Why the empty atom picture misunderstands quantum theory

The camera zooms in on the person’s arm to reveal the cells, then a cell nucleus. A DNA strand grows on the screen. The camera focuses on a single atom within the strand, dives into a frenetic cloud of rocketing particles, crosses it, and leaves us in oppressive darkness. An initially imperceptible tiny dot grows smoothly, revealing the atomic nucleus. The narrator lectures that the nucleus of an atom is tens of thousands of times smaller than the atom itself, and poetically concludes that we are made from emptiness.

How often have you seen such a scene or read something equivalent to it in popular science? I am sure plenty, if you are fans of this genre like me. However, the narrative is wrong. Atomic nuclei in a molecule are not tiny dots, and there are no empty spaces within the atom.

The empty atom picture is likely the most repeated mistake in popular science. It is unclear who created this myth, but it is sure that Carl Sagan, in his classic TV series Cosmos (1980), was crucial in popularising it. After wondering how small the nuclei are compared with the atom, Sagan concluded that

(M)ost of the mass of an atom is in its nucleus; the electrons are by comparison just clouds of moving fluff. Atoms are mainly empty space. Matter is composed chiefly of nothing.

I still remember how deeply these words spoke to me when I heard them as a kid in the early 1980s. Today, as a professional theoretical chemist, I know that Sagan’s statements failed to recognise some fundamental features of atoms and molecules.

Yet his reasoning is still influential. While preparing this essay, I ran a poll on Twitter asking whether people agreed with Sagan’s quote above. Of the 180 voters, 43 per cent answered that they mostly agreed, and 27 per cent fully agreed. Google ‘atoms empty space’, and you will find tens of essays, blog posts and YouTube videos concluding that atoms are 99.9 per cent empty space. To be fair, you will also find a reasonable share of articles debunking the idea.

Misconceptions feeding the idea of the empty atom can be dismantled by carefully interpreting quantum theory, which describes the physics of molecules, atoms and subatomic particles. According to quantum theory, the building blocks of matter – like electrons, nuclei and the molecules they form – can be portrayed either as waves or particles. Leave them to evolve by themselves without human interference, and they act like delocalised waves in the shape of continuous clouds. On the other hand, when we attempt to observe these systems, they appear to be localised particles, something like bullets in the classical realm. But accepting the quantum predictions that nuclei and electrons fill space as continuous clouds has a daring conceptual price: it implies that these particles do not vibrate, spin or orbit. They inhabit a motionless microcosmos where time only occasionally plays a role.

Most problems surrounding the description of the submolecular world come from frustrated attempts to reconcile conflicting pictures of waves and particles, leaving us with inconsistent chimeras such as particle-like nuclei surrounded by wave-like electrons. This image doesn’t capture quantum theory’s predictions. To compensate, our conceptual reconstruction of matter at the submolecular level should consistently describe how nuclei and electrons behave when not observed – like the proverbial sound of a tree falling in the forest without anyone around.

Here’s a primer on how to think of the fundamental components of matter: a molecule is a stable collection of nuclei and electrons. If the collection contains a single nucleus, it is called an atom. Electrons are elementary particles with no internal structure and a negative electric charge. On the other hand, each nucleus is a combined system composed of several protons and a roughly equal number of neutrons. Each proton and neutron is 1,836 times more massive than an electron. The proton has a positive charge of the same magnitude as an electron’s negative charge, while neutrons, as their name hints, have no electric charge. Usually, but not necessarily, the total number of protons in a molecule equals the number of electrons, making molecules electrically neutral.

The interior of the protons and neutrons is likely the most complex place in the Universe. I like to consider each of them a hot soup of three permanent elementary particles known as quarks boiling along inside, with an uncountable number of virtual quarks popping into existence and disappearing almost immediately. Other elementary particles called gluons hold the soup within a pot of 0.9 femtometres radius. (A femtometre, abbreviated fm, is a convenient scale that measures systems tens of thousands of times smaller than an atom. Corresponding to 10‑15 m, we must juxtapose 1 trillion femtometres to make one millimetre.)

Instead of localised bullets in empty space, matter delocalises into continuous quantum clouds

Particles with the same electric charge sign repel each other. So additional interactions are required to hold protons close-packed in the nucleus. These interactions arise from quark and antiquark pairs called pions that constantly spill out of each proton and neutron to be absorbed by another such particle nearby. The energy exchanged in this transfer is big enough to compensate for the electric repulsion between protons and, thus, bind together protons and neutrons, storing the immense energy that may be released in nuclear fission processes.

