2. The ambiguity around chemical bonds

The chemical bond is a central concept whose explanatory and descriptive value in science cannot be easily overstated. However, despite its wide use, it is unclear what exactly the chemical bond is. As Weisberg states, “Once one moves beyond introductory textbooks to advanced treatments, one finds many theoretical approaches to bonding, but few if any definitions or direct characterizations of the bond itself. While some might attribute this lack of definitional clarity to common background knowledge shared among all chemists, I believe this reflects uncertainty or maybe even ambivalence about the status of the chemical bond itself” (Reference Weisberg2008, 932–33).

This is not just a philosophical worry. Chemists and quantum chemists also admit that there is an ambiguity around chemical bonds: “Later studies showed that the nature of the chemical bond is far more complicated than initially thought and that the connection between the Lewis model and the physical nature of chemical bonding is quite intricate” (Zhao et al. Reference Zhao, Pan, Holzmann, Schwerdtfeger and Frenking2019, 8782). One way to illustrate this ambiguity is by looking at how the chemical bond is defined and classified in science. ^{Footnote 1} For example, the International Union for Pure and Applied Chemistry (IUPAC), which is the leading authority on chemical nomenclature and terminology, defines the chemical bond as follows: “When forces acting between two atoms or groups of atoms lead to the formation of a stable independent molecular entity, a chemical bond is considered to exist between these atoms or groups” (IUPAC 2014, 257).

This definition does not state what chemical bonds are; rather it states the conditions that hold when a chemical bond is considered to exist. This unclarity about the exact nature of bonds is not resolved by the definitions of the types of bonds posited in chemistry. Even though most definitions identify a specific referent for chemical bonds, the referent is each time something else, further confusing what chemical bonds are. ^{Footnote 2} For example, the covalent bond is defined as a “region of relatively high electron density between nuclei” (IUPAC 2014, 344); the ionic bond “refers to the electrostatic attraction experienced between the electric charges of a cation and an anion” (2014, 767); the hydrogen bond refers to a “form of association” (2014, 697); whereas multicenter bonds refer to “electron pairs” that “occupy orbitals encompassing three or more atomic centers” (2014, 968; italics added).

One would expect that quantum chemistry resolves this ambiguity by revealing what underwrites all types of bonds. However, this is not the case because “the need for approximate (quantum) methods greatly complicates the explanatory relationship” (Hendry Reference Hendry2010b, 124).

Specifically, the complication stems from the fact that there is no unique way of solving the molecular Schrödinger equation. The Schrödinger equation provides a quantum mechanical description of atoms and molecules in terms of the interactions between their composing parts. However, the equation cannot be solved analytically for almost any molecule. So, different computational methods are developed that solve the equation by following different mathematical strategies and making different assumptions.

Two are the main methods of solving the Schrödinger equation: the Valence Bond (VB) and the Molecular Orbital (MO) approach. Each approach encompasses more than one methods of solving the Schrödinger equation. ^{Footnote 3} More relevantly, each implies a different understanding of what chemical bonds are. Under the VB approach, chemical bonds are taken to refer to the “change in electron distribution and the resulting energetic stabilization from this change” (Weisberg Reference Weisberg2008, 935). The methods that fall under the VB approach understand chemical bonds as directional, identify them with subatomic regions of high electron density, and take the electrons that participate in a bond to exhibit increased delocalization (Weisberg Reference Weisberg2008, 939). However, the MO approach takes chemical bonds to refer to phenomena that, due to delocalization effects, are neither directional nor subatomic (Weisberg Reference Weisberg2008, 941–43).

In addition to solving the Schrödinger equation through these approaches, quantum chemists have developed methods that connect the numerical results of quantum calculations to chemistry’s tools of describing bonds (Zhao et al. Reference Zhao, Pan, Holzmann, Schwerdtfeger and Frenking2019, 8790). Each of these methods is used in conjunction with the most appropriate method of solving the Schrödinger equation. In this way, quantum chemists acquire additional information about the chemical bonds posited in a system. Specifically, there are three main methods used for such purposes: the Natural Bond Orbital Method (NBO), the Quantum Theory of Atoms in Molecules (QTAIM), and the Energy Decomposition Analysis and Natural Orbitals for Chemical Valence (EDA- NOCV). ^{Footnote 4}

