# Fundamental forces behind covalent bonding

I understand that covalent bonding is an equilibrium state between attractive and repulsive forces, but which one of fundamental forces actually causes atoms to attract each other?

Also, am I right to think that "repulsion occurs when atoms are too close together" comes from electrostatic interaction?

I understand that covalent bonding is an equilibrium state between attractive and repulsive forces, but which one of fundamental forces actually causes atoms to attract each other?

The role of Pauli Exclusion in bonding

It is an unfortunate accident of history that because chemistry has a very convenient and predictive set of approximations for understanding bonding, some of the details of why those bonds exist can become a bit hard to discern. It's not that they aren't there -- they most emphatically are! -- but you often have to dig a bit deeper to find them. They are found in physics, in particular in the concept of Pauli exclusion.

Chemistry as avoiding black holes

Let's take your attraction question first. What causes that? Well, in one sense that question is easy: it's electrostatic attraction, the interplay of pulls between positively charged nuclei and negatively charged electrons.

But even in saying that, something is wrong. Here's the question that points that out: If nothing else was involved except electrostatic attraction, what would be the most stable configuration of two or more atoms with a mix of positive and negative charges?

The answer to that is a bit surprising. If the charges are balanced, the only stable, non-decaying answer for conventional (classical) particles is always the same: "a very, very small black hole." Of course you could modify that a bit by assuming that the strong force is for some reason stable, in which case the answer becomes "a bigger atomic nucleus," one with no electrons around it.

Or maybe atoms as Get Fuzzy?

At this point some of you reading this should be thinking loudly "Now wait a minute! Electrons don't behave like point particles in atoms, because quantum uncertainty makes them 'fuzz out' as they get close to the nucleus." And that is exactly correct -- I'm fond of quoting that point myself in other contexts!

However, the issue here is a bit different, since even "fuzzed out" electrons provide a poor barrier for keeping other electrons away by electrostatic repulsion alone, precisely because their charge is so diffuse. The case of electrons that lack Pauli exclusion is nicely captured by Richard Feynman in his Lectures on Physics, in Volume III, Chapter 4, page 4-13, Figure 4-11 at the top of the page. The outcome Feynman describes is pretty boring, since atoms would remain simple, smoothly spherical, and about the same size as more and more protons and electrons get added in.

While Feynman does not get into atoms how such atoms would interact, there's a problem there too. Because the electron charges would be so diffuse in comparison to the nuclei, the atoms would pose no real barrier to each other until the nuclei themselves begin to repel each other. The result would be a very dense material that would have more in common with [neutronium[(http://en.wikipedia.org/wiki/Neutronium) than with conventional matter.

For now I'll just forge ahead with a more classical description, and capture the idea of the electron cloud simply by asserting that each electron is selfish and likes to capture as much "address space" (see below) as possible.

Charge-only is boring!

So, while you can finagle with funny configurations of charges that might prevent the inevitable for a while by pitting positive against positive and negative against negative, positively charged nuclei and negatively charged electrons with nothing much else in play will always wind up in the same bad spot: either as very puny black holes, or as tiny boring atoms that lack anything resembling chemistry.

A universe full of nothing but various sizes of black holes or simple homogenous neutronium is not very interesting!

Preventing the collapse

So, to understand atomic electrostatic attraction properly, you must start with the inverse issue: What in the world is keeping these things from simply collapsing down to zero size -- that is, where is the repulsion coming from?

And that is your next question:

Also, am I right to think that "repulsion occurs when atoms are too close together" comes from electrostatic interaction?

No; that is simply wrong. In the absence of "something else," the charges will wiggle about and radiate until any temporary barrier posed by identical charges simply becomes irrelevant... meaning that once again you will wind up with those puny black holes.

What keeps atoms, bonds, and molecules stable is always something else entirely, a "force" that is not traditionally thought of as being a force at all, even thought it is unbelievably powerful and can prevent even two nearby opposite electrical charges from merging. The electrostatic force is enormously powerful at the tiny separation distances within atoms, so anything that can stop charged particles from merging is impressive!

