“Orbital shell” redirects here. For the collection of spaceflight orbits, see Orbital shell (spaceflight) .

The shapes of the first five atomic orbitals are 1s, 2s, 2p x , 2p y , and 2p z . The two colors show the phase or sign of the wave function in each region. Each picture is domain coloring of a ψ( x ,  y ,  z ) function which depends on the coordinates of one electron. To see the elongated shape of ψ( x ,  y ,  z ) 2 functions that show probability density more directly, see pictures of d-orbitals below.

In the grand, often frustrating, theater of quantum mechanics , an atomic orbital (pronounced /ˈɔːrbɪtəl/ — if you must make noise ⓘ ) isn’t some quaint little planetary path. No, it’s a sophisticated mathematical function that attempts to describe the probable whereabouts and inherent wave-like behavior of an electron as it exists within the confines of an atom . Think of it as a cosmic whisper, detailing where an electron might be, rather than a definitive address. This function, in its elegant complexity, sketches out an electron’s charge distribution around the atom’s nucleus , granting mere mortals the ability to calculate the elusive probability of encountering an electron in any given volumetric sliver of space surrounding that nucleus. It’s less “here it is” and more “it’s probably in this general vicinity, but don’t ask for specifics.”

Each distinct orbital within an atom is, rather meticulously, defined by a specific set of three fundamental quantum numbers : n, ℓ, and mℓ. These aren’t just arbitrary labels; they correspond to deeply intrinsic properties of the electron. Specifically, n dictates the electron’s energy level, ℓ defines its orbital angular momentum —a measure of its rotational motion—and mℓ quantifies the projection of that orbital angular momentum along a predetermined axis, often referred to as the magnetic quantum number . Orbitals possessing a precisely defined magnetic quantum number are, more often than not, expressed as complex-valued mathematical entities. However, for the sake of human comprehension (and perhaps sanity), real-valued orbitals can be constructed by forming linear combinations of these complex mℓ and −mℓ orbitals. These simplified, real-valued orbitals are frequently given labels derived from their associated harmonic polynomials (e.g., xy, x² − y²), which elegantly capture their characteristic angular structure and spatial orientation.

A single orbital, despite its vastness in concept, has a rather strict occupancy limit: a maximum of two electrons. Each of these electrons must possess its own unique projection of spin , denoted as m_s. This adherence to the Pauli exclusion principle is fundamental. The familiar and rather unimaginative names – s orbital, p orbital, d orbital, and f orbital – are simply shorthand for orbitals characterized by an angular momentum quantum number ℓ equal to 0, 1, 2, and 3, respectively. These designations, paired with their principal quantum number (n) values, form the bedrock for describing the intricate electron configurations of atoms. Their origins, surprisingly, trace back to the early days of spectroscopy, where pioneering scientists, observing the distinct series of alkali metal spectroscopic lines , whimsically labeled them as sharp , principal , diffuse , and fundamental . As if the universe needed more complexity, orbitals for ℓ > 3 continue this alphabetical progression (g, h, i, k, …), with a curious omission of the letter ‘j’ [^3] [^4] [^5] – apparently, some languages struggle to differentiate between ‘i’ and ‘j’, and the scientific community, in its infinite wisdom, decided to accommodate this linguistic quirk. [^6]

Atomic orbitals serve as the fundamental conceptual units within the atomic orbital model (also known as the electron cloud or wave mechanics model), providing a modern and remarkably effective framework for visualizing the otherwise elusive submicroscopic behavior of electrons within matter. In this model, the electron cloud that envelops an atom can be mentally constructed (as an approximation, of course) from an electron configuration that is, in essence, a mathematical product of simpler, hydrogen-like atomic orbitals. The repeating patterns and inherent periodicity observed in the blocks of 2, 6, 10, and 14 elements across the various sections of the periodic table emerge quite naturally from the maximum number of electrons that can populate a complete set of s, p, d, and f orbitals, respectively. However, for higher values of the principal quantum number n, particularly when an atom bears a positive charge, the energy levels of certain sub-shells can become remarkably similar. This similarity means that the conventional order in which electrons are said to populate these orbitals (for example, chromium as [Ar]4s¹3d⁵ versus Cr²⁺ as [Ar]3d⁴) can only be rationalized with a degree of arbitrariness, a testament to the subtle complexities that even our best models struggle to fully capture with simple rules.

Cross-sections of atomic orbitals of the electron in a hydrogen atom at different energy levels. The probability of finding the electron is given by the color, as shown in the key at upper right.

Electron properties

With the revolutionary advent of quantum mechanics and a deluge of compelling experimental findings (such as the rather inconvenient, yet undeniable, two-slit diffraction of electrons), it became starkly apparent that electrons, those supposedly simple particles orbiting a nucleus, could not be adequately described as mere point-like entities. Their behavior stubbornly demanded an explanation rooted in wave–particle duality . In this rather unsettling quantum landscape, electrons exhibit a peculiar blend of wave-like and particle-like characteristics:

Wave-like properties:

• Electrons, it turns out, are not tiny planets dutifully tracing elliptical paths around a star-like nucleus. Instead, they manifest as ethereal standing waves . Consequently, the lowest possible energy state an electron can occupy is analogous to the fundamental frequency of a vibrating string. Higher energy states, in a similar vein, correspond to the harmonics of that fundamental frequency, each a more complex vibrational pattern.
• The notion of an electron being confined to a single, precise point in space is, frankly, a quaint relic of classical physics. While the probability of interacting with an electron at a specific point can be derived from its wave function , the electron itself is never truly at a single point. Instead, its charge behaves as if it’s continuously smeared out across space, its distribution at any given point directly proportional to the squared magnitude of its wave function. It’s less a solid object and more a persistent hum.

Particle-like properties:

• Despite their wave-like diffusion, the number of electrons orbiting a nucleus remains stubbornly integer-based. You won’t find half an electron, no matter how much you wish for it.
• Electrons execute discrete “jumps” between orbitals, behaving, in these moments, much like particles undergoing a sudden relocation. For instance, if a single photon impinges upon a collection of electrons, only one electron will undergo a change in its energy state as a direct result. A single, decisive action.
• Electrons steadfastly retain certain particle-like attributes. Each wave state, for example, carries the exact same electric charge as an individual electron particle. Furthermore, each wave state possesses a distinct, discrete spin (either “spin up” or “spin down”), though it can exist in a superposition of these states until measured.

Therefore, the simplistic notion of electrons as mere solid particles is, to put it mildly, insufficient. A more apt, if still imperfect, analogy might be to envision a vast, often strangely sculpted “atmosphere” (representing the electron) enveloping a comparatively minuscule planet (the nucleus). However, it’s crucial to remember that atomic orbitals precisely describe the shape of this “atmosphere” only when a single electron is present. When additional electrons are introduced, their mutual repulsions and interactions cause them to distribute themselves more evenly throughout the space around the nucleus. The resulting collective “electron cloud” [^7] then tends towards a more generally spherical zone of probability, describing the aggregate likelihood of finding an electron, a phenomenon partly governed by the inherent limitations imposed by the uncertainty principle .

One should, of course, always bear in mind that these orbital ‘states’, as meticulously described here, are merely eigenstates of an electron operating within its orbital. A real electron, in its natural, unobserved state, exists in a superposition of these states. This is not simply a weighted average , mind you, but involves complex number weights, adding a layer of mathematical elegance that most people conveniently ignore. For instance, an electron might be found in a pure eigenstate like (2, 1, 0), or it could be in a more… ambiguous mixed state, such as ⁠1/2⁠(2, 1, 0) + • ⁠1/2⁠

i

{\displaystyle i}

(2, 1, 1), or even the less aesthetically pleasing mixed state • ⁠2/5⁠(2, 1, 0) + • ⁠3/5⁠

i

{\displaystyle i}

(2, 1, 1). For each eigenstate , a specific property will yield a definite eigenvalue . Thus, for the three states just mentioned, the principal quantum number (n) is unequivocally 2, and the azimuthal quantum number (ℓ) is definitively 1. However, for the second and third states, the magnetic quantum number (mℓ) exists as a superposition of 0 and 1. As a superposition, its value is inherently ambiguous – it is either exactly 0 or exactly 1, never some convenient intermediate or average fraction like • ⁠1/2⁠. A superposition encompassing eigenstates such as (2, 1, 1) and (3, 2, 1) would similarly render both n and ℓ ambiguous, yet mℓ would, with a touch of cosmic irony, remain definitively 1. Eigenstates are, in essence, a mathematical convenience, simplifying the otherwise daunting calculations. One can always opt for a different basis of eigenstates by superimposing eigenstates from any other valid basis (a concept we’ll touch upon later when discussing Real orbitals ).

