Abstract
In this paper I expand Eric Scerri’s notion of Popper’s naturalised approach to reduction in chemistry and investigate what its consequences might be. I will argue that Popper’s naturalised approach to reduction has a number of interesting consequences when applied to the reduction of chemistry to physics. One of them is that it prompts us to look at a ‘bootstrap’ approach to quantum chemistry, which is based on specific quantum theoretical theorems and practical considerations that turn quantum ‘theory’ into quantum ‘chemistry’ proper. This approach allows us to investigate some of the principles that drive theory formation in quantum chemistry. These ‘enabling theorems’ place certain limits on the explanatory latitude enjoyed by quantum chemists, and form a first step into establishing the relationship between chemistry and physics in more detail.
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Notes
As suggested by one of the referees of this paper.
They also associate the SIS requirement with the provisoes of Hempel (1988) as an assumption about completeness. There is a considerable amount of literature in the philosophy of quantum chemistry on the topic of the ‘isolation’ of molecular systems and the consequences that this assumption has for a principled explanation of chemical phenomena, most of which is in origin due to the work of Primas (1975), or Primas (1983). In recent times, the ‘emergence of a classical world’ in the theory of quantum physics is also studied through decoherence Schlosshauer (2007).
Popper (1957) states in the Preface: ‘I tried to show, in The Poverty of Historicism, that historicism is a poor method—a method which does not bear any fruit. But I did not actually refute historicism’ (p. v). In this sense Popper’s criterion of ‘poor’ is weaker than is generally understood as in the sense of ‘incorrect’. Beyond this contingency, Popper’s argument is of course completely different from mine.
As it stands in regards to scientific practice, however, van Brakel’s argument is not that convincing, as one of the referees of this paper pointed out. Quantum chemical packages, for better or worse, presently are an important tool for many chemical researchers, such as designers of new chemicals including catalysts, nano-materials and drugs.
I wish to note that van Brakel (2000)’s comments on the various editions of Coulson’s Valence (and especially his critique on the last edition edited by McWeeny (1979) are in my view justified. A historical study of the various editions of Valence would make for an interesting research topic outlining how the perception of explanation through quantum chemistry has evolved over time.
For instance, (1) is not strictly speaking a problem, and there are many examples in physical theories where nonlinear equations for higher order properties are developed on the basis of linear equations. (2) could be represented as an internal problem for physics with little bearing on the nature of the relationship between physics and chemistry, while it is hard to claim (3) for instance for short lived transition states. Woolley’s claims about the difficulties with molecular structure have bee the subject of considerable discussion in the literature, and cannot be accepted without significant qualifications. Even so, providing detailed rebuttals of these arguments is not the purpose of this paper.
I would like to note that I disagree with Woody that the MO diagrams share their mathematical structure with that underlying configuration interaction. They do not, at least not the same configuration interaction as it is traditionally understood in quantum chemistry. Much less than configuration interaction is needed to create the MO diagrams. A charitable interpretation of Woody might take it to mean that she intends the comment as an indication of the interaction of the (electronic) configuration of individual atoms, and such a reading might indeed support the MO diagrams as resulting from configuration interaction. Such a reading would be highly discontinuous with current practice however and is very confusing in the present context.
A historical fact which may underpin this contention is that the early quantum chemists tend to be less well-known than the early pioneers of quantum theory. While in the early stages there was considerable overlap in the activities of these two groups, around 1929 there was a distinct difference between the work of early quantum chemists like Hylleraas in Norway, Hellman in Germany and later Russia, Eyring, van Vleck, Pauling and others in the USA, and quantum theorists like Dirac who had a passing brush with chemistry, but moved on to relativistic quantum mechanics in the late 1920s. In the 1930s foundational issues tended to take centre stage as the frontier of development. It is little surprise, therefore, that a number of the theorems we will discuss in this section are closely associated with quantum chemists. This history still needs a considerable amount of fleshing out at this point in time and is in no sense of the word complete.
Proof: we can write the resolution of the identity as
$$ \hat 1 = \sum_n | \psi_n \rangle \langle \psi_n | $$Inserting this into the expectation value 〈H 〉ϕ yields
$$ \langle H \rangle_\phi = \frac{\sum_n {\langle \phi | H | \psi_n \rangle \langle \psi_n | \phi \rangle}} {\sum_n \langle \phi | \psi_n \rangle \langle \psi_n | \phi \rangle} = \frac{\sum_n E_n {|\langle \phi | \psi_n \rangle |^2}} {\sum_n | \langle \phi | \psi_n \rangle |^2} $$Since E 0 < E 1 < E 2 … and | 〈ϕ| ψ n 〉|2 ≥ 0:
$$ \sum_n E_n |\langle \phi | \psi_n \rangle |^2 \geq E_0\sigma_n {|\langle \phi | \psi_n \rangle |^2} $$and thus 〈H 〉ϕ ≥ E 0.
