1 The Potential Energy Surface in Molecular Quantum Mechanics
7
where k B is Boltzmann’s constant, H is the Hamiltonian function for the molecule,
and G #
+ is the Gibb’s free energy of the active molecules relative to the mean energy
of the reactant molecules. It is independent of the canonical variables. There is an
analogous expression for the reverse reaction involving G #
− . Marcelin quoted a formula due to Gibbs [9] for the number of molecules dN crossing a surface element
ds in the critical surface in the neighbourhood of q, p, in time dt, which may be
written in shorthand as
dN = dtf (q, p, t)J (˙ q, ˙
p, q, p)
where ˙
p, ˙
q are regarded as functions of q, p by virtue of Hamilton’s equations of
motion. The total rate is
v =
ddf (q, p)J δ
S(q, p)
(1.5)
where the delta function confines the integration to the critical surface S . Equation (1.2) results from taking the difference between this expression for the forward
and reverse reactions, and factoring out the terms in G #
± ; the remaining integration,
which Marcelin did not evaluate, defines the multiplying factor M.
1.2.2 Molecular Spectroscopy and the Old Quantum Theory
Although the discussion in the previous section looks familiar, it does so only because of the modern interpretation we put upon it. 3 It is important to note that
nowhere did Marcelin elaborate on how the canonical variables were to be chosen, nor even how n could be fixed in any given case. The words ‘atom’, ‘electron’,
‘nucleus’ do not appear anywhere in his thesis, in which respect he seems to have
followed the scientific philosophy of his countryman Duhem [11]. On other pages
in the thesis Marcelin referred to the ‘structure’ (also ‘architecture’) of a molecule
and to molecular ‘oscillations’ but never otherwise invoked the atomic structural
conception of a molecule due to e.g. van ’t Hoff, although he was very well aware
of van ’t Hoff’s Physical Chemistry.
Contemporary with Marcelin’s investigation of chemical reaction rates was the
introduction of a completely novel model of an atom due to Rutherford. However
3 Nevertheless it seems proper to regard Marcelin’s introduction of phase-space variables and a
critical reaction surface into chemical dynamics as the beginning of a formulation of the Transition
State Theory that was developed by Wigner in the 1930’s [12–15]. The 2n phase-space variables
q, p were identified with the n nuclei specified in the chemical formula of the participating species,
and the Hamiltonian H was that for classical nuclear motion on a Potential Energy Surface; this
dynamics was assumed to give rise to a critical surface which was such that reaction trajectories
cross the surface precisely once. The classical nature of the formalism was quite clear because
the Uncertainty Principle precludes the precise specification of position on the critical surface
simultaneously with the momentum of the nuclei.
7
where k B is Boltzmann’s constant, H is the Hamiltonian function for the molecule,
and G #
+ is the Gibb’s free energy of the active molecules relative to the mean energy
of the reactant molecules. It is independent of the canonical variables. There is an
analogous expression for the reverse reaction involving G #
− . Marcelin quoted a formula due to Gibbs [9] for the number of molecules dN crossing a surface element
ds in the critical surface in the neighbourhood of q, p, in time dt, which may be
written in shorthand as
dN = dtf (q, p, t)J (˙ q, ˙
p, q, p)
where ˙
p, ˙
q are regarded as functions of q, p by virtue of Hamilton’s equations of
motion. The total rate is
v =
ddf (q, p)J δ
S(q, p)
(1.5)
where the delta function confines the integration to the critical surface S . Equation (1.2) results from taking the difference between this expression for the forward
and reverse reactions, and factoring out the terms in G #
± ; the remaining integration,
which Marcelin did not evaluate, defines the multiplying factor M.
1.2.2 Molecular Spectroscopy and the Old Quantum Theory
Although the discussion in the previous section looks familiar, it does so only because of the modern interpretation we put upon it. 3 It is important to note that
nowhere did Marcelin elaborate on how the canonical variables were to be chosen, nor even how n could be fixed in any given case. The words ‘atom’, ‘electron’,
‘nucleus’ do not appear anywhere in his thesis, in which respect he seems to have
followed the scientific philosophy of his countryman Duhem [11]. On other pages
in the thesis Marcelin referred to the ‘structure’ (also ‘architecture’) of a molecule
and to molecular ‘oscillations’ but never otherwise invoked the atomic structural
conception of a molecule due to e.g. van ’t Hoff, although he was very well aware
of van ’t Hoff’s Physical Chemistry.
Contemporary with Marcelin’s investigation of chemical reaction rates was the
introduction of a completely novel model of an atom due to Rutherford. However
3 Nevertheless it seems proper to regard Marcelin’s introduction of phase-space variables and a
critical reaction surface into chemical dynamics as the beginning of a formulation of the Transition
State Theory that was developed by Wigner in the 1930’s [12–15]. The 2n phase-space variables
q, p were identified with the n nuclei specified in the chemical formula of the participating species,
and the Hamiltonian H was that for classical nuclear motion on a Potential Energy Surface; this
dynamics was assumed to give rise to a critical surface which was such that reaction trajectories
cross the surface precisely once. The classical nature of the formalism was quite clear because
the Uncertainty Principle precludes the precise specification of position on the critical surface
simultaneously with the momentum of the nuclei.