Formal quantum mechanical definition

Atomic orbitals can be articulated with greater precision using the rigorous language of formal quantum mechanics . They are, in essence, approximate solutions to the formidable Schrödinger equation for electrons that are bound within an atom, held captive by the powerful electric field emanating from the atom’s nucleus . More specifically, within the framework of quantum mechanics, the precise state of an atom – which is to say, an eigenstate of the atomic Hamiltonian – is typically approximated by an expansion (see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products, famously known as Slater determinants , which are composed of one-electron functions. The spatial components of these individual one-electron functions are precisely what we refer to as atomic orbitals . (When one also includes their spin component, the more precise term is atomic spin orbitals.) While an atom’s true state is a function of the coordinates of all its electrons, implying a complex, correlated motion, this is frequently approximated by an independent-particle model that relies on products of single electron wave functions. [^8] (It’s worth noting that phenomena like the London dispersion force , for example, are a direct consequence of these subtle correlations in electron motion, proving that approximations, while useful, always leave something out.)

In the specialized realm of atomic physics , the distinct atomic spectral lines that we observe correspond to precise transitions, often dramatically referred to as quantum leaps , between the various quantum states of an atom. These states are meticulously categorized and labeled by a specific set of quantum numbers , neatly summarized in a term symbol , and are typically associated with particular electron configurations – essentially, the schemes by which electrons occupy atomic orbitals (for instance, 1s² 2s² 2p⁶ for the ground state of neon , with its term symbol: ¹S₀).

This particular notation implies that the corresponding Slater determinants carry a significantly higher weight within the configuration interaction expansion. Consequently, the concept of the atomic orbital becomes an absolutely pivotal tool for visualizing the excitation process that underpins a given transition . One might, for example, assert that a specific transition corresponds to the excitation of an electron from an occupied orbital to a vacant one. However, it’s crucial to retain the fundamental understanding that electrons are fermions , rigidly governed by the Pauli exclusion principle , and are, by their very nature, indistinguishable from one another. [^9] Furthermore, it occasionally happens that the configuration interaction expansion converges at an excruciatingly slow pace, making it impossible to accurately describe the system with a simple one-determinant wave function. This situation arises precisely when electron correlation is particularly significant, reminding us that the universe rarely conforms to our desire for simplicity.

Fundamentally, an atomic orbital is conceived as a one-electron wave function, a convenient simplification, even though most atoms are teeming with multiple electrons, not just one. Thus, this one-electron perspective is inherently an approximation. When we conjure images of orbitals in our minds, we are often presented with visualizations heavily influenced by the Hartree–Fock approximation – a mathematical sleight of hand designed to reduce the staggering complexities inherent in molecular orbital theory to something vaguely manageable.

Types of orbital

3D views of some hydrogen-like atomic orbitals showing probability density and phase ( g orbitals and higher not shown)

Atomic orbitals can manifest in various forms, depending on the context and the level of approximation required. The simplest and most analytically precise are the hydrogen-like “orbitals,” which represent the exact solutions to the Schrödinger equation for a truly hydrogen-like “atom” – that is, an atom possessing only a single electron. Such a system could be the lone hydrogen atom itself, or any other element ionized to the point where only one electron remains (think He⁺, Li²⁺, and so on). These systems behave remarkably similarly to hydrogen, and their orbitals adopt the same fundamental mathematical form.

Alternatively, the term “atomic orbital ” can also refer to functions that, while still dependent on the coordinates of a single electron, serve as crucial starting points for approximating the more complex wave functions that describe the simultaneous coordinates of all electrons within a multi-electron atom or molecule. The choice of coordinate systems for these orbitals is typically spherical coordinates (r, θ, φ) when dealing with individual atoms, and Cartesian (x, y, z) for the more intricate geometries of polyatomic molecules. The inherent advantage of spherical coordinates in atomic contexts is that an orbital wave function can be elegantly expressed as a product of three independent factors, each dependent on only one coordinate: ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ). The angular factors of these atomic orbitals , Θ(θ)Φ(φ), are responsible for generating the characteristic s, p, d, etc., functions, often represented as real combinations of spherical harmonics Yℓm(θ, φ) (where ℓ and m are the familiar quantum numbers).

There are, generally speaking, three primary mathematical forms for the radial functions R(r), which researchers can select as a foundation for calculating the properties of atoms and molecules teeming with multiple electrons:

• The hydrogen-like orbitals : These are, as mentioned, derived directly from the exact solutions of the Schrödinger equation for a single electron interacting with a nucleus. The portion of this function that describes the electron’s behavior as a function of its distance r from the nucleus is characterized by distinct radial nodes and exhibits an exponential decay proportional to e⁻ᵃʳ.
• The Slater-type orbital (STO): This form, while lacking the radial nodes characteristic of hydrogen-like orbitals, still accurately captures the exponential decay from the nucleus, mimicking the behavior of its hydrogenic counterparts.
• The Gaussian type orbital (Gaussians): These functions also forego radial nodes but exhibit a more rapid, Gaussian-like decay, proportional to e⁻ᵃʳ².

While hydrogen-like orbitals continue to serve as invaluable pedagogical tools for teaching fundamental concepts (they are, after all, elegantly simple), the relentless march of computational power has rendered Slater-type orbitals (STOs) more practical for detailed calculations involving atoms and diatomic molecules. This is because sophisticated combinations of STOs can effectively reproduce the nodal structures found in hydrogen-like orbitals. Gaussian orbitals , on the other hand, are the workhorses for molecules comprising three or more atoms. Though a single Gaussian orbital might not be as intrinsically accurate as an STO, the sheer power of combining numerous Gaussians allows for the attainment of accuracy comparable to, or even exceeding, that of hydrogen-like orbitals. It’s a testament to brute-force computation, really.

History

• Main article: Atomic theory

The term “orbital” itself, a rather concise descriptor, was coined by the illustrious Robert S. Mulliken in 1932. It served as a handy abbreviation for the more cumbersome “one-electron orbital wave function.” [^10] [^11] Prior to this, in the nascent days of quantum understanding, Niels Bohr had, around 1913, put forth the then-revolutionary idea that electrons might indeed revolve around a compact nucleus, albeit with specific, quantized values of angular momentum. [^12] Bohr’s model was a significant leap forward, building upon the earlier, more rudimentary explanations provided by Ernest Rutherford in 1911, which simply posited electrons moving around a nucleus. Even earlier, the Japanese physicist Hantaro Nagaoka had, as far back as 1904, published an orbit-based hypothesis concerning electron behavior. [^13] These theories, like many scientific endeavors, were incrementally constructed upon fresh observations, beginning with simplistic notions and gradually evolving into more accurate and intricate understandings. The relentless quest to explain the enigmatic behavior of these electron “orbits” became, in essence, one of the primary catalysts driving the entire development of quantum mechanics . [^14]

Early models

With J. J. Thomson ’s groundbreaking discovery of the electron in 1897 [^15], it became glaringly obvious that atoms were not, in fact, the smallest building blocks of nature , as had been previously assumed. Instead, they were revealed to be complex, composite particles, possessing an internal structure. This newly unearthed complexity within atoms naturally spurred many to ponder how these constituent parts might interact with one another. Thomson, in his conceptual zeal, theorized that multiple electrons might revolve in orbit-like rings, embedded within a positively charged, rather amorphous, jelly-like substance [^16]. From the electron’s discovery until approximately 1909, this rather charming, if ultimately flawed, “plum pudding model ” stood as the most widely accepted, albeit short-lived, explanation of atomic structure.

Not long after Thomson’s revelation, Hantaro Nagaoka ventured to predict a distinctly different model for electronic structure [^13]. Diverging sharply from the prevailing plum pudding hypothesis, Nagaoka’s “Saturnian Model” proposed that the positive charge was, in fact, concentrated into a dense, central core. This core, he suggested, gravitationally pulled the electrons into circular orbits, evoking a striking resemblance to the majestic rings of Saturn. At the time, Nagaoka’s work garnered precious little attention [^17]. Even Nagaoka himself, with a prescient understanding, recognized a fundamental flaw inherent in his theory from its very inception: a classical charged object, by the immutable laws of electromagnetism, cannot sustain orbital motion indefinitely, as it would constantly accelerate and thus continuously lose energy through electromagnetic radiation [^18]. Nevertheless, with the benefit of hindsight, the Saturnian model proved to possess more conceptual commonality with modern atomic theory than any of its contemporary rivals.

