Do we know the chemical bond? A case for the ethical teaching of undefined paradigms

2.1 1st axiom: the paradigm

It is a good homework to ruminate on how the existence of the chemical bond is our biggest paradigm. Indeed, the idea of the bond provides the scientific basis of chemistry and its jargon, it tells us how to carry out and interpret experiments, and it even defines what are the valid questions that we can ask when carrying out a chemical project. Or paraphrasing a famous Argentinian quote: “Within the bond everything, outside the bond nothing”.

Let us focus on the issue of what is a valid question in chemistry. For instance, when studying a chemical reaction, the fundamental (and sometimes the only) question to formulate is how the chemical bonds are reshuffled during the reaction. If we do not have chemical bonds, would that question make sense? What question would we ask if not? How is phlogiston released? How does the water:air:earth:fire ratio change?

Every question about structure, reactivity, nanochemistry, analytical chemistry, DNA, organometallics, the ozone layer, IR spectroscopy, or astrochemistry only makes sense in terms of having bonds, which gives us the language to describe chemistry. Is there a chemical field where the chemical bond is not its obvious starting point? Chemistry students are usually not aware of the weight of this chemical paradigm, and how much it affects the whole building of their studies. It is a good practice to start examining this concept from introduction to chemistry, and never stop pondering about it.

To be fair, many consider as the biggest paradigm in chemistry not the chemical bond, but the atom as the basic building block of matter. The existence of atoms, championed originally by Dalton (and maybe Democritus if we extend ourselves), applied theoretically by Boltzmann, and then finally proven by Einstein and Perrin, is at the heart of chemistry. From there it was natural to think of a “field” that bring those atoms together into molecules, even if its reason was hardly understood [it was considered a kind of blending of the “atmospheres of caloric” in the early 19th century (Grossman, 2017)]. However, in our opinion the essence of chemistry is the qualitative and quantitative way that the elements bind, while atoms “merely” are the basic components of chemistry. Or in metaphorical terms, mortar maketh the house, not bricks. In any case, the two ideas are not mutually exclusive, and we are also allowed to consider a greater chemical paradigm as “matter made by atoms joined by their chemical bonds”.

2.2 2nd axiom: the real chemist

The second epistemological axiom is about the vital importance of knowing, teaching, and learning the concept of the chemical bond. If it is the central paradigm of chemistry, it is not only important; it is inevitable. The chemical bond, with its orbitals, electrostatic potential, electron density, and energy patterns, provides the ABC of chemistry work, research, and education, and nothing can be built without it. Therefore, no chemist is a real chemist without understanding the chemical bond.

The only thing that stands for debate is how deep are we going to teach the subject (mostly in terms of how much quantum mechanics can we cover), or how broad are we going to go (how many types of bonds and effects can we cover). Undoubtedly, a better knowledge of chemical bond theory makes us better chemists, but the teaching time and the students’ ability to digest information is limited. Moreover, the approach that a lecturer chooses might depend on the target, whether it is directed to physical chemistry graduate students or if it is a general chemistry course. In our specific chemical bond course, which as explained above was directed to the general chemistry student, the focus was on the broad but relatively shallow conceptual and quantum mechanical description of as many types of bonds and effects as possible. Nevertheless, many articles, techniques and novel ideas were brought as examples, a way of showing the deepness of state-of-the-art chemical bond research and how they tackle its knowledge holes.

2.3 3rd axiom: we know nothing

The last axiom clearly is the most controversial and subjective. Simply put, it is our belief that no chemist really comprehend the chemical bond in its totality (and probably we will never will). According to a strict use of logic, of course the second and third axiom taken together imply that there are no real chemists; but needless to say, herein we are speaking figuratively.

The fact that we do not fully comprehend the bond does not mean that we know nothing about the bonds. On the contrary, we know a lot; we have a firm theoretical basis of their existence and nature based on a plethora of experimental and computational studies. For example, we know that the bond mostly is a subtle equilibrium between electrons-nuclei attractions, nuclei–nuclei and e⁻–e⁻ Coulomb and Pauli repulsions, and kinetic contributions that reach a minimum in the potential energy surface. However, the point is that:

1. We certainly do not know everything. There is still much to learn.

2. We are unsure of the real nature of many chemical bonds. Some bonds are hotly debated, with some people being sure of one explanation, and others being sure of the opposite explanation.

3. Although the existence of the chemical bond is not debated (it is a paradigm), its definition is very far from uncontested (as we shall see below).