However, the extremely short lifetime of the pions limits how far protons and neutrons may be from each other, curbing the nucleus size to a 1 to 10 fm radius. Thus, from a particle perspective, the nucleus is tiny compared with an atom. A nitrogen nucleus, composed of seven protons and seven neutrons, has a radius of about 3 fm. In contrast, nitrogen’s atomic radius is 179,000 fm. At the scale of atoms and molecules, nuclei are no more than heavy, point-like positive charges without any apparent internal structure. So are the electrons: they are just light, point-like negative charges.

If atoms and molecules remained a collection of point-like particles, they would be mostly empty space. But at their size scale, they must be described by quantum theory. And this theory predicts that the wave-like picture predominates until a measurement disturbs it. Instead of localised bullets in empty space, matter delocalises into continuous quantum clouds.

Matter is fundamentally quantum. Molecules cannot be assembled under the rules of classical physics. The classical electrical interactions between nuclei and electrons are insufficient to build a stable molecule. Due to the electric attraction of charges of opposite signs, the negatively charged electrons would quickly spiral toward the positively charged nuclei and glue to them. The resulting combined particles with no net charge would fly apart, preventing any molecule from forming.

Two quantum properties avoid this bleak fate.

The first property arises from the Heisenberg uncertainty principle, which holds that a quantum particle cannot simultaneously be at a precise position and also have zero speed. This implies that an electron cannot glue to a nucleus because both particles would be in a well-defined place and at rest to each other – defying a central rule of the quantum world.

The second quantum property is the Pauli exclusion principle. The fundamental components of matter are split into two types, bosons and fermions. The gluons inside the proton are examples of bosons. We can have as many of them as we want, sharing the same position simultaneously. On the other hand, fermions – such as electrons, quarks, protons and neutrons – obey a much more restrictive rule named the Pauli exclusion principle: no two identical fermions can simultaneously occupy the same space and have the same spin (a quantum property analogous to a classical rotation of a particle about its axis).

In the quantum world, the wave function represents more than a mere lack of knowledge

With all those effects encoded into the Schrödinger equation, the master equation of quantum theory, it predicts that our point-like nuclei and electrons must, in fact, behave like waves. They delocalise in quantum clouds much bigger than their particle-picture size to satisfy the Heisenberg uncertainty principle, with electrons shaped into different clouds to satisfy the Pauli exclusion principle. The lighter the particles are, the bigger the delocalisation. Thus, a single electron cloud may spread over multiple nuclei, forming a chemical bond and stabilising the molecule.

Take an ammonia molecule, NH3, illustrated below. The small blue smudge in the middle is the nitrogen nucleus cloud, while the three green blobs are the proton (hydrogen nuclei) clouds. The 10 electrons of the ammonia molecule delocalise into the fat yellow cloud, tying the party together.

Figure 1: Electronic and nuclear quantum clouds in an ammonia molecule. The yellow cloud represents the 10 electrons in this molecule. The small blue cloud is the nitrogen nucleus, while the three green clouds indicate each hydrogen nucleus. Electronic points in front of the nuclei were made transparent so as not to hide the nuclear clouds. Technical details are explained in Toldo et al 2023. Courtesy the author

A particle-like nitrogen nucleus has a 3 fm radius. However, in the ammonia molecule, the nitrogen nucleus grows to a respectable 3,000 fm radius due to delocalisation. The delocalisation of the hydrogen nuclei is even more impressive. They grow from a radius of 0.9 fm when seen as particles to clouds of about 23,000 fm. But the electrons take the cake. Due to their tiny mass, they grow from particles much smaller than a nucleus into a cloud that defines the molecular volume.

Nuclei and electrons, however, are not atomic giants. If the nitrogen nucleus is measured (for instance, by throwing fast electrons against it and observing them bounce back), the nuclear cloud would immediately collapse into the initial 3 fm dot. The same is true for each electron.

Indeed, quantum theory prescribes a precise relationship between the wave and particle pictures. The clouds of the wave picture are mathematically described by a wave function, essentially an equation that attributes an intensity to every point in space and how these intensities change with time. The wave function is analogous to mathematical functions describing conventional sound or water waves, but with the peculiarity that it has an imaginary-number component, which is negative when squared.

