1 The Potential Energy Surface in Molecular Quantum Mechanics
11
for the calculation. Formally the rotational motion of the system can be dealt with
by requiring the coordinates for the reference structure to satisfy 7 what were later to
become known as ‘the Eckart conditions’ [37]. Then with a suitable set of internal
variables and
λ =
m
M
1
2
as the expansion parameter, the Hamiltonian was expressed as a series
H = H o + λ
2 H 2 + · · ·
(1.7)
to be treated by the action-angle perturbation theory Born had developed. The ‘unperturbed’ Hamiltonian H o is the full Hamiltonian for the electrons with the nuclei
fixed at the equilibrium structure, H 2 is quadratic in the nuclear variables (harmonic
oscillators) and also contains the rotational energy, 8 while . . . stands for higher order anharmonic vibrational terms. H 1 may be dropped because of the equilibrium
condition. With considerable effort there follows the usual separation of molecular energies, although of course no concrete calculation was possible within the
Old Quantum Theory framework. It is noteworthy that their calculation gives the
electronic energies at a single configuration because the perturbation calculation requires the introduction of the (assumed) equilibrium structure. This is different from
the adiabatic approach Nordheim tried (unsuccessfully) to get the electronic energy
at any separation of the nuclei [33].
1.3 Quantum Theory
With the completion of quantum mechanics in 1925–1926, the old problems in
atomic and molecular theory were reconsidered and considerable success was
achieved. The idea that the dynamics of the electrons and the nuclei should be
treated to some extent as separate problems was generally accepted. Thus the electronic structure calculations of London [39–41] can be seen as a successful reformulation of the approach Nordheim had tried in terms of the older quantum theory, and
the idea of ‘adiabatic separation’ is often said to originate in this work. It is however
also implied in the closing section of Slater’s early He atom paper where he sketches
(but does not carry through) a perturbation method of approximate calculation for
molecules in which the nuclei are first held fixed, and the resulting electronic eigenvalue(s) then act as the potential energy for the nuclei [42]. A quantum mechanical
7 This also deals with the uninteresting overall translation of the molecule.
8 The rotational and vibrational energies occur together because of the choice of the parameter λ;
as is well-known, Born and Oppenheimer later showed that a better choice is to take the quarter
power of the mass ratio as this separates the vibrational and rotational energies in the orders of the
perturbation expansion [38].
11
for the calculation. Formally the rotational motion of the system can be dealt with
by requiring the coordinates for the reference structure to satisfy 7 what were later to
become known as ‘the Eckart conditions’ [37]. Then with a suitable set of internal
variables and
λ =
m
M
1
2
as the expansion parameter, the Hamiltonian was expressed as a series
H = H o + λ
2 H 2 + · · ·
(1.7)
to be treated by the action-angle perturbation theory Born had developed. The ‘unperturbed’ Hamiltonian H o is the full Hamiltonian for the electrons with the nuclei
fixed at the equilibrium structure, H 2 is quadratic in the nuclear variables (harmonic
oscillators) and also contains the rotational energy, 8 while . . . stands for higher order anharmonic vibrational terms. H 1 may be dropped because of the equilibrium
condition. With considerable effort there follows the usual separation of molecular energies, although of course no concrete calculation was possible within the
Old Quantum Theory framework. It is noteworthy that their calculation gives the
electronic energies at a single configuration because the perturbation calculation requires the introduction of the (assumed) equilibrium structure. This is different from
the adiabatic approach Nordheim tried (unsuccessfully) to get the electronic energy
at any separation of the nuclei [33].
1.3 Quantum Theory
With the completion of quantum mechanics in 1925–1926, the old problems in
atomic and molecular theory were reconsidered and considerable success was
achieved. The idea that the dynamics of the electrons and the nuclei should be
treated to some extent as separate problems was generally accepted. Thus the electronic structure calculations of London [39–41] can be seen as a successful reformulation of the approach Nordheim had tried in terms of the older quantum theory, and
the idea of ‘adiabatic separation’ is often said to originate in this work. It is however
also implied in the closing section of Slater’s early He atom paper where he sketches
(but does not carry through) a perturbation method of approximate calculation for
molecules in which the nuclei are first held fixed, and the resulting electronic eigenvalue(s) then act as the potential energy for the nuclei [42]. A quantum mechanical
7 This also deals with the uninteresting overall translation of the molecule.
8 The rotational and vibrational energies occur together because of the choice of the parameter λ;
as is well-known, Born and Oppenheimer later showed that a better choice is to take the quarter
power of the mass ratio as this separates the vibrational and rotational energies in the orders of the
perturbation expansion [38].