The "repulsive force that is not a force" is the Pauli exclusion I mentioned earlier. A simple way to think of Pauli exclusion is that identical material particles (electrons, protons, and neutrons in particular) all insist on having completely unique "addresses" to tell them apart from other particles of the same type. For an electron this address includes: where the electron is located in space, how fast and in what direction it is moving (momentum), and one last item called spin, which can only have on of two values that are usually called "up" or "down."

You can force such material particles (called fermions) into nearby addresses, but with the exception of that up-down spin part of the address, doing so always increases the energy of at least one of the electrons. That required increase in energy is a nutshell is why material objects push back when you try to squeeze them. Squeezing them requires minutely reducing the available space of many of the electrons in the object, and those electrons respond by capturing the energy of the squeeze and using it to push right back at you.

Now, take that thought and bring it back to the question about where repulsion comes from when to atoms bond at a certain distance, but no closer. They are the same mechanism!

That is, two atoms can "touch" (move so close, but no closer) only because they both have a lot of electrons that require separate space, velocity, and spin addresses. Push them together and they start hissing like cats from two households who have suddenly been forced to share the same house. (If you own multiple cats, you'll know exactly what I mean by that.)

So, what happens is that the overall set of plus-and-minus forces of the two atoms is trying really hard to crush all of the charges down into a single very tiny black hole -- not into some stable state! It is only the hissing and spitting of the overcrowded and very unhappy electrons that keeps this event from happening.

Orbitals as juggling acts

But just how does that work?

It's sort of a juggling act, frankly. Electrons are allowed to "sort of" occupy many different spots, speeds, and spins (mnemonic $s^3$, and no, that is not standard, I'm just using it for convenience in this answer only) at the same time, due to quantum uncertainty. However, it's not necessary to get into that here beyond recognizing that every electron tries to occupy as much of its local $s^3$ address space as possible.

Juggling between spots and speeds requires energy. So, since only so much energy is available, this is the part of the juggling act that gives atoms size and shapes. When all the jockeying around wraps up, the lowest energy situations keep the electrons stationed in various ways around the nucleus, not quite touching each other. We call those special solutions to the crowding problem orbitals, and they are very convenient for understanding and estimating how atoms and molecules will combine.

Orbitals as specialized solutions

However, it's still a good idea to keep in mind that orbitals are not exactly fundamental concepts, but rather outcomes of the much deeper interplay of Pauli exclusion with the unique masses, charges, and configurations of nuclei and electrons. So, if you toss in some weird electron-like particle such as a muon or positron, standard orbital models have to be modified significantly and applied only with great care. Standard orbitals can also get pretty weird just from having unusual geometries of fully conventional atomic nuclei, with the unusual dual hydrogen bonding found in boron hydrides such as diborane probably being the best example. Such bonding is odd if viewed in terms of conventional hydrogen bonds, but less so if viewed simply as the best possible "electron juggle" for these compact cases.

"Jake! The bond!"

Now on to the part that I find delightful, something that underlies the whole concept of chemical bonding.

Recall that it takes energy to squeeze electrons together in terms of the main two parts of their "addresses," the spots (locations) and speeds (momenta)? I also mentioned that spin is different in this way: the only energy cost for adding two electrons with different spin addresses is that of conventional electrostatic repulsion. That is, there is no "forcing them closer" Pauli exclusion cost like you get for locations and velocities.

Now you might think "but electrostatic repulsion is huge!", and you would be exactly correct. However, compared to the Pauli exclusion "non-force force" cost, the energy cost of this electrostatic repulsion is actually quite small -- so small that it can usually be ignored for small atoms. So when I say that Pauli exclusion is powerful, I mean it, since it even makes the enormous repulsion of two electrons stuck inside the same tiny sector of a single atom look so insignificant that you can usually ignore its impact!

But that's secondary, because the real point is this: When two atoms approach each other closely, the electrons start fighting fierce energy-escalation battles that keep both atoms from collapsing all the way down into a black hole. But there is one exception to that energetic infighting: spin! For spin and spin alone, it become possible to get significantly closer to that final point-like collapse that all the charges want to do.