Bohr atom

In 1909, Ernest Rutherford delivered a scientific bombshell, revealing that the vast majority of an atom’s atomic mass was not diffusely spread, but rather tightly condensed into a minuscule, positively charged nucleus. His subsequent analysis in 1911 made it unequivocally clear that the “plum pudding model ” was utterly incapable of explaining this newfound atomic structure. Two years later, in 1913, Rutherford’s brilliant post-doctoral student, Niels Bohr , proposed a revolutionary new model of the atom. Bohr’s model posited that electrons orbited the nucleus with classical periods, but, crucially, were permitted to possess only discrete, specific values of angular momentum, which he famously quantized in units of ħ [^12]. This elegant constraint automatically restricted electrons to only certain allowable energy levels. The Bohr model of the atom ingeniously sidestepped the thorny issue of energy loss due to radiation from a ground state (by simply declaring that no state existed below it) and, perhaps more significantly, provided the first coherent explanation for the mysterious origin of spectral lines .

The Rutherford–Bohr model of the hydrogen atom, particularly after Bohr’s astute application of Einstein ’s explanation of the photoelectric effect to link atomic energy levels with the precise wavelength of emitted light, solidified the connection between the electronic structure of atoms and their observed emission and absorption spectra . This connection rapidly evolved into an increasingly indispensable tool for understanding the behavior of electrons within atoms. The most striking characteristic of these emission and absorption spectra (a phenomenon known experimentally since the mid-19th century) was their composition of distinct, discrete lines. The profound significance of the Bohr model lay in its ability to directly correlate these spectral lines with the specific energy differences between the allowed orbits that electrons could occupy around an atom. This groundbreaking achievement, however, was not attained by endowing electrons with wave-like properties; the radical idea that electrons could behave as matter waves would not be proposed until a full eleven years later. Nevertheless, the Bohr model’s pioneering use of quantized angular momenta, and by extension, quantized energy levels, represented an monumental leap forward in our comprehension of electrons in atoms. It also served as a critical precursor to the full development of quantum mechanics , powerfully suggesting that quantized restraints were absolutely essential to account for all discontinuous energy levels and discrete spectra observed in atoms.

With Louis de Broglie ’s audacious suggestion in 1924 regarding the existence of electron matter waves , and for a brief, glorious period preceding the comprehensive 1926 Schrödinger equation treatment of hydrogen-like atoms , a Bohr electron’s “wavelength” could be conceptualized as a direct function of its momentum. Thus, a Bohr orbiting electron was envisioned as tracing a circular path where its circumference was a precise multiple of its half-wavelength. For a fleeting moment, the Bohr model could be seen as a classical model, subtly enhanced by an additional constraint derived from this “wavelength” argument. However, this period of semi-classical grace was almost immediately superseded by the full, three-dimensional wave mechanics that emerged in 1926. In our contemporary understanding of physics, the Bohr model is now aptly categorized as a semi-classical model, primarily due to its pivotal quantization of angular momentum, rather than its retrospective association with electron wavelength, which, with the benefit of hindsight, appeared a full dozen years after Bohr initially proposed his model.

The Bohr model, for all its brilliance, was adept at explaining the emission and absorption spectra of hydrogen alone. The energy values for electrons in the n = 1, 2, 3, etc. states, as predicted by the Bohr model, remarkably align with those derived from modern physics. However, its limitations became apparent when attempting to explain the striking similarities observed between different atoms, particularly those expressed in the periodic table. For instance, it failed to account for the similar chemical inertness displayed by helium (two electrons), neon (10 electrons), and argon (18 electrons). Modern quantum mechanics elegantly resolves this by introducing the concepts of electron shells and subshells, each capable of accommodating a specific number of electrons determined by the immutable Pauli exclusion principle . Thus, the n = 1 state can house either one or two electrons, while the n = 2 state can accommodate up to eight electrons, distributed across its 2s and 2p subshells. In helium, all n = 1 states are fully occupied, leading to its inertness. The same holds true for both n = 1 and n = 2 states in neon. In argon, the 3s and 3p subshells are similarly fully occupied by eight electrons. While quantum mechanics permits a 3d subshell, in argon, this subshell resides at a higher energy level than the 3s and 3p (a stark contrast to the situation in hydrogen) and consequently remains empty, further illustrating the nuanced complexity of electron behavior in multi-electron atoms.

Modern conceptions and connections to the Heisenberg uncertainty principle

Immediately following Werner Heisenberg ’s earth-shattering discovery of his uncertainty principle [^19], Niels Bohr astutely observed that the very existence of any discernible wave packet inherently implies an uncertainty in both its wave frequency and wavelength. This is because a spread, or superposition, of frequencies is fundamentally required to construct such a localized packet in the first place [^20]. In the realm of quantum mechanics , where the momentum of every particle is inextricably linked to a corresponding wave, it is precisely the formation of such a wave packet that serves to localize the wave, and by extension, the particle, within space. In quantum mechanical states where a particle is bound – that is, confined within a potential well – it must exist as a localized wave packet. The very presence of this packet, and its inherent minimum spatial extent, necessitates a corresponding spread and minimal value in the particle’s wavelength, which, by extension, dictates a spread in its momentum and energy. In simpler terms, as a quantum particle is forced into an ever-smaller region of space, the associated wave packet becomes increasingly compressed, which in turn demands a progressively larger range of momenta, and consequently, a greater kinetic energy. This implies that the binding energy required to contain or trap a particle within a shrinking spatial region increases without any upper bound. Therefore, particles cannot, under any circumstances, be confined to a geometric point in space, as this would necessitate an infinite amount of particle momentum – a theoretical impossibility.

In the practical domain of chemistry, luminaries such as Erwin Schrödinger , Linus Pauling , Mulliken, and others quickly grasped the profound implication of Heisenberg’s relation: the electron, understood as a wave packet, could no longer be considered to possess an exact, precisely defined location within its orbital. Max Born then stepped forward with the crucial insight that the electron’s position demanded description not by a single point, but by a probability distribution . This distribution was intrinsically linked to the likelihood of finding the electron at any given point within the wave-function that characterized its associated wave packet. The new quantum mechanics, it became clear, did not offer deterministic, exact results, but rather provided only the probabilities for the occurrence of a spectrum of possible outcomes. Heisenberg himself maintained that the very concept of a moving particle’s path holds no meaningful significance if we are fundamentally unable to observe it – a predicament precisely true for electrons within the confines of an atom.

Orbital names

Orbital notation and subshells

Orbitals, in their infinite variety, have been bestowed with names, typically presented in a format that’s meant to be descriptive, if a bit dry:

X

t y p e

{\displaystyle X,\mathrm {type} \ }

Here, X is a numerical value representing the energy level, which directly corresponds to the principal quantum number n. The “type” is a lowercase letter, a cryptic symbol denoting the specific shape or subshell of the orbital, which in turn corresponds to the angular momentum quantum number ℓ.

For instance, the designation “1s” (pronounced, with appropriate reverence, as “one ess”) signifies the absolute lowest energy level available (where n = 1) and an angular quantum number of ℓ = 0, which is universally denoted by the letter ’s’. Orbitals with ℓ = 1, 2, and 3 are, in a similar fashion, denoted as p, d, and f, respectively. One might wonder about the origin of these letters; they’re a remnant from early spectroscopy, where observed spectral lines were categorized as ‘sharp’, ‘principal’, ‘diffuse’, and ‘fundamental’. A charmingly archaic legacy, wouldn’t you say?

A complete collection of orbitals sharing a common n and ℓ value is collectively termed a subshell. This is often represented as:

X

t y p e

y

{\displaystyle X,\mathrm {type} ^{y}\ }

.

In this notation, the superscript y indicates the precise number of electrons that currently reside within that subshell. For example, the notation 2p⁴ tells us that the 2p subshell of a particular atom is home to 4 electrons. This specific subshell, by its very nature, comprises 3 distinct orbitals, each characterized by n = 2 and ℓ = 1. Simple, if you ignore the quantum weirdness.

X-ray notation

• Main article: X-ray notation

There exists yet another, less frequently encountered, system of notation, which stubbornly persists within the domain of X-ray science: the X-ray notation . This system is a direct relic, a continuation of the nomenclature employed before the intricate tapestry of orbital theory was fully understood. In this archaic system, the principal quantum number is not given a numerical value but rather assigned a letter. For n = 1, 2, 3, 4, 5, and so on, the corresponding letters are K, L, M, N, O, respectively. It’s a bit like using Roman numerals when Arabic ones are perfectly adequate, but tradition, I suppose, holds a certain sway, even over rationality.