4. We know how to compute bonds, and we know the impact of each component of the Hamiltonian when creating molecules. But this gives us the “how much” information on the bond, while we still debate what are bonds in general terms.

5. For a surprisingly large number of intrinsic concepts of the chemical bond there are also multiple definitions that play against the clarity and unambiguity that one would expect from chemistry as an exact science (such as bond strength and energy, bond order, atomic charge, molecular orbitals, etc.).

How is it possible that we know so much about the chemical bond, and yet we know so little? Moreover, how is it possible that we do not share these uncertainties with students, and unrealistically teach the chemical bond as if it were a closed topic? Is this not a kind of soft lying to the students, artificially shielding them from the reality of the chemical world and therefore leaving them unprepared?

A chemical bond course should begin by deconstructing it into its multiple definitions (Ayers et al., 2018; Hendry, 2008, 2022]; IUPAC, 2024a; Seifert, 2022b; Weisberg, 2008). We will describe here three approaches, their usefulness, their problems, and their unknowns. Considering their convenience, but also their limited range, they can also be labelled as chemical models (Schummer, 2014), as we shall discuss below. These are the energetic, structural, and orbital viewpoints of the bond.

3.1 A force acting between two atoms

Part (a) of IUPAC’s definition says that the chemical bond simply is a force that binds atoms. That is, if energy must be spent to stretch apart two atoms, then they are bonded. The definition does not expand on which kind of force, and therefore any force will be valid, from strong covalent to feeble London. Interestingly, the force must be exactly zero in a “stable independent molecular entity”, or the bonds would continuously shrink until reaching a singularity (see Figure 1). Mathematically, with V being the potential energy and r the distance between two atoms:

That is, a stable geometry involves a minimum in V, the potential energy surface, and a zero force. In fact, a molecule is defined as a multiatomic entity in a minimum of the potential energy surface (IUPAC, 2024b). Therefore, the definition of chemical bonds in a molecule in its ground state in terms of forces should be paradoxically expressed when the molecule is outside equilibrium, with stretched bonds. Note that the potential energy term here is not the Hamiltonian potential energy (obtained from the V ˆ operator in the H ˆ = − ℏ / 2 m ∇ 2 + V ˆ Schrödinger equation), but the potential energy as in a stretched or compressed classical spring or in the movement of a pendulum. In other words, the ‘V’ in eq. (1) is the ‘E’ from H ˆ Ψ = E Ψ at different molecular geometries.

Figure 1:

Three examples of potential energies (solid lines, left axis in kJ mol⁻¹) and forces acting in a bond [dashed lines, right axis in kJ mol⁻¹ Å⁻¹, see eq. (1) (Gaussian16, 2024)]. In a stationary geometry the forces are equal to zero by definition, since the potential energy of the molecule must be at its minimum. Note that by IUPAC’s definition the hydrogens of ethene would be bonded, as there is a force hindering their separation.

It would be more accurate to define a chemical bond as a restoring force after they are stretched. The elongating and shrinking of the interatomic distance creates harmonic potentials (Figure 1), and therefore many people consider a chemical bond in a stable geometry by considering not the force (F in eq. (1)), but the force constant (f in eq. (2)) of the hypothetical spring between the atoms, i.e. the second derivative of the potential energy as a function of a small distortion of the bond distance (relaxing all other coordinates) from the molecule in its stable geometry (Brandhorst & Grunenberg, 2008; Kraka et al., 2020; Zhao et al., 2022).

One advantage of this definition is that it also gives us a measure of the bond strength. The larger f, the harder it is to stretch the bond, and therefore the stronger it will be. For instance, f for a single covalent C–C bond is around 400 N m⁻¹, for the triple bond of N₂ (one of the strongest bonds in nature) is ∼2,300 N m⁻¹, and for non-covalent interactions it is usually lower than 100 N m⁻¹.

The bigger drawback brought by this energy/force-based mathematical definition of the chemical bond is the fact that if you stretch any pair of atoms in any molecule you will observe a force that will attract them back to the optimal geometry. This is a necessary conclusion of defining the molecule as a minimum in the potential energy surface, which makes any geometrical distortion raise the molecular energy. It will happen even if these two atoms are far apart and removed by many ‘proper’ bonds. In other words, according to this definition, there is a pairwise attraction between all the atoms of a molecule, and therefore we will erroneously infer that all its atoms would be directly bonded, even the ones on opposite sides of the molecule. This would also be technically valid between the atoms of a large molecular cluster several nm apart, an idea that not many chemists would approve.