The NBO and the QTAIM are charge-partitioning methods (Zhao et al. Reference Zhao, Pan, Holzmann, Schwerdtfeger and Frenking2019, 8790–92). The NBO divides the molecular wave function into atomic regions in a way that corresponds to chemistry’s understanding in terms of localized bonds and lone pairs of electrons (IUPAC 2014, 980; Zhao et al. Reference Zhao, Pan, Holzmann, Schwerdtfeger and Frenking2019, 8791). The QTAIM offers a topological analysis of electron density and identifies chemical bonds in terms of “well-defined atomic regions and interatomic bond paths” (Zhao et al. Reference Zhao, Pan, Holzmann, Schwerdtfeger and Frenking2019, 8791). Both methods illuminate the role of subatomic charge in the description of chemical bonds. In contrast to the NBO and the QTAIM, the EDA-NOCV analysis is an energy-partitioning method that focuses on the interaction energy of a bond and on the bond dissociation energy (Zhao et al. Reference Zhao, Pan, Holzmann, Schwerdtfeger and Frenking2019, 8792). As such, this analysis illuminates the role of energy in the formation and stability of chemical bonds.

It is also worth looking at the history of quantum chemistry, in particular with respect to how the chemical bond has been understood throughout the field’s development. Scientists, historians, and philosophers have pointed out that the way in which the MO and VB methods were developed has affected not only how the chemical bond is understood but also explains the persistent ambiguity around its nature (Gavroglu and Simões Reference Gavroglu and Simões2011; Needham Reference Needham2014).

Very briefly, the VB and MO methods were initially developed in competition and scientists participating in their development took their preferred quantum method to infer a substantially different understanding of the chemical bond. The VB approach got a head start through Heitler and London’s work who were the first to solve Schrödinger’s equation for specific chemical systems (Reference Heitler and London1927). They developed a mathematical method of solving the Schrödinger equation for the hydrogen ion and hydrogen molecule and showed that in the hydrogen molecule there is a covalent bond that is manifested by the sharing of two electrons between the two hydrogen atoms.

Pauling took up the work of Heitler and London and set to show how, through the VB approach, quantum mechanics can become relevant to chemists (Pauling Reference Pauling1960). ^{Footnote 5} In particular, Pauling’s goal was to show that the VB approach retains chemistry’s image of chemical bonds as real parts of molecules that partially determine their structural and chemical properties (Needham Reference Needham2014, 2). This project was strongly motivated by the work of Lewis whose discovery and explanation of covalent bonds (in terms of an electron pair connecting two atoms) was (and still is) considered as one of the most important achievements in chemistry. ^{Footnote 6} Lewis’s view on the nature of chemical bonds illuminates the spirit underlying the development of the VB approach: “in the mind of the organic chemist the chemical bond is no mere abstraction; it is a definite physical reality, a something that binds atom to atom” (Lewis [Reference Lewis1923] Reference Lewis1966, 67). However, the MO approach was taken to support a completely different understanding of chemical bonds. Some of its central proponents, Coulson and Mulliken, took the development of the MO approach to challenge not only the Lewisian understanding of chemical bonds but also its reality (Weisberg Reference Weisberg2008, 933). As Coulson stated, “a chemical bond is not a real thing: it does not exist: no-one has ever seen it, no-one ever can. It is a figment of our own imagination” (Reference Coulson1955, 2084). This was not without empirical warrant as the development of MO methods lead to higher predictive accuracy and revealed previously unknown factors that explained chemical behavior. For example, modern versions of the MO approach showed that the repulsion of electrons, the ionic character of chemical bonds, and the mixing of higher energy states play an important role in correctly describing the structure of specific types of molecules (Weisberg Reference Weisberg2008, 939–43; Needham Reference Needham2013, 54).

To recapitulate, different ideas about the chemical bond have resulted from the use of different quantum chemical methods and chemical classifications. That different methods and types of bonds have been developed is in turn explained by the fact that each accommodates the description of specific molecules and serves specific aims (such as predictive success, agreement with chemistry’s conceptual understanding of bonds, etc.).

Of course, these features of scientific practice are absolutely reasonable, and it is not my intention to contest such practice. Nevertheless, when it comes to understanding chemical bonds this situation is not particularly helpful. First, if we take ontologically literally what existing understandings of bonds state, then we arrive at the impossible result that the chemical bond is at the same time a region of high electron density (as per the covalent bond’s definition), an electrostatic attraction (as per the ionic bond), a form of association between atoms (as per hydrogen bonds), and the overlapping atomic wavefunction (as per the VB approach). Secondly, none of the existing definitions are general enough to capture all cases of bonding. For example, some definitions require for there to be two electrons for a bond to exist between two atomic centers, even though there are cases of bonding between two atomic centers with just one electron (de Sousa and Nascimento Reference de Sousa and Nascimento2019). Additionally, certain types of bonds (such as covalent bonds) imply that the bond is a localizable thing materialized in a particular region, even though there are molecules that exhibit resonance structures due to electron delocalization. ^{Footnote 7} So despite the many ideas on the chemical bond, it is an open question how to spell out its nature in a way that accommodates all existing accounts of it.

