1 The Potential Energy Surface in Molecular Quantum Mechanics
15
the electronic wave equation, regarded as a function of the internuclear distances,
occurs as a potential function. After some remarks about the coordinates they say
The first step in the treatment of a molecule is to solve this electronic wave equation for all
configurations of the nuclei. It is found that the characteristic values U n (ξ ) of the electronic
energy are continuous functions of the nuclear coordinates ξ . For example, for a free diatomic molecule the electronic energy function for the most stable electronic state (n = 0)
is a function only of the distance r between the two nuclei, and it is a continuous function
of r, such as shown in Fig. 34-2.
Figure 34-2 referred to here is a Morse potential function. Later in the book where
they give a brief introduction to activation energies of chemical reactions they explicitly cite London [41] as the origin of the idea of adiabatic nuclear motion on
a Potential Energy Surface, though there is also a nod back towards Chap. X. Although it is now almost universal practice to refer to treating the nuclei as classical particles that give rise to an electronic energy surface as ‘making the BornOppenheimer approximation’ it is our opinion that the justification for such a strategy is not to be found in The Quantum Theory of Molecules, [38]. Nor is it to be
found in the early papers of London [39–41] where it was simply assumed as a reasonable thing to do. And it is certainly the case that Born and Oppenheimer did not
show the electronic energy to be a continuous function of the nuclear coordinates;
that was first demonstrated for a diatomic molecule forty years after Pauling and
Wilson’s book was published (see Sect. 1.3.4).
1.3.2 Born and the Elimination of Electronic Motion
Many years after his work with Heisenberg and Oppenheimer, Born returned to the
subject of molecular quantum theory and developed a different account of the separation of electronic and nuclear motion [44, 49]. It is to this method that the expression ‘Born-Oppenheimer approximation’ usually refers in modern work. Consider
the unperturbed electronic Hamiltonian H o (x, X f ) at a fixed nuclear configuration
X f that corresponds to some molecular structure (not necessarily an equilibrium
structure). The Schrödinger equation for H o is
H o (x, X f ) − E
o (X f ) m
ϕ(x, X f ) m = 0.
(1.13)
This equation can have both bound-state and continuum eigenfunctions; the boundstate eigenvalues considered as functions of the X f are the molecular Potential
Energy Surfaces. Born proposed to solve the full molecular Schrödinger equation,
(1.10) by an expansion
ψ(x, X) =
m
Φ(X) m ϕ(x, X) m
(1.14)
with coefficients {Φ(X) m } that play the role of nuclear wavefunctions. As in the
original calculation (Sect. 1.3.1) a crucial step is to assign the nuclear coordinates
15
the electronic wave equation, regarded as a function of the internuclear distances,
occurs as a potential function. After some remarks about the coordinates they say
The first step in the treatment of a molecule is to solve this electronic wave equation for all
configurations of the nuclei. It is found that the characteristic values U n (ξ ) of the electronic
energy are continuous functions of the nuclear coordinates ξ . For example, for a free diatomic molecule the electronic energy function for the most stable electronic state (n = 0)
is a function only of the distance r between the two nuclei, and it is a continuous function
of r, such as shown in Fig. 34-2.
Figure 34-2 referred to here is a Morse potential function. Later in the book where
they give a brief introduction to activation energies of chemical reactions they explicitly cite London [41] as the origin of the idea of adiabatic nuclear motion on
a Potential Energy Surface, though there is also a nod back towards Chap. X. Although it is now almost universal practice to refer to treating the nuclei as classical particles that give rise to an electronic energy surface as ‘making the BornOppenheimer approximation’ it is our opinion that the justification for such a strategy is not to be found in The Quantum Theory of Molecules, [38]. Nor is it to be
found in the early papers of London [39–41] where it was simply assumed as a reasonable thing to do. And it is certainly the case that Born and Oppenheimer did not
show the electronic energy to be a continuous function of the nuclear coordinates;
that was first demonstrated for a diatomic molecule forty years after Pauling and
Wilson’s book was published (see Sect. 1.3.4).
1.3.2 Born and the Elimination of Electronic Motion
Many years after his work with Heisenberg and Oppenheimer, Born returned to the
subject of molecular quantum theory and developed a different account of the separation of electronic and nuclear motion [44, 49]. It is to this method that the expression ‘Born-Oppenheimer approximation’ usually refers in modern work. Consider
the unperturbed electronic Hamiltonian H o (x, X f ) at a fixed nuclear configuration
X f that corresponds to some molecular structure (not necessarily an equilibrium
structure). The Schrödinger equation for H o is
H o (x, X f ) − E
o (X f ) m
ϕ(x, X f ) m = 0.
(1.13)
This equation can have both bound-state and continuum eigenfunctions; the boundstate eigenvalues considered as functions of the X f are the molecular Potential
Energy Surfaces. Born proposed to solve the full molecular Schrödinger equation,
(1.10) by an expansion
ψ(x, X) =
m
Φ(X) m ϕ(x, X) m
(1.14)
with coefficients {Φ(X) m } that play the role of nuclear wavefunctions. As in the
original calculation (Sect. 1.3.1) a crucial step is to assign the nuclear coordinates