Hydrogen-like orbitals

• Main article: Hydrogen-like atom

The most straightforward atomic orbitals – and thus, the easiest for us to calculate with any degree of certainty – are those derived for systems containing a single electron. The quintessential example, of course, is the hydrogen atom itself. Any atom of another element that has been ionized down to a solitary electron (such as He⁺, Li²⁺, and so on) behaves remarkably similarly to hydrogen, and consequently, its orbitals adopt the same fundamental mathematical form. In the Schrödinger equation tailored for this elegantly simple system of one negative and one positive particle, the atomic orbitals emerge as the eigenstates of the Hamiltonian operator for the energy. These solutions can be obtained analytically, meaning they can be expressed precisely through mathematical formulas. The resulting orbitals are therefore products of a polynomial series, along with exponential and trigonometric functions (a rather beautiful mathematical dance, as detailed in the treatment of the hydrogen atom ).

For atoms burdened with two or more electrons, the governing equations become significantly more complex, defying direct analytical solution. In these multi-electron systems, the equations can only be tackled through the laborious application of iterative approximation methods. While the orbitals of these multi-electron atoms are qualitatively similar to their hydrogenic counterparts, adopting roughly the same form in the simplest models, truly rigorous and precise analysis demands the use of sophisticated numerical approximations. It seems the universe prefers to keep its secrets under lock and key, only yielding to computational brute force.

Each specific (and idealized) hydrogen-like atomic orbital is uniquely identified by a distinct combination of three quantum numbers : n , ℓ , and mℓ. The strict rules governing the permissible values of these quantum numbers, along with their corresponding energy levels (which we’ll delve into shortly), are what ultimately explain the intricate electron configuration of atoms and, by extension, the entire structure of the periodic table .

It’s important to clarify that the stationary states – the quantum states – of a hydrogen-like atom are precisely its atomic orbitals . However, in the broader, more complex reality, an electron’s behavior is rarely, if ever, fully encapsulated by a single, isolated orbital. More accurately, electron states are best represented as time-dependent “mixtures” or linear combinations of multiple orbitals. This concept is central to approaches like the Linear combination of atomic orbitals molecular orbital method , which attempts to paint a more complete, if still approximate, picture.

The principal quantum number n first made its appearance in the rather charmingly simplistic Bohr model , where it defined the radius of each circular electron orbit. In the more sophisticated framework of modern quantum mechanics , however, n takes on a slightly different, yet equally crucial, role: it determines the mean distance of the electron from the nucleus. Consequently, all electrons sharing the same value of n are said to reside at the same average distance from the nucleus. For this reason, orbitals with identical n values are grouped together and said to constitute an “electron shell ”. Furthermore, orbitals that share both the same n value and the same ℓ value are considered even more intimately related and are thus said to form a “subshell ”. It’s a hierarchical system, much like any other attempt to categorize the universe’s inherent messiness.

Quantum numbers

• Main article: Quantum number

Due to the inherently quantum mechanical nature of electrons as they cavort around a nucleus, atomic orbitals can be uniquely and precisely defined by a specific set of integers, universally known as quantum numbers . These quantum numbers do not, however, occur in any arbitrary combination; they adhere to strict rules, and their physical interpretation can subtly shift depending on whether one employs real or complex mathematical versions of the atomic orbitals .

Complex orbitals

Energetic levels and sublevels of polyelectronic atoms

In the realm of physics, the most common descriptions of orbitals are rooted in the solutions derived for the elegant hydrogen atom , where orbitals are expressed as the product of a radial function and a pure spherical harmonic . The quantum numbers , along with the rather strict rules governing their permissible values, are as follows:

The principal quantum number n is the grand arbiter of an electron’s energy and is, without exception, a positive integer . In theory, it could be any positive integer, but in practice, you’ll rarely encounter truly enormous numbers for n, for reasons that become painfully obvious when you consider the scale of atomic energies. Each atom, generally speaking, possesses a multitude of orbitals associated with each value of n; these orbitals, when grouped, are sometimes referred to as electron shells .

The azimuthal quantum number ℓ, sometimes called the orbital angular momentum quantum number, describes the orbital angular momentum of each electron and is always a non-negative integer. Within a given shell where n holds a specific integer value, n₀, ℓ is permitted to range across all integer values that satisfy the condition:

0 ≤ ℓ ≤

n

0

− 1

{\displaystyle 0\leq \ell \leq n_{0}-1}

. For instance, the n = 1 shell is rather limited, containing only orbitals with

ℓ

0

{\displaystyle \ell =0}

. The n = 2 shell , however, is slightly more expansive, accommodating orbitals with both

ℓ

0

{\displaystyle \ell =0}

and

ℓ

1

{\displaystyle \ell =1}

. The collection of orbitals associated with a particular value of ℓ is, rather predictably, collectively designated as a subshell.

The magnetic quantum number , mℓ, provides insight into the projection of the orbital angular momentum along a specifically chosen axis. It dictates the magnitude of the “current” that circulates around this axis and, through the Ampèrian loop model, contributes to the overall magnetic moment of an electron [^21]. Within a given subshell, defined by ℓ, the magnetic quantum number mℓ can adopt any integer value within the range:

− ℓ ≤

m

ℓ

≤ ℓ

{\displaystyle -\ell \leq m_{\ell }\leq \ell }

.

These rather intricate results can be conveniently summarized in the following table, if you’re inclined to such neat categorizations. Each cell within the table represents a distinct subshell, meticulously listing the permissible values of mℓ available within that subshell. Any empty cells are, quite simply, placeholders for subshells that, in the grand scheme of things, do not exist.

Subshells are, with predictable regularity, identified by their n- and ℓ-values. The principal quantum number n is represented by its straightforward numerical value, but ℓ is, for historical reasons, represented by a letter as follows: 0 is denoted by ’s’, 1 by ‘p’, 2 by ’d’, 3 by ‘f’, and 4 by ‘g’. Thus, one might casually refer to the subshell with n = 2 and ℓ = 0 as a ‘2s subshell’. It’s all rather systematic, once you get past the initial arbitrary assignments.

To further complicate matters, each electron also possesses an intrinsic angular momentum, a purely quantum mechanical spin , denoted by s = • ⁠1/2⁠. The projection of this spin along a specified axis is given by the spin magnetic quantum number , m_s, which can only take one of two discrete values: + • ⁠1/2⁠ or − • ⁠1/2⁠. These values are, perhaps unimaginatively, referred to as “spin up” or “spin down,” respectively.

The immutable Pauli exclusion principle dictates that no two electrons within a single atom can ever possess the identical set of all four quantum numbers . This means that if two electrons happen to occupy the same orbital – sharing the same values for n, ℓ, and mℓ – they must differentiate themselves by having opposing spin projections, m_s. It’s the universe’s way of enforcing individuality, even at the subatomic level.

These conventions, you’ll notice, implicitly assume the existence of a preferred axis (for example, the z-direction in a Cartesian coordinate system ), and indeed, a preferred direction along this axis. Without such assumptions, distinguishing m = +1 from m = −1 would be an exercise in futility. Consequently, this model proves most useful when applied to physical systems that naturally exhibit these symmetries. The Stern–Gerlach experiment , wherein an atom is exposed to an external magnetic field, provides a classic, compelling example of such a system [^22].

Real orbitals

Animation of continuously varying superpositions between the p 1 and the p x orbitals. This animation does not use the Condon–Shortley phase convention.

Instead of wrestling with the often abstract complex orbitals just described, it’s quite common, particularly within the more practical domain of chemistry, to employ real atomic orbitals . These real orbitals are not some entirely different beast; rather, they emerge from straightforward linear combinations of their complex counterparts. When adhering to the venerable Condon–Shortley phase convention , real orbitals bear a direct relationship to complex orbitals, mirroring the way real spherical harmonics are derived from complex ones. Let ψ_n,ℓ,m denote a complex orbital characterized by the quantum numbers n, ℓ, and m. Then the real orbitals, ψ^real_n,ℓ,m, can be formally defined as [^23]:

$$ \psi_{n,\ell,m}^{\text{real}} = \begin{cases} \sqrt{2}(-1)^{m}{\text{Im}}\left{\psi_{n,\ell,|m|}\right} & {\text{ for }}m<0 \[2pt] \psi_{n,\ell,|m|} & {\text{ for }}m=0 \[2pt] \sqrt{2}(-1)^{m}{\text{Re}}\left{\psi_{n,\ell,|m|}\right} & {\text{ for }}m>0 \end{cases} $$

which can also be expressed as:

$$ = \begin{cases} {\frac{i}{\sqrt{2}}}\left(\psi_{n,\ell,-|m|}-(-1)^{m}\psi_{n,\ell,|m|}\right) & {\text{ for }}m<0 \[2pt] \psi_{n,\ell,|m|} & {\text{ for }}m=0 \[4pt] {\frac{1}{\sqrt{2}}}\left(\psi_{n,\ell,-|m|}+(-1)^{m}\psi_{n,\ell,|m|}\right) & {\text{ for }}m>0 \end{cases} $$

If the complex orbital ψ_n,ℓ,m(r, θ, φ) is given by R_nl(r)Y^m_ℓ(θ, φ), where R_nl(r) is the radial component of the orbital, then this definition is equivalent to:

$$ \psi_{n,\ell,m}^{\text{real}}(r,\theta,\phi) = R_{nl}(r)Y_{\ell m}(\theta,\phi) $$

where Y_ℓm represents the real spherical harmonic that is directly related to either the real or imaginary part of its complex counterpart, Y^m_ℓ.

