Introduction

What I should like to do in this presentation is to provide a succinct account of what can properly be said at present about the way that we usually approach electronic structure calculations in terms of a clamped nuclei Hamiltonian involving only Coulomb forces and the way that permutational symmetry impinges on that relationship. Wherever possible I shall cite secondary sources but there are, unfortunately, no such sources for the relation of the clamped nuclei Coulomb problem to the full Coulomb problem, including nuclear motion, and the original literature here is rather mathematically detailed.

I shall begin by trying to summarise what is known about bound state solutions to the full molecular problem, nuclear motion included, posed in terms of the standard Schrödinger equation with just Coulomb forces acting, and how the solutions may be classified according to the permutational symmetries of the particles involved. I then want to present what is known in a mathematically acceptable way, about the Born-Oppenheimer and Born-Huang approaches to the separation of electronic and nuclear motions. Finally I'd like to look at the idea of a potential energy surface and attempt the outline of a discussion on why the idea might not be as secure as we could hope.

My aim is to be brief but, I hope, convincing in my views of where there are as yet unsolved problems that, it seems to me, make Dirac's celebrated dictum 1929 [1]

The fundamental laws necessary for the mathematical treatment of large parts of physics and the whole of chemistry are thus fully known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.

at best still aspirational, or even ironical and at worst, probably misleading.

The solutions of the full molecular problem

The Coulomb Hamiltonian operator for a system of N electrons and A atomic nuclei may be written as

This operator is essentially self-adjoint and bounded from below. It has, however, a completely continuous spectrum [0, ∞]. This is because of the centre-of-mass motion and to see any discrete spectrum this motion must be removed as

Since the centre-of-mass variable does not enter the potential energy term, the centre-of-mass motion may be separated off completely so that the eigenfunctions of H are of the form

where Ψ(t) is a wavefunction for the Hamiltonian H'(t) which we will refer to as the translationally invariant Hamiltonian. It is this Hamiltonian that we must use when considering the separation of nuclear from electronic motion.

There are infinitely many possible choices of translationally invariant coordinates, so that the form of H' is not determined, but whatever coordinates are chosen the essential point is that all H' have the same spectrum. More detailed accounts of the spectral properties of the Coulomb Hamiltonian that will be cited below can be found in [2], [3] and briefly but fully in [4].

There are various ways in which the spectrum σ(A) of a self-adjoint operator A may be classified. From the point of view of measure theory the natural decomposition is into pure point, absolutely continuous and singular continuous parts. The sets are closed but need not be disjoint. The discrete spectrum σ_d(A) is the subset of the pure point spectrum that consists of isolated eigenvalues of finite multiplicity. The essential spectrum σ_ess(A) is the complement of the discrete spectrum and is the infinite dimensional part.

The discrete spectrum and the essential spectrum are, by definition, disjoint; however, although the essential spectrum is always closed, the discrete spectrum need not be. σ_ess(A) includes the absolutely continuous spectrum σ_ac(A) and the singular continuous spectrum, σ_sc(A) this last consisting of infinitely degenerate eigenfunctions.

The Coulomb Hamiltonian has an empty singular continuous spectrum.

The absolutely continuous spectrum describes states in which the particles leave a bounded region in a finite time; these are the scattering states of the system. The discrete spectrum describes bound states in which the particles stay infinitely long in a bounded region. This is the part of the spectrum that is most usefully investigated in the search for the molecule. If one considers the spectrum of the Coulomb Hamiltonian the location of the bottom of the essential spectrum is defined by the famous Hunzicker-Van Winter-Zhislin (HVZ) theorem. It is obtained by looking at the lowest energy of all possible separated clusters. It can be shown that the lowest possible energy arises from a two-cluster decomposition.

The problem is: what is the extent of the discrete spectrum in any particular case?

For a neutral or positively charged “atomic” system the discrete spectrum is infinite and ends at the first ionization energy. This result is independent of the “nuclear” mass. The essential spectrum consists then of any remaining pure point states (the hydrogen atom-like discrete states in the case of the helium atom, for example) and the true continuum.

For a negatively charged “atomic” system there are, at most, a finite number of bound states. ( H^- has just one bound state.)

