1 The Potential Energy Surface in Molecular Quantum Mechanics
9
For an electron this yields r o ≈ 2.8 × 10 −15 m and an even smaller value for any
nucleus. It was clear that this was far too small to be relevant to an atomic theory;
of course the Bohr radius a o ≈ 0.5 × 10 −10 m is of just the right dimension.
Bohr’s theory developed into the Old Quantum Theory which was based on a
phase-space description of an atomic-molecular system and theoretical techniques
originally developed in celestial mechanics. These came from the application of the
developing quantum theory to molecular band spectra by Schwarzschild [19] and
Heurlinger [20] who used it to describe the quantized vibrational and rotational energies of small molecules (diatomic and symmetric top structures). Schwarzschild,
an astrophysicist, was responsible for the introduction of action-angle methods as
a basis for quantization in atomic/molecular theory. Heurlinger assumed a quantization of the energy of the nuclear vibration analogous to that used by Planck for
his ideal linear oscillators, with the possibility of anharmonic behaviour. Thus a
force-law or potential energy depending on the separation of the nuclei, for a given
arrangement of the electrons, was required.
The basic calculational tool was a perturbation theory approach developed enthusiastically by Born [21] and Sommerfeld [22] with their research assistants. The solution of the Hamiltonian equations of motion could be attempted via the HamiltonJacobi method based on canonical transformations of the action-angle variables.
This leads to an expression for the energy that is a function of the action integrals
only. The action (or ‘phase’) integrals are constants of the motion, and are also adiabatic invariants [23], and as such are natural objects for quantization according to
the ‘quantum conditions’. Thus for a separable system with k degrees of freedom
and action integrals {J i , i = 1, . . . , j ≤ k}, the quantum conditions according to
Sommerfeld are
J i ≡
p i dq i = n i h, i = 1, . . . , j
(1.6)
where the n i are non-negative integers (j < k in case of degeneracy). Here it is
assumed that each p i is a periodic function of only its corresponding conjugate coordinate q i , and the integration is taken over a period of q i . An important principle,
due to Bohr, was that slow, continuous (‘adiabatic’) deformations of an atomic system kept the system in a stationary state [24, 25]. Thus the action integrals for a
Hamiltonian depending on parameters that vary slowly in time are conserved under
slow changes of the parameters. 4 This could be applied to the problem of chemical
bonding by treating the nuclear positions as the slowly varying parameters in an
adiabatic transformation of the Hamiltonian for the electrons in the presence of the
nuclei.
We now know that systems of more than 2 particles with Coulomb interactions
may have very complicated dynamics; Newton famously struggled to account quantitatively for the orbit of the moon in the earth-moon-sun problem (n = 3). The
underlying reason for his difficulties is the existence of solutions carrying the signature of chaos [27] and this implies that there are classical trajectories to which
4 This is strictly true only for integrable Hamiltonians [26].
9
For an electron this yields r o ≈ 2.8 × 10 −15 m and an even smaller value for any
nucleus. It was clear that this was far too small to be relevant to an atomic theory;
of course the Bohr radius a o ≈ 0.5 × 10 −10 m is of just the right dimension.
Bohr’s theory developed into the Old Quantum Theory which was based on a
phase-space description of an atomic-molecular system and theoretical techniques
originally developed in celestial mechanics. These came from the application of the
developing quantum theory to molecular band spectra by Schwarzschild [19] and
Heurlinger [20] who used it to describe the quantized vibrational and rotational energies of small molecules (diatomic and symmetric top structures). Schwarzschild,
an astrophysicist, was responsible for the introduction of action-angle methods as
a basis for quantization in atomic/molecular theory. Heurlinger assumed a quantization of the energy of the nuclear vibration analogous to that used by Planck for
his ideal linear oscillators, with the possibility of anharmonic behaviour. Thus a
force-law or potential energy depending on the separation of the nuclei, for a given
arrangement of the electrons, was required.
The basic calculational tool was a perturbation theory approach developed enthusiastically by Born [21] and Sommerfeld [22] with their research assistants. The solution of the Hamiltonian equations of motion could be attempted via the HamiltonJacobi method based on canonical transformations of the action-angle variables.
This leads to an expression for the energy that is a function of the action integrals
only. The action (or ‘phase’) integrals are constants of the motion, and are also adiabatic invariants [23], and as such are natural objects for quantization according to
the ‘quantum conditions’. Thus for a separable system with k degrees of freedom
and action integrals {J i , i = 1, . . . , j ≤ k}, the quantum conditions according to
Sommerfeld are
J i ≡
p i dq i = n i h, i = 1, . . . , j
(1.6)
where the n i are non-negative integers (j < k in case of degeneracy). Here it is
assumed that each p i is a periodic function of only its corresponding conjugate coordinate q i , and the integration is taken over a period of q i . An important principle,
due to Bohr, was that slow, continuous (‘adiabatic’) deformations of an atomic system kept the system in a stationary state [24, 25]. Thus the action integrals for a
Hamiltonian depending on parameters that vary slowly in time are conserved under
slow changes of the parameters. 4 This could be applied to the problem of chemical
bonding by treating the nuclear positions as the slowly varying parameters in an
adiabatic transformation of the Hamiltonian for the electrons in the presence of the
nuclei.
We now know that systems of more than 2 particles with Coulomb interactions
may have very complicated dynamics; Newton famously struggled to account quantitatively for the orbit of the moon in the earth-moon-sun problem (n = 3). The
underlying reason for his difficulties is the existence of solutions carrying the signature of chaos [27] and this implies that there are classical trajectories to which
4 This is strictly true only for integrable Hamiltonians [26].