Before concluding this section, it is worth considering how the philosophy of chemistry has dealt with this ambivalence around the chemical bond. I focus here on one of the clearest expressions of this ambivalence, given by Hendry’s two conceptions of the chemical bond. Hendry distinguishes between what he calls the structural and the energetic conception of the chemical bond (Reference Hendry2008). These two conceptions highlight the main features of the chemical bond (as outlined in the preceding text), and group them in terms of two candidate understandings of the chemical bond.

The structural conception takes chemical bonds to be “material parts of the molecule that are responsible for spatially localized submolecular relationships between individual atomic centers” (Hendry Reference Hendry2008, 917). It encompasses the postulations of chemical theory about there being a bond between two atoms that share electrons, without always necessitating such a pairwise relation. It encompasses all chemical classifications, including the different types of chemical bonds mentioned in the preceding text. More importantly, this conception implies that chemical bonds are “a submolecular phenomenon, confined to regions between the atom”; “[t]his eliminates the possibility that bonds are a molecule-wide phenomenon” (Weisberg Reference Weisberg2008, 935; italics in original).

The energetic conception takes chemical bonding to signify “facts about energy changes between molecular or supermolecular states” (Hendry Reference Hendry2008, 919). It explains why a bond is formed in terms of the energetic stabilization of the molecule, rather than defines what the chemical bond is. It doesn’t require that there are bonds in a molecule but rather remains agnostic with respect to their existence. The molecule is regarded as a set of nuclei and electrons that all together interact and form a stable entity. The structural features of the chemical bond are no longer required. Instead, what is necessary is a loose notion of bonding that is described in terms of the energetic stabilization of the whole system.

Hendry’s two conceptions don’t provide a way out of the ambiguity around chemical bonds. Instead, they summarize and highlight the main points of disagreement around the chemical bond. The structural conception is consonant with chemistry’s standard way of defining, classifying, and representing the chemical bond. As such, it is in line with the VB approach and how it recovers the chemical bond (namely as a localized material thing between atoms). The energetic conception is in line with modern approaches in quantum chemistry (namely the MO approach and its variants) and is consistent with views that reject the existence of chemical bonds (at least) as material parts of molecules (see Coulson Reference Coulson1955). ^{Footnote 8}

Another reason why the two conceptions aren’t suitable candidates to account for chemical bonds is because neither does justice to the results of both chemistry and quantum chemistry. On the one hand, the structural conception implies that bonding is always confined in a submolecular region despite evidence from quantum chemistry that this is not always the case. ^{Footnote 9} On the other hand, the energetic conception dismisses some of chemistry’s most accepted insights and says nothing of the nature of chemical bonds: As Hendry states, it is more of “a theory of chemical bonding than a theory of bonds” (Reference Hendry2008, 919).

So, it is an open question what chemical bonds are and even though one could explore this question by arguing in favor of the structural or energetic conception, I follow a different strategy. The next section draws inspiration from Dennett’s paper Real Patterns and argues that chemical bonds are patterns of subatomic interactions. ^{Footnote 10} This proposal offers a novel solution to the ambivalence around chemical bonds.

3. Chemical bonds are real patterns

In his paper Real Patterns, Dennett presents six frames each made of black and white dots (figure 1). ^{Footnote 11} According to Dennett, one way to describe each frame (say frame A in figure 1) is to state its bit map: that is, specify the position of each and every black and white dot that makes up the frame. The bit map is the least efficient description of the frame because it specifies all the properties of all the dots that make up that frame (it identifies “each dot seriatim” [Dennett Reference Dennett1991, 32]). There are however alternative ways of describing the frame that don’t require identifying the position of each and every dot. For example, one can describe the frame by specifying the number, size, and position of five black boxes. Given that such a description can be offered then, according to Dennett, this means that there is a pattern in how the black and white dots are positioned in the frame. Put differently, because frame A can be described in a manner that is more efficient than the bit map, this suffices to argue that a real pattern exists.

Figure 1. Dennett’s Six Frames (Reference Dennett1991, 31).