The clamped nuclei electronic Hamiltonian is just like an “atomic” Hamiltonian but the form of the discrete spectrum depends upon the nuclear geometry.

Almost nothing is known about “molecular” systems. It is known that the hydrogen molecule has at least one bound state. It is also known that if a system becomes “too” negative (or positive) it has no bound states. If a trial wave function can be found whose expected energy is below the start of the essential spectrum, then the system has at least one bound state.

If it is known that if the beginning of the essential spectrum is at a point where the two clusters have opposite charges, then there are an infinite number of bound states: if the two clusters are neutral then there are only a finite number of bound states.

The origin of the essential spectrum has been located only for the hydrogen molecule.

The results recorded above were proved with no consideration of any symmetry restrictions upon the solutions. But subsequent work, especially by Balslev [5], established that the atomic results remained true even after requiring the electronic states to obey the Pauli Principle and to be angular momentum eigenfunctions.

In extending these considerations to molecules, however, certain technical problems arise in showing that the HVZ theorem can be maintained allowing for permutational symmetry. This is because the separating clusters may well contain identical particles.

These technical problems arise because from the N+A variables x defined in the laboratory fixed frame, only N+A-3 translationally invariant variables t may be used to define H'(t). It is natural to wish to define a subset comprising N of these variables such that they can be thought of as describing electronic motion. As such they should be invariant to any permutations made of the nuclear variables. This could be done by expressing the N “electronic” t with respect to the centre of nuclear mass, for example.

However, it can be shown that the clusters in “atomic” form do not arise directly as asymptotes in such a coordinate system. To get the “atomic” asymptotes one must use a coordinate system in which the “electronic” coordinate origins are particular nuclei. But such “electronic” coordinates are not invariant to permutations made of the nuclear variables.

Such problems of “coordinate mixing” are tricky technical ones but they can be surmounted. The HVZ theorem remains valid allowing for permutational symmetry. The spectrum of H'(t) is independent of any particular coordinate choice and if one knew the exact solutions then one could express them in any chosen coordinate set, to taste. However one has to make a coordinate choice in ignorance and what these results show is that it is not always possible to keep “electronic” and “nuclear” variables separate in a uniformly useful way.

Thus if one regards H'(t) as the starting Hamiltonian for a discussion of electron-nucleus separation, the “natural” description of the electronic variables will not be useful at the asymptotes.

Permutational Symmetry in the Coulomb Hamiltonian

We shall speak of irreps of the translationally invariant Hamiltonian which correspond to physically realisable states as permutationally allowed. In general such irreps will be many dimensional and so we would expect to have to deal with degenerate sets of eigenfunctions in attempting to identify a molecule in the solutions to the translationally invariant problem.

It should be emphasised that the variables x(t) simply specify field points, and cannot actually be particle coordinates because of the indistinguishability of sets of identical particles. Weyl stresses that, in the case of sets of identical particles, in addition to supporting the canonical quantum conditions, the space on which quantum mechanical operators act must be confined to a sub-space of the full Hilbert space of definite permutational symmetry. This means that the effect of any operator on a function in this sub-space must be to produce another function in the subspace. Multiplication of a properly symmetrised function by a single coordinate variable produces a new function which is not in the symmetrised sub-space. Thus only operators symmetric in all the coordinates of identical particles can properly be deployed in the calculation of expectation values that represent observables.

Weyl discusses this in Section C 9 of Chapter IV of The Theory of Groups and Quantum Mechanics [6]. He says of the two particle case:

Physical quantities have only an objective significance if they depend symmetrically on the two individuals.

and he then goes on to generalise this conclusion to the symmetrical form for the quantities constructed from the variables of N identical particles.

He closes his discussion by looking at the two electron problem. He says that although it might be supposed that the electrons as a pair of twins could be named “Mike” and “Ike”

it is impossible for either of these individuals to retain his identity so that one of them will always be able to say “I'm Mike” and the other “I'm Ike”. Even in principle one cannot demand an alibi of an electron! In this way the Leibnizian principle of coincidentia indiscernibilium holds in quantum mechanics.

