14
B. Sutcliffe and R.G. Woolley
The crucial observation that makes the calculation successful is the choice of X o ;
the Schrödinger equation for the unperturbed Hamiltonian H o can be solved for any
choice of the nuclear parameters X, and yields 11 an unperturbed energy E(X) for
the configuration X. For the consistency of the whole scheme however it turns out
(cf. footnote 9) that X o in (1.11) cannot be arbitrarily chosen, but must correspond
to a minimum of the electronic energy. That there is such a point is assumed to be
self-evident for the case of a stable molecule. The result of the calculation was a
triumph; the low-lying energy levels of a stable molecule can be written in the form
E Mol = E Elec + κ
2 E Vib + κ
4 E Rot + · · ·
(1.12)
in agreement with a considerable body of spectroscopic evidence. The eigenfunctions that correspond to these energy levels are simple products of an electronic
wavefunction obtained for the equilibrium geometry and suitable vibration-rotation
wavefunctions for the nuclei.
The Born and Heisenberg calculation [36] had been performed while Heisenberg
was a student with Born; Kragh [35] quotes Heisenberg’s later view of it in the
following terms
As an exasperated Heisenberg wrote to Pauli, “The work on molecules I did with
Born. . . contains bracket symbols [Klammersymbole] with up to 8 indices and will probably
be read by no one.” Certainly, it was not read by the chemists.
Curiously that may have initially been the fate of Born and Oppenheimer’s paper. As
noted by one of us many years ago, a survey of the literature up to about 1935 shows
that the paper was hardly if ever mentioned, and when it was mentioned, its arguments were used as a posteriori justification for what was being done anyway [47].
What was being done was the general use in molecular spectroscopy and chemical
reaction theory of the idea of Potential Energy Surfaces on which the nuclei moved.
As we have seen, that idea is not to be found in the approach taken by Born and
Oppenheimer which used (and had to use) a single privileged point in the nuclear
configuration space—the assumed equilibrium arrangement of the nuclei [38].
In 1935 a significant event was the publication of the famous textbook Introduction to Quantum Mechanics [48] which was probably the first textbook concerned
with quantum mechanics that addressed in detail problems of interest to chemists.
Generations of chemists and physicists took their first steps in quantum theory with
this book, which is still available in reprint form. Chapter X of the book is entitled
The Rotation and Vibration of Molecules; it starts by summarizing the empirical results of molecular spectroscopy which are consistent with (1.12). The authors then
turn to the wave equation for a general collection of electrons and nuclei and remark
that its Schrödinger wave equation may be solved approximately by a procedure
that they attribute to Born and Oppenheimer; first solve the wave equation for the
electrons alone, with the nuclei in a fixed configuration, and then solve the wave
equation for the nuclei alone, in which a characteristic energy value [eigenvalue] of
11 W (X) in the notation of the above quotation.
B. Sutcliffe and R.G. Woolley
The crucial observation that makes the calculation successful is the choice of X o ;
the Schrödinger equation for the unperturbed Hamiltonian H o can be solved for any
choice of the nuclear parameters X, and yields 11 an unperturbed energy E(X) for
the configuration X. For the consistency of the whole scheme however it turns out
(cf. footnote 9) that X o in (1.11) cannot be arbitrarily chosen, but must correspond
to a minimum of the electronic energy. That there is such a point is assumed to be
self-evident for the case of a stable molecule. The result of the calculation was a
triumph; the low-lying energy levels of a stable molecule can be written in the form
E Mol = E Elec + κ
2 E Vib + κ
4 E Rot + · · ·
(1.12)
in agreement with a considerable body of spectroscopic evidence. The eigenfunctions that correspond to these energy levels are simple products of an electronic
wavefunction obtained for the equilibrium geometry and suitable vibration-rotation
wavefunctions for the nuclei.
The Born and Heisenberg calculation [36] had been performed while Heisenberg
was a student with Born; Kragh [35] quotes Heisenberg’s later view of it in the
following terms
As an exasperated Heisenberg wrote to Pauli, “The work on molecules I did with
Born. . . contains bracket symbols [Klammersymbole] with up to 8 indices and will probably
be read by no one.” Certainly, it was not read by the chemists.
Curiously that may have initially been the fate of Born and Oppenheimer’s paper. As
noted by one of us many years ago, a survey of the literature up to about 1935 shows
that the paper was hardly if ever mentioned, and when it was mentioned, its arguments were used as a posteriori justification for what was being done anyway [47].
What was being done was the general use in molecular spectroscopy and chemical
reaction theory of the idea of Potential Energy Surfaces on which the nuclei moved.
As we have seen, that idea is not to be found in the approach taken by Born and
Oppenheimer which used (and had to use) a single privileged point in the nuclear
configuration space—the assumed equilibrium arrangement of the nuclei [38].
In 1935 a significant event was the publication of the famous textbook Introduction to Quantum Mechanics [48] which was probably the first textbook concerned
with quantum mechanics that addressed in detail problems of interest to chemists.
Generations of chemists and physicists took their first steps in quantum theory with
this book, which is still available in reprint form. Chapter X of the book is entitled
The Rotation and Vibration of Molecules; it starts by summarizing the empirical results of molecular spectroscopy which are consistent with (1.12). The authors then
turn to the wave equation for a general collection of electrons and nuclei and remark
that its Schrödinger wave equation may be solved approximately by a procedure
that they attribute to Born and Oppenheimer; first solve the wave equation for the
electrons alone, with the nuclei in a fixed configuration, and then solve the wave
equation for the nuclei alone, in which a characteristic energy value [eigenvalue] of
11 W (X) in the notation of the above quotation.