Real spherical harmonics take on particular physical relevance when an atom finds itself embedded within a crystalline solid. In such environments, there are often multiple preferred symmetry axes, but no single, overarching preferred direction [^needed]. Consequently, real atomic orbitals are far more commonly encountered in introductory chemistry textbooks and are the forms typically depicted in popular orbital visualizations [^24]. In these real hydrogen-like orbitals , the principal quantum number n and the azimuthal quantum number ℓ retain their original interpretation and significance from their complex counterparts. However, the magnetic quantum number m is no longer considered a “good” quantum number (though its absolute value still holds significance).

Some real orbitals are, rather charmingly, given specific names that extend beyond the simple ψ_n,ℓ,m designation. Orbitals characterized by an azimuthal quantum number ℓ = 0, 1, 2, 3, 4, 5, 6… are known as s, p, d, f, g, h, i, … orbitals, respectively. With this established nomenclature, one can already assign names to complex orbitals, such as 2p_±1 = ψ_2,1,±1. Here, the first symbol, ‘2’, represents the n quantum number ; the second character, ‘p’, is the symbol for that particular ℓ quantum number ; and the subscript, ‘±1’, denotes the m quantum number .

As a rather illustrative example of how the full orbital names are generated for these real orbitals, one might calculate ψ^real_n,1,±1. Consulting the table of spherical harmonics , we find that ψ_n,1,±1 = R_n,1Y^±1₁ = ∓ R_n,1√(3/8π) ⋅ (x ± iy)/r, where r = √(x² + y² + z²). From this, the real orbitals are derived as:

$$ \psi_{n,1,+1}^{\text{real}} = R_{n,1}{\sqrt{\frac{3}{4\pi }}}\cdot {\frac{x}{r}} $$

$$ \psi_{n,1,-1}^{\text{real}} = R_{n,1}{\sqrt{\frac{3}{4\pi }}}\cdot {\frac{y}{r}} $$

Similarly, ψ_n,1,0 = R_n,1√(3/4π) ⋅ z/r. As a slightly more intricate example, consider:

$$ \psi_{n,3,+1}^{\text{real}} = R_{n,3}{\frac{1}{4}}{\sqrt{\frac{21}{2\pi }}}\cdot {\frac{x\cdot (5z^{2}-r^{2})}{r^{3}}} $$

In all these instances, we construct a Cartesian label for the orbital by carefully examining, and then abbreviating, the polynomial in x, y, and z that appears in the numerator. We, rather conveniently, disregard any terms in the z, r polynomial except for the term possessing the highest exponent in z. The abbreviated polynomial then serves as a subscript label for the atomic state , employing the same nomenclature as above to indicate the n and ℓ quantum numbers [^needed].

$$ \psi_{n,1,-1}^{\text{real}} = n{\text{p}}{y} = {\frac{i}{\sqrt{2}}}\left(n{\text{p}}{-1}+n{\text{p}}_{+1}\right) $$

$$ \psi_{n,1,0}^{\text{real}} = n{\text{p}}{z} = 2{\text{p}}{0} $$

$$ \psi_{n,1,+1}^{\text{real}} = n{\text{p}}{x} = {\frac{1}{\sqrt{2}}}\left(n{\text{p}}{-1}-n{\text{p}}_{+1}\right) $$

$$ \psi_{n,3,+1}^{\text{real}} = nf_{xz^{2}} = {\frac{1}{\sqrt{2}}}\left(nf_{-1}-nf_{+1}\right) $$

The expressions presented above all adhere to the Condon–Shortley phase convention , a standard approach particularly favored by quantum physicists [^25] [^26]. However, it’s worth noting that other conventions exist for the phase of the spherical harmonics [^27] [^28]. Under these alternative conventions, the p_x and p_y orbitals might, for instance, appear as the sum and difference of p₊₁ and p_-1, which, as you can see, directly contradicts what is depicted above. It’s a small detail, but one that can cause much confusion if you’re not paying attention.

Below, for your convenience, is a list of these Cartesian polynomial names for the atomic orbitals [^29] [^30]. There seems to be a curious lack of consensus in the academic literature regarding how one should abbreviate the lengthy Cartesian spherical harmonic polynomials for ℓ > 3. Consequently, there appears to be no established agreement on the naming conventions for g orbitals or higher, according to this nomenclature [^needed]. A rather glaring oversight, wouldn’t you say?

Shapes of orbitals

Transparent cloud view of a computed 6s ( n = 6, ℓ = 0, m = 0) hydrogen orbital. The s orbitals, though spherically symmetric, have radially placed wave-nodes for n > 1. Only s orbitals invariably have a center anti-node; the other types never do.

These “simple pictures” that purportedly illustrate orbital shapes are, in reality, merely attempts to depict the angular forms of regions in space where electrons, if they bother to occupy a particular orbital, are likely to be found. It’s crucial to understand that these diagrams cannot, and do not, portray the entire region where an electron might exist. After all, according to the immutable laws of quantum mechanics , there remains a non-zero, albeit often infinitesimally small, probability of encountering an electron (almost) anywhere in the vast expanse of space. Instead, these diagrams serve as rather crude, approximate representations of boundary or contour surfaces where the probability density , denoted as | ψ(r, θ, φ) |², maintains a constant value. This value is typically chosen such that there’s a predefined, significant probability (say, 90%) of locating the electron within the confines of that contour. While | ψ |², being the square of an absolute value , is inherently non-negative everywhere, the sign of the wave function ψ(r, θ, φ) itself is frequently indicated within each subregion of the orbital illustration, usually with different colors.

Occasionally, the ψ function itself is graphically represented to explicitly show its phases, rather than the more common | ψ(r, θ, φ) |² which, while depicting probability density , inherently loses phase information (a consequence of taking the absolute value, as ψ(r, θ, φ) is a complex number ). Graphs of | ψ(r, θ, φ) |² orbitals generally exhibit less spherical, more attenuated lobes compared to the ψ(r, θ, φ) graphs. However, they consistently possess the same number of lobes in the same spatial locations and are otherwise recognizably similar. This article, for the sake of clarity and to convey the crucial wave function phase, predominantly features ψ(r, θ, φ) graphs.

These “lobes,” if you care to peer deeper, can be conceptualized as standing wave interference patterns arising from the interaction between two counter-rotating, ring-resonant traveling wave modes, specifically those corresponding to m and −m. The projection of the orbital onto the xy plane, in this interpretation, exhibits a resonant m wavelength along its circumference. Although rarely depicted, the traveling wave solutions themselves can be visualized as rotating banded tori, with these bands representing the intricate phase information. For each value of m, there exist two distinct standing wave solutions: ⟨m⟩ + ⟨−m⟩ and ⟨m⟩ − ⟨−m⟩. If m = 0, the orbital aligns vertically, any counter-rotating information becomes indeterminate, and the orbital itself displays z-axis symmetry. If ℓ = 0, there are, rather predictably, no counter-rotating modes whatsoever; only radial modes exist, resulting in a perfectly spherically symmetric shape.

Nodal planes and nodal spheres are, rather poetically, surfaces within the orbital where the probability density completely vanishes. The precise number of these nodal surfaces is meticulously governed by the quantum numbers n and ℓ. An orbital with an azimuthal quantum number ℓ will inherently possess ℓ radial nodal planes that conspicuously pass through the origin. For example, the s orbitals (where ℓ = 0) are perfectly spherically symmetric and are entirely devoid of nodal planes, whereas the p orbitals (where ℓ = 1) famously exhibit a single nodal plane situated precisely between their two lobes. The number of nodal spheres, a distinct type of node, is equal to n−ℓ−1, a relationship entirely consistent with the restriction that ℓ ≤ n−1 for the quantum numbers. The principal quantum number n ultimately dictates the total number of nodal surfaces, which is always n−1. Loosely, and perhaps overly simplistically speaking, n represents the energy, ℓ is somewhat analogous to eccentricity , and m denotes the orientation.

