arXiv:1206.4239v4 [quant-ph] 25 Sep 2012 h oeua Cuob aitna u tta iesc ca such time that at but Hamiltonian (Coulomb) molecular the nrysraecntuto smr prpitl regarde appropriately more argu is is It construction surface eigensolutions. energy the from derivable assumed those initio hmclpurposes. chemical ue acltdett n oo oepanhwtransition how explain to on go pre the and to entity surface” energy calculated “potential sumed name the give then They say: authors molecule the atom, hydrogen hydrogen the a between and reaction [1 the t 1944 of describing in 16 published When Chapter Kimball, in and Walter described Eyring, of is by work basis textbook contemporary Its almost 1930s. the the in and Polanyi Eyring of work the with surface. this 3 dimension opoieasraewih o ytmof system a for which, surface functiona a a provide to en to fitted energies repulsion “total” Coulomb resulting the classical and ergy clamped-nuclei are the energies to electronic These added a positions. energy nuclear electronic of the choice clamped-nuc gives the which for Hamiltonian surface electronic equation Schr¨odinger energy the potential a of calculate solutions to is calculation ical Introduction 1. coul surface a such that assumed was accou It to (PES). 1930s surface the energy in introduced was theory state Transition Abstract rpitsbitdt ora fCeia hsc;accepte Physics; Chemical of Journal to submitted Preprint h dao oeta nrysraei uharl began role a such in surface energy potential a of idea The h rnia i fmc fcneprr unu chem- quantum contemporary of much of aim principal The ∗ mi address: Email author Corresponding einl eue oseiytecngrto fthe of configuration con- the specify can system. mo- to used nuclear distances... be slower internuclear veniently the three elec- for The the field that tions.) static rapid a so is form electrons trons the that here of assume motion (We is the energy. dimension the additional specify one to required nuclei, relative the the of express to positions space: required dimensional are four may dimensions in itself three point system a The by represented eigenfunctions. be its it, by governing is that equation Schr¨odinger the by termined lcrncsrcuecluain r otn n rmthe from and routine are calculations structure electronic h rpriso hssse r opeeyde- completely are system this of properties The A − [email protected] .Ncermto stetda curn on occurring as treated is motion Nuclear 6. a evc eCii uniu tPoohsqe Universit´e Photophysique, et quantique Chimie de Service b colo cec n ehooy otnhmTetUnivers Trent Nottingham Technology, and Science of School BinT Sutcli T. (Brian nteQatmTer fMolecules of Theory Quantum the On A ra .Sutcli T. Brian > o publication for d ff uli sof is nuclei, 2 e) form l from then salgtmt n e and legitimate a as d tfrceia ecin.Cnrlt hster steidea the is theory this to Central reactions. chemical for nt dhr htti eifi none.I ssgetdta th that suggested is It unfounded. is belief this that here ed a t lei he ff ecntutduigegnouin fteSchr¨odinger e the of eigensolutions using constructed be d ]. rrslsPS a ecntutdwihaeblee oappr to believed are which constructed be can PESs results ir - - e cltoswr o osil.Nwdy unu mechanica quantum Nowadays posssible. not were lculations a, ∗ .GyWoolley Guy R. , limvdte ce saibtcprmtr nteelectro the calle in parameters actually adiabatic wavefunction. London as the acted that as they pa- note that moved assumed 1928 clei to he that his interesting saying in “adiabatic” the is approach approach to this It an nuclei such the [5]. used fixing per who of involvi London idea reaction of the chemical work attribute a they of atoms description three their diat of a In rotation for function”, system. energy and functi “electronic energy vibration the potential call the they the nu- which introduce describe a then and to They electronic order molecules. an wave- in of a product part of simple clear use a their is justifying which as function quoted is Oppenheimer and equi from departures 3]. that