{ "version": "https://jsonfeed.org/version/1.1", "title": "Keiran Rowell", "home_page_url": "https://keiran-rowell.github.io/", "feed_url": "https://keiran-rowell.github.io/feed.json", "items": [ { "id": "https://keiran-rowell.github.io/oxygen/2026-04-02-the-oxygen-apocalypse/", "url": "https://keiran-rowell.github.io/oxygen/2026-04-02-the-oxygen-apocalypse/", "title": "The Oxygen Apocalypse: how symmetry saves us from the flames", "content_html": "

An artist’s impression of Earth in the early Archean with a purple hydrosphere. Oleg Kuznetsov [CC BY-SA 4.0]

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That everything is on fire, slow fire, and we’re all less than a million breaths away from an oblivion more total than we can even bring ourselves to even try to imagine…

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— The Pale King, David Foster Wallace

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Diradical by Nature

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We are protected at every moment by a quantum mechanical safety catch. The most dangerous element of our atmosphere normally can’t react with us. It is physically verboten. Oxygen doesn’t immediately burn our organic matter to a crisp because our biomolecules are a different spin state to O2.

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An insuperable barrier of electronic symmetry is our shield (expand the briefing below).

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\n \n ▶ Technical Briefing: The Quantum Safety Catch\n \n\n
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Technical Briefing: The Quantum Safety Catch\n\"The

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The anatomy of a forbidden reaction — The two \\(\\pi^{*}\\) anti-bonding orbitals of O2 are shown. As the frontier orbitals, these determine the reactivity of oxygen. Two electrons of the same spin can’t occupy the same orbital, so nature has three choices. The same-spin triplet state option (\\(^3\\Sigma_g^-\\)) that makes up 20% of our atmosphere has its reactivity “locked”; the diradical spin density restrained from reacting with typical matter due forbidden spin-interaction rules between molecules. For the opposite-spin paired singlet state options (\\(^1\\Delta_g\\) & \\(^1\\Sigma_g^+\\)), both are highly reactive, but the rarity of the latter attests to a large energy penalty to accessing higher electron spin states. Essentially, “regular” oxygen has two electrons ‘trapped’ in parallel spins, which can’t ‘break into’ regular molecules to cause a reaction — all their orbitals are filled with paired electrons.

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The molecular oxygen in the air is a diradical — it has two unpaired electrons: •O–O•. That makes it a molecule in a ‘triplet’ state. For almost all other molecules on earth, they live in a ‘singlet’ state where all electrons are ‘paired’ off to avoid opening a new shell. They can’t react with each other because doing so would mean the total angular momentum wouldn’t be conserved. This safety catch prevents the atmosphere from incinerating the biosphere.

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Singlet oxygen is its triplet twin let loose. Be afraid

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They captured the sun…

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\"Chroococcidiopsis

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Light harvesting extremophiles, Chroococcidiopsis thermalis, that can live off lower energy light than most cyanobacteria. T. Darienko [CC BY-SA 4.0]

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Before we had the green world as we know it, lifeforms were reliant on chemically ‘reducing’ conditions. Among the first inhabitants of the seas were the archaea (literally, “the ancients”). Some species of ancients could power themselves from hydrogen and abundant carbon dioxide, emitting methane into the atmosphere. Other ancestral organisms eked out a diet from sulfur sources. This was the hypothesised Purple Earth, and was sustained by a chemical environment fundamentally different from our own for billions of years. Until, in the late Archean eon, the first single-cellular shoot of green appeared.

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A biological cousin had become a stranger — they hoisted an antenna of chlorophyll towards the sun. The archaea had their own antenna, a simpler retinal light gate: it used the centre of the solar spectrum to power primordial pumps. There had already been new life among the ancients; early bacteria, sulfur dependent, had begun using a new proto-antenna of a ruddy purplish hue, but still coexisting with the archaea without threat — the seas were peacefully doubly purple. The chlorophyll antenna of the green bacterial newcomers, in contrast, could harvest the high energy blue and deep red spectral bands. Forced to occupy the solar fringes as the centre was taken, they had hit upon a more potent use of the distance-dimmed energy radiating off the fusion furnace they orbited. They had developed the technology to generate high-voltage electrons from the sun’s rays. The Earth would never be the same again.

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…and began to exhale

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These chlorophyll wielding bacteria were living in ways previously inaccessible. Their industrious photosynthesis leaked onto the globe a new alien molecule: oxygen gas. O2 had never been a component of the atmosphere before, it was a highly exotic molecule only created in brief flashes when extreme UV radiation smashed apart water or CO2. This new gas was to eventually snuff out the ancients.

