Structure of Atom: The Quantum Mechanical Model

In 1924, French physicist Louis de Broglie proposed something genuinely audacious: if light, normally understood as a wave, could also behave like a stream of particles, as the photoelectric effect had shown, then perhaps matter, normally understood as particles, could also behave like a wave. He gave this idea a precise mathematical form, lambda = h/mv, relating a particle's wavelength directly to its momentum (mass times velocity). The prediction was confirmed experimentally soon after, when a beam of electrons was shown to undergo diffraction, a behaviour that only waves exhibit, a discovery that led directly to the electron microscope, a device now capable of magnifications around 15 million times by exploiting exactly this wave behaviour. Crucially, de Broglie's wavelength shrinks as mass grows, which is why a thrown cricket ball shows no detectable wave behaviour at all, while a single electron, thousands of times lighter than an atom, shows it clearly.

Werner Heisenberg, in 1927, stated a principle that follows directly from matter's dual wave-particle nature: it is fundamentally impossible to determine an electron's exact position and exact momentum simultaneously, not due to any instrument's imperfection, but as a basic feature of nature. Mathematically, delta x times delta p is greater than or equal to h/4pi: the more precisely position is known (small delta x), the less precisely momentum can be known (large delta p), and vice versa. The effect is utterly negligible for everyday objects; applying the principle to a one-milligram object gives an uncertainty so small it has no practical consequence. But for an electron, with its vastly smaller mass, the same calculation gives a genuinely enormous uncertainty, large enough that the classical picture of an electron following a fixed, precisely defined path simply cannot hold.

Put de Broglie's wave nature and Heisenberg's uncertainty together, and Bohr's entire picture of the atom collapses at its foundation. An orbit, by definition, is a precisely defined path, and a precisely defined path can only be traced if both an electron's exact position and exact velocity are known at every instant, exactly what the uncertainty principle rules out categorically. Bohr's model ignored the electron's wave character entirely, treating it as a simple charged particle tracing a clean circular path, and it directly contradicted the uncertainty principle it predates. This isn't a minor technical objection; it means the very concept of an electron 'orbit' has no real physical meaning, and cannot ever be demonstrated experimentally, no matter how precise the instrument. What was needed instead was a framework built from the ground up around probability rather than precise trajectories.

Quantum mechanics, developed independently by Heisenberg and Erwin Schrodinger in 1926, replaces Bohr's precise orbits with something fundamentally different. Schrodinger's equation, incorporating de Broglie's wave-particle duality and consistent with the uncertainty principle, is solved for a given system to yield two things: a set of allowed, quantised energy levels, and a corresponding wave function, psi, for each one. Psi itself carries no direct physical meaning, it's simply a mathematical function of the electron's coordinates, but its square, |psi|^2, gives the probability density of finding the electron at any given point. A wave function describing one electron in an atom is called an atomic orbital, and when an electron is described by a particular wave function, it's said to occupy that orbital. Solving Schrodinger's equation exactly for hydrogen successfully predicts every feature of its spectrum, including subtleties Bohr's model never could.

Every atomic orbital is precisely identified by a set of three quantum numbers arising naturally from Schrodinger's equation. The principal quantum number (n = 1, 2, 3...) determines an orbital's size and, to a large extent, its energy; it also identifies the shell, with all orbitals sharing one n value constituting a single shell (labelled K, L, M, N...) containing n^2 total orbitals. The azimuthal quantum number (l), ranging from 0 to n-1, determines an orbital's three-dimensional shape and identifies its subshell, with l=0, 1, 2, 3 corresponding to the familiar s, p, d, f notation; a shell with principal quantum number n contains exactly n subshells. The magnetic quantum number (ml) determines an orbital's spatial orientation relative to a chosen axis, taking 2l+1 possible values from -l to +l for a given subshell, which is exactly why there's one s orbital, three p orbitals, and five d orbitals per relevant subshell.

Three quantum numbers still weren't enough to explain certain fine details in multi-electron spectra, lines that split into closely spaced doublets and triplets. In 1925, George Uhlenbeck and Samuel Goudsmit proposed a fourth quantum number: an electron spins on its own axis, much like Earth spinning while it orbits the sun, and this spin angular momentum can take only two orientations relative to a chosen axis, described by the spin quantum number ms, either +1/2 (spin up) or -1/2 (spin down). This fourth number, combined with the Pauli exclusion principle, that no two electrons in an atom can share an identical set of all four quantum numbers, directly explains why any single orbital can hold at most two electrons, and why those two electrons must have opposite spins: they already share identical n, l, and ml values, so only opposite ms values keep them distinct.