This is exemplified in Figure 1, which shows that if you elongate the distance between two hydrogens in ethene, the energy goes up and a force will be created that will push back the hydrogens to their lowest energy. However, it is inconceivable to think under our typical conceptualization that there is a proper chemical bond between these two hydrogens separated by two carbons. What would you call the ‘force’ between the hydrogens? Certainly not a covalent or ionic bond, not even a van der Waals interaction [although it might be considered a ‘through-bond interaction’ (Albright et al., 2013) or in certain cases a ‘mechanical bond’ (Bo, 2017)]. Without the carbons holding the hydrogens in their positions (as can be done in a model H₂ dimer) they would repel each other, clearly indicating that there is no chemical bond between hydrogens in ethene, no matter what the stretching force tells us.

A possible solution to this problematic definition of the bond as a force might be to indicate that it works only when the atoms are already connected. However, this will fail due to a circular reasoning fallacy, as we will have to explain what makes this connection in the first place.

Another solution might be to model the pairwise interaction with a simplified system that removes all the molecular constraints; if they keep the attraction, then the bond was real. This can be done, for instance, by mimicking ethane with an H₂ dimer (which would show that the H–H bond is spurious).

Another case is the short SiH–HSi interaction of the cage molecule shown in Figure 2 (Mandal et al., 2017; Mandal & Datta, 2020; Zong et al., 2013), a hydrogen-hydrogen ‘bond’ is kept in a very tight compartment in a kind of ‘iron maiden’ (Pascal, 2004) system. A strong force is required to separate both hydrogens, but that is an artificial effect brought by the constricted cage. If we release the constraints, which can be easily done with a silane dimer model (on the right of Figure 2), we will see that the bonding force is extrinsic to the hypothetical bond, completely beyond the H–H region. As a matter of fact, some orbital, energy decomposition, and topographical quantum mechanical methods indicate that there may be a weak force binding the hydrogens (Mandal et al., 2017), but it is completely overpowered by the Pauli repulsion without the external constraint. And so, what shall we say about this H–H interaction? Is there a chemical bond between the two hydrogens? Is a mathematical definition of the bond through the force enough to declare the existence of a bond? (Berson, 2008)

Figure 2:

In this cage molecule (Mandal et al., 2017; Mandal & Datta, 2020; Zong et al., 2013), the confined internal hydrogens depict an ultra-short non-covalent interaction. Since it is in its optimal geometry, any intent to separate the hydrogens will require to employ a significant force, meaning that they have a chemical bond between them according to IUPAC’s definition. However, if we release the tension brought by the cage, such as with a silane dimer (but still keeping the symmetry), the bond is utterly broken (Gaussian16, 2024).

I will leave another strawman fallacy for the reader to analyse: Adenine and thymine are a base pair that are bound through two H-bonds (N–H and O–H), which are undisputed chemical bonds, even if they are not covalent. If we break the N–H bond by a rotation (lower part of Figure 3), the O–H bond is elongated to 2.02 Å. This implies that without the external force of the N–H bond, the O–H bond would be repulsive at the original bond length of 1.94 Å. If an attractive force is proof of a chemical bond, is a repulsive force proof of the lack of a chemical bond?

Figure 3:

Adenine-thymine base pair. When breaking the N–H hydrogen bond the O–H bond is elongated, which indicates that there was a repulsive force between the oxygen and the hydrogen that was relaxed when the external forces were removed (Gaussian16, 2024). Does this mean that there was no chemical bond between the O and the H?

3.2 Everything goes

There are more issues with the energy/force definition of the chemical bond. One critical problem is that “This definition does not state what chemical bonds are; rather it states the conditions that hold when a chemical bond is considered to exist” (Seifert, 2022b). There is no solution to this problem when working only with raw energy values.

Parts (b) and (c) of IUPAC’s definition (see above) are also not free of challenges. The discussion about the potential and kinetic energies (part b of the definition) is a thorny and open topic of debate, as we try to comprehend which one of these two components of the bond energy is the real culprit of the chemical bonds.