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\"Oxygenation

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The upper (red) and lower (green) bounds on the oxygen levels in Earth’s atmosphere, on a billions of years timescale. Heinrich D. Holland [CC BY-SA 3.0]

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Many ancient microbes were hydrogen sulfide metabolisers. H2S was a more readily split chemical metabolite than water — primordial photosystems could generate a potential capable of powering the metabolic enzymes to process it. But oxygen was harder to crack: it was locked in a molecular safe, water’s O–H bound 100 kJ/mol more strongly than S–H down a period. H2O required photon-driven electrical impulses and quantum spin constraints to handle unpicking the bond.

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Enter the cyanos.

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Cyanobacteria had staged a metabolic coup with the light energy from their more powerful antenna. The cyanos departed from their ancient ancestors who were constrained to metabolic hydrogen and sulfur sources. For the cyanos, the abundant water they were suspended in became fuel. They prospered in resource-starved shallows, but were able to spread more widely, splitting the most plentiful fuel source on the planet.

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…and the sea boiled

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These green cyanobacteria had developed spin-forbidden alien technology. Their oxygen evolving complex (OEC) would radically terraform the Earth. At its heart: a distorted cube of manganese and oxygen, with a jagged edge of calcium. This was their ‘quantum lockpick’ to access the forbidden triplet states. Driven by light power, an ‘S-state’ clock pumped charges from the manganese core. Once electrons were stripped from the core, it became charged enough to fire—rapidly ripping H2O apart into the powering particles for the cyanobacteria.

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\n \n ▶ Technical Briefing: The Quantum Lockpick\n \n\n
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Technical Briefing: The Quantum Lockpick\n\"The

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[Left] The Oxygen Evolving Complex (OEC): “Mimicking the Oxygen-Evolving Center in Photosynthesis”, Chen, Y.; et al., Frontiers of Plant Science, 13, #929532, 2022. \n
[Right] The “spin diode” S-clock that enables the generation of triplet oxygen from water: “Closing Kok’s cycle of nature’s water oxidation catalysis”, Guo, Y.; et al., Nature Communications, 15, #5982, 2024

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Nestled within photosystem II at the start of the photosynthesis machinery is the oxygen evolving complex (OEC) — the ‘lockpick’ that splits apart water to release the protons and electrons used to power the cell. A photon from sunlight knocks an electron off a set of four chlorophyll antennae (P680). To fill the leftover electron hole, an electron is shuffled off the OEC, progressively raising the charge of the manganese (Mn) metals.

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This light driven process pushes the OEC up the charge states (the “S-clock” cycle). Eventually the OEC is pushed to its limits; from the fourth (S4) state water is ripped apart, pushing the cluster back to resting charge and resetting the S-clock.

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\n Arming the S-clock\n
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\n \n S1 \n [Mn4III,III,IV,IV]\n\n \n hv\n ⟶\n \n\n S2 \n [Mn4III,IV,IV,IV]\n\n \n hv\n ⟶\n \n\n S3 \n [Mn4IV,IV,IV,IV]\n\n \n hv\n ⟶\n \n\n S4\n [Mn4IV,IV,IV,V]\n\n
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\n Firing the S-clock\n
\n S4 + 2H2O ⟶ S0 + 4H+ + 3Σg− O2\n
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\n \n ▶ Technical Briefing: The S-clock Timebomb\n \n\n
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The S-clock is not set to zero. It is self-priming—set one tick ahead to S1. Rather than the clock resting in a ‘ground-state’—where all manganese metals hold their stable electron count—electrons are transferred out of the cluster to tyrosine residues placed nearby. The oxygen evolving complex is capable of self-priming in the complete dark: a powerful cluster of manganese ions that have already been destabilised from their +II configuration by donating electrons to the cluster oxygens and surrounding protein. A primed cannon to bring forth the apocalypse of a furnace world.

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Lying in wait until morning, when the next bombardment of photons can push the cluster to higher and higher charge states, and pass a redline. Some propose a ‘firing pin’ manganese ion reaches an outrageous +V charge in the S4 state. Detailed computational work suggests the water molecules are progressively deprotonated and S4 creates an ‘armed bullet’ of an oxyl radical (O\\(^{\\bullet-}\\)). The S4 state is so fleeting not even kilometer-long femtosecond X-ray Free-Electron Lasers (XFELs) can fully follow it.

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Pushing past S4 triggers a violently reactive reset; ripping four electrons from water to replenish the depleted cluster within milliseconds; joining oxo-oxyl groups, each with a missing spin-aligned electron, to form the triplet •O–O• bond, and exhaling oxygen into the environment. The manganese oxide cluster’s magnetic moment acts as a spin template, ensuring the cell doesn’t incinerate itself by creating a singlet. This process is tightly coordinated to avoid creation of cell-destroying peroxide (H2O2) or superoxide (•O2–) species.