The existence of a lowest eigenvalue is an assumption which does not hold for Dirac Hamiltonians. The mathematically rigorous proof of the existence of the lowest eigenvalue for Schrödinger Hamiltonians was only given in Kato (1951a, b).
This is the description given in Helgaker et al. (2000) on pp. 267 and 870. Eh is one Hartree, the absolute value of the electronic potential energy of a hydrogen atom in the ground state. It is equal to 2625.5 kJ/mol.
Proof: The proof is again quite simple and hinges on the fact that since \(S_2 \subset S_1\) all functions \(|\psi_i^{S_2}\rangle\) can be expanded in S 1 as follows
$$ |\psi_i^{S_2}\rangle = \sum_n^{|S_1|} c_n |\psi_i^{S_1}\rangle $$The energy \(E_0^{S_2}\) can be written as
$$ \begin{aligned} E_0^{S_2} =& \langle\psi_i^{S_2}| H | \psi_i^{S_2}\rangle \\ =& \sum_{nm}^{|S_1|} c_n^* c_m \langle\psi_n^{S_1}| H | \psi_m^{S_1}\rangle \\ =& \sum_{nm}^{|S_1|} c_n^* c_m E_m^{S_1} \delta_{nm} \\ &= \sum_n |c_n|^2 E_n^{S_1}. \end{aligned} $$This quantity is larger than \(E_0^{S_1}\).
This should be read to apply to molecular systems. There is no indication throughout the paper that Feynman only has atoms in mind, and there are no reasons why the theorem should be restricted to atoms as opposed to molecules. It is equally valid for molecules as well.
Proof: Consider the steady state where
$$ H (\lambda) | \psi_\lambda \rangle = E_\lambda | \psi_\lambda \rangle $$Then, if we assume that ψλ is normalised, so that 〈ψλ| ψλ〉 = 1:
$$ \begin{aligned} \left. \frac{d E_\lambda} {d\lambda}\right|_{\lambda = 0} &= \frac{\partial} {\partial \lambda} \langle \psi_\lambda|H (\lambda)|\psi_\lambda\rangle \\ &= \left\langle \left.\frac{\partial\psi_\lambda}{ \partial \lambda}\right|_{\lambda=0} | H (\lambda) | \psi_\lambda\right\rangle + \left\langle \psi_\lambda \left| \frac{\partial H (\lambda)}{\partial \lambda} \right| \psi_\lambda \right\rangle +\left\langle \psi_\lambda | H (\lambda) | \left.\frac{\partial\psi_\lambda} {\partial \lambda}\right|_{\lambda=0} \right\rangle \\ &= E_\lambda\left[\left\langle \left.\frac{\partial\psi_\lambda}{\partial \lambda}\right|_{\lambda=0} | \psi\right\rangle + \left\langle \psi \left| \frac{\partial\psi_\lambda}{\partial \lambda}\right|_{\lambda=0} \right\rangle \right] +\left\langle \psi_\lambda \left| \frac{\partial H (\lambda)} {\partial \lambda} \right| \psi_\lambda \right\rangle \\ &= \left\langle \psi_\lambda \left| \frac{\partial H (\lambda) }{\partial \lambda} \right| \psi_\lambda \right\rangle \\ &= \left\langle\psi_\lambda | V | \psi_\lambda \right\rangle\\ \end{aligned} $$The term in square brackets vanishes because
$$ \left[\left\langle \left.\frac{\partial\psi_\lambda}{\partial \lambda}\right|_{\lambda=0} | \psi\right\rangle + \left\langle \psi \left| \frac{\partial\psi_\lambda}{ \partial \lambda}\right|_{\lambda=0} \right\rangle \right] = \frac{\partial} {\partial \lambda} \left\langle \psi_\lambda | \psi_\lambda \right\rangle = 0. $$The proof is given in Feynman (1939). The Hellman–Feynman theorem has since become a staple of quantum mechanics courses and the proof is also given in for instance Helgaker et al. (2000) and Cohen-Tannoudji et al. (1977) to mention only a few.