small [2, and only librium minimum involved unique motion a nuclear justifi had the be surface book, could energy nuclei their potential the the of of if treatment publication a Oppen- the such and that Born before argued of had years work 17 the mention which, posit not whose heimer do particles and classical fixed as be treated can obviou be as can it take nuclei They the cal shape. be characteristic its can using reaction lated exchange hydrogen the for probabilities o h oeua olm aitna;rte t appearan its rather equat Schr¨odinger Hamiltonian; the Coulomb of molecular solution the for the from naturally arise consid be will a that regarded work is that here. is surface it energy and justification potential its a owing of idea general more t motion general o nuclear more originally and a electronic that presented of 1951 he separation this which the in from of in and account [6] seen Born paper be by a recognized can published he indeed as was mis-attribution This a discussion. of something approximation is Born-Oppenheimer it the “making often as is surface to energy referred potential a generate to space in fixed lhuhtetn h ulia lsia atce htma that particles classical as nuclei the treating Although Born of work the [4] Wilson and Pauling by textbook the In eamt hwta h oeta nrysraede not does surface energy potential the that show to aim We ir eBuels -00Buels Belgium Bruxelles, B-1050 Bruxelles, de Libre t,Ntiga G18S U.K. 8NS, NG11 Nottingham ity, ff b ciemdfiaino unu ehnc for mechanics quantum of modification ective ff rd ti oti ae okb onta the that Born by work later this to is It ered. etme 6 2012 26, September fapotential a of uto for quation potential e oximate that s omic l now ered ions be y han ab- nu- cu- ion on, nic ng ed ce d ” s - 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 . We may by U(x, X). We further introduce the abbreviation suppose the {ϕm} are orthonormalized independently of X f
∂ ∗ Te + U = Ho x, , X . (3) dx ϕ(x, X f ) ϕ(x, X f )m = δnm. ∂x ! Z n 2 In the absence of degeneracies (“curve-crossing”) they may be a wider application than predicted by the original theory, and he chosen to be real; otherwise there is a phase factor to be consid- proposed an alternativeformulation [6, 7]. It is assumed that the o ered. For every X f , Ho is self-adjoint on the electronic Hilbert functions E (X)m and ϕ(x, X)m arising from equation (8) which space H(X f ), and therefore the set of states {ϕ(x, X f )m} form a represent the energy and wavefunction of the electrons in the complete set for the electronic Hilbert space indexed by X f . state m for a fixed nuclear configuration X, are known. Born The clamped-nuclei Hamiltonian can be analysed with the proposed to solve the wave equation (7) by an expansion HVZ theorem which shows that it has both discrete and contin- uous parts to its spectrum [8–12], ψ(x, X) = Φ(X)m ϕ(x, X)m (11) Xm σ(X ) ≡ σ(H (x, X )) = Eo(X ) ,... Eo(X ) Λ(X ), ∞ f o f f 0 f m f with coefficients {Φ(X) } that play the role of nuclear wave- [ m (9) functions. Substituting this expansion into the full Schr¨odinger where the {Eo(X ) } are isolated eigenvalues of finite multiplic- ∗ f k equation (7), multiplying the result by ϕ(x, X)n and integrating ities. Λ(X f ) is the bottom of the essential spectrum marking the over the electronic coordinates x leads to a system of coupled lowest continuum threshold. In the case of a diatomic molecule equations for the nuclear functions {Φ}, the electronic eigenvalues depend only on the internuclear sep- o aration r, and have the form of the familiar potential curves TN + E (X)n − E Φ(X)n + C(X, P)nn′ Φ(X)n′ = 0 (12) shown in Fig.1. For the general polyatomic molecule, the dis- Xnn′
where the coupling coefficients {C(X, P)nn′ } have a well-known form which we need not record here [7]. In this formulation the adiabatic approximation consists of retaining only the diagonal terms in the coupling matrix C(X, P), for then AD ψ(x, X) ≈ ψ(x, X)n = ϕ(x, X)n Φ(X)n. (13)
