32
B. Sutcliffe and R.G. Woolley
This destroys completely the concept of a single internuclear potential in diatomic
molecules because it is not possible to introduce on the basis of non-adiabatic potentials
a single, approximate, mean potential that would describe well more than one vibrational
level.
It is obvious that in the case of rotations the situation is even more complex.
Wilson suggested [88] that using the clamped-nuclei Hamiltonian instead of the
full one in (1.48) to define the potential might avoid the spikes but Hunter in [86]
showed why this was unlikely to be the case, and Cassam-Chenai [89] repeated the
work of Czub and Wolniewicz using a clamped-nuclei electronic Hamiltonian and
showed that exactly the same spiky behaviour occurred.
Another approach to this problem is in [84]; there is reason to believe however
that this sort of difficulty is bound to arise whatever the approach. To see this, simply
rewrite (1.45) to recognise that the exact states will actually have definite quantum
numbers according to their symmetry, so that it would be more realistic to write
ψ J Mpri
t
n , t
e
= χ J MP ri
t
n
φ J Mpri
t
n , t
e
.
(1.49)
In the H 2 study cited the first four quantum numbers are of no relevance, only i
remains and here i labels the vibrational states. There is thus every reason to expect
that the best that can be done from this approach is a distinct PES for each nuclearmotion state.
This anticipated behaviour seems to be confirmed in very accurate calculations
on H 2 [90] for the electronic Σ ground state of the molecule assumed to dissociate
into two hydrogen atoms in their ground states. That work shows that, for example,
the J = 0 state supports just 14 vibrational states while the J = 15 state supports
10 and the J = 31 supports only 1 state. Of course in a diatomic molecule, states
of different k are states which differ in the electronic angular momentum and these
results cannot be regarded as typifying the results for a polyatomic system. However work on H
+
3 shows that in the case of J = 0 there are 1280 vibrational states
below dissociation [91] and that 46 is the highest value of J for which at least one
vibrational state exists [92]. At this level then it cannot be assumed that the potential
surface calculated in the usual way is an approximation to anything exact.
The eigenstates of the full molecular Hamiltonian (the Coulomb Hamiltonian for
the electrons and nuclei specified by a chemical formula)—a theory of an Isolated
Molecule modeled on the quantum theory of the atom which we call the Isolated
Molecule model—are reasonably well understood and might have some utility in
a limited area of high-resolution experiments on very small molecules where questions of isomerism do not arise [93]. Their computation poses formidable problems,
and really belongs to few-body physics. If it is to be taken as underlying Quantum
Chemistry then it is worth exploring the consequences of the model without regard
to approximations made for practical utility which are a quite separate matter. In this
paper we have attempted to discuss the Born-Oppenheimer and Born approaches to
the quantum theory of molecules in terms first set out by Combes [61]. The essential
point is that the decomposition of the molecular Hamiltonian (with centre-of-mass
contribution removed) into the nuclear kinetic energy, proportional to κ 4 and a remainder, is specified by (1.26), not by (1.9), or in other words, (1.9) cannot be
B. Sutcliffe and R.G. Woolley
This destroys completely the concept of a single internuclear potential in diatomic
molecules because it is not possible to introduce on the basis of non-adiabatic potentials
a single, approximate, mean potential that would describe well more than one vibrational
level.
It is obvious that in the case of rotations the situation is even more complex.
Wilson suggested [88] that using the clamped-nuclei Hamiltonian instead of the
full one in (1.48) to define the potential might avoid the spikes but Hunter in [86]
showed why this was unlikely to be the case, and Cassam-Chenai [89] repeated the
work of Czub and Wolniewicz using a clamped-nuclei electronic Hamiltonian and
showed that exactly the same spiky behaviour occurred.
Another approach to this problem is in [84]; there is reason to believe however
that this sort of difficulty is bound to arise whatever the approach. To see this, simply
rewrite (1.45) to recognise that the exact states will actually have definite quantum
numbers according to their symmetry, so that it would be more realistic to write
ψ J Mpri
t
n , t
e
= χ J MP ri
t
n
φ J Mpri
t
n , t
e
.
(1.49)
In the H 2 study cited the first four quantum numbers are of no relevance, only i
remains and here i labels the vibrational states. There is thus every reason to expect
that the best that can be done from this approach is a distinct PES for each nuclearmotion state.
This anticipated behaviour seems to be confirmed in very accurate calculations
on H 2 [90] for the electronic Σ ground state of the molecule assumed to dissociate
into two hydrogen atoms in their ground states. That work shows that, for example,
the J = 0 state supports just 14 vibrational states while the J = 15 state supports
10 and the J = 31 supports only 1 state. Of course in a diatomic molecule, states
of different k are states which differ in the electronic angular momentum and these
results cannot be regarded as typifying the results for a polyatomic system. However work on H
+
3 shows that in the case of J = 0 there are 1280 vibrational states
below dissociation [91] and that 46 is the highest value of J for which at least one
vibrational state exists [92]. At this level then it cannot be assumed that the potential
surface calculated in the usual way is an approximation to anything exact.
The eigenstates of the full molecular Hamiltonian (the Coulomb Hamiltonian for
the electrons and nuclei specified by a chemical formula)—a theory of an Isolated
Molecule modeled on the quantum theory of the atom which we call the Isolated
Molecule model—are reasonably well understood and might have some utility in
a limited area of high-resolution experiments on very small molecules where questions of isomerism do not arise [93]. Their computation poses formidable problems,
and really belongs to few-body physics. If it is to be taken as underlying Quantum
Chemistry then it is worth exploring the consequences of the model without regard
to approximations made for practical utility which are a quite separate matter. In this
paper we have attempted to discuss the Born-Oppenheimer and Born approaches to
the quantum theory of molecules in terms first set out by Combes [61]. The essential
point is that the decomposition of the molecular Hamiltonian (with centre-of-mass
contribution removed) into the nuclear kinetic energy, proportional to κ 4 and a remainder, is specified by (1.26), not by (1.9), or in other words, (1.9) cannot be