Proof: We assume that the time dependent Schrödinger equation is satisfied, so that
$$ H \psi (t) = i \hbar \frac{\partial \psi }{\partial t} $$so that
$$ \begin{aligned} \frac{\hbox{d}}{{\hbox{d}}t}\langle \psi (t) | A | \psi (t) \rangle &= \left\langle \frac{\partial \psi (t)}{\partial t} | A | \psi (t) \right\rangle + \left\langle \psi (t)| A | {\partial \psi (t) \over \partial t} \right\rangle + \left\langle \psi (t) \left| \frac{\partial A}{\partial t} \right| \psi (t) \right\rangle\\ &= \frac{1}{i \hbar} \left\langle \psi (t) | (AH - HA) | \psi (t) \right\rangle + \left\langle \psi (t) \left| \frac{\partial A}{\partial t} \right| \psi (t) \right\rangle \\ &= \frac{1}{i \hbar} \left\langle \psi (t) | \left[ A , H \right]| \psi (t) \right\rangle + \left\langle \psi (t) \left| \frac{\partial A}{\partial t} \right| \psi (t) \right\rangle \\ \end{aligned} $$where the transition from the first to the second line is because ψ(t) satisfied the Schrödinger equation and the transition to the last line is through the definition of the commutator
$$ [A,H] = AH - HA $$The proof is given in a number of places, for instance in Cohen-Tannoudji et al. (1977).
I will not present the proof here, details are found in a number of textbooks such as for instance in Cohen-Tannoudji et al. (1977) on p. 242.
I had a number of interesting discussions with Paul Wormer, Josef Paldus and Bogomil Jeziorski on the somewhat confusing historical situation surrounding the original reference to Wigner’s 2n + 1 theorem and the interesting discussions via email about whether Wigner’s 1935 paper actually contains the (proof of the) theorem. These discussions did not lead to a clear cut conclusion, but I think I can sum the situation up fairly if I say that the 1935 paper did contain the theorem though not the proof, with the proof of the theorem appearing in the 1950s. In addition, Egil Hylleraas already stated the theorem for third order wave functions in 1930, as outlined below.
The ‘art’ of basis set creation quite possibly requires a paper of its own given this very common misconception in the philosophy of chemistry. The criteria that basis sets must meet in order to qualify as a ‘good’ basis set are manifold, but in general a fit to experimental data tends to be one of the lesser considerations. Basis sets are generally evaluated on a number of criteria, such as whether they are ‘even-tempered’, have sufficient ‘polarisation functions’ and such like. A detailed discussion of this would make an interesting paper in the philosophy of chemistry on its own and falls outside the scope of the current paper.
This is not to say that the UHF wave function is the ‘correct’, or even the ‘better’ one, since the UHF wave function fails in other important respects. Specifically, UHF wave functions are not eigenfunctions of S 2 which is in a way the saving grace in the description of the energetic curve but also breaks the spin symmetry of the overall wave function.
In terms of a diagrammatic evaluation of the energetic property this is due to the presence of so-called unlinked diagrams. This analysis also leads to a number of formulae that can be used to correct for size consistency, such as the Davidson formula. See for instance Paldus (1981, 1992) for a detailed discussion of these issues.
The issue of convergence of a perturbation series is commonly settled by considering the convergence radius of the series, which is hard to establish in detail for cases other than the most simple model cases. In practice, the issue of convergence tends to be settled on the basis of common knowledge about ‘where a method does well’. In practice, the convergence depends on the type of molecule, the type of bond, the internuclear distance and such like.
Note that the remainder of this discussion only applies insofar philosophical criticisms of quantum chemistry are technically accurate. As one of the referees of this paper pointed out, this is not always the case.
As an aside, it should in this context be noted that the abstraction of intermediate laws is also what drives the difference between Nagel (1961) and Kemeny and Oppenheim (1956) reduction. The latter claim that a Nagelian reduction scheme is a special case of a more general reduction scheme in which the intermediate laws can be eliminated to establish an (in the words of Kemeny and Oppenheim 1956) ‘indirect’ connection between two theories. The point then seems to be that if the indirectness of the connection can be established to sufficient degree, there is room to argue for significant explanatory independence at the higher theoretical level (in this case that of chemistry). Such explanatory independence can than take place in an entirely autonomous fashion (which seems to be the viewpoint advocated by van Brakel 2000), or in some intermediate fashion (which seems to be the viewpoint advocated by Woody 2000).
I would like to thank one of the referees of this paper for pointing out these examples.
The notion of ‘finalisation’ is used here in the context of the Starnbergers, where it means the ‘application’ phase of a research programme in the sense of Lakatos. The key distinguishing feature of a finalised programme is that its problem shifts are dictated by external, rather than internal, considerations.
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Acknowledgments
I would like to thank Paul Wormer, Josef Paldus and Bogomil Jeziorski for elucidating the somewhat confusing historical situation surrounding the original reference to Wigner’s 2n + 1 theorem and the interesting discussions via email about whether Wigner’s 1935 paper actually contains the (proof of the) theorem. I would also like to thank Paul Wormer and an anonymous referee of this journal for many helpful comments.
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Hettema, H. Explanation and theory formation in quantum chemistry. Found Chem 11, 145–174 (2009). https://doi.org/10.1007/s10698-009-9075-8
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DOI: https://doi.org/10.1007/s10698-009-9075-8