As Dennett has famously stated, “a pattern exists in some data—is real—if there is a description of the data that is more efficient than the bit map, whether or not anyone can concoct it” (Reference Dennett1991, 34). ^{Footnote 12}

A similar claim can be made about chemical bonds. Specifically, I argue that Dennett’s method of identifying real patterns applies to how subatomic interactions are described in terms of chemical bonds, thus showing that chemical bonds are real patterns of subatomic interactions. ^{Footnote 13}

I provide support to this claim by examining how a common molecule—methane (CH₄)—is described in chemistry and quantum chemistry. Methane consists of one carbon atom and four hydrogen atoms (i.e., 10 electrons and 5 nuclei). In principle, methane can be described by solving the Schrödinger equation from first principles. Solving the Schrödinger equation from first principles amounts to identifying the interactions of each and every electron and nucleus that comprises the molecule; this implies that none of the entities or interactions that are postulated at the quantum scale are disregarded. As such, solving the Schrödinger equation from first principles amounts to providing the bit map description because this description identifies “each dot seriatim” (Dennett Reference Dennett1991, 32). ^{Footnote 14} Specifically, the bit map is the description of the quantum state that is produced by solving the Schrödinger equation from first principles. ^{Footnote 15}

However, there are more efficient ways of describing methane. For example, it can be described using the QTAIM, which calculates the electric field at carbon nuclei, the electron density, and the energy density of the carbon and hydrogen bonds (Macedo and Haiduke Reference Macedo and Haiduke2020). Others, like Mendoza et al. (Reference Mendoza, Sauro and Herrmann2013), calculate methane’s electron charge distributions by applying the kinetic theory to the DFT method (figure 2).

Figure 2. A pictorial representation of methane where the blue and red surfaces denote the low and high electron charge distributions that are calculated by applying a version of the DFT (Mendoza et al. Reference Mendoza, Sauro and Herrmann2013, 1). The red surfaces correspond to chemical bonds.

In general, there are different quantum methods that describe subatomic interactions and identify chemical bonds in terms of electron clouds, bond paths and electron charge distributions. These represent a set of descriptions that, in the spirit of Dennett’s account, are more efficient than the bit map: They are more efficient ways of describing subatomic interactions than by solving the Schrödinger equation from first principles.

Similarly, chemistry has its own method of identifying subatomic interactions that is more efficient than using the bit map. ^{Footnote 16} This method involves positing specific numbers and types of bonds and stating their properties (such as bond angles and bond length). For example, methane’s subatomic interactions are described by positing four covalent bonds between the carbon atom and each hydrogen atom (figure 3).

Figure 3. A three-dimensional representation of methane that shows the number, type, and orientation of four covalent bonds. Image from Wikipedia: https://en.wikipedia.org/wiki/Methane.

So, there are chemical and quantum chemical descriptions that are more efficient ways of describing subatomic interactions than by solving the Schrödinger equation from first principles (i.e., solving the bit map). Therefore, Dennett’s account is satisfied and there are real patterns of subatomic interactions that are identified as chemical bonds. ^{Footnote 17}

For this claim to be convincing, one needs to specify efficiency. Dennett employs “compressibility” as a means to specify efficiency, by explaining how compression algorithms are efficient ways of describing a system (Reference Dennett1991, 34). For example, frame D (in figure 1) “can be described as ‘ten rows of ninety: ten black followed by ten white, etc., with the following exceptions: dots 57, 88,…’” (Reference Dennett1991, 32–33). This is a more efficient description than specifying the position of every black and white dot because it is much shorter than its bit map in the sense that it requires less storage space in a computer (i.e., number of bits) (Reference Dennett1991, 33).

In the case of chemical bonds, calculating the number of bits of chemical and quantum chemical descriptions doesn’t seem like a sensible way to understand (let alone measure) their efficiency. ^{Footnote 18} So I present two alternative (and compatible to each other) understandings that capture Dennett’s general idea about efficiency and can be easily grasped for the case of chemical bonds.

First, efficiency can be understood in a naive heuristic way. One could argue that chemical and quantum chemical methods are more efficient methods than the bit map if they are regarded as such by the scientific community. Indeed, chemical and quantum chemical descriptions are used almost exclusively compared to solving the Schrödinger equation from first principles. This is because chemical and quantum chemical descriptions are computationally less complex and more tractable. On this view, one needs not to specify efficiency more precisely. The exact way in which scientists regard these methods as being more efficient is not important. As long as scientists regard them as such, this suffices to accept them as more efficient descriptions and thus as descriptions that identify real patterns.