In general, for a given nucleus, n primarily determines both the size and energy of the orbital. As n increases, the spatial extent of the orbital expands proportionally. However, the greater nuclear charge (Z) found in heavier elements causes their orbitals to contract when compared to lighter elements, resulting in the rather counter-intuitive observation that the overall size of an atom remains remarkably constant, even as the number of electrons it contains increases dramatically.

Experimentally imaged 1s and 2p core-electron orbitals of Sr, including the effects of atomic thermal vibration and excitation broadening, retrieved from energy dispersive x-ray spectroscopy (EDX) in scanning transmission electron microscopy (STEM). [^32]

Furthermore, in general terms, ℓ largely dictates an orbital’s fundamental shape, while mℓ governs its orientation in space. However, because some orbitals are described by equations involving complex numbers , their precise shape can, at times, also be subtly influenced by mℓ. Collectively, the entire ensemble of orbitals for a given ℓ and n conspires to fill space as symmetrically as possible, albeit with increasingly intricate sets of lobes and nodes.

The singular s orbitals (where ℓ = 0) are, rather straightforwardly, shaped like perfect spheres. For n = 1, it resembles a roughly solid ball , with the highest electron density at its center, decaying exponentially outwards. However, for n ≥ 2, each individual s orbital is composed of distinct spherically symmetric surfaces, which manifest as nested shells (meaning the “wave-structure” is radial, following a sinusoidal radial component). Observe the illustration of a cross-section of these nested shells to the right. The s orbitals, for all n values, are the only orbitals that possess an anti-node (a region of maximal wave function density) directly at the center of the nucleus. All other orbital types (p, d, f, etc.) are endowed with angular momentum and thus, rather elegantly, avoid the nucleus, exhibiting a wave node precisely at the nuclear core. Recently, in a testament to advanced imaging, a dedicated effort has been made to experimentally visualize the 1s and 2p core-electron orbitals within a SrTiO₃ crystal, utilizing scanning transmission electron microscopy coupled with energy dispersive x-ray spectroscopy [^32]. Given that the imaging was conducted with an electron beam, the resulting image (see figure to the right) inherently incorporates the effects of Coulombic beam-orbital interaction, often referred to as the impact parameter effect.

The shapes of p, d, and f orbitals, while described verbally here, are more eloquently illustrated graphically in the “Orbitals table” below. The three p orbitals for n = 2 adopt a form reminiscent of two ellipsoids that meet at a single point of tangency precisely at the nucleus . This characteristic two-lobed shape is often, somewhat crudely, likened to a “dumbbell ” – a pair of lobes extending in diametrically opposite directions from each other. The three p orbitals within each shell are, by quantum decree, oriented at perfect right angles to one another, their specific spatial alignment determined by their respective linear combinations of mℓ values. The net effect is a lobe pointing definitively along each of the three primary axes of a Cartesian coordinate system .

Four of the five d orbitals for n = 3 share a similar aesthetic, each featuring four distinctive pear-shaped lobes. These lobes are arranged such that each is tangent at right angles to two others, with the centers of all four residing within a single plane. Three of these planes align with the familiar xy, xz, and yz planes – the lobes nestle between the pairs of primary axes. The fourth d orbital, however, positions its lobes directly along the x and y axes themselves. The fifth and final d orbital presents a rather unique configuration, consisting of three regions of high probability density : a torus (a donut shape, for the uninitiated) nestled between two pear-shaped regions, which are symmetrically positioned along its z-axis. The overall result, when considering all d orbitals, is a grand total of 18 directional lobes, pointing in every primary axis direction and elegantly occupying the spaces between every pair.

Beyond these, there are seven f orbitals, each boasting shapes even more intricate and complex than their d orbital predecessors. Frankly, trying to describe them verbally is an exercise in futility; one simply has to see them to believe them.

Furthermore, mirroring the behavior observed in the s orbitals, individual p, d, f, and g orbitals with n values greater than their lowest possible value exhibit an additional radial node structure. This intricate pattern is reminiscent of higher-order harmonic waves of the same type, when compared to the lowest (or fundamental) mode of the wave. For instance, a 3p orbital, when contrasted with the fundamental 2p orbital, will possess an additional node within each of its lobes. Progressively higher values of n only serve to further increase the number of these radial nodes for each specific type of orbital, a relentless march of increasing complexity.

The shapes of atomic orbitals in a single-electron atom are intrinsically linked to three-dimensional spherical harmonics . These shapes, however, are not immutable or unique; any linear combination of them is mathematically valid. This allows for transformations, such as to cubic harmonics , which can, in fact, generate sets where all the d orbitals share an identical shape, much like how the p_x, p_y, and p_z orbitals are fundamentally the same shape, merely oriented differently [^33] [^34]. The universe, it seems, offers choices, even in fundamental physics.

The 1s, 2s, and 2p orbitals of a sodium atom. Notice how they coexist around the nucleus.

While individual orbitals are most frequently depicted in isolation, giving a rather misleading sense of singularity, the reality is that all orbitals coexist simultaneously around the nucleus. Moreover, in 1927, Albrecht Unsöld provided a profound insight: if one sums the electron density of all orbitals belonging to a particular azimuthal quantum number ℓ within the same electron shell n (e.g., all three 2p orbitals, or all five 3d orbitals), assuming each orbital is either occupied by a single electron or a pair of electrons, then all angular dependence completely disappears. That is to say, the resulting total probability density of all the atomic orbitals within that subshell (those sharing the same ℓ) becomes perfectly spherical. This elegant phenomenon is universally known as Unsöld’s theorem .

Orbitals table

This table, a rather comprehensive overview, presents the real hydrogen-like wave functions for all atomic orbitals extending up to 7s. Consequently, it encompasses all occupied orbitals found in the ground state of every element currently known in the periodic table , up to and including radium . The “ψ” graphs are meticulously rendered, with the negative and positive phases of the wave function clearly distinguished by two different colors (arbitrarily, but consistently, red and blue). The p_z orbital is, in essence, identical to the p₀ orbital. However, the p_x and p_y orbitals are not fundamental eigenstates but rather formed by taking linear combinations of the p₊₁ and p_-1 orbitals (which explains why they are grouped under the m = ±1 label). It’s also worth noting that the p₊₁ and p_-1 orbitals do not share the same shape as the p₀ orbital, as they represent pure spherical harmonics with distinct angular dependencies.

* No elements with 6f, 7d or 7f electrons have been discovered yet. † Elements with 7p electrons have been discovered, but their electronic configurations are only predicted – save the exceptional Lawrencium , which fills 7p¹ instead of 6d¹. ‡ For the elements whose highest occupied orbital is a 6d orbital, only some electronic configurations have been confirmed. (Meitnerium , Darmstadtium , Roentgenium and Copernicium are still missing).

These are the real-valued orbitals, the ones most commonly employed in the pragmatic world of chemistry. It’s important to note that only the m = 0 orbitals are true eigenstates of the orbital angular momentum operator, L̂_z. The columns labeled with m = ±1, ±2, … are, in fact, merely linear combinations of two distinct eigenstates. For a clearer visual comparison, one might consult the accompanying image: Atomic orbitals spdf m-eigenstates (right) and superpositions (left) .

Qualitative understanding of shapes

The shapes of atomic orbitals can be understood, at least qualitatively, by drawing an analogy to the rather more tangible case of standing waves on a circular drum [^35]. To truly grasp this analogy, one must consider the average vibrational displacement of each segment of the drum membrane from its equilibrium position over numerous cycles – a proxy for the average drum membrane velocity and momentum at that specific point – and relate this to that point’s distance from the center of the drum head. If this displacement is then conceptually equated to the probability of locating an electron at a particular distance from the nucleus, it becomes strikingly apparent that the myriad modes of the vibrating disk eerily trace out the diverse shapes of atomic orbitals . The fundamental reason for this uncanny correspondence lies in the profound truth that the distribution of kinetic energy and momentum within a matter-wave is inherently predictive of where the particle associated with that wave is most likely to be found. That is to say, the probability of finding an electron at a given location is also a direct function of the electron’s average momentum at that point. High electron momentum at a specific position tends to “localize” the electron there, a consequence of the properties of electron wave-packets (for the unsettling details of this mechanism, one need only consult the Heisenberg uncertainty principle ).

This intricate relationship implies that certain key features are observable in both drum membrane modes and atomic orbitals . For instance, in all the modes that are analogous to s orbitals (the uppermost row in the animated illustration below), it’s clear that the very center of the drum membrane vibrates with the greatest amplitude. This corresponds directly to the antinode found in all s orbitals within an atom. This central antinode signifies that the electron is most likely to be found at the physical location of the nucleus (which, rather counter-intuitively, it passes straight through without scattering or direct impact), precisely because it is moving (on average) with the greatest rapidity at that point, thereby possessing maximal momentum.

