On the Quantum Theory of Molecules
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On the Quantum Theory of Molecules Brian T. Sutcliffea,∗, R. Guy Woolleyb aService de Chimie quantique et Photophysique, Universit´e Libre de Bruxelles, B-1050 Bruxelles, Belgium bSchool of Science and Technology, Nottingham Trent University, Nottingham NG11 8NS, U.K. Abstract Transition state theory was introduced in the 1930s to account for chemical reactions. Central to this theory is the idea of a potential energy surface (PES). It was assumed that such a surface could be constructed using eigensolutions of the Schr¨odinger equation for the molecular (Coulomb) Hamiltonian but at that time such calculations were not posssible. Nowadays quantum mechanical ab- initio electronic structure calculations are routine and from their results PESs can be constructed which are believed to approximate those assumed derivable from the eigensolutions. It is argued here that this belief is unfounded. It is suggested that the potential energy surface construction is more appropriately regarded as a legitimate and effective modification of quantum mechanics for chemical purposes. 1. Introduction probabilities for the hydrogen exchange reaction can be calcu- lated using its characteristic shape. They take it as obvious that The principal aim of much of contemporary quantum chem- the nuclei can be treated as classical particles whose positions ical calculation is to calculate a potential energy surface from can be fixed and do not mention the work of Born and Oppen- solutions of the Schr¨odinger equation for the clamped-nuclei heimer which, 17 years before the publication of their book, electronic Hamiltonian which gives the electronic energy at a had argued that such a treatment of the nuclei could be justified choice of nuclear positions. These electronic energies are then if the potential energy surface had a unique minimum and that added to the clamped-nuclei classical Coulomb repulsion en- the nuclear motion involved only small departures from equi- ergy and the resulting “total” energies fitted to a functionalform librium [2, 3]. to provide a surface which, for a system of A > 2 nuclei, is of In the textbook by Pauling and Wilson [4] the work of Born dimension 3A − 6. Nuclear motion is treated as occurring on and Oppenheimer is quoted as justifying their use of a wave- this surface. function which is a simple product of an electronic and a nu- The idea of a potential energy surface in such a role began clear part in order to describe the vibration and rotation of with the work of Eyring and the almost contemporary work of molecules. They then introduce the potential energy function, Polanyi in the 1930s. Its basis is described in Chapter 16 of the which they call the “electronic energy function”, for a diatomic textbook by Eyring, Walter and Kimball, published in 1944 [1]. system. In their description of a chemical reaction involving When describing the reaction between the hydrogen molecule three atoms they attribute the idea of fixing the nuclei to the and a hydrogen atom, the authors say: work of London who used such an approach in his 1928 pa- The properties of this system are completely de- per [5]. It is interesting to note that London actually called termined by the Schr¨odinger equation governing it, this approach “adiabatic” saying that he assumed that as the nu- that is by its eigenfunctions. The system itself may clei moved they acted as adiabatic parameters in the electronic be represented by a point in four dimensional space: wavefunction. arXiv:1206.4239v4 [quant-ph] 25 Sep 2012 three dimensions are required to express the relative Although treating the nuclei as classical particles that maybe positions of the nuclei, one additional dimension is fixed in space to generatea potentialenergy surfaceis often now required to specify the energy. (We assume here that referred to as “making the Born-Oppenheimer approximation” the motion of the electrons is so rapid that the elec- it is something of a mis-attribution as can be seen from this trons form a static field for the slower nuclear mo- discussion. This was indeed recognized by Born and in 1951 tions.) The three internuclear distances... can con- he published a paper [6] in which he presented a more general veniently be used to specify the configuration of the account of the separation of electronic and nuclear motion than system. that originally offered. It is to this later work by Born that the more general idea of a potential energy surface is regarded as They then give the name “potential energy surface” to the pre- owing its justification and it is that work that will be considered sumed calculated entity and go on to explain how transition here. We aim to show that the potential energy surface does not ∗Corresponding