This discussion holds for identical particles of any kind that are to be described by quantum mechanics and it precludes the specification of, for example, the expected value of a particular coordinate chosen from a set describing many identical particles.

If a particular irreducible representation of the symmetric group of the electronic coordinates be denoted as [λ]^N and let the conjugate representation be denoted as ^~[λ]^N. For electrons (or any spin 1/2 particles) the representation of the symmetric group carried by the spin-eigenfunctions Θ_{S, M_S, i} must be one described by a no more than two-rowed Young diagram, that is [λ]^N≡ [λ₁, λ₂] where

The representations are independent of the choice of M_S and i labels the rows (columns) of the representation. The dimension of the representation is given by the Wigner number

Assuming that the translationally invariant part of the Coulomb Hamiltonian for the chosen system has eigenfunctions in the discrete spectrum then, among them, there will be a degenerate set that provides a basis for the representation conjugate to that for the chosen spin-eigenfunctions. The representation and the conjugate representation have the same dimension and a basis of space-spin products can be formed which belongs to the antisymmetric representation of the symmetric group and hence satisfies the Pauli Principle.

For example, suppose that a 10 electron system, such as ammonia, was being considered and it was hoped to identify a singlet state. In this case (ignoring for the moment the nuclear variables) one would be looking for a set of 42 degenerate eigenfunctions of the Coulomb Hamiltonian which provided a basis for the irrep ^~[5,5] under permutations of electronic variables. These would be functions of the kind earlier called permutationally allowed.

One could provide precisely similar arguments to deal with protons and so to describe ammonia, for example, one might look for a nuclear spin doublet arising from the protons and then [λ]^A would just be [2,1] and one would be looking for a pair of degenerate functions to provide a basis for the irrep ^~[2,1] under permutation of the protons. Thus to describe ammonia in a singlet electronic spin state with a doublet nuclear spin state one would have to find a degenerate set of 84 eigenfunctions among the eigenfunctions of the Coulomb Hamiltonian to provide a basis for the permutationally allowed irrep of S_N(10)×S_A(3).

This sort of argument could be extended to particles with spins other than 1/2 and with Bose rather than Fermi statistics. To do so is, however, much more difficult. The difficulties arise because it is much harder in the general case than it is with particles of spin 1/2, to associate the spin functions with their space parts to produce functions of appropriate symmetry. Although it is true that particles of spin s can provide a basis for representations of the symmetric group corresponding to Young diagrams with, at most, only 2s+1 rows, it is not in general possible to determine the lengths of these rows simply from the total S and N values in the problem.

If one moves from ammonia to a larger molecule, for example, the simple hydrocarbon with empirical formula C₈H₈, then a host of other problems begin to emerge. Consider first the 56 electrons. It is easy, though tedious, to show that the dimension of the permutationally allowed representation ^~[28, 28] for the singlet state, given by the Wigner formula f^N_S with N=56 and S=0, is

53×47×44×43×41×37×35×34×31≈ 2.6 10¹⁴

so that for this rather simple system one can expect the eigenfunctions, if any, in the discrete spectrum of H'(t) to be very extensively degenerate even without considering any degeneracies arising from the nuclear variables.

Interestingly enough such a possibility may have troubled Born and Oppenheimer. In their discussion of equation (15) in Part I of the paper they write V_n as the sum of the electronic and nuclear repulsion energy for the nth electronic state and say

Moreover we assume that V_n is a non-degenerate eigenvalue. As a matter of fact, this is never the case, since, because of the indistinguishability of the electrons the resonance degeneracy, discovered by Heisenberg and Dirac, enters; … But since we are concerned here only with the systematics of the approximation procedure, we will not consider these degeneracies. Their consideration would require higher approximations in the secular equation.

Although it is never considered explicitly in the standard arguments for electron nucleus separation, the permutational degeneracy arising from the electronic part of the problem, can be dealt with without too much trouble. It is sufficient to incorporate spin variables into the electronic part of the problem and to assume proper antisymmetrisation. This will remove some of the degeneracy, but it will leave behind those degeneracies arising from rotation-reflection symmetry and it will also leave unconsidered degeneracy arising from permutational symmetry in the nuclear part of the problem. To see the sort of problems to which this can lead consider an example of the way in which isomers are usually accounted for.