The square of the wave function modulus (a mathematical operation that always yields positive numbers) gives the probability of finding the particle at each point in space if we attempt to observe it. The denser the cloud, the bigger the odds of observing the particle there. Thus, if we try to measure the point-like nitrogen nucleus, we are sure that it will be somewhere in the region of the delocalised nitrogen nucleus cloud, the blue smudge in the figure.

However, interpreting the quantum cloud as probability does not mean it is just a measure of a lack of knowledge about the system. If I left my keys in one of my jacket’s two pockets, but I am unsure which one, I may write a probability function with a 50 per cent value at each pocket and zero value at every other point of my office. This function obviously does not imply that my keys are delocalised over the two pockets. It just states my ignorance, which can be easily fixed by checking the jacket.

In the quantum world, the wave function represents more than a mere lack of knowledge. Delocalised systems – like nuclear and electronic clouds – cause phenomena that localised particles cannot explain. The existence of chemical bonds forming molecules is a direct example of the effect of electronic delocalisation. In the case of nuclear delocalisation, one of its main effects is to boost the chances of a hydrogen nucleus (a single proton) flowing from one molecule to another nearby. This kind of enhanced proton transfer has dramatic biological consequences, like increasing the acidity of specific enzymes compared with how acidic they would be if hydrogen nuclei behaved as particles.

Although electron clouds are commonly depicted in popular science and chemistry, delocalisation of the nucleus is often interpreted as vibrations and rotations. But these are only classical, albeit helpful, analogies. From a quantum perspective and for conceptual consistency, nuclei should be depicted on the same footing as electrons, as clouds as well.

Yet another misconception is that atoms are empty because their mass is in their nucleus. The atomic mass is indeed highly localised. In an ammonia molecule, 82 per cent of the mass is in the blue smudge of the nitrogen nucleus shown in Figure 1 above. If we add the masses of the three green proton clouds, they account for 99.97 per cent of the total. Thus, the big yellow cloud of the electrons carries only 0.03 per cent of the mass.

The association between this mass concentration and the idea that atoms are empty stems from a flawed view that mass is the property of matter that fills a space. However, this concept does not hold up to close inspection, not even in our human-scale world. When we pile objects on top of each other, what keeps them separated is not their masses but the electric repulsion between the outmost electrons at their touching molecules. (The electrons cannot collapse under pressure due to the Heisenberg uncertainty and Pauli exclusion principles.) Therefore, the electron’s electric charge ultimately fills the space.

Anyone taking Chemistry 101 is likely to be faced with diagrams of electrons orbiting in shells

In atoms and molecules, electrons are everywhere! Look how the yellow cloud permeates the entire molecular volume in Figure 1. Thus, when we see that atoms and molecules are packed with electrons, the only reasonable conclusion is that they are filled with matter, not the opposite.

Despite all this, anyone taking Chemistry 101 is likely to be faced with diagrams of electrons orbiting in shells, like concentric and separated layers with empty space between them. The idea that these diagrams represent physical reality is a third common misconception. Electrons do not literally orbit around the atomic nucleus in the shape of these shells.

In atoms and molecules, electrons must have specific energies, each energy associated with a particular cloud shape. Consider, for example, an atom with a single electron. In the lowest possible energy, the ground energy level, this electron delocalises into a spherical cloud, dense at the centre of the atom and gradually fading out. The single-electron wave functions describing these clouds are called orbitals.

At higher energy levels, the single electron delocalises into more complex clouds with nested spheres, multiple blobs or even doughnut shapes. Thus, when speaking of atoms and molecules, electrons are not little particles chaotically rocketing around the nuclei until they become a fuzzy cloud, as often depicted. And electrons are not in the orbitals, nor do they populate them. Electrons are the orbitals. They are delocalised clouds.

With multiple electrons, which have been terra incognita in popular science, things get much more complicated. This is hardly a surprise since even professional theoretical chemists are uncomfortable describing them, despite their exceptional competence in predicting the properties of multi-electron systems.

