12
B. Sutcliffe and R.G. Woolley
proof of Ehrenfest’s adiabatic theorem for time-dependent perturbations was given
by Born and Fock [43]. Most famously though, the quantum mechanical basis for
the idea of electronic Potential Energy Surfaces is commonly attributed to Born and
Oppenheimer, and it is to a consideration of their famous paper [38] that we now
turn.
1.3.1 Born and Oppenheimer’s Quantum Theory of Molecules
Much of the groundwork for Born and Oppenheimer’s treatment of the energy levels of molecules was laid down in the earlier attempt by Born and Heisenberg [36].
The basic idea of both calculations is that the low-lying excitation spectrum of
a molecule can be obtained by regarding the nuclear kinetic energy as a ‘small’
perturbation of the energy of the electrons for stationary nuclei in an equilibrium
configuration. The physical basis for the idea is the large disparity between the
mass of the electron and the masses of the nuclei; classically the light electrons
undergo motions on a ‘fast’ timescale (τ e ≈ 10 −16 → 10 −15 s), while the vibrationrotation dynamics of the much heavier nuclei are characterized by ‘slow’ timescales
(τ N ≈ 10 −14 → 10 −12 s).
Consider a system of electrons and nuclei and denote the properties of the former by lower-case letters (mass m, coordinates x, momenta p) and of the latter by
capital letters (mass M, coordinates X, momenta P ). The small parameter for the
perturbation expansion must clearly be some power of m/M o , where M o can be
taken as any one of the nuclear masses or their average. In contrast to the earlier
calculation they found the correct choice is
κ =
m
M o
1
4
rather than Born and Heisenberg’s λ = κ 2 . In an obvious shorthand notation using a
coordinate representation the kinetic energy of the electrons is then 9
T e = T e
∂
∂x
while the nuclear kinetic energy depends on κ
T N = κ
4 H 1
∂
∂X
.
9 The details can be found in the original paper [38], and in various English language presentations,
for example [44–46].
B. Sutcliffe and R.G. Woolley
proof of Ehrenfest’s adiabatic theorem for time-dependent perturbations was given
by Born and Fock [43]. Most famously though, the quantum mechanical basis for
the idea of electronic Potential Energy Surfaces is commonly attributed to Born and
Oppenheimer, and it is to a consideration of their famous paper [38] that we now
turn.
1.3.1 Born and Oppenheimer’s Quantum Theory of Molecules
Much of the groundwork for Born and Oppenheimer’s treatment of the energy levels of molecules was laid down in the earlier attempt by Born and Heisenberg [36].
The basic idea of both calculations is that the low-lying excitation spectrum of
a molecule can be obtained by regarding the nuclear kinetic energy as a ‘small’
perturbation of the energy of the electrons for stationary nuclei in an equilibrium
configuration. The physical basis for the idea is the large disparity between the
mass of the electron and the masses of the nuclei; classically the light electrons
undergo motions on a ‘fast’ timescale (τ e ≈ 10 −16 → 10 −15 s), while the vibrationrotation dynamics of the much heavier nuclei are characterized by ‘slow’ timescales
(τ N ≈ 10 −14 → 10 −12 s).
Consider a system of electrons and nuclei and denote the properties of the former by lower-case letters (mass m, coordinates x, momenta p) and of the latter by
capital letters (mass M, coordinates X, momenta P ). The small parameter for the
perturbation expansion must clearly be some power of m/M o , where M o can be
taken as any one of the nuclear masses or their average. In contrast to the earlier
calculation they found the correct choice is
κ =
m
M o
1
4
rather than Born and Heisenberg’s λ = κ 2 . In an obvious shorthand notation using a
coordinate representation the kinetic energy of the electrons is then 9
T e = T e
∂
∂x
while the nuclear kinetic energy depends on κ
T N = κ
4 H 1
∂
∂X
.
9 The details can be found in the original paper [38], and in various English language presentations,
for example [44–46].