Spin thus becomes a major "hole" -- the only such major hole -- in the ferocious armor of repulsion produced by Pauli exclusion. If you interpret atomic repulsion due to Pauli exclusion as the norm, then spin-pairing two electrons becomes another example of a "force that is not a force," or a pseudo force. In this case, however, the result is a net attraction. That is, spin-pairing allows two atoms (or an atom and an electron) to approach each other more closely that Pauli exclusion would otherwise permit. The result is a significant release of electrostatic attraction energy. That release of energy in turn creates a stable bond, since it cannot be broken unless that same energy is returned.

Sharing (and stealing) is cheaper

So, if two atoms (e.g. two hydrogen atoms) each have an outer orbital that contains only one electron, those two electrons can sort of look each other over and say, "you know, if you spin downwards and I spin upwards, we could both share this space for almost no energy cost at all!" And so they do, with a net release of energy, producing a covalent bond if the resulting spin-pair cancels out positive nuclear charges equally on both atoms.

However, in some cases the "attractive force" of spin-pairing is so overwhelming greater for one of the two atoms that it can pretty much fully overcome (!) the powerful electrostatic attraction of the other atom for its own electron. When that happens, the electron is simply ripped away from the other atom. We call that an ionic bond, and we act as it if it's no big deal. But it is truly an amazing thing, one that is possible only because the pseudo force of spin-pairing.

Bottom line: Pseudo forces are important!

My apologies for having given such a long answer, but you happened to ask a question that cannot be answered correctly without adding in some version of Pauli "repulsion" and spin-pair "attraction." For that matter, the size of an atom, the shape of its orbitals, and its ability to form bonds similarly all depend on pseudo forces.

• I would have expected mention of Slater's treatment using the quantum virial theorem, and the important result that it is electrostatics that keeps atoms together. Also, your answer seems to ignore two important observations 1) nuclear fusion is possible and 2) the Pauli principle leads to the vanishing of the wave function when electrons of the same quantum number share the same region of space, again cf. Slater determinants for antisymmetrized wave functions. (I do not deny the importance of the Pauli principle in specific solutions to $H \Psi=E \Psi$, but that is not the issue here.) – Eric Brown Jun 2 '14 at 6:07
• I don't like this explanation. To start off with the Pauli Exclusion principle puts the cart before the horse. The Pauli Exclusion principle is in the middle of quantum mechanics. You have to explain the standard model of particle physics to explain why the Pauli Exclusion principle keeps electrons and protons apart. To me it seems that the OP doesn't understand why Coulomb's law isn't a sufficient description of atoms. To make a poor analogy the OP's question is like asking how do you install a kitchen sink before the house is built. So (1) atom (2) ionic bonding (3) covalent bonding – MaxW Apr 19 '16 at 15:54
• I always thought I have a firm grasp of quantum theory, but this left me dazed and confused. – Martin - マーチン Apr 22 '16 at 12:44
• My issue with this answer is that by starting with Pauli repulsion etc. you ignore that all covalent bonding characteristics should be exhibited in the simplest bound molecule - H2+. Thus any explanation based on Pauli repulsion - necessarily involving 2e- - can only give detail on other forms of bonding, and not the core fundamentals of covalent bonding. – Swedish Architect May 12 '17 at 21:17
• Sure! You are right to point out the the case of $H^{2+}$ bonding is both unique and for that reason especially interesting. It is to the best of my knowledge the one and only particle configuration that depends solely on electrostatic attraction, unmodified by the complex energy and geometric constraints (just look at orbital diagrams!) added by Pauli exclusion. So, it is a very interesting and well worth pointing out, even if it does not generalize to other situations... hmm, hold on: Do species such as $Li^{2±}$ exist, say maybe in deep space?? Those too would be examples of your bond... – Terry Bollinger May 14 '17 at 10:35

Thinking about this like a physicist, there are four fundamental forces: the strong nuclear force, the weak nuclear force, the electromagnetic force, and gravity. The strong nuclear force holds the protons and neutrons in the nucleus together. The weak nuclear force causes beta decay. Those two might be considered chemistry. I don't, but some people do. Gravity is much too weak to have any effect on chemistry. So that leaves the electromagnetic force to control nearly all of chemistry.