A mental “planetary orbit” picture that comes closest to describing the behavior of electrons in s orbitals, all of which are famously devoid of angular momentum , might be that of a Keplerian orbit with an orbital eccentricity of 1, yet possessing a finite major axis. While such a trajectory is physically impossible for particles that would inevitably collide, it can be conceptualized as a mathematical limit of orbits sharing equal major axes but with ever-increasing eccentricity.

Below, you’ll find a selection of drum membrane vibration modes juxtaposed with the corresponding wave functions of the hydrogen atom . A conceptual correspondence can be established where the wave functions of a vibrating drum head operate within a two-coordinate system, ψ(r, θ), while the wave functions for a vibrating sphere – a more accurate analogy for an atom – employ a three-coordinate system, ψ(r, θ, φ).

• s-type drum modes and wave functions
• Drum mode u₀₁
• Drum mode u₀₂
• Drum mode u₀₃
• Wave function of 1s orbital (real part, 2D-cut, r_max = 2a₀)
• Wave function of 2s orbital (real part, 2D-cut, r_max = 10a₀)
• Wave function of 3s orbital (real part, 2D-cut, r_max = 20a₀)

None of the other sets of modes observed in a drum membrane, conspicuously, possess a central antinode; in all of them, the very center of the drum remains motionless. These correspond precisely to the node found at the nucleus for all non-s orbitals within an atom. These orbitals, by their very nature, all possess some amount of angular momentum . In the classical planetary model analogy, they correspond to particles orbiting with an eccentricity less than 1.0, ensuring that they do not traverse directly through the center of the primary body, but rather maintain a certain distance from it.

Furthermore, the drum modes that are analogous to p and d modes in an atom conspicuously display spatial irregularity along different radial directions originating from the center of the drum. This stands in stark contrast to all the modes analogous to s modes, which are perfectly symmetrical in their radial distribution. These non-radial-symmetry properties of non-s orbitals are absolutely essential for localizing a particle that possesses angular momentum and exhibits a wave nature within an orbital, where it must inherently tend to remain distanced from the central attractive force. After all, any particle localized precisely at the point of central attraction could, by definition, possess no angular momentum . For these more complex modes, the waves in the drum head demonstrably tend to avoid the central point, a testament to their inherent properties. Such features, once again, underscore the profound truth that the intricate shapes of atomic orbitals are a direct, unavoidable consequence of the inherent wave nature of electrons.

• p-type drum modes and wave functions
• Drum mode u₁₁
• Drum mode u₁₂
• Drum mode u₁₃
• Wave function of 2p orbital (real part, 2D-cut, r_max = 10a₀)
• Wave function of 3p orbital (real part, 2D-cut, r_max = 20a₀)
• Wave function of 4p orbital (real part, 2D-cut, r_max = 25a₀)
• d-type drum modes
• Drum mode u₂₁
• Drum mode u₂₂
• Drum mode u₂₃

Orbital energy

• Main article: Electron shell

In the relatively simple case of atoms possessing a single electron (the idealized hydrogen-like atom ), the energy of an orbital – and, consequently, any electron residing within that orbital – is primarily determined by the principal quantum number n. The n = 1 orbital, quite predictably, represents the absolute lowest possible energy state within the atom. Each successively higher value of n corresponds to a higher energy level, though the energy difference between adjacent levels diminishes as n increases. For sufficiently high values of n, the electron’s energy becomes so elevated that it can, with remarkable ease, escape the atom’s gravitational pull entirely. In these single-electron atoms, all energy levels associated with different azimuthal quantum numbers ℓ within a given n are considered degenerate in the Schrödinger approximation , meaning they possess the exact same energy. This approximation, however, is subtly broken by the more refined solution of the Dirac equation (where energy becomes dependent on n and another quantum number, j), and further perturbed by the minute effects of the nucleus’s magnetic field and the complexities of quantum electrodynamics . The latter, in particular, induce tiny but measurable binding energy differences, especially for s electrons that venture closer to the nucleus, as these electrons experience a very slightly altered nuclear charge, even in single-electron atoms – a phenomenon famously known as the Lamb shift .

In the far more common and complex scenario of atoms containing multiple electrons, the energy of an electron is not merely a function of its orbital. It is also inextricably linked to its intricate interactions with all the other electrons present. These interactions, in turn, are highly dependent on the nuanced details of the electron’s spatial probability distribution . Consequently, the energy levels of orbitals in multi-electron atoms are determined not only by the principal quantum number n but also, quite significantly, by the azimuthal quantum number ℓ. Higher values of ℓ are consistently associated with higher energy values. For instance, the 2p state invariably sits at a higher energy than the 2s state. When ℓ reaches 2 (the d orbitals), the increase in the orbital’s energy becomes so substantial that it can push the energy of that orbital above the energy of the s orbital in the next higher shell. When ℓ climbs to 3 (the f orbitals), this energy increase is even more dramatic, pushing the orbital’s energy into the shell two steps higher. This explains why the filling of the 3d orbitals, for example, does not commence until the 4s orbitals have already been fully occupied, and similarly, the 4f orbitals are not completely filled until the 6s orbitals have taken their electrons (for the exhaustive, and often perplexing, details, consult the Electron configurations of the elements (data page) ).

This observed increase in energy for subshells with increasing angular momentum in larger atoms is a direct consequence of complex electron–electron interaction effects. More specifically, it is tied to the remarkable ability of low angular momentum electrons (like s electrons) to penetrate more effectively towards the nucleus, where they experience significantly less screening from the repulsive charge of intervening electrons. Thus, in atoms possessing a higher atomic number , the azimuthal quantum number ℓ of electrons progressively becomes a more dominant factor in determining their energy, while the principal quantum number n of electrons becomes, rather ironically, less and less crucial in dictating their precise energy placement. It’s almost as if the universe enjoys adding layers of complexity just to keep us on our toes.

The energy sequence for the first 35 subshells (e.g., 1s, 2p, 3d, etc.) is presented for your perusal in the following table. Each cell represents a distinct subshell, with its n and ℓ values clearly indicated by its row and column indices, respectively. The numerical value within each cell denotes that subshell’s position within the overall energy filling sequence. For a more linear, sequential listing of these subshells in terms of increasing energies in multi-electron atoms, you can refer to the section below.

Note: Empty cells signify non-existent sublevels, which is rather convenient. Numbers displayed in italics indicate sublevels that could, hypothetically, exist but are not observed to hold electrons in any element currently known. The universe, it seems, has its limits, even for theoretical possibilities.

Electron placement and the periodic table

Electron atomic and molecular orbitals. The chart of orbitals ( left ) is arranged by increasing energy (see Madelung rule ). Atomic orbits are functions of three variables (two angles, and the distance r from the nucleus). These images are faithful to the angular component of the orbital, but not entirely representative of the orbital as a whole. Atomic orbitals and periodic table construction. A rather convenient visual aid, assuming you can decipher it.

• Main articles: Electron configuration and Electron shell

The placement of electrons into orbitals, a process known as electron configuration , is governed by a few fundamental, immutable rules. The first, and perhaps most crucial, dictates that no two electrons within a single atom can ever possess the identical set of values for their quantum numbers – this is, of course, the venerable Pauli exclusion principle . These quantum numbers encompass the three that define the orbital itself (n, ℓ, mℓ), as well as the spin magnetic quantum number m_s. Consequently, a maximum of two electrons may occupy any given orbital, provided they differentiate themselves by having opposing values of m_s. Since m_s is restricted to only two possible values (+ • ⁠1/2⁠ or − • ⁠1/2⁠), this elegantly limits each orbital to a maximum occupancy of two electrons.

Furthermore, an electron, in its inherent laziness, perpetually seeks to fall into the lowest possible energy state available. While it is theoretically possible for an electron to occupy any orbital without violating the Pauli exclusion principle , if lower-energy orbitals are vacant, this higher-energy configuration is inherently unstable. The electron will, eventually, shed its excess energy (typically by emitting a photon ) and descend into a lower-energy orbital. Thus, electrons populate orbitals in a predictable sequence, following the energy order previously outlined. It’s a cosmic game of musical chairs, where everyone wants the lowest seat.

This predictable behavior is precisely what underpins the elegant and recurring structure of the periodic table . The table can be neatly segmented into several horizontal rows, aptly termed ‘periods’, which are numbered sequentially starting with 1 at the very top. The elements currently known to humanity occupy seven such periods. If a given period bears the number i, it signifies that the outermost electrons of the elements within that period reside in the i-th electron shell . Niels Bohr , with his characteristic insight, was the first to propose (in 1923) that the striking periodicity in the chemical properties of the elements could be elegantly explained by the systematic, periodic filling of electron energy levels, which in turn dictates the electronic structure of the atom [^36].