However, one may find such an understanding of efficiency anthropocentric, contingent on science’s changing computational abilities or just ambiguous (see, e.g., discussion in Ross [Reference Ross, Ross, Brook and Thompson2000, 160]). After all, scientists may in the future develop their computational means to such an extent that they solve the Schrödinger equation from first principles just as easily as they do through approximations.

To avoid such objections, I propose further specifying efficiency in terms of the degrees of freedom of the relevant descriptions. I base this idea on Wilson’s account of degrees of freedom, though there are other accounts one could alternatively employ. ^{Footnote 19} A degree of freedom is “an independent parameter needed to characterise an entity as being in a state functionally relevant to its law-governed properties and behaviour” (Wilson Reference Wilson2010, 281). Wilson claims that there are three ways that the degrees of freedom required to describe special sciences entities relate to the degrees of freedom for physical ones: she calls these relations “restriction,” “reduction,” and “elimination” (Reference Wilson2010, 281). ^{Footnote 20} Based on this, I take a description to be more efficient than the bit map if it “restricts, eliminates or reduces” the number of variables required to describe a system sufficiently. That is, if a description can be formulated by specifying less variables than those required by the bit map, then the former is more efficient than the latter.

There are two advantages to understanding efficiency this way. First, this is not an anthropocentric understanding. The number of variables required to formulate a description is constrained by the system under investigation (namely, by the properties and the number of entities that comprise it). While scientists chose the entities and/or properties that will be disregarded, such a choice is always justified on empirical grounds and is constrained by what is physically possible. For example, almost all quantum chemical methods make the Born–Oppenheimer (BO) approximation. This approximation assumes that nuclei hold fixed positions relevant to the electrons, and thus disregards certain of the interactions that the bit map description would specify. Scientists justify this approximation “on the fact that the ratio of electronic to nuclear mass (m/M ≅ 5 x 10^-4) is sufficiently small and the nuclei, as compared to the rapidly moving electrons, appear to be determinate” (IUPAC 2014, 179).

The second advantage of this understanding of efficiency is that it is not connected to a subjective notion of simplicity. Simplicity is often conflated with efficiency and this is the source of many apparent ambiguities around Dennett’s account (see Willard Reference Willard2014; Beni Reference Beni2017). Very crudely, what is simple in one context may be complicated in another or regarded as such by someone else. So, if we understand efficiency in such terms then it becomes just as amenable to subjective interpretations. However, in the context of degrees of freedom, there is no room for this. It is a matter of fact that the number of variables required by one description is smaller or larger than the number of variables required by another. Thus understood, efficiency is not vulnerable to problems raised against simplicity.

In the context of my proposed understanding of efficiency, it is evident why chemical and quantum chemical descriptions are more efficient than the bit map. Solving the Schrödinger equation from first principles requires specifying the interactions of each and every subatomic particle that makes up a molecule. However, quantum chemical descriptions solve the equation by disregarding some of the interactions between certain pairs of subatomic particles, thus reducing the number of variables requiring specification. The most widely used assumption in these descriptions is the BO approximation. It leads to the reduction of the degrees of freedom because it disregards the interactions between the nuclei that comprise the examined molecule.

Of course, applying the BO approximation doesn’t suffice to specify quantum chemically a molecule’s bonds. This approximation is just one of the multiple steps taken so as to arrive at a description that, among other things, identifies chemical bonds. The previous section explains how different quantum chemical methods identify bonds by calculating the bond dissociation energy, bond paths, and so forth. Given that the BO approximation is applied in all these models, and that the BO leads to a reduction in degrees of freedom, this suffices to argue that these models are more efficient descriptions of molecules and bonds.

Similarly, chemistry doesn’t specify the interactions of each and every subatomic particle but instead employs a smaller number of variables to formulate its description. Specifically, chemistry focuses on the electrons that occupy the outer shell orbital of each composing atom. The rest of the electrons that make up each atom are disregarded because it is the outer shell electrons that determine how the specific atom tends to bond. For example, carbon has four electrons in its outer shell. Given that hydrogen has just one, chemists infer that in methane carbon shares one of its outer shell electrons with the electron of each hydrogen atom, thus forming four covalent bonds. So, chemistry describes methane more efficiently than the bit map because it disregards the electrons that occupy the inner orbitals of methane’s constituting atoms.

In sum, neither chemical nor quantum chemical methods require specifying the interactions between all subatomic particles. In contrast, the bit map requires specifying the interactions between all subatomic particles. This suffices to show that the former descriptions are more efficient than the bit map.