The Coulomb Hamiltonian H'(t) for C₈H₈ is the molecular Hamiltonian for cubane, cyclooctatetrene, vinylbenzene and many other compounds too. Indeed it is even the molecular Hamiltonian for a system with optical isomers, 3-vinyl hexa-1,4-diyne, which has the molecular formula

				H
				|
CH2	=	CH	—	C	&mdash	C	≡	CH
				|
				C	≡	C	—	CH3

and in which the central carbon is clearly chiral by the conventional rules.

The irrep of the nuclear permutation group that corresponds to an antisymmetric singlet state for the protons and a symmetric singlet state for the carbons is of dimension 14, that which corresponds to a triplet state for the protons with same carbon spin state is of dimension 28 and so on.

This apparent “confusion” of molecules might not seem too serious a matter for it might be argued that the different isomers corresponded simply to different eigenstates of the same Hamiltonian H', perhaps with different nuclear spin states. But in classical structural chemistry, different isomers mean different geometries and it is the idea of a distinct geometry that is problematic in quantum mechanics, given that the wave function must belong to a subspace of definite permutational symmetry.

If we write the variables corresponding to the carbon nuclei in C₈H₈ as xⁿ_j, j=1,…8 and those corresponding to the protons as xⁿ_i+8, i=1,…8 then a particular CH interparticle distance is

One might be tempted to suppose that the calculation of the expected values of such interparticle distances with a particular eigenfunction of H' would determine the geometry. However we have seen that x^CH_ij is not a proper observable. Its product with a function lying in a particular permutational subspace carries the function outside the subspace and so out of quantum mechanical utility.

The only possible operator incorporating these distances is the symmetrical sum

and all that can be inferred from its expectation value is that, on average, all the CH interparticle distances are the same.

This is not to suppose that this average value is the same for all the eigenfunctions of H' that might be investigated in a search for isomers, it is simply that what differences there might be, cannot support the detailed geometrical interpretation which is characteristic of isomer identification in classical chemical structure theory.

Of course usually when one computes an electronic energy using the clamped nuclei Hamiltonian, one simply treats the nuclei as identifiable particles and one happily computes energy differences between various isomeric forms of the molecule under consideration and constructs isomerization paths between them and so on. And one usually assumes that what one has done can be received into the church of solutions of the full problem. The relevant baptismal words for reception into this church are, of course, “Born and Oppenheimer”.

But if the nuclei are treated as indistinguishable particles then it is not too clear how one can work back from the position that one is left in at the end of a clamped nuclei calculation to a solution of the full problem.

So an examination of the usual approaches is now in order.

Separating electronic and nuclear motion

The classical 1927 work of Born and Oppenheimer [7] and the 1950s approach by Born as presented in App. VIII of Born and Huang [8] do not consider permutational symmetry at all. It is this last approach which seems to have overtaken the classical approach in people's minds generally and tends now to be called the Born-Oppenheimer approach. But in fact they are quite distinct approaches and I'll consider them separately.

The standard “proofs” as presented in both of the original papers are purely formal. They are not at all mathematically precise and attempts to make them so involve a huge amount of extremely sophisticated of effort.

One can make neither of the approaches precise unless one assumes that the electronic function is is a properly antisymmetrised function of space and spin variables. Otherwise the assumption of the electronic ground state being non-degenerate is utterly implausible.

In none of the efforts so far made has there been any consideration of nuclear permutational symmetry. What the work that has been done to date establishes is that, provided that the nuclei can be treated as distinguishable particles, the restricted (product form) solutions are asymptotic solutions to the full problem. They are not convergent solutions.

The classical B-O approach

The mathematically secure work for diatomics the is summarised in J.-M. Combes, P. Duclos and R. Seiler [9] For polyatomics the secure work is M. Klein, A. Martinez, R. Seiler, and X. P. Wang, [10]

What is shown rigorously in this series of papers is that around a local minimum in the potential formed from the sum of the electronic and nuclear repulsion energy the wave function can be written as the product of an electronic, vibrational and rotational part and that to a first approximation the vibrational part is a simple harmonic oscillator and the rotational part a (symmetric or asymmetric) top.