On a simple conceptual level, that's all there is. The nuclei are both positively charged, so they repel each other. The electrons are negatively charged, so they are attracted to their respective nuclei. When the electron clouds get close enough to interact with both nuclei, then they begin to pull the nuclei together.

The deeper explanation requires quantum mechanics. When the atoms are separated, you can use the Schrödinger equation with the electric potential from the nucleus. That gives you the electron orbitals for an atom all by itself. When the two atoms get close together, you use the electric potential for both nuclei in the Schrödinger equation. The solution is then the molecular orbital rather than the atomic orbitals. Because the Schrödinger equation is impossible to solve exactly for a molecule, chemists need an approximation. The usual approximation is to build the molecular orbital out of the atomic orbitals by adding and subtracting the atomic orbitals. This is where the ideas of $sp$-, $sp^2$-, and $sp^3$-hybridizaton, and $\pi$- and $\sigma$-bonding come from.

For further information, most introductory college-level general chemistry texts should discus this. As an example, I pulled most of the above explanation from Zumdahl's Chemistry. In the 5th edition, this is in chapters 8 and 9 (the current edition appears to be the 7th). This is a much more important idea in organic chemistry, so those textbooks usually review it in the first one or two chapters. The organic chemistry book I have in front of me at the moment is McMurray's Organic Chemistry. This is discussed in chapter 1 of the 3rd edition of that book (current edition is the 8th).

• The first paragraph answers the OP's first question nicely. To answer the second question I'd moved stuff in 2nd and 3rd paragraphs around to explain why electrons of an atom don't collapse into its nucleus. I'd then discussed ionic bonding and then gone into covalent bonding. To me this is a much nicer explanation than Terry's. – MaxW Apr 19 '16 at 17:39
• @MaxW, think carefully about what Colin's first paragraph really says. First it brings in forces that are not really relevant to question. Very few people reading that excellent question would seriously think the answer involves gravity, so why even bring it up? But worse, the paragraph ends by simply repeating the question as if it was an answer: "It's all electromagnetics." Well... no, it just is not. Even invoking Schrodinger's equation sidesteps the real issue of where the repulsive forces originate. So sorry you don't like Fermi exclusion... but that without it, it's all hand waving. – Terry Bollinger Apr 20 '16 at 4:04

I understand that covalent bonding is an equilibrium state between attractive and repulsive forces, but which one of fundamental forces actually causes atoms to attract each other?

Thinking about this like a physicist, there are four fundamental forces: the strong nuclear force, the weak nuclear force, the electromagnetic force, and gravity. The strong nuclear force holds the protons and neutrons in the nucleus together. The weak nuclear force causes beta decay. ... Gravity is much too weak to have any effect on chemistry. So that leaves the electromagnetic force to control nearly all of chemistry. [From Colin's answer where he nailed this part of the answer.]

Side Note - There is a overlapping field of physics and chemistry called Nuclear Chemistry which involves working with radioisotopes which require special handling to avoid radioactive contamination. For the most part the effect of different isotopic masses of an element on the element's chemistry can be ignored. Nuclear properties, such as radioactivity and nuclear transmutation, are really more physics than chemistry.

Also, am I right to think that "repulsion occurs when atoms are too close together" comes from electrostatic interaction?

Not really. There is a significant twist here. Coulomb's Law predicts an attractive force between an oppositely charged bodies (whatever the bodies may be...), but offers no explanation as to why the two bodies don't combine. The explanation of why a $\ce{Na+}$ cation and a $\ce{Cl^-}$ anion don't collapse first requires an understanding of why a proton and electron don't combine to form a neutron. That explanation requires electrons to form orbitals around the nucleus of an atom. The first explanation of orbitals was the Bohr model which has a number of shortcomings and which has now been replaced by quantum mechanics.

Although quantum mechanics offers a far better analysis of the behavior of a hydrogen atom via the Schrödinger equation, there is a serious limitation. The Schrödinger equation can only be exactly solved for one electron system. (This is much like the three body problem with gravity.) So multielectron atoms are built-up by using combinations of one electron orbitals.