4. The challenge from pluralism and instrumentalism

As with any account in philosophy, objections have been raised against Dennett’s real patterns. A central objection is that it is not a genuinely realist thesis (most notably raised by Fodor Reference Fodor1985). Dennett’s account seems to allow all efficient descriptions to identify real patterns and critics interpret this to imply either instrumentalism or pluralism about patterns. ^{Footnote 21} This section formulates this objection with respect to chemical bonds.

For a clear formulation of this objection, it is instructive to return to the example of methane. Chemists describe subatomic interactions by identifying four covalent bonds between carbon and each hydrogen atom. This means that chemists only identify as bonds some of the interactions within the molecule; the interactions between hydrogen atoms aren’t identified as bonds, despite the fact that chemists acknowledge that such interactions take place (see Bader Reference Bader2009).

Given this, it is sensible to expect that there is a principled difference between those interactions that are referred to as bonds and those that aren’t. One might even expect that Dennett’s account provides a criterion that discerns between the two. However, this is not the case. Dennett’s account is consistent with there being more than one ways to describe a frame efficiently and admits that there are different patterns within a single frame (Reference Dennett1991, 33). ^{Footnote 22}

This feature of Dennett’s account becomes apparent via the notion of noise. Noise refers to the information about a frame which is disregarded or eliminated by an efficient description (Dennett Reference Dennett1991, 32–35). According to Dennett, noise is to be expected as not all information about a frame can be compressed by an efficient description, and there will inevitably be some information that is left out. Moreover, each description describes the same frame by compressing the information differently. So, what is regarded as noise under one description, may be part of a pattern by another. ^{Footnote 23}

A similar case can be made about methane. For example, the DFT method describes surfaces of electron charge distributions not only between carbon and hydrogen atoms, but also between the hydrogen atoms (recall the blue surfaces in figure 2). This implies that unlike the chemical description, the DFT describes methane (i.e., the frame) by identifying a pattern not only between carbon and hydrogen but also between the hydrogen atoms. It is only after calculating the intensity of electron charge distributions that one posits a bond just in the regions with high electron charge (namely between carbon and hydrogen). However, in Dennett’s account, this is irrelevant: There is also a pattern between the hydrogen atoms because there is a method that describes more efficiently than the bit map the interactions between them.

Critics to Dennett interpret this situation in two alternative ways (e.g., Brading Reference Brading2010; McAllister Reference McAllister2010, 810–12). On the one hand, if something is part of a pattern under one description and noise under another, then the patterns aren’t real. In the context of chemical bonds this could be stated as follows: There is no principled reason why specific subatomic interactions qualify as patterns, therefore there are no real patterns and thus no chemical bonds. Alternatively, one could interpret this situation from a pluralist perspective. Specifically, all subatomic interactions that are efficiently described are patterns irrespective of whether chemical and quantum chemical descriptions identify them as bonds. While pluralism maintains that chemical bonds are real patterns, it undermines their special status relevant to subatomic interactions that aren’t regarded as bonds.

Neither the instrumentalist nor the pluralist interpretation is appealing for a scientific realist. This is because both interpretations dismiss (in different ways) the special status that chemical bonds hold in scientific practice. Bonds are essential in the explanation of chemical, biological, and physical phenomena. Disregarding their success in science is not in line with the commitment to take seriously the success of scientific concepts when evaluating the existence of relevant entities.

So even though I don’t offer evidence against instrumentalist and pluralist interpretations, there is an incentive to clarify Dennett’s account in a way that does justice to the scientific success of bonds. To do so, I incorporate ideas about real patterns that have been presented in structural realist positions.

5. Structural realism to the rescue

Among realist accounts in the literature, structural realism is the only position that explicitly draws inspiration from Dennett’s account of real patterns. ^{Footnote 24} So it is natural to examine it to overcome an objection that undermines the reality of bonds as patterns. This section shows how incorporating ideas from structural realism circumvents instrumentalism and pluralism about chemical bonds. ^{Footnote 25}

The most detailed contribution to clarifying the realist feature of Dennett’s account is by Ross who, on his own and jointly with Ladyman, has argued for an amendment to Dennett’s account to overcome instrumentalist objections and advocated a universal thesis about the reality of special science entities (Ross Reference Ross1995, Reference Ross, Ross, Brook and Thompson2000; Ladyman et al. Reference Ladyman, Ross, Collier, Spurrett and Collier2007). Ladyman and Ross, based (in part) on their analysis of Dennett’s account, defend a form of nonreductive unity that doesn’t eliminate special science entities from our ontology (called ontic structural realism [OSR]). ^{Footnote 26} OSR circumvents pluralist and instrumentalist interpretations of real patterns, by understanding them as follows: “real patterns are those that indispensably figure in projectable generalisations that allow us to predict and explain the behaviour of the world” (Ladyman Reference Ladyman2017, 157).