Part (c) indicates that any type of interaction is valid to enlist it as a chemical bond, as long as it creates an attraction force: from the Na⁺ to Cl^- ionic to the O₂ dimer London interactions, everything goes. Wherever there is a force, no matter its nature or how weak it is, there is a bond. This is not only ambiguous, but also goes against our conventional understanding of the chemical bond. Do we have a chemical bond in every case of London forces? What about He₂, a dimer 25,000 times weaker than the benzene dimer? What is the limit where we can safely say there is no bond anymore? (Berson, 2008)

The convenient energetic definition can represent and partly explain the bond, and it can predict where a bond should occur; but as seen above, its utility must be framed within limits. In this sense, we can consider it not only as a definition, but also as a model (Schummer, 2014) of the chemical bond. This has the advantage that students are used to working with chemical models of all types (VSEPR, ideal gases, crystal field, etc.), and therefore it is a point worthy of teaching. Another characteristic of models is that they can accommodate other models that may explain the same phenomenon with completely different mathematical or heuristic tools, each one of them having explanation and prediction power, as well as their own limits [or as the ironic aphorism says, “all models are wrong, but some are useful” (Box, 1979)]. Still, there are many models of the bond that do not work as definitions of the chemical bond (for example Lewis electron pair and Langmuir octet rule, hypervalency, σ-bonding/π-backbonding, etc.), and therefore we can conclude that most chemical definitions can work as models, but most models are not definitions. In this sense, with the understanding that they are also models, we will continue with two other definitions of the bond.

5.1 The omnipresent but cryptic molecular orbital

Although the concept of a wave function is intrinsic to physics, the molecular orbital (MO) resides in the realm of chemistry (Fortin & Jaimes Arriaga, 2022; Weisberg, 2008), especially when building them from atomic orbitals (MO-LCAO). What is in the realm of philosophy is the question of whether the MO really exists or not, considering that it is not a quantum observable (Hettema, 2017; Krylov, 2020; Labarca & Lombardi, 2010; Mulder, 2011; Pham & Gordon, 2017; Scerri, 2000, 2001]; Truhlar et al., 2019; Zuo et al., 1999). Even if they may not be real, analyses such as photoelectron spectroscopy, Koopmans theorem, Fukui function, Dyson orbitals, and atomic resolution spectroscopy techniques indicate that molecules behave as if MOs were real. This is a fascinating debate, but here we will focus only on its utility as a quantitative and qualitative chemical tool (Jenkins, 2003).

The MOs approach to the chemical bond is simple. We count the orbitals, and if the number of bonding ones is larger than antibonding ones, then there is a chemical bond, as shown in the interaction diagram of Figure 9 for the triple bond of CO. Since each MO can hold up to two electrons, we have a direct connection with Lewis’ shared electron pair model. Can this work as a general definition of the chemical bond? Well, most chemistry books think so. It certainly has a lot of appeal in terms of explanatory and predictive power, much more than the previously discussed structural and energetic approaches, models that show the existence of bonds but have many difficulties explaining their raison d’être (Seifert, 2022b). For example, we can build the final orbitals from their fragments in interaction diagrams, allowing better comprehension of the interaction (Albright et al., 2013; Rauk, 2001). The structural and energetic definitions only consider what happens in the final product, or in other words, they work on the chemical bond, but not on the chemical bonding.

Figure 9:

Interaction diagram for CO. There is a chemical bond since there are more bonding orbitals than antibonding ones. In this case, CO has a triple bond, with two covalent π and one dative (Nandi & Kozuch, 2020) σ bonding MOs.

MOs can be of very different styles, including canonical, Dyson, NTO, NBO, and others. These orbital schemes can be constructed from linear combinations of the other orbital styles, and they are all legitimate as long as they provide the correct electronic density (Krylov, 2020; Truhlar et al., 2019; Weinhold & Landis, 2005). Each one of these MO representations gives a useful depiction for a specific use (energetic, spectroscopic, reactivity, bonding, etc.), but they will disappoint if trying to use them for another property. For instance, despite what we learn in college, most canonical MOs do not provide any clear bonding pattern like the ‘clean’ CO orbitals of Figure 9. The canonical MOs (which are orthogonal to each other, and are the ones produced by most quantum chemistry methods and programs) are mostly delocalized all around the molecule, and therefore can be bonding between one pair of atoms, antibonding between another pair, and nonbonding for the rest (see Figure 10). While bond orders can be mathematically obtained from the canonical MOs (Mayer, 2007), these methods loose the simple visual connection between a bond and the orbitals. However, their localization can do the trick (see Figure 10), although the localization process may sound too artificial and begging the question for the taste of some physical chemistry purists, plus they lose any spectroscopical insight. In essence, each style of molecular orbital representation must be used for the purpose it was designed, for instance the orbitals localized in the bond can be used to define, quantify, and understand each bond (see the NBO and Boys localization in Figure 10).