Because of the problems that arise in the asymptotic behaviour of solutions expressed in terms of coordinates in which electrons and nuclei are separately identified, it has not proved possible so far to provide a satisfactory account of scattering states in this sort of approach.

In summary, just so long as nuclei are identifiable particles, we have a secure basis for the way in which spectroscopy is usually interpreted.

The Born-Huang approach

There has been no completely satisfactory work starting from the time-independent Born-Huang equations. However a time dependent approach using the coherent-states method was begun by Hagedorn in 1980 [11] and it is thus that a potential surface away from the minimum can be given a clear meaning in the context of solutions to the full problem. The nuclei are considered as identifiable particles.

Because of the way that the coherent-states method works, essentially using wave-packets and remaining in the laboratory fixed coordinate system, the problems of asymptotic behaviour do not arise so acutely in this approach. It is thus possible to give an account of the potential energy surface at arbitrary nuclear separations. It is not, however, possible to do more with its aid than to describe asymptotic solutions on distinct and well-separated surfaces.

The latest work that I know of here, again using coherent-states theory, is by G. Hagedorn and A. Joye in 2001 [12]. What it shows, again assuming identifiable nuclei, is that there is a secure basis for elastic and much of inelastic scattering as usually done but, unsurprisingly, the approach fails for reactive scattering.

An exact potential surface?

At present we seem to be stuck and until we find out how to put in the permutations we are not quite sure that we are connecting properly to the full problem. But even if it proves possible to sort this out there is one aspect of the problem that might make us wonder if we were going about things the right way anyway.

The anxiety stems from some work by Hunter and later developed by Czub and Wolniewicz. The papers are [13] and [14].

They consider what would happen if one knew the exact solution to the full problem and rewrote it as a product of a nuclear and electronic part.

The quantitative work is confined to a consideration of the J=0 singlet state of the hydrogen molecule. An exact wave function for the lower states of this system can be expressed in terms of the internuclear distance r and two electronic coordinates z_i whose origin is half-way between the nuclei.

The full problem here can be posed as

Now suppose that ψ were actually known and square integrable, and we were to write

with the accessory condition that

for all values of r in the domain.

This last requirement avoids the trivial factorisation f(r)=1. It also means that

This sort of factorisation is well known in probability theory as expressing a total probability in terms of the product of a marginal and a conditional probability. In the present case φ^*φ is the conditional probability density and f^*f the marginal probability density.

Formally at least then

and if f(r) were strictly positive, that is, without nodes, then to within a sign factor, the electronic wave function could be written as

For a neutral diatomic system it was shown by Czub and Wolniewicz that the nuclear wave function in this definition was strictly positive.

The prospects here are really very attractive for if one can determine a suitable electronic function φ(r,z), then one can define an exact nuclear motion problem in terms of a single potential.

The actual analysis involved in working things out is a bit tricky but it is possible to formulate a nuclear motion problem in terms of a potential

where H^elec is a pretty standard electronic Hamiltonian, Vⁿ is the nuclear repulsion term and g⁰ a correction term arising from electron-nucleus interactions integrated out over the electronic coordinate.

This equation is quite useless, of course, because we don't have the faintest idea what φ is. However, it might reasonably be supposed that the exact electronic function, whatever it is, differed very little in squared modulus from that determined as an eigenvalue E(r), of the electronic Hamiltonian. In that case the potential would become

and the approximate eigenfunctions for the full problem would be:

where χ would be a vibrational wave function of standard form. Identifying a particular state, one would expect |ψ|² and |ψ^ad|² for it, to differ only in second-order and hence for |f|² and |χ|² to do similarly.

However, when determined in an adiabatic potential, the wavefunction for an excited vibrational state will have nodes. That is, unlike the exact function, it will not be strictly positive. If the exact and the adiabatic nuclear motion functions are to differ only negligibly in squared modulus, then wherever the adiabatic function has a node, the exact function must have a minimum. Thus f(r) must have a large curvature near a node in the adiabatic function and this in turn implies that the potential in the exact problem, U(r), must have large peaks centred about the nodes of the adiabatic function.