Like ill-fitting clothes, chemistry vernacular is filled with awkward analogies and descriptions. Chemists may say that an electron occupies or populates an orbital as if orbitals were pre-existing places where electrons are put. Chemists often draw diagrams where orbitals are represented as short horizontal lines and electrons as small vertical arrows on those lines, like objects on shelves. All these verbal and visual metaphors fail to translate what quantum theory tells us about atoms and molecules.

When dealing with multi-electron systems (encompassing virtually all molecules), quantum theory no longer distinguishes between each electron; they are all described by a single wave function, a single cloud. Nevertheless, single electron orbitals are still a valid approximation that chemists constantly use to rationalise chemical reactions. The multi-electron wave function resembles a composition of these individual clouds overlapping within the volume defining the molecule. They feel each other; they recombine into new shapes; some bulge and others shrink; the clouds skew, stretch and twist until they comfortably adapt, occupying every available space. It may look like a messy sock drawer.

For a fraction of a picosecond, the tempest rages and reshapes the molecular landscape until stillness is restored

A molecule is a static object without any internal motion. The quantum clouds of all nuclei and electrons remain absolutely still for a molecule with a well-defined energy. Time is irrelevant. Quantum theory does not predict vibrating nuclei or orbiting and spinning electrons; those dynamic features are classical analogues to intrinsic quantum properties. Angular momentum, for instance, which in classical physics quantifies rotational speed, manifests as blobs in the wave function. The more numerous the blobs, the bigger the angular momentum, even though nothing rotates.

Time, however, comes into play when a molecule collides with another one, triggering a chemical reaction. Then, a storm strikes. The quantum steadiness bursts when the sections of the electronic cloud pour from one molecule upon another. The clouds mix, reshape, merge, and split. The nuclear clouds rearrange to accommodate themselves within the new electronic configuration, sometimes even migrating between molecules. For a fraction of a picosecond (10-12 seconds or a billionth of a millisecond), the tempest rages and reshapes the molecular landscape until stillness is restored in the newly formed compounds.

In the Flammarion engraving (Figure 2 below), a person at the edge of Earth dares to look beyond the firmament dome to uncover the marvellous machinery of clouds controlling the heavens. They could well be looking at a molecule instead. Then, this non-disturbing observer would find that nuclei and electrons are majestic, stable, structured, closed-packed clouds, driving every aspect of matter as we know it.

Figure 2: Wood engraving from L’atmosphère: météorologie populaire (1888) by Camille Flammarion. Courtesy Wikipedia

My criticism of the empty atom picture isn’t meant to shame people’s previous attempts to describe atoms and molecules to the public. On the contrary, I applaud their effort in this challenging enterprise. Our common language, intuitions and even basic reasoning processes are not adapted to face quantum theory, this alien world of strangeness surrounded by quirky landscapes we mostly cannot make sense of.

And there is so much we do not understand. We have yet to learn how to reconcile the dual wave-like and particle-like behaviour of matter. We do not even know whether wave functions have objective reality. Our brains melt, facing the multiple potential interpretations of quantum theory to the point that outstanding scientists seemingly gave up hope that we may reach a scientific consensus. We turn a blind eye to the dirty tricks we carry from the conceptual construction of quantum theory to the actual predictions.

The account of the quantum molecular world I presented is on comfortably safe grounds

We could conform to the unsatisfying ‘Shut up and calculate!’ attitude that has accompanied the increasingly weird predictions of quantum theory, which enabled the outstanding technological advancements of the past 100 years, from lasers to microprocessors. However, we do not want to make only useful predictions. Our ultimate goal is to tell stories about our Universe. Thus, we calculate but do not shut up. Generations of scientists and science popularisers do their best to translate all this strangeness into friendly metaphors of a theoretical body still full of mystery. We build new mental images of the quantum world one step at a time, even under the risk of tripping up here and there.

The account of the quantum molecular world I presented is on comfortably safe grounds. It is based on a quantum theory domain that is highly consensual among specialists. It is the town square of what the Nobel laureate Frank Wilczek called the Core Theory, the physics framework describing fundamental particles, their interactions and Albert Einstein’s general relativity. Physicists are so confident about this core’s stability that they believe it should persist within any new theories of matter developed in the future.

Breathing this confidence and realising we are not made of empty space may be a soothing thought.

This Essay was made possible through the support of a grant to Aeon+Psyche from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the Foundation. Funders to Aeon+Psyche are not involved in editorial decision-making.

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