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Erschienen in:
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2013 | OriginalPaper | Buchkapitel

1. The Potential Energy Surface in Molecular Quantum Mechanics

verfasst von : Brian Sutcliffe, R. Guy Woolley

Erschienen in: Advances in Quantum Methods and Applications in Chemistry, Physics, and Biology

Verlag: Springer International Publishing

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Abstract

The idea of a Potential Energy Surface (PES) forms the basis of almost all accounts of the mechanisms of chemical reactions, and much of theoretical molecular spectroscopy. It is assumed that, in principle, the PES can be calculated by means of clamped-nuclei electronic structure calculations based upon the Schrödinger Coulomb Hamiltonian. This article is devoted to a discussion of the origin of the idea, its development in the context of the Old Quantum Theory, and its present status in the quantum mechanics of molecules. It is argued that its present status must be regarded as uncertain.

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Nächstes Kapitel A Comment on the Question of Degeneracies in Quantum Mechanics
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Fußnoten
1
René Marcelin was killed in action fighting for France in September 1914.
 
2
That a molecule must reach a certain region of space at a suitable angle, that its speed must exceed a certain limit, that its internal structure must correspond to an unstable configuration etc.; …
 
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 https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-01529-3_1/311421_1_En_1_IEq11_HTML.gif 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.
 
4
This is strictly true only for integrable Hamiltonians [26].
 
5
The difficulties for action-angle quantization posed by the existence of chaotic motions in non-separable systems [30] were recognized by Einstein at the time the Old Quantum Theory was developed [31].
 
6
This is the earliest reference we know of where the idea of adiabatic separation of the electrons and the nuclei is proposed explicitly.
 
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].
 
9
The details can be found in the original paper [38], and in various English language presentations, for example [44‐46].
 
10
If one sets κ=0…one obtains a differential equation in the x k alone, the X l appearing as parameters:…. Evidently, this represents the electronic motion for stationary nuclei.
 
11
W(X) in the notation of the above quotation.
 
12
The reader may find it helpful to refer to the Appendix which summarizes some mathematical notions that are needed here, and illustrates them in a simple model of coupled oscillators with two degrees of freedom.
 
13
It is always possible to split off the kinetic energy of the centre-of-mass without any approximation; with this choice we retain the separation of the electronic and nuclear kinetic energies as well, as in (1.24). Explicit formulae are given in e.g. [3] where it is shown that the nuclear kinetic energy terms involve reciprocals of the nuclear masses, so that overall, the nuclear kinetic energy is proportional to κ 4.
 
14
After the elimination of the centre-of-mass variables \(\hat{\mathsf{H}}^{\mathrm{elec}}\) is playing the role of \(\hat{H}_{o}\) in (1.20).
 
15
We assume that the centre-of-mass contributions are eliminated as usual.
 
16
The multiminima case can also be treated in this way.
 
17
The Lorentz Theory of the electron for example [77].
 
18
A similar requirement must be placed on the denominator in (12) of [85] for the equation to provide a secure definition.
 
19
In its original form b=b o , the equilibrium configuration, on the right-hand side of (1.50).
 
20
The variables are expressed in dimensionless form for simplicity. The quantum oscillator \(\hat{\mathsf{h}}=\frac{1}{2}(\hat{\mathsf{p}}^{2}+\hat{\mathsf{q}}^{2})\) has eigenvalues \(n+\frac{1}{2}\).
 
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Metadaten
Titel
The Potential Energy Surface in Molecular Quantum Mechanics
verfasst von
Brian Sutcliffe
R. Guy Woolley
Copyright-Jahr
2013
Buch
Advances in Quantum Methods and Applications in Chemistry, Physics, and Biology

Print ISBN: 978-3-319-01528-6

Electronic ISBN: 978-3-319-01529-3

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-319-01529-3_1