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The metal cluster core functions as a highly calibrated spin diode. \nThe intermediary materiel on the way to this spin diode was abundant, as free manganese in seawater and mixed with iron (another intermediary) in nodules on the ocean floor; a source of “dark oxygen”.

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\"Ferromanganese

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Ferromanganese nodules on the sea floor. North Atlantic Stepping Stones expedition. NOAA, [Public domain]

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This new engine has a mineral heart. Nature had formed birnessite deposits on the ocean floor — a cubic manganese oxide that catalytically produces oxygen from water in the presence of calcium. As a bubbling rock, birnessite wasn’t a threat to survival. The oxygen evolving complex had reconfigured what manganese could carry out, wielded by an exponentially growing cyano population it would extinguish the ancients and leave behind a furnace world that would come to fuel unrecognisably morphed descendants.

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\n \n ▶ Technical Briefing: The Cluster Spin Diode\n \n\n
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Technical Briefing: The Cluster Spin Diode

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\"OEC

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[Left & Middle] The manganese-oxide–calcium cluster of the oxygen evolving complex, and an artificial mimic: “Mimicking the Oxygen-Evolving Center in Photosynthesis”, Chen, Y.; et al., Frontiers of Plant Science, 13, #929532, 2022.\n
[Right] The natural mineral birnessite that generates oxygen from water in the presence of calcium: “Birnessite: A Layered Manganese Oxide To Capture Sunlight for Water-Splitting Catalysis”, Lucht, K.; Mendoza-Cortez, J., The Journal of Physical Chemistry C, 119, 22838–22846, 2015

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The OEC cluster is a distorted manganese-oxide cube, with calcium edge. Though artificial mimics have been created, it is still a somewhat open question if had the OEC had direct mineral origin, and if so, which?

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Manganese is the natural metal for water splitting due to its versatile oxidation state and favourable redox energetics. Minerals like birnessite also display oxygen evolution, but at far higher overpotentials, and are prone to create reactive oxygen species as side products which would shred a biological cell.

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The asymmetric cuboid arrangement helps the OEC function as a “spin diode”. The cluster’s four metal centers are in communication to tightly coordinate a series of spin flips which guide the generation of forbidden triplet oxygen. The sequential spin flips are crucial to this near-alien technology developed by the cyanos: they’re what allow the OEC to overcome the verboten spin-barrier, previously unbroken at biospheric scale.

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The cyanobacteria had broken nature’s rule of closed-shell ground states, pushing through forbidden symmetry regimes with electromagnetic clockwork. From captured spiralling lightwaves the flow of energy was controlled and constrained; the spin gate of the asymmetrically distorted cluster ensured departing electrons would pass through the spin-state barrier. The cyanos could take the substance the world was drowning in —singlet water— and, at ambient temperature, pump out triplet oxygen gas. The cyanos held a quantum lockpick and could now consume the previously forbidden.

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…and the oceans bled

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There had been other microbial light-eaters, but their atmospheric and oceanic impact was marginal, since previous phototrophs still took electrons from limited dissolved material — hydrogen sulfide or Fe2+ (consumed by the photoferrotrophs). The cyanos, in contrast, could on their own unbalance terrestrial feedback loops, rapidly proliferating away from specific chemical niche locales. While their S-clocks ratcheted through firing cycles at a velocity never before accessed in sun-driven technology, barely any oxygen initially escaped to the atmosphere. The build up in the air was delayed for nearly two billion years. The oceans had a natural buffer to the exotic spin toxin — they contained iron.

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Instead of terraforming the atmosphere, oxygen reshaped the oceans first. The oceans held vast reserves of dissolved Fe2+ (making them a vibrant green), that formed iron-hydroxide minerals when oxidised to rust-brown Fe3+. Steadily, the buffer iron was worn through, the bacteria (both oxygen-forming and ferrotrophic) shifted geological masses of dissolved oceanic iron into solids in the form of banded iron formations in the Precambrian.

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\"Jaspilite

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Banded iron formation in Soudan Underground Mine State Park, MN. James St. John, [CC BY 2.0]

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By unpicking oxygen from its water surrounds, bacteria had begun to inexorably rust the oceans. The iron deposited in solid scabs, layered in puff-pastry sheets of blood red and dull metal hues. Rolled, pressed, and folded by the tectonic slippage below.

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The Hamersley Range in the Pilbara region of Western Australia preserves some of the most arresting evidence of this process; resurfaced testaments to this planetary-wide haemorrhaging.

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\"Hamersley

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Banded iron formation in the Hamersley Range, WA. Dated from the Neoarchaean to the Paleoproterozoic era. James St. John, [CC BY 2.0]

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During this purge of iron from