The periodic table can also be conceptually divided into several numbered, rectangular ‘blocks ’. Elements belonging to a specific block share a common characteristic: their highest-energy electrons all reside within the same ℓ-state. However, the principal quantum number (n) associated with that particular ℓ-state will, predictably, vary depending on the period. For instance, the two leftmost columns constitute the ’s-block’. The outermost electrons of lithium and beryllium , for example, belong to the 2s subshell, while those of sodium and magnesium occupy the 3s subshell. It’s a beautifully organized system, if you squint hard enough.

The following is the standard, rather verbose, order for filling these “subshell” orbitals, which, perhaps unsurprisingly, also dictates the arrangement of the “blocks” within the periodic table :

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

The inherently “periodic” nature of orbital filling, along with the distinct emergence of the s, p, d, and f “blocks,” becomes even more graphically apparent if this filling order is presented in a matrix format. In this representation, increasing principal quantum numbers initiate new rows (or “periods”) within the matrix. Each subshell (defined by the first two quantum numbers ) is then replicated as many times as necessary to accommodate each pair of electrons it can contain. The result is a rather compressed, yet informative, depiction of the periodic table , where each entry conceptually represents two successive elements:

While this matrix provides the general order of orbital filling according to the widely accepted Madelung rule , it’s important to acknowledge that exceptions invariably exist. The actual electronic energies of each element are also subtly influenced by additional, finer details of the atoms themselves (for a more thorough exploration, consult the section on Electron configuration § Atoms: Aufbau principle and Madelung rule ). The universe, it seems, enjoys its nuances.

The number of electrons within an electrically neutral atom, quite simply, increases directly with its atomic number . The electrons residing in the outermost shell , known rather dramatically as valence electrons , tend to be the primary architects of an element’s distinct chemical behavior. Consequently, elements that possess the identical number of valence electrons can be grouped together, and, rather predictably, they will exhibit similar chemical properties. It’s a convenient shortcut, if nothing else.

Relativistic effects

• Main article: Relativistic quantum chemistry
• See also: Extended periodic table

For elements burdened with a high atomic number Z, the subtle, yet profound, effects of relativity become increasingly pronounced. This is particularly true for s electrons, which, owing to their unique orbital characteristics, move at truly relativistic velocities as they penetrate the screening electrons near the dense core of these high-Z atoms. This relativistic surge in momentum for high-speed electrons precipitates a corresponding decrease in their wavelength and a noticeable contraction of the 6s orbitals relative to the 5d orbitals (when compared to their analogous s and d electrons in lighter elements occupying the same column of the periodic table ). The net result of this relativistic dance is that the 6s valence electrons experience a significant lowering of their energy.

Examples of the tangible, physical consequences of this rather esoteric effect are quite striking. They include the anomalously low melting temperature of mercury (a direct result of its 6s electrons being less available for effective metal bonding) and the distinct golden hue of both gold and caesium [^37]. It’s almost as if the universe is showing off its fundamental laws in rather flashy ways.

In the somewhat simplistic, yet historically significant, Bohr model , an n = 1 electron is assigned a velocity given by the formula v = Zαc, where Z represents the atomic number , α is the ubiquitous fine-structure constant , and c is the immutable speed of light. Consequently, in the framework of non-relativistic quantum mechanics , any hypothetical atom possessing an atomic number greater than 137 would, by this calculation, require its 1s electrons to be traveling faster than the speed of light – a rather inconvenient impossibility. Even within the more sophisticated Dirac equation , which meticulously accounts for relativistic effects , the wave function of the electron for atoms with Z > 137 becomes oscillatory and unbounded , signaling a breakdown of the standard model. The profound significance of element 137, sometimes informally dubbed untriseptium , was first highlighted by the legendary physicist Richard Feynman [^38]. This hypothetical element is occasionally, and rather affectionately, referred to as feynmanium (with the proposed symbol Fy). However, Feynman’s initial approximation, insightful as it was, fails to precisely predict the exact critical value of Z. This discrepancy arises due to the non-point-charge nature of the nucleus and the incredibly small orbital radius of the innermost electrons, which collectively result in a potential experienced by these inner electrons that is effectively less than Z. The true critical Z value, which would render an atom unstable with respect to the high-field breakdown of the vacuum and the subsequent production of electron–positron pairs, doesn’t actually occur until Z is approximately 173. These extreme conditions are rarely observed, except transiently in the energetic collisions of exceptionally heavy nuclei, such as lead or uranium, within particle accelerators, where some claims of electron–positron production stemming from these effects have, indeed, been made.

Curiously, in the realm of relativistic orbital densities , the familiar concept of nodes vanishes entirely, even though individual components of the wave function will still exhibit nodal structures [^39]. It’s another reminder that the universe doesn’t always conform to our non-relativistic intuitions.

pp hybridization (conjectured)

In the far reaches of the extended periodic table , specifically among the late period 8 elements , a peculiar phenomenon known as “pp hybridization” is conjectured to exist [^40]. This theoretical construct posits the formation of a hybrid orbital from the 8p₃/₂ and 9p₁/₂ subshells. Here, the “3/2” and “1/2” refer to the total angular momentum quantum number of the electron, incorporating both its orbital and spin angular momenta. This rather exotic “pp” hybrid is hypothesized to be responsible for the distinctive characteristics of the p-block within that period, exhibiting properties remarkably similar to the more conventional p subshells found in ordinary valence shells . The energy levels of the 8p₃/₂ and 9p₁/₂ subshells are expected to converge due to the powerful relativistic spin–orbit effects . Furthermore, the 9s subshell is also anticipated to participate in this complex hybridization, as these elements are predicted to exhibit analogous chemical behavior to their respective 5p counterparts, ranging from indium through xenon . It’s a fascinating glimpse into the extreme physics that awaits at the edges of our known elements.

Transitions between orbitals

• Main article: Atomic electron transition

Bound quantum states , by their very nature, possess discrete and quantized energy levels . When this fundamental principle is applied to atomic orbitals , it implies that the energy differences between these states are also, unequivocally, discrete. Consequently, a transition between these states – that is, an electron either absorbing or emitting a photon – can only occur if the photon in question possesses an energy that precisely matches the exact energy difference separating the initial and final states. It’s a rather strict cosmic bouncer, only letting in those with the exact ticket.

Consider, for a moment, two specific states of the hydrogen atom :

• State 1: n = 1, ℓ = 0, mℓ = 0, and m_s = + • ⁠1/2⁠
• State 2: n = 2, ℓ = 0, mℓ = 0, and m_s = − • ⁠1/2⁠

According to the immutable tenets of quantum theory , State 1 possesses a fixed energy of E₁, and State 2 possesses a fixed energy of E₂. Now, let’s ponder what would transpire if an electron initially residing in State 1 were to attempt a transition to State 2. For this transition to be successful, the electron would need to acquire an energy precisely equal to E₂ − E₁. If the electron receives energy that is either less than or greater than this exact value, it simply cannot jump from State 1 to State 2. It’s not a suggestion; it’s a rule. Now, imagine we bombard the atom with a broad spectrum of light. Photons that arrive at the atom possessing an energy exactly equal to E₂ − E₁ will be absorbed by the electron in State 1, prompting that electron to make the quantum leap to State 2. However, photons with energies greater or lesser than this precise value cannot be absorbed by the electron, because the electron is constrained to jump only to one of the allowed orbitals; it cannot occupy a state that lies between orbitals. The profound consequence of this behavior is that only photons of a very specific frequency will be absorbed by the atom. This selective absorption creates a distinct, dark line in the observed spectrum, known as an absorption line , which directly corresponds to the precise energy difference between State 1 and State 2.

The atomic orbital model thus elegantly predicts the existence of line spectra , a phenomenon that is consistently observed experimentally. This predictive power stands as one of the primary and most compelling validations of the atomic orbital model.

Nevertheless, it’s crucial to remember that the atomic orbital model is, ultimately, an approximation of the full, unadulterated quantum theory , which truly only recognizes many-electron states in their full, correlated complexity. While the predictions of line spectra are qualitatively useful, offering valuable conceptual insights, they are not quantitatively accurate for atoms and ions that contain more than a single electron. The universe, in its infinite wisdom, rarely grants us perfectly simple answers.

See also

• Atomic electron configuration table
• Condensed matter physics
• Electron configuration
• Energy level
• Hund’s rules
• Molecular orbital
• Orbital overlap
• Quantum chemistry
• Quantum chemistry computer programs
• Solid-state physics
• Wave function collapse
• Wiswesser’s rule