1 The Potential Energy Surface in Molecular Quantum Mechanics
29
and looking at the relations between the electronic and nuclear parts of a wavepacket [72]. This is essentially a use of standard coherent state theory where again
the nuclei are treated as distinguishable particles and the method is that of asymptotic expansion.
1.4 Discussion
Quite generally one needs to make a distinction between an hypothetical Isolated
Molecule, and a real observable individual molecule. There are no strictly ‘isolated’
systems of course, but what is striking is that an approach based on the stationary state eigensolutions of the appropriate Coulomb Hamiltonian works so well for
atoms and diatomic molecules, but fails with three or more nuclei. We have always
been clear that for most of chemistry, molecular eigenstates (‘stationary states’) are
of no relevance since metastability is an essential aspect of isomerism. The interesting question is how to get from the quantum theory of an Isolated Molecule to
a quantum theory of an individual molecule by rational mathematics. It is as well
to remember that the generic molecule is sufficiently complex that the quantum
mechanical permutation symmetry of identical nuclei is a feature that cannot be ignored, if one is doing quantum mechanics. The Isolated Molecule model doesn’t
capture isomerism, nor optical activity. We see no reason at all for Löwdin’s optimistic assertion (Sect. 1.1) that molecular symmetry must be contained somehow in
the Coulomb Hamiltonian.
If a molecule is not isolated it must be interacting with something; that something is loosely referred to as the ‘environment’. It might be other molecules, the
(macroscopic) substance the molecule finds itself in, or quantized electromagnetic
radiation. Blackbody radiation is all pervasive and charges are always coupled to
the photon vacuum state in QED and so ‘dressed’ with clouds of virtual photons.
A crucial feature of ‘environments’ in quantum theory is that generally they are described by Hamiltonians with purely continuous spectra. This is important because
a quantum system with a finite number, n, of degrees of freedom described by the
usual linear Schrödinger equation does not yield ‘broken symmetry’ solutions if
n < ∞. Such matters were discussed at length thirty years ago in the context of
molecular structure and quantum theory [73–75]. The characteristic feature of such
discussions, and this also applies to more modern formulations under the chic heading of ‘decoherence’, is that they start with some primitive notion of structure built
in: two-state systems, potential energy wells, wavefunctions associated with distinct isomers etc. The ‘environment’ is modelled in the simplest possible way (for
example, a free boson quantum field). These crucial ideas are put in by hand at the
outset. We don’t see that as a ‘problem’ or ‘difficulty’; it is a characteristic feature of
many-body physics (condensed matter, nuclei, chemistry) and results in remarkably
powerful and fruitful theoretical formalisms; see, for example, Anderson’s discussion of what he calls ‘adiabatic continuity’ [76]. But one can hardly avoid noticing
that the models of molecules used are caricatures that contain just the right features
29
and looking at the relations between the electronic and nuclear parts of a wavepacket [72]. This is essentially a use of standard coherent state theory where again
the nuclei are treated as distinguishable particles and the method is that of asymptotic expansion.
1.4 Discussion
Quite generally one needs to make a distinction between an hypothetical Isolated
Molecule, and a real observable individual molecule. There are no strictly ‘isolated’
systems of course, but what is striking is that an approach based on the stationary state eigensolutions of the appropriate Coulomb Hamiltonian works so well for
atoms and diatomic molecules, but fails with three or more nuclei. We have always
been clear that for most of chemistry, molecular eigenstates (‘stationary states’) are
of no relevance since metastability is an essential aspect of isomerism. The interesting question is how to get from the quantum theory of an Isolated Molecule to
a quantum theory of an individual molecule by rational mathematics. It is as well
to remember that the generic molecule is sufficiently complex that the quantum
mechanical permutation symmetry of identical nuclei is a feature that cannot be ignored, if one is doing quantum mechanics. The Isolated Molecule model doesn’t
capture isomerism, nor optical activity. We see no reason at all for Löwdin’s optimistic assertion (Sect. 1.1) that molecular symmetry must be contained somehow in
the Coulomb Hamiltonian.
If a molecule is not isolated it must be interacting with something; that something is loosely referred to as the ‘environment’. It might be other molecules, the
(macroscopic) substance the molecule finds itself in, or quantized electromagnetic
radiation. Blackbody radiation is all pervasive and charges are always coupled to
the photon vacuum state in QED and so ‘dressed’ with clouds of virtual photons.
A crucial feature of ‘environments’ in quantum theory is that generally they are described by Hamiltonians with purely continuous spectra. This is important because
a quantum system with a finite number, n, of degrees of freedom described by the
usual linear Schrödinger equation does not yield ‘broken symmetry’ solutions if
n < ∞. Such matters were discussed at length thirty years ago in the context of
molecular structure and quantum theory [73–75]. The characteristic feature of such
discussions, and this also applies to more modern formulations under the chic heading of ‘decoherence’, is that they start with some primitive notion of structure built
in: two-state systems, potential energy wells, wavefunctions associated with distinct isomers etc. The ‘environment’ is modelled in the simplest possible way (for
example, a free boson quantum field). These crucial ideas are put in by hand at the
outset. We don’t see that as a ‘problem’ or ‘difficulty’; it is a characteristic feature of
many-body physics (condensed matter, nuclei, chemistry) and results in remarkably
powerful and fruitful theoretical formalisms; see, for example, Anderson’s discussion of what he calls ‘adiabatic continuity’ [76]. But one can hardly avoid noticing
that the models of molecules used are caricatures that contain just the right features