An obvious defect in this presentation, recognized by Born and Huang [7], is that the kinetic energy of the overall center- of-mass is retained in the Schr¨odingerequation (7); this is easily corrected, either by the explicit separation of the center-of-mass kinetic energy operator (see §3), or implicitly, as in the compu- tational scheme proposed by Handy and co-workers [13–15]. For many years now these equations have been regarded in the theoretical molecular spectroscopy/quantum chemistry litera- ture as defining the “Born-Oppenheimer approximation”, the Figure 1: The spectrum σ(r) for a diatomic molecule [8]. original perturbation method being relegated to the status of his- torical curiosity. Commonly they are said to provide an exact crete eigenvalues are molecular potential energy surfaces. (in principle) solution [16–20] for the stationary states of the Born and Oppenheimer used the set {ϕm} to calculate ap- molecular Schr¨odinger equation (7), it being recognized that in proximate eigenvalues of the full molecular Hamiltonian H on practice drastic truncation of the infinite set of coupled equa- the assumption that the nuclear motion is confined to a small tions (12) is required. In the next section we make a critical 0 vicinity of a special (equilibrium) configuration X f . Their great evaluation of this conventional account. success was in establishing that the energy levels of the low- lying states typical of small polyatomic molecules obtained from molecular spectroscopy could be written as an expansion 3. The molecular Schrodinger¨ equation in powers of κ2 We now start again and reconsider the Hamiltonian for a col- (0) 2 (2) 4 (4) lection of electrons and nuclei, assuming their interactions are EnvJ ≈ Vn + κ Env + κ EnvJ + ... (10) restricted to the usual Coulombic form. The key idea in §2 (0) where Vn is the minimum value of the electronic energy is the decomposition of the molecular Hamiltonian (4) into a (2) which characterises the molecule at rest, Env is the energy of part containing all contributions of the nuclear momenta, and (4) the nuclear vibrations, and EnvJ contains the rotational energy a remainder. This must be done in conjunction with a proper [2]. The corresponding approximate wavefunctions are simple treatment of the center-of-mass motion. These two ideas guide products of an electronic function ϕm and a nuclear wavefunc- the following discussion. tion; this is known as the adiabatic approximation. In the orig- Let the position variables for the particles in a laboratory inal perturbation formulation the simple product form is valid fixed frame be designated as {xi}. When it is necessary to distin- through κ4, but not for higher order terms. guish between electrons and nuclei, the position variables may About 25 years later Born observed that the results of molec- be split up into two sets, one set consisting of N coordinates, e ular spectroscopy suggest that the adiabatic approximation has xi , describing the electrons with charge −e and mass m, and 3 n the other set of A coordinates, xg, describing the nuclei with mixing; the explicit equations for this choice were given in Sut- charges +Zge and masses mg, g = 1,... A; then n = N + A. We cliffe and Woolley [24]. There are A−1 translationally invariant n n define corresponding canonically conjugate momentum vari- coordinates ti expressed entirely in terms of the original xg that ables {pi} for the electrons and nuclei. With this notation, and may be associated with the nuclei, and there are N translation- e after the usual canonical quantization, the Hamiltonian operator ally invariant coordinates for the electrons ti which are simply e fora systemof N electrons and A atomic nuclei with Coulombic the original electronic coordinates xi referred to the center-of- interactions may be written as nuclear-mass. There are corresponding canonically conjugate internal momentum operators. The classical total kinetic en- A 2 2 A 2 N pg e ZgZh e 1 H = + + ergy still separates in the form 2m 4πǫ r 4πǫ r Xg g 0 Xg