author arise naturally from the solution of the Schr¨odinger equation Email address: [email protected] (Brian T. Sutcliffe) for the molecular Coulomb Hamiltonian; rather its appearance Preprint submitted to Journal of Chemical Physics; accepted for publication September 26, 2012 requires the additional assumption that the nuclei can at first be The full Hamiltonian for the molecule is then treated as classical distinguishable particles and only later (af- ter the potential energy surface has materialized) as quantum H = Te + U + TN = Ho + TN. (4) particles. In our view this assumption, although often very suc- cessful in practice, is ad hoc. The fundamental idea of Born and Oppenheimer is that the The outline of the paper is as follows. In the next section low-lying excitation spectrum of a typical molecule can be cal- we review the main features of the standard Born-Oppenheimer culated by regarding the nuclear kinetic energy TN as a small and Born adiabatic treatments following Born and Huang’s perturbation of the Hamiltonian Ho. The physical basis for the well-known book [7]. The key idea is that the nuclear kinetic idea is the large disparity between the mass of the electron and energy contribution can be treated as a small perturbation of the all nuclear masses. The expansion parameter must clearly be electronic energy; the small parameter (κ) in the formalism is some power of m/Mo, where Mo can be taken as any one of obtained from the ratio of the electronic mass (m) to the nuclear the nuclear masses or their mean. They found that the correct = 4 choice is mass (Mo): m/Mo κ . The argument leads to the expres- 1 sion of the molecular Hamiltonian as the sum of the “clamped- m 4 κ = nuclei” electronic Hamiltonian (independent of κ) and the nu- Mo ! clear kinetic energy operator (∝ κ4), equation (6). and therefore In §3 we attempt a careful reformulation of the conventional Born-Oppenheimer argument drawing on results from the mod- ~2 2 4 ∂ Mo ∂ ern mathematical literature. The calculation is essentially con- TN = κ H1 = . (5) ∂X ! M 2m ∂X2 ! cerned with the internal motion of the electrons and nuclei so X we require the part of the molecular Hamiltonian that remains Thus the total Hamiltonian may be put in the form after the center-of-mass contribution has been removed. We 4 show that it is possible to express the internal motion Hamilto- H = Ho + κ H1 (6) nian in a form analogousto equation (6); howeverthe electronic part, independent of κ, is not the clamped-nuclei Hamiltonian. with Schr¨odinger equation Instead, the exact electronic Hamiltonian can be expressed as a direct integral of clamped-nuclei Hamiltonians and necessar- H − E ψ(x, X) = 0. (7) ily has a purely continuous spectrum of energy levels; there are no potential energy surfaces. This continuum has nothing to do In the original paper Born and Oppenheimer say at this point with the molecular centre-of-mass, by construction. The paper in their argument that [3]: concludes (§4) with a discussion of our finding. If one sets κ = 0... oneobtainsadifferential equa- tion in the x alone, the X appearing as parameters: 2. The Born-Oppenheimer approximation ∂ The original Born and Oppenheimer approximation [2, 3] is H x, , X − W ψ = 0. o ∂x summarized in the famous book by Born and Huang, and the " ! # later Born adiabatic method [6] is given in an appendix to that This represents the electronic motion for stationary book [7]. Born and Huang use the same notation for both for- nuclei. mulations and it is convenient to follow initially their presen- tation; the following is a short account focusing on the main and it is perhaps to this statement that the idea of an elec- ideas. They work in a position representation and for simplicity tronic Hamiltonian with fixed nuclei as arising by letting the suppress all individual particle labels. Let us consider a system nuclear masses increase without limit, can be traced. In modern of electrons and nuclei and denote the properties of the former parlance Ho is customarily referred to as the “clamped-nuclei by lower-case letters (mass m, coordinates x, momenta p) and Hamiltonian”. of the latter by capital letters (mass M, coordinates X, momenta Consider the unperturbed electronic Hamiltonian Ho(x, X f ) P). The kinetic energy of the nuclei is the operator at a fixed nuclear configuration X f that corresponds to some ~2 2 molecular structure. The Schr¨odinger equation for Ho is 1 2 ∂ TN = P = − (1) 2M 2M ∂X2 ! X X o Ho(x, X f ) − E (X f )m ϕ(x, X f )m = 0. (8) and that of the electrons 1 ~2 ∂2 = 2 = − . This Hamiltonian’s natural domain, Do, is the set of square in- Te p 2 (2) X 2m X 2m ∂x ! tegrable electronic wavefunctions {ϕm} with square integrable The total Coulomb energy of the electrons will be represented first and second derivatives; Do is independent of X f .