28
B. Sutcliffe and R.G. Woolley
The singular nature of the microscope transformation for λ = 0 is demonstrated
by the modification of the spectrum associated with the limit κ → 0. The spectrum of the Coulomb Hamiltonian for a molecule can be discussed in terms of the
Hunziker-van Winter-Zhislin theorem [66–68]; for the diatomic molecule, σ ess ( ˆ
H )
starts at the lowest two-body threshold Σ = λ A (m A ) + λ B (m B ) given by the minimal value of the sums of pairs of binding energies for atoms A and B with finite masses m A and m B respectively. On the other hand the spectrum of the electronic Hamiltonian, ˆ
H elec , is purely continuous, σ ( ˆ
H elec ) = [V 0 , ∞). In the limit
m A , m B → ∞, Σ does not generally converge to V 0 ; instead the missing part of
the continuous spectrum [V 0 , λ A (∞) + λ B (∞)] is provided by an accumulation of
bound states in this interval [62]. The microscope transformation is formally applicable to the polyatomic case but it may not be sufficient to control the asymptotic
behaviour, and has not been used for general molecules.
Since the initial work of Combes, a considerable amount of mathematical work
has been published using both time-independent and time-dependent techniques
with developments for the polyatomic case; for a recent review of rigorous results
about the separation of electronic and nuclear motions see Hagedorn and Joye [69]
which covers the literature to 2006. The Hamiltonian (1.26) is the one used by Klein
et al. [70] in their consideration of the precise formulation of the Born-Oppenheimer
approximation for polyatomic systems. Their work was based on a powerful symbolic operator method, the pseudodifferential calculus [70, 71] and a formalism related to the partitioning technique described above. In [70] it is assumed that (1.26)
has a discrete eigenvalue which has a minimum as a function of the t n in the neighborhood of some values t n
i = b i . If it can be assumed that (a) the electronic wavefunction vanishes strongly outside a region close to a particular nuclear geometry
and (b) that the electronic energy at the given geometry is an isolated minimum, then
it is possible to present a rigorous account of the separation of electronic and nuclear motion which corresponds in some measure to the original Born-Oppenheimer
treatment.
A novel feature arises from the requirement that the inversion symmetry of the
original problem be respected. If the geometry at the minimum energy configuration
is either linear or planar then inversion can be dealt with in terms of a single minimum in the electronic energy. If the geometry at the minimum is other than these
two forms, inversion produces a second potential minimum and the problem must
be dealt with as a two-minimum problem; then extra consideration is necessary to
establish whether the two wells have negligible interaction so that only one of the
wells need be considered for the nuclear motion. The nuclei are treated as distinguishable particles that can be numbered uniquely. The symmetry requirements on
the total wavefunction that would arise from the invariance of the Hamiltonian operator under the permutation of identical nuclei are not considered. Because of the
continuous spectrum of the electronic Hamiltonian ˆ
H elec , it is not possible to use
regular perturbation theory in the analysis; instead asymptotic expansion theory is
used so that the result has essentially the character of a WKB approximation [70].
Similar functional analytic techniques have been used to consider such phenomena as Landau-Zener crossing by using a time-dependent approach to the problem
B. Sutcliffe and R.G. Woolley
The singular nature of the microscope transformation for λ = 0 is demonstrated
by the modification of the spectrum associated with the limit κ → 0. The spectrum of the Coulomb Hamiltonian for a molecule can be discussed in terms of the
Hunziker-van Winter-Zhislin theorem [66–68]; for the diatomic molecule, σ ess ( ˆ
H )
starts at the lowest two-body threshold Σ = λ A (m A ) + λ B (m B ) given by the minimal value of the sums of pairs of binding energies for atoms A and B with finite masses m A and m B respectively. On the other hand the spectrum of the electronic Hamiltonian, ˆ
H elec , is purely continuous, σ ( ˆ
H elec ) = [V 0 , ∞). In the limit
m A , m B → ∞, Σ does not generally converge to V 0 ; instead the missing part of
the continuous spectrum [V 0 , λ A (∞) + λ B (∞)] is provided by an accumulation of
bound states in this interval [62]. The microscope transformation is formally applicable to the polyatomic case but it may not be sufficient to control the asymptotic
behaviour, and has not been used for general molecules.
Since the initial work of Combes, a considerable amount of mathematical work
has been published using both time-independent and time-dependent techniques
with developments for the polyatomic case; for a recent review of rigorous results
about the separation of electronic and nuclear motions see Hagedorn and Joye [69]
which covers the literature to 2006. The Hamiltonian (1.26) is the one used by Klein
et al. [70] in their consideration of the precise formulation of the Born-Oppenheimer
approximation for polyatomic systems. Their work was based on a powerful symbolic operator method, the pseudodifferential calculus [70, 71] and a formalism related to the partitioning technique described above. In [70] it is assumed that (1.26)
has a discrete eigenvalue which has a minimum as a function of the t n in the neighborhood of some values t n
i = b i . If it can be assumed that (a) the electronic wavefunction vanishes strongly outside a region close to a particular nuclear geometry
and (b) that the electronic energy at the given geometry is an isolated minimum, then
it is possible to present a rigorous account of the separation of electronic and nuclear motion which corresponds in some measure to the original Born-Oppenheimer
treatment.
A novel feature arises from the requirement that the inversion symmetry of the
original problem be respected. If the geometry at the minimum energy configuration
is either linear or planar then inversion can be dealt with in terms of a single minimum in the electronic energy. If the geometry at the minimum is other than these
two forms, inversion produces a second potential minimum and the problem must
be dealt with as a two-minimum problem; then extra consideration is necessary to
establish whether the two wells have negligible interaction so that only one of the
wells need be considered for the nuclear motion. The nuclei are treated as distinguishable particles that can be numbered uniquely. The symmetry requirements on
the total wavefunction that would arise from the invariance of the Hamiltonian operator under the permutation of identical nuclei are not considered. Because of the
continuous spectrum of the electronic Hamiltonian ˆ
H elec , it is not possible to use
regular perturbation theory in the analysis; instead asymptotic expansion theory is
used so that the result has essentially the character of a WKB approximation [70].
Similar functional analytic techniques have been used to consider such phenomena as Landau-Zener crossing by using a time-dependent approach to the problem