Now that we have an atom we can consider $\ce{Na+}$ cation to be a hard sphere with the electronic configuration of a neon atom which has a closed subshell and a $\ce{Cl^-}$ anion to be a hard sphere with the electronic configuration of a argon atom which also has a closed subshell. The two ions then form a ionic bond. The ionic bond between the two ions can sort be fudged in a solid by the use of the Madelung constant. However an ionic bond doesn't explain the behavior of organic molecules like methane, or even a permanganate anion.

So extending the concept of combining single electron orbitals to form multielectron atoms, we can combine multiple atomic orbitals of different atoms to form molecular orbitals. The basic notion is that when the atomic orbitals from two different atoms combine then a molecular bonding orbital with lower energy is formed as well as an molecular antibonding orbital of higher energy. If each atom only contributes one electron, then the net result is that the bond formation releases energy as heat. If both atoms contribute two electrons then both the bonding and the antibonding orbitals are occupied and there is no net bond strength and no heat is produced. (Combing multiple atomic orbitals on an atom is where the ideas of $sp$-, $sp^2$-, and $sp^3$-hybridizaton comes from.)

Now it seems that we're reached a point to cut-off drilling down into the problem. The gist is in that using a linear combination of atomic orbitals (LCAO) to form a molecular orbital the overlap depends on the distance between the atoms. There is a distance for which the orbitals have a maximum overlap, and hence the strongest bond strength.

• Overlap is a terrible indicator for bond strength, bigger overlap does not necessarily conclude in stronger bonding, and it often does not. – Martin - マーチン Apr 22 '16 at 12:36

Both the attraction and repulsion are the result of the electromagnetic interaction. At long distances, two atoms attract each other because of induced dipole-induced dipole interactions. When they get close enough together, the exchange interaction acts on the non-valence electrons of the atoms and forces them into higher energy states. This makes the atoms repel each other at short distances.

• This is just wrong. In chemistry an atom doesn't have a dipole but rather molecules. Also Coulombic repulsion of the nuclei isn't the primary reason that keeps atoms in molecules some distance apart. – MaxW Apr 19 '16 at 17:31
• @MaxW: Atoms do have dipole moments when near other atoms. The other error has been corrected. – Dan Apr 19 '16 at 18:59
• An Induced dipole-induced dipole interaction is called the London dispersion force en.wikipedia.org/wiki/London_dispersion_force – MaxW Apr 19 '16 at 20:41

I think this question is generally poorly answered in textbooks - It seems in most chemistry undergraduate books this is "glossed over". Explaining it in terms of MO theory is sort of great - the orbitals overlap, electron experiences both potentials, and so bonding orbital is lower in energy etc. But this is ignoring some of the subtlety in covalent bond which is present. Covalent bonding is also clearly a quantum phenomena - describing it in terms of Coulombic interactions, purely classically, cannot lead to any meaningful result.

If covalent bonding is a general principle, then it stands that the principles involved should be exemplified by the simplest bound species - $\ce{H2+}$ Considered from a classical viewpoint purely, the electron is not stable in the inter-atomic space (middle of the atoms). It is trivial to show the potential energy is lower at one of the nuclei, and so all matter should just condense - electron loses all kinetic energy by radiating, and so orbit collapses - this is precisely what drove the founders of quantum mechanics - the simple stability of matter.

Let's examine $\ce{H2+}$. At infinite separation the electron is in one of the potential wells, but due to the infinite separation, cannot tunnel from one potential well to the other potential well provided by the $\ce{H+}$ nuclei. When the distance is decreased to a finite distance, but still very long, the electron is able to tunnel from one potential well to the other. This means the volume of space the electron occupies is larger - the electron's wavefunction is more spread out. This corresponds to a smaller curvature, and thus a smaller Laplacian, leading to a lower expected kinetic energy term $<T>$. However what happens to the potential energy? If the electron is no longer in one potential well, but tunnels to the other potential well does the expected potential energy term, $<V>$ increase or decrease? It rises - the electron's expected distance from the nucleus increases, as wavefunction grows in the internuclear distance, facilitating the tunneling of the electron from one well to the other. You could also see it as:

1)If a wavefunction is highly concentrated, it is highly curved -> Large value of the Laplacian, and so high kinetic energy. By being concentrated around the nucleus it also has a large expectation value of $<1/r>$ and consequently a large $<V>$ term.