Boundary surface diagrams, drawn to enclose the region where an electron's probability of being found is about 90%, give a practical picture of each orbital's shape. Every s orbital is a perfect sphere centred on the nucleus, growing larger as n increases (4s > 3s > 2s > 1s), with the electron equally likely to be found in any direction at a given distance. Every p orbital, by contrast, consists of two lobes on either side of a plane through the nucleus, oriented along the x, y, or z axis (hence 2px, 2py, 2pz), with zero probability density exactly on that dividing plane. The five d orbitals are more elaborate still: four share a similar four-lobed shape, while the fifth, dz2, looks different but carries identical energy to the others. Anywhere probability density drops to exactly zero is called a node; radial nodes occur at specific distances from the nucleus (n-l-1 of them), while angular nodes occur along specific planes or directions (l of them), together totalling n-1 nodes for any orbital.

In hydrogen, a single electron feels only the nucleus's pull, so orbital energy depends solely on n, and orbitals sharing an n value are degenerate, identical in energy. Multi-electron atoms are different: every electron also feels repulsion from every other electron, and inner-shell electrons partially shield outer electrons from the nucleus's full positive charge, an effect called shielding, leaving outer electrons exposed to a reduced effective nuclear charge (Zeff) rather than the full nuclear charge. Because a spherical s orbital keeps its electron closer to the nucleus on average than a p orbital does, and a p orbital closer than a d orbital, s electrons shield more effectively and feel a higher Zeff than p electrons, which in turn feel more than d electrons, splitting what would otherwise be one energy level into several. This is captured by the (n+l) rule: the lower an orbital's (n+l) value, the lower its energy; when two orbitals tie on (n+l), the one with lower n wins, which is precisely why the 4s orbital, (n+l) = 4, fills before the 3d orbital, (n+l) = 5, despite belonging to an earlier shell.

Three rules together govern exactly how electrons fill an atom's orbitals. The Aufbau principle (German for 'building up') states that electrons occupy the lowest-energy available orbital first, filling higher orbitals only once lower ones are full, following the practical order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s. The Pauli exclusion principle caps every orbital at exactly two electrons, and those two must have opposite spins. Hund's rule of maximum multiplicity governs how electrons fill a set of degenerate orbitals within one subshell, like the three p orbitals or five d orbitals: electrons occupy each orbital singly, with parallel spins, before any pairing begins at all, meaning pairing in a p subshell only starts with the fourth electron, in a d subshell only with the sixth, and in an f subshell only with the eighth.

Electron configurations can be written two ways: the compact s^a p^b d^c notation, listing each occupied subshell with the electron count as a superscript, and orbital diagrams, drawing each orbital as a box with arrows showing individual electrons and their spins. Hydrogen is 1s1, helium 1s2, and from lithium (1s2 2s1) through neon (1s2 2s2 2p6), electrons fill the second shell in order. Since fully filled inner shells rarely change chemically, they're often abbreviated using the preceding noble gas's symbol in brackets, so sodium becomes [Ne]3s1 rather than writing out 1s2 2s2 2p6 3s1 in full. The electrons inside that bracketed noble-gas core are called core electrons; everything beyond it, added to the shell with the highest principal quantum number, are the valence electrons, the ones that actually determine an atom's chemistry, why elements combine, which are metals, and why noble gases resist reacting at all.

Following the standard filling order strictly predicts chromium's configuration as [Ar]3d4 4s2 and copper's as [Ar]3d9 4s2, but both are observed experimentally to be [Ar]3d5 4s1 and [Ar]3d10 4s1 instead, one electron short of the prediction in 4s, one electron over in 3d. The explanation is that 4s and 3d orbitals sit close enough in energy that shifting a single electron from 4s to 3d becomes favourable whenever that shift completes a half-filled or fully-filled 3d subshell, since half-filled and fully-filled configurations carry genuine extra stability. Chromium 'sacrifices' its second 4s electron to reach the exceptionally stable d5 configuration rather than settling for d4, and copper does the same to reach d10 rather than d9. These aren't arbitrary exceptions to memorise blindly; they follow directly from the same stability principle that governs the rest of the periodic table.

The extra stability of half-filled and fully-filled subshells comes from two compounding effects. First, a symmetrical, evenly spread electron distribution shields the nucleus's electrons from each other only weakly, so each electron feels a stronger effective pull from the nucleus and sits at lower energy. Second, and more significant, is exchange energy: whenever two or more electrons with parallel spin occupy degenerate orbitals within the same subshell, they can effectively swap positions with each other, and each possible exchange releases a small amount of stabilising energy. The number of such possible exchanges is maximised precisely when a subshell is either exactly half-filled or completely filled, since both configurations maximise the count of same-spin electron pairs available to swap. This same exchange energy is what underlies Hund's rule in the first place, electrons filling singly with parallel spins isn't an arbitrary bookkeeping rule, it's the electron configuration that genuinely minimises the atom's total energy.