Figure 10:

Porphyrin molecular orbitals. The canonical MOs (HOMO and HOMO-1 in the figure) are completely delocalized, losing the chemical intuition that localization provides. The lower images show pairs of C–N bonding MOs in NBO and boys localization flavours. NBO produces the σ and π conventional MOs, boys generates the pauling-style ‘banana’ double bonds.

Here is another issue when taking the MO-based chemical bond in a simplistic way: According to the basic view of the model it is possible to formally have a single, double, triple, quadruple (Cotton & Harris, 1965), quintuple (Brynda et al., 2006; Frenking, 2005), or up to a sextuple (Frenking & Tonner, 2007; Roos et al., 2007) covalent bond; you might even have half a covalent bond (like in the 3-center-4 electron cases such as XeF₂ or with F₂ ⁻), one and a half covalent bond (as in benzene), or another option if an odd number of electrons are involved. In this formal language, the covalent bond order is always a function of the number of electrons and the orbitals they occupy. However, this integer formality can disappear, producing fractional bond orders, for instance when there is superposition of localized orbitals, or when using multi-reference methods that result in fractional orbital occupancy. In other words, integer bond orders are only a formal idealization of real bonds. In contrast, when speaking about chemical bonds from their structural or energetic perspectives, we generally do not speak in terms of bond orders, but in terms of their quantitative strengths, just stating that a bond is weak or strong (Brandhorst & Grunenberg, 2008; Shaik et al., 2016).

But maybe the biggest disadvantage of the MO as a diagnostic of the chemical bond is, similar to the structural method, its difficulty in describing non-covalent bonds. MO theory as a computational approximation to the solution of Schrödinger equation can certainly compute the electrostatic and van der Waals bonding through ab initio or DFT methods, and there are orbital-based explanations for these interactions (Schneider et al., 2016; Truhlar, 2019). Nevertheless, no significant density between atoms implies that there are no significant bonding orbitals between them, limiting the model. We cannot say that orbitals make the bonds when there are no orbitals in these bonds.

5.2 Valence bond structures, the wave function in the image and likeness of the chemical bond

A final thought is briefly dedicated to valence bond theory (VB), a method with less computational use, less fame, and less flashy pictures compared to MO theory, but with a longer history (Pauling, 1960) and a better ability to define chemical bonds as chemists like to think about them (Shaik & Hiberty, 2008; Weisberg, 2008). Indeed, if à la Lewis we consider the shared electron pair as the definition of the chemical bond (Lewis, 1916), VB generates mathematical structures with a one-to-one correspondence to these pairs. If in MO theory the ground state MO of H₂ is written as σ = s_A + s_B, in VB theory the covalent bond is written as ϕ _HL = s_A × s_B (to write the physically correct expressions we must add spins and antisymmetrize the wave functions, but for the sake of simplicity we show here only its basic term). For MO we create an orbital to put the electrons as the positive linear combination of atomic orbitals (in this case the σ MO from the 2s AOs). In VB we create a covalent structure for that purpose, called ‘Heitler-London’ in honor of the authors of the first quantum description of a chemical bond (Heitler & London, 2000).

While with enough refinements VB and MO theory in the end converge to the same energy value, a great advantage of VB is that it can clearly define in conceptual and mathematical manner our most basal ideas of the bond in chemical terms, such as the bond order, resonance, covalent, and ionic characters. Therefore, if we qualitatively consider the chemical bond as shared electron pairs (or in Lewis’ words: “…to express this idea of chemical union in symbols I would suggest the use of a colon … to represent the two electrons which act as the connecting links between the two atoms” (Lewis, 1916)), valence bond theory provides its quantitative quantum mechanical justification. While VB computations are harder than MO theory computations, VB competes head-to-head in terms of explanatory and predictive power (Shaik & Hiberty, 2008). Still, even if high-level VB can compute weak interactions, it does not have any kind of van der Waals structure. In these cases, like with orbitals, we cannot say that shared electron pairs make the bonds when there are no pairs in these bonds. Nothing is perfect, but quoting Voltaire, “the best is the enemy of good”. And both VB and MO models are really good.