Thus if, for example, we could determine f(r) for a state whose energy corresponded to an approximate state that would be described as the molecule in its ground electronic state with rotational eigenvalue J=0 and vibrational eigenvalue ν=3 one would expect to see large peaks in U(r) where χ(r) vanished.

In work of Czub and Wolniewicz, absolutely the best ever “adiabatic” wave functions are used and they argue, in my view completely convincingly, that the form that they use to approximate f(r), though constructed and not arising from the exact solution by integration, must have all the analytic properties that f(r) has in the region around the equilibrium geometry. From this they construct an approximation to U(r) and, low and behold, it has peaks in it exactly where one might expect.

This means that there cannot be a common exact (or non-adiabatic) potential for all the nuclear motion states of a given problem. One must therefore regretfully agree with Czub and Wolniewicz that “it is not possible to define an ab initio non-adiabatic potential that would have practical importance.” This in turn implies that the idea of a single potential energy surface or, of a collection of potential energy surfaces, on which nuclear motion can be described, is an inherently approximate one. There is no reason to suppose that a description of phenomena given in terms of such surfaces would survive refinement of the computational techniques to include coupling (non-adiabatic) effects more and more accurately.

Conclusions

It is argued here that it is still not clear how properly to connect the usual clamped nuclei electronic structure calculations to solutions of the full Schrödinger Coulomb Hamiltonian including nuclear motion. The difficulties seem to arise because the solutions to the full problem are a basis for irreps of the permutation groups of all the sets of identical nuclei while those of the clamped nuclei problem are not. If a rationale could be found for treating the nuclei as identifiable particles still described by the Schrödinger Coulomb Hamiltonian then it seems likely that a satisfactory connection could be found, at least close to a potential minimum.

It is generally supposed that such a rationale can be provided by using the idea of feasible permutations introduced by Christopher Longuet-Higgins in 1963 [15] and discussed at length in [16]. There are two difficulties here however. Whether a permutation is feasible or not depends upon the height of the potential barrier between a configuration and its permuted partner. But, as has been seen, to define a potential energy surface, even locally, one must first make the Born-Oppenheimer approximation and this one cannot at present justify, unless the nuclei are treated as distinguishable particles. Not only is there this internal circularity but the definition of feasibility seems to treat a permutation as a physical operation according to whether it can or cannot be easily achieved. However a permutation in the theory is simply a mathematical operation and there is nothing in the theory that endows a permutation with any physical character. It is not, for example, an Hermitian operator.

Of course it is perfectly possible that if exact eigenfunctions were known, then it might be that the ones which were relevant to a conventional molecule, comprised sets which formed a basis for a representation of the full group that behaved in a recognizable and regular way under those permutations regarded as feasible. So it might be that the idea of feasibility has a deeper significance than it is at present possible to demonstrate. Such a demonstration would be very welcome for the idea proves a very fruitful one in interpreting the spectra of floppy molecules.

But even supposing that all this worked out, the status of the potential energy surface, as usually calculated, for general nuclear motion, must still be a cause of some anxiety.

There is another symmetry property which poses as big a problem to the idea of a definite molecular geometry as does the permutational symmetry that has been considered here. The Hamiltonian for the Schrödinger Coulomb problem is, as noted earlier, invariant under any orthogonal transformation of coordinates. In particular, it is invariant under inversion (sign change of all coordinates) so exact solutions have definite parity: they go into either plus or minus themselves under inversion. But this means that no such eigenfunction can display a permanent dipole moment neither can it be chiral. Both of these observations militate strongly against recognising a molecular geometry in such an eigenfunction. It was this aspect of the problem of molecular shape that so intrigued theoretical physicists from very early times. It was discussed by Fermi and by Wigner in the late 1940s and an account of some of the ways that theoretical physicists have approached this problem, together with some reminiscences of the early days of their consideration, can be found in the paper by Wightman [17]. To put matters briefly, if in a somewhat over-simplified manner. It is generally agreed among the physicists, that molecular shape can never emerge from a solution of the isolated molecule Coulomb Hamiltonian. It is necessary to consider the environment as shaping it. Thus the problem becomes a problem in the quantum mechanics of open systems and of quantum statistical mechanics. If the problem is posed in this way, it becomes of interest to applied mathematicians and for an example of their approach, the paper [18] is worth looking at.