2) If wavefunction is diffuse and spread out, it has a small curvature, small Laplacian, and so a small $<T>$ term, but the expectation value $<1/r>$ is small, and so $<V>$ is small.

Note that this process of electron delocalisation brings about a degree of delocalisation of electron density into the internuclear region - IE the charge delocalisation (leading to a decrease in KE) funds the redistribution of electron density away from the nucleus into the internuclear space (resulting in an increase in potential energy)

As the internuclear distance becomes smaller, this trend continues - the smaller distance facilitates enhanced tunneling of the electron from one potential well to the other, and so a decrease in the kinetic energy, but a rise in the potential energy. Overall though, the total energy is negative, with a positive gradient, IE value of $(dE/dR)$ is positive, so as R decreased, E also decreases. However this trend slows - as internuclear distance becomes smaller, tunneling this facilitates is no longer much, and so kinetic energy term decreases more slowly. In the case of $\ce{H2+}$ the minimum of kinetic energy term occurs at around $2r_0$. At this distance we can presume the potential well's are so close that no increase in tunneling can occur, and instead by reducing bond length, the volume over which the electron can "wander" is reduced, and so kinetic energy starts to increase again.

At this stage of bonding, the classical picture starts to kick back in again - the potential energy term, until now has been positive, and so disfavours bonding, now becomes negative. In the previous "stage" up until the KE term's minimum, electron density moves away from nucleus to the internuclear region. This now allows for orbital contraction - the electron's wavefunction now shrinks, and becomes more concentrated around the nuclei. This leads to an decrease in potential energy, a rather sharp decrease, taking $<V>$ term from positive to quite negative. At the same time, the volume over which the electron wanders is reduced, and so the kinetic energy term $<T>$ increases, and becomes positive - IE level of delocalisation is reduced. However the potential energy term wins out. This can be seen from the virial equations:

$<V>+2<T> = 0$

Which applies at equilibrium bond length. This can be rearranged to give:

$<T>/-<V> = 1/2$

Thus at the equilibrium bond distance, the kinetic energy term is 1/2 magnitude of potential energy term, leading to a binding energy: $E = <T> + <V> = -<T> = 1/2 <V>$

For interest the amount of electron density donated to the internuclear space is 16%, and so the typical view that the bonding interaction is due to this is shown to be wrong. Also note that the potential energy term has a minima at a much shorter R distance than $r_0$ and is still steeply decreasing at the equilibrium bond distance. This shows that the common fallacy that an equilibrium bond distance is reached due to an equilibrium between nuclear-nuclear repulsion and nuclear-electron attraction is wrong.

Also shown to be wrong is the classical view of covalent bonding - covalent bonding is purely a quantum phenomena - it cannot be explained by classical electrostatics + mechanics. This can be seen simply from considering the potential well created by 2 $\ce{H+}$ nuclei - the maxima of such a system is found at the halfway bond point, and so a classical electron will simply decay it's orbit until on top of one of the nuclei. It is therefore found that the only stable classical arrangement of charges is a singularity. Consider for example a lattice of positive and negative charges. A single perturbation from perfect lattice geometry would see the entire lattice collapse - such as system could only be metastable. Thus the stability of matter owes itself to quantum physics.

While this picture is undoubtedly the most simple system, it also should, and has, exemplified the characteristics of covalent bonding in all systems. Interestingly for some systems - Notably $\ce{F2}$ - this breaks down, due the presence of compact lone pairs. This is the phenomena of charge shift bonding, and there are some excellent answers over at:What is charge shift bonding?

Sometimes you may also come across the idea of covalent bonding as being the resonance energy of two different spin states: $\ce{A\uparrow B\downarrow <->A\downarrow B\uparrow}$

This is precisely the picture we have - the electrons tunnel from one potential well to the other, and we simply use their spin as "labels" so we can say this electron tunneled to this nuclei, and the other went the other way. Thus both have expanded the "volume" of their wavefunctions etc. as above.