[1] H. Eyring, J. Walter and G.K. Kimball Quantum Chemistry, John Wiley, London, (1944) [2] M. Born and J.R. Oppenheimer, Ann. Phys. 84, 457 (1927) [3] An English language translation of the original paper can be found at www.ulb.ac.be/cpm/people/scientists/bsutclif/main.html. [4] L. Pauling and E. Bright Wilson Introduction to Quantum Mechanics, McGraw-Hill, New York, (1935) [5] F. London in Probleme der modernen Physik Ed. P. Debye, S. Hirsel, Leipzig, 104, (1928) [6] M. Born, Nachr. Akad. Wiss. Goett. II. Math.-Phys. K1, 6, 1 (1951) [7] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Claren- don, Oxford, (1954) [8] J.-M. Combes and R. Seiler in Quantum Dynamics of Molecules, Ed. R. G. Woolley, NATO ASI B57, Plenum Press, New York, 435 (1980) [9] M. Reed and B. Simon, B. Methods of Modern Mathematical Physics, IV, Analysis of Operators, Academic Press, New York, (1978) [10] G.M. Zhislin, Tr. Mosk. Mat. Obs. 9, 81 (1960); the full Russian text can be obtained from http://mi.mathnet.ru/eng/mmo/v9/p81. [11] J. Uchiyama, J. Pub. Res. Inst. Math. Sci. Kyoto 2, 117 (1966) [12] W. Thirring, W. A Course in Mathematical Physics, 3, Quantum Me- chanics of Atoms and Molecules, Tr. E.M. Harrell, Springer-Verlag, New York (1981) [13] N.C. Handy, Y. Yamaguchi and H.F. Schaefer III, J. Chem. Phys. 84, 4481 (1986) [14] N.C.Handy and A.M. Lee, Chem. Phys. Letters, 252, 425 (1996) [15] W. Kutzelnigg, Mol. Phys. 90, 909 (1997) [16] A. Messiah, Quantum Mechanics, Vol. 2, North-Holland (1960) [17] C.A. Mead and A. Moscowitz, Int. J. Quant. Chem. 1, 243 (1967) [18] T. F. O’Malley, Adv. At. Mol. Phys. 7, 223 (1971) [19] H. J. Monkhorst, Phys. Rev. A36, 1544 (1987) [20] L. Lodi and J. Tennyson, J. Phys. B At. Mol. Opt. Phys. 43, 133001 (2010) [21] This is the analogue for operators with continuous spectra of the famil- iar result in linear algebra that the eigenspaces of an n × n matrix can be decomposed as a direct sum of a n−dimensional vector space. Alter- natively, note that each of the spaces H(P) transforms under a different representation of the translation group in 3-dimensions — under transla- tion by an amount a, an element of H(P) is multiplied by the factor eiPa. Thus the direct integral is the infinite-dimensional analogue of the de- composition of a finite-dimensional vector space on which a group acts as a direct sum of irreducible representations. [22] M. Loss, T. Miyao and H. Spohn, J. Funct. Anal. 243, 353 (2007) [23] P.-O. L¨owdin, in Molecules in Physics, Chemistry and Biology, Vol. 2, Ed. J. Maruani, Kluwer Academic Publishers, Dordrecht (1988) [24] B.T. Sutcliffe and R.G. Woolley, Phys. Chem. Chem. Phys. 7, 3664 (2005) [25] A. M. Frolov, Phys. Rev. A., 59, 4270 (1999) [26] T. Kato Trans. Amer. Math. Soc. 70, 212 (1951) [27] J.-M. Combes, International Symposium on Mathematical Physics (Ky- oto University, Kyoto), Lecture Notes in Physics 39, 467 (1975) [28] J.-M. Combes, Acta Phys. Austriaca Suppl., XVII, 139 (1977) [29] G.A. Hagedorn and A. Joye in Spectral Theory and Mathematical Physics. A Festschrift in Honor of Barry Simon’s 60th Birthday, Eds. F. Gesztezy, P. Deift, C. Galvez, P. Perry and G.W. Schlag, Oxford Uni- versity Press, London, 203 (2007) [30] M. Klein, A. Martinez, R. Seiler, and X. P. Wang, Commun. Math. Phys., 143, 607 (1992) [31] G.A. Hagedorn and A. Joye, Revs. Math. Phys. 11, 41 (1999) [32] P.-O. L¨owdin, Pure and Appl. Chem. 61, 2065 (1989) [33] R.G. Woolley, Adv. Phys. 25, 27 (1976) [34] R.G. Woolley and B.T. Sutcliffe, Chem. Phys. Letters, 45, 393 (1977) [35] R.G. Woolley, Structure and Bonding, Springer-Verlag, Berlin, 52, 1 (1982) [36] R.G. Woolley and B.T. Sutcliffe, in Fundamental World of Quantum Chemistry, Eds. E.J. Br¨andas and E.S. Kryachko, Kluwer Academic Publishers, Dordrecht, 1, 21 (2003) [37] B.T. Sutcliffe and R.G. Woolley, Chem. Phys. Letters, 408, 445 (2005) [38] I. Prigogine and C. George, Proc. Natl. Acad. Sci. U.S.A. 80, 4590 (1983) 8