If the theoretical physicists are correct, then to tie quantum theory to traditional chemistry will be an even more difficult undertaking than is supposed here. The only comfortable thing to emerge from the theoretical physicists musings is that they all seem to think that the Born-Oppenheimer approximation is just fine. But precisely why they think so is not clear, at least not in any of the papers that I have read.

It would seem prudent at present for Quantum Chemists to supplement Dirac's dictum by regarding Quantum Chemistry as a distinct enterprise within Quantum Mechanics to which a characteristic chemical contribution is made by allowing nuclei to be treated as identifiable, though not classical, particles. This is not at all to separate Quantum Chemistry from most areas of Quantum Physics, for example condensed matter theory, in which precisely such an identification of nuclei is made. Nor is it to separate Quantum Chemistry from its origins in Quantum Physics as exemplified by the foundational work of Heitler and London on electronic structure, of Eckart on molecular spectra and of Wigner on the origin of point group symmetry, in all of which work, the nuclei were treated as identifiable particles.

Acknowledgments

I am grateful to Guy Woolley with whom I have worked on such matters as I have presented here, on and off for the past 25 years. I remain as grateful now as I was at the beginning of our collaboration, for his insights and his help. Some of the ideas that appear in this contribution are developed more fully in a paper by us both [19] dedicated to the memory of P.-O. Löwdin.

References

[1] P. A. M. Dirac, Proc. Roy. Soc. A 123, 714 (1929).

[2] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of Operators, Academic Press, New York, 1978.

[3] W. Thirring, A Course in Mathematical Physics, 3, Quantum Mechanics of Atoms and Molecules, Tr. E. M. Harrell, Springer-Verlag, New York, 1981.

[4] W. Thirring in Schrödinger, centenary celebrations of a polymath, Ed. C. W. Kilminster CUP, Cambridge, 1987, p 65.

[5] E. Balslev, Ann. Phys.(N.Y.), 73, 49 (1972).

[6] H. Weyl, The Theory of Groups and Quantum Mechanics, 2nd Edition, Tr. H. P. Robertson, Dover, New York, 1931.

[7] M. Born and J. R. Oppenheimer, Ann. der Phys., 84, 457 (1927).

[8] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1955.

[9] J.-M. Combes, P. Duclos and R. Seiler in Rigorous atomic and molecular physics, Eds. G. Velo and A. Wightman, Plenum, New York, 1981, p. 185.

[10] M. Klein, A. Martinez, R. Seiler, and X. P. Wang, Commun. Math. Phys., 143, 607 (1992).

[11] G. Hagedorn, Commun. Math. Phys., 77, 1 (1980).

[12] G. Hagedorn and A. Joye, Commun. Math. Phys.,223, 583 (2001)

[13] G. Hunter, Int. J. Quant. Chem., 9, 237 (1975)

[14] J. Czub and L. Wolniewicz, Mol. Phys. 36, 1301 (1978)

[15] H. C. Longuet-Higgins, Mol. Phys., 6, 445. (1963)

[16] P. R. Bunker and P Jensen, Molecular Symmetry and Spectroscopy, 2nd Edition, National Research Council (Canada), 1998.

[17] A. S. Wightman, Il Nuovo Cimento 110 B, 751 (1995)

[18] E. B. Davies, J. Phys A: Math. Gen. 28, 4025 (1995)

[19] R. G. Woolley and B. T. Sutcliffe, in Fundamental World of Quantum Chemistry: A Tribute Volume to the Memory of Per-Olov Löwdin, 1, Ed. E.J. Brandäas and E.S. Kryachko , Kluwer, Dordrecht, 2002. Ch.3.

This paper formed the basis of a talk given at the VIth Girona Seminar on Molecular Similarity hosted by the Institute of Computational Chemistry, University of Girona, Spain, July 21-24th 2003.