although there are no analytically solved molecular
problems, the work of Frolov [55] provides extremely accurate numerical solutions
for a problem with two nuclei and a single electron. Frolov investigated what happens when the masses of one and then two of the nuclei increase without limit in his
calculations. To appreciate his results, consider a system with two nuclei; the natural nuclear coordinate is the internuclear distance which will be denoted here simply
as t. When needed to express the electron-nuclei attraction terms, x n
i is simply of the
form α i t where α i is a signed ratio of the nuclear mass to the total nuclear mass; in
the case of a homonuclear system α i = ±
1
2 .
The di-nuclear electronic Hamiltonian after the elimination of the centre-of-mass
contribution as described in Sect. 1.3.3 is
ˆ
H
elec
t
e , t
= −
2
2m
N
i=1
∇
2
t
e
i
−
2
2(m 1 + m 2 )
N
i,j =1
∇
t
e
i
· ∇
t
e
j
−
e 2
4πε 0
N
j =1
Z 1
|t e
j + α 1 t|
+
Z 2
|t e
j + α 2 t|
+
e 2
8πε 0
N
i,j =1
1
|t e
i − t e
j |
+
Z 1 Z 2
R
, R = |t|
(1.33)
while the nuclear kinetic energy part is:
ˆ
T N (t) = −
2
2
1
m 1
+
1
m 2
∇
2 (t) ≡ −
2
2μ
∇
2 (t).
(1.34)
The full internal motion Hamiltonian for the three-particle system is then
ˆ
H
t
e , t
= ˆ
H
elec
t
e , t
+ ˆ
T N (t)
(1.35)
which is of the same form as (1.26).
It is seen from (1.34), that if only one nuclear mass increases without limit then
the kinetic energy term in the nuclear variable remains in the full problem and so the
Hamiltonian (1.35) remains essentially self-adjoint. Frolov’s calculations showed
that when one mass increased without limit (the atomic case), any discrete spectrum persisted but when two masses were allowed to increase without limit (the
molecular case), the Hamiltonian ceased to be well-defined and this failure led to
what he called adiabatic divergence in attempts to compute discrete eigenstates of
(1.35). This divergence is discussed in some mathematical detail in the Appendix to
Frolov [55]. It does not arise from the choice of a translationally invariant form for
the electronic Hamiltonian; rather it is due to the lack of any kinetic energy term to
dominate the Coulomb potential.
B. Sutcliffe and R.G. Woolley
of the internal Hamiltonian, ˆ
H , would actually be those that would have been obtained from (1.10) after separation of the centre-of-mass term, by letting the nuclear
masses increase without limit. Although there are no analytically solved molecular
problems, the work of Frolov [55] provides extremely accurate numerical solutions
for a problem with two nuclei and a single electron. Frolov investigated what happens when the masses of one and then two of the nuclei increase without limit in his
calculations. To appreciate his results, consider a system with two nuclei; the natural nuclear coordinate is the internuclear distance which will be denoted here simply
as t. When needed to express the electron-nuclei attraction terms, x n
i is simply of the
form α i t where α i is a signed ratio of the nuclear mass to the total nuclear mass; in
the case of a homonuclear system α i = ±
1
2 .
The di-nuclear electronic Hamiltonian after the elimination of the centre-of-mass
contribution as described in Sect. 1.3.3 is
ˆ
H
elec
t
e , t
= −
2
2m
N
i=1
∇
2
t
e
i
−
2
2(m 1 + m 2 )
N
i,j =1
∇
t
e
i
· ∇
t
e
j
−
e 2
4πε 0
N
j =1
Z 1
|t e
j + α 1 t|
+
Z 2
|t e
j + α 2 t|
+
e 2
8πε 0
N
i,j =1
1
|t e
i − t e
j |
+
Z 1 Z 2
R
, R = |t|
(1.33)
while the nuclear kinetic energy part is:
ˆ
T N (t) = −
2
2
1
m 1
+
1
m 2
∇
2 (t) ≡ −
2
2μ
∇
2 (t).
(1.34)
The full internal motion Hamiltonian for the three-particle system is then
ˆ
H
t
e , t
= ˆ
H
elec
t
e , t
+ ˆ
T N (t)
(1.35)
which is of the same form as (1.26).
It is seen from (1.34), that if only one nuclear mass increases without limit then
the kinetic energy term in the nuclear variable remains in the full problem and so the
Hamiltonian (1.35) remains essentially self-adjoint. Frolov’s calculations showed
that when one mass increased without limit (the atomic case), any discrete spectrum persisted but when two masses were allowed to increase without limit (the
molecular case), the Hamiltonian ceased to be well-defined and this failure led to
what he called adiabatic divergence in attempts to compute discrete eigenstates of
(1.35). This divergence is discussed in some mathematical detail in the Appendix to
Frolov [55]. It does not arise from the choice of a translationally invariant form for
the electronic Hamiltonian; rather it is due to the lack of any kinetic energy term to
dominate the Coulomb potential.