1 The Potential Energy Surface in Molecular Quantum Mechanics
33
written with an = sign if the conventional interpretation of the X acting as parameters is made. Allowing the nuclear masses to increase without limit in ˆ
H elec does
not produce an operator with a discrete spectrum since this would just cause the
mass polarisation term to vanish and the effective electronic mass to become the
rest mass. As we have seen it leads to ‘adiabatic divergence’ [55].
It is thus not possible to reduce the molecular Schrödinger equation to a system
of coupled differential equations of classical type for nuclei moving on Potential
Energy Surfaces as suggested by Born. An extra choice of fixed nuclear positions
must be made to give any discrete spectrum and normalizable L 2 eigenfunctions. In
our view this choice, that is, the introduction of the clamped-nuclei Hamiltonian into
the molecular theory as in Sect. 1.3.1, is the essence of what is commonly meant by
the expression, 19 the ‘Born-Oppenheimer approximation’
ˆ
H
elec
=
⊕
X
ˆ
K
b, t
e
o
db → ˆ
K
b, t
e
.
(1.50)
If the molecular Hamiltonian H were classical as in [36], the removal of the nuclear
kinetic energy terms would indeed leave a Hamiltonian representing the electronic
motion for stationary nuclei, as claimed by Born and Oppenheimer [38, 46]. As we
have seen, quantization of H changes the situation drastically, so an implicit appeal
to the classical limit for the nuclei is required. The argument is a subtle one, for
subsequently, once the classical energy surface has emerged, the nuclei are treated as
quantum particles for the determination of the vibration-rotation spectrum (though
indistinguishability is rarely carried through); this can be seen from the complexity
of the mathematical account given by Klein and co-workers [70].
This qualitative modification of the internal Hamiltonian, the extra choice of
fixed nuclear positions in the ‘electronic’ Hamiltonian, is ad hoc in the same sense
that Bohr’s quantum theory of the atom is an ad hoc modification of classical mechanics. An essential feature of the answer is put in by hand. We know that both
modifications have been tremendously useful and our point is not that something
else must be done in practical calculations on molecules. The point is how the successful description of molecules involving the clamped-nuclei modification at some
stage can best be understood in terms of quantum mechanics. In the case of the Bohr
atom the resolution of the inconsistency in mechanics applied to the microscopic
realm was achieved quite quickly with the formulation of quantum mechanics; in
the molecular case, no such resolution is at present known.
Appendix
In this appendix we give an heuristic account of the mathematical notion of the
direct integral of Hilbert spaces, and then study a simple model problem to illustrate
the general ideas discussed in the paper.
19 In its original form b = b o , the equilibrium configuration, on the right-hand side of (1.50).
33
written with an = sign if the conventional interpretation of the X acting as parameters is made. Allowing the nuclear masses to increase without limit in ˆ
H elec does
not produce an operator with a discrete spectrum since this would just cause the
mass polarisation term to vanish and the effective electronic mass to become the
rest mass. As we have seen it leads to ‘adiabatic divergence’ [55].
It is thus not possible to reduce the molecular Schrödinger equation to a system
of coupled differential equations of classical type for nuclei moving on Potential
Energy Surfaces as suggested by Born. An extra choice of fixed nuclear positions
must be made to give any discrete spectrum and normalizable L 2 eigenfunctions. In
our view this choice, that is, the introduction of the clamped-nuclei Hamiltonian into
the molecular theory as in Sect. 1.3.1, is the essence of what is commonly meant by
the expression, 19 the ‘Born-Oppenheimer approximation’
ˆ
H
elec
=
⊕
X
ˆ
K
b, t
e
o
db → ˆ
K
b, t
e
.
(1.50)
If the molecular Hamiltonian H were classical as in [36], the removal of the nuclear
kinetic energy terms would indeed leave a Hamiltonian representing the electronic
motion for stationary nuclei, as claimed by Born and Oppenheimer [38, 46]. As we
have seen, quantization of H changes the situation drastically, so an implicit appeal
to the classical limit for the nuclei is required. The argument is a subtle one, for
subsequently, once the classical energy surface has emerged, the nuclei are treated as
quantum particles for the determination of the vibration-rotation spectrum (though
indistinguishability is rarely carried through); this can be seen from the complexity
of the mathematical account given by Klein and co-workers [70].
This qualitative modification of the internal Hamiltonian, the extra choice of
fixed nuclear positions in the ‘electronic’ Hamiltonian, is ad hoc in the same sense
that Bohr’s quantum theory of the atom is an ad hoc modification of classical mechanics. An essential feature of the answer is put in by hand. We know that both
modifications have been tremendously useful and our point is not that something
else must be done in practical calculations on molecules. The point is how the successful description of molecules involving the clamped-nuclei modification at some
stage can best be understood in terms of quantum mechanics. In the case of the Bohr
atom the resolution of the inconsistency in mechanics applied to the microscopic
realm was achieved quite quickly with the formulation of quantum mechanics; in
the molecular case, no such resolution is at present known.
Appendix
In this appendix we give an heuristic account of the mathematical notion of the
direct integral of Hilbert spaces, and then study a simple model problem to illustrate
the general ideas discussed in the paper.
19 In its original form b = b o , the equilibrium configuration, on the right-hand side of (1.50).