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Austere quantum mechanics as a reductive basis for chemistry

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Abstract

This paper analyses Richard Bader’s ‘operational’ view of quantum mechanics and the role it plays in the the explanation of chemistry. I argue that QTAIM can partially be reconstructed as an ‘austere’ form of quantum mechanics, which is in turn committed to an eliminative concept of reduction that stems from Kemeny and Oppenheim. As a reductive theory in this sense, the theory fails. I conclude that QTAIM has both a regulatory and constructive function in the theories of chemistry.

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Notes

  1. The ‘no parallel universes’ comment is derived from the title of Bader (2011a) As an aside, it is interesting to note that the elimination of concepts in this way precisely follows the two horns of Hempel’s Theoretician’s Dilemma (Hempel 1958).

  2. This argument from analogy can for instance already be seen in Dirac’s paper ‘Quantum Mechanics and the Hydrogen Atom’ (Dirac 1926).

  3. An example of such an early ‘atoms in molecules’ theory is found in the article by Moffitt (1951).

  4. The term ‘complete set of commuting observables’ stems from (Dirac 1958) and is also a key element of the development of quantum mechanics in (Cohen-Tannoudji et al. 1977).

  5. For a brief illumination of the different formulations of quantum mechanics, see for instance the paper by Styer et al. (2002).

  6. This issue has has been discussed at length in the literature. To address it, Nasertayoob and Shahbazian (2009) define a concept of ‘Quantum Divided Basins’ (QDBs), and to propose these QDBs as an alternative structure which may be amenable to theoretical investigation. However, that still leaves open the question of how much of the theory of atoms in molecules actually deals with atoms as opposed to arbitrary zero-flux regions. As an example, in the final analysis given in Nasertayoob and Shahbazian (2009) topological atoms are defined on the basis of ‘chemical intuition’ rather than robust mathematical analysis.

  7. She does not expand on this statement in detail.

  8. Note that in another chapter Bokulich argues that there is room to suppose that classical trajectories of particles in a particular class of quantum system influence the (semi-classical) quantum trajectories, and hence she argues that in this sense the classical structure may determine the quantum solution. For details, see her Chapter 5.

  9. For a discussion of the Nagelian perspective in the reduction relationship between chemistry and physics, see Hettema (2012).

  10. Specifically, as summarised in Bader and Matta (2012), MOs have a role to play in the explanation of the relative ordering of excited states, and considerations where symmetry plays an important role. Similarly, orbitals play a role in understanding the phenomenon of polarisability.

  11. It is also worthwhile to note in this context that orbitals themselves are unique only up to a particular set of transformations—for instance in the Hartree-Fock scheme a sufficient requirement for a correct HF solution is the Brillouin condition, which stipulates that matrix elements of the Hamiltonian between the ground state and single-excited states must vanish. This conditions does not fully determine all orbitals.

  12. The specific example Pap mentions, that the framework of mechanics stipulates that the mass of the electron, the mass of the sun and the mass of a football all refer to the same quantity ‘mass’ serves a key role in the explanatory strength of mechanics. Yet in a purely operationalist framework, these three different masses are measured in a different manner and should therefore refer to different quantities.

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Acknowledgments

I would like to thank Richard Bader for a fruitful discussion about some of the topics discussed in this paper. The present paper contains my considered response to some of the topics we discussed during his visit to Auckland in September 2010. I would like to thank two referees of an earlier draft of this paper for constructive comments.

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  1. Department of Philosophy, The University of Auckland, Auckland, New Zealand

    Hinne Hettema

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Appendix: Possible implications of the Dirac perspective in the philosophy of chemistry

Appendix: Possible implications of the Dirac perspective in the philosophy of chemistry

Austere quantum mechanics is based on a significant restriction of the Dirac perspective on theories, one in which ‘analogies’ become ‘identities’, one in in which prediction trumps everything, and, most importantly, one in which there is no room for ‘open theories’. Hence it makes sense to ask whether adopting Dirac’s particular perspective to the relationship between chemistry and physics can save both the positive achievements of Bader’s QTAIM as well as a significant portion of classical chemistry.

To determine this question, it makes sense to ask what Dirac used the analogy for. Its main purpose was determining the form of operators that correspond to particular observables. In the specific sections of The Principles of Quantum Mechanics that deal with the action principle, Dirac uses the analogy between the unitary transformations of quantum mechanics and the contact transformations of classical mechanics to derive dynamical behaviour in both systems that is of the same form. Dirac also notes, that the analogy is not complete, however, since in classical mechanics the action S has to be real while there is no corresponding requirement in quantum mechanics (p. 130).

In chemistry, there are no clear equations of motion that can be cast in terms of either Poisson brackets or quantum mechanical commutators, and there is no ‘chemistry’ that clearly corresponds to a CSCO. Hence the direct application of austere quantum mechanics to chemistry is not possible. It is precisely in this sense that austere quantum mechanics is eliminative: it claims that the lack of such a direct application is the result of the non-existence of the theoretical entities of chemistry.

In the main text of the paper I have argued that austere quantum mechanics plays a regulatory role vis a vis the concepts of chemistry—i.e. that it assists in interpreting the results of quantum chemical calculations in terms of a mechanical model of the molecule. At the same time, to leave room for chemistry, this mechanical model is not ‘all there is’. ‘Parallel universes’ are possible, but in a limited sense.

However, relaxing the notion of identity of contact transformations and unitary transformations to one of analogy seems to suggest that the introduction of chemical concepts is possible in a more direct way. For instance, it would be interesting to consider the unitary group characterisation of open shell systems and the concept of valence along these lines.

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Hettema, H. Austere quantum mechanics as a reductive basis for chemistry. Found Chem 15, 311–326 (2013). https://doi.org/10.1007/s10698-012-9173-x

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