On this view, special science descriptions identify real patterns not only because they are more efficient than the bit map but because they make counterfactual and nomological generalizations that are highly successful in explaining and predicting phenomena (Ladyman Reference Ladyman2011, 100; Ladyman Reference Ladyman2017, 154). ^{Footnote 27} This is what qualifies only some efficient descriptions as correctly identifying real patterns and stands as the criterion to discern those patterns.

That this is the appropriate way to pick out real patterns becomes also apparent by Wallace’s similar understanding of them: “Dennett’s criterion: A macro-object is a pattern, and the existence of a pattern as a real thing depends on the usefulness—in particular, the explanatory power and predictive reliability—of theories which admit that pattern in their ontology” (Reference Wallace, Saunders, Barrett, Kent and Wallace2010, 6).

So Ladyman, Ross, and Wallace take the explanatory, predictive, and heuristic success of the special sciences to be the appropriate means to discern descriptions that identify real patterns. And indeed, applied to chemical bonds, this criterion explains why only some patterns of subatomic interactions are identified as chemical bonds in science.

Consider for example covalent bonds. By positing such bonds, chemistry describes a large class of molecules and makes counterfactual and nomological generalizations about how these molecules react and why they are stable. For example, carbon atoms are taken to be bonded to each other using covalent bonds. This explains the stability of the resulting molecules and the formation of large macromolecules. Moreover, that large amounts of energy are produced when hydrocarbon molecules react with oxygen is also explained by the covalent bonds that make up these molecules.

So only some sets of subatomic interactions are referred to as chemical bonds because only these figure in empirically successful counterfactual and nomological generalizations. That aromatic compounds are unusually stable is explained by the covalent bonds formed between the carbon atoms; that water boils at a high temperature is explained by the hydrogen bonds that are formed between H₂O molecules; and that metals conduct electricity is explained by their ionic bonds.

Only specific interactions figure in counterfactual and nomological generalizations because it is those that play the largest role in a molecule’s behavior. For example, the subatomic interactions between carbon and hydrogen determine to the largest extent how methane is structured and reacts. This doesn’t mean that other subatomic interactions don’t occur within methane, nor that these don’t have some effect on its behavior. However, such interactions are negligible and therefore rightly don’t figure in counterfactual and explanatory generalizations.

This also becomes apparent if we consider why similar subatomic interactions are identified in certain cases as patterns and in others as noise. For example, chemical bonds usually refer to subatomic interactions among entities that form part of the same molecule. However, there are specific atoms which are posited to form a bond (called hydrogen bond) even if they are parts of neighboring molecules. So, while chemistry doesn’t standardly identify subatomic interactions among neighboring molecules as bonds (because they play no substantial role in explaining and predicting molecules’ behavior), the case of hydrogen bonds illustrates that such interactions aren’t always negligible. Hydrogen bonds determine how some molecules behave and this is why they are included in explanations and predictions and are identified as bonds.

One could accept the preceding text as an accurate rendering of how science posits bonds yet maintain that this illustrates that bonds are just useful tools for describing specific phenomena. For example, Beni argues against Ladyman and Ross’s understanding of real patterns because the usefulness of a specific method is not an “objective notion, and the decision about usefulness or uselessness of patterns may well depend on the cognitive and practical interests of the investigator” (Reference Beni2017, 298).

In the case of bonds, the previous examples show that the usefulness of positing chemical bonds and the choice of which subatomic interactions are identified as such are based on the explanatory and predictive success that is produced by their use. Their explanatory and predictive success is in turn evaluated in terms of well-confirmed empirical evidence drawn from the experimental manipulation and measurement of molecules. So, even when there is subjectivity about which method is employed and what assumptions are made, the usefulness of positing chemical bonds is evaluated by looking at how their use fares with empirical evidence. ^{Footnote 28}

6. Advantages

Requiring efficient descriptions to be part of successful counterfactual and nomological generalizations is a sufficient amendment to Dennett’s account so as to circumvent instrumentalism and pluralism about chemical bonds as patterns. However, there are additional advantages to accepting such an understanding of patterns, which this section briefly presents.

First, the proposed account implies an understanding of composition which is in line with the physical underpinning of chemical bonds. Ladyman states that in the context of real patterns, composition: “is a real feature of the world and is in general diachronic, dynamical, and domain specific, since it depends on the relevant kinds of interaction among parts” (Reference Ladyman2017, 152; italics added).

This “diachronic and dynamical” feature of real patterns is particularly appealing because it is consonant with the scientific understanding of chemical bonds. Electrons and nuclei that form a bond are dynamic: Subatomic particles move around the molecule and occupy different positions and orbitals at different times. ^{Footnote 29} By advocating a nonstatic understanding of the composing entities of a pattern, we illuminate an important (yet neglected) feature of chemical bonding; namely that it is a diachronic and dynamic process during which subatomic particles interact through (mostly) Coulomb forces.

Secondly, this proposal resolves the ambiguity around chemical bonds without undermining or dismissing any scientific classification or method of describing them. Under the framework of real patterns, all scientifically admissible methods and classifications of bonds identify real patterns. The proliferation of methods is partially due to the specific aims that are in play but also due to the characteristics of the molecule that is under examination. Given that each molecule differs in the number and mass of subatomic particles that constitute it, it is only natural that the resulting patterns of interactions exhibit different characteristics, even though all interactions are the result of Coulomb forces among negatively and positively charged entities (as well as of their relativistic effects). For example, some patterns of interactions involve transferring an electron to one nucleus (as per the ionic bond), while others involve sharing two (or more) electrons between two nuclei (as per the covalent bond or multicenter bonds). Each type identifies the unique way a chemical bond is instantiated in a molecule, without this undermining that all types correspond to patterns of interactions among subatomic particles.

Regarding the structural and energetic conceptions, when it comes to specifying the metaphysical nature of bonds both conceptions become obsolete within the proposed account. Chemical bonds are real patterns of subatomic interactions; they are neither material things (as per the structural conception) nor energetic facts (as per the energetic conception). However, there is still value in talking of bonds in terms of the two conceptions. Each conception illuminates features of chemical bonds that are relevant to the description and explanation of some set of molecules. For example, the structural conception is helpful in understanding the role of bonds in chemical reactions: Given that reactions are standardly explained in terms of the formation and breaking of bonds, the structural conception helps understand the result of a chemical reaction but also the mechanism by which it is realized. Nevertheless, this doesn’t mean that the structural conception captures what chemical bonds are. As stated in section 2, both conceptions are incomplete descriptions that identify some (though not all) features of chemical bonds.

Moreover, the proposed account explains the use of approximations and idealizations in chemical and quantum chemical descriptions, without undermining the reality of bonds. Recall that when a description is more efficient than the bit map, it inevitably disregards some of the entities and/or properties that are part of the system. The entities, properties and interactions that are disregarded through idealizations and approximations correspond to what Dennett calls noise. In the case of chemical bonds, some descriptions involve idealizations or approximations that don’t affect the accuracy with which they describe bonds (i.e., the pattern). ^{Footnote 30} This is because they disregard factors that are negligible with respect to how the molecule is bonded (e.g., this is the case with the BO approximation). Other descriptions make idealizations that include information a more efficient description would regard as part of a pattern. This explains why the MO approach is preferred over the VB (at least for a class of molecules), as the former doesn’t disregard interactions that play a determinant role on how some molecules are bonded (such as the delocalization of electrons). However, it is not the case that a description accompanied by any amount of noise is qualified as identifying real patterns. Science decides on empirical and theoretical grounds whether a specific idealization is acceptable and whether the relevant description adequately agrees with empirical evidence.

All this shows that the proposed account is constrained, compatible, and informed by scientific practice. That chemical bonds are real patterns is justified by the relevant chemical and quantum chemical methods and classifications, and by how chemical bonds are understood in science. In addition, this proposal is a novel contribution to the question of the reality of chemical entities. This becomes apparent if one takes into account that the reality of chemical entities is standardly defended within antireductionist, pluralist, and emergentist contexts (e.g., Hendry Reference Hendry2006).

Lastly, this proposal opens new avenues for understanding how chemical bonds are related to their lower-level constituents. ^{Footnote 31} While Dennett does not provide a complete account of intertheory relations, subsequent talk of real patterns (such as within OSR or Wilson’s NRP) explores how patterns are related to their constituents. While some initial conclusions are drawn by my present analysis, much more has to be said about how chemical bonds (as patterns) relate to subatomic interactions. Nevertheless, understanding bonds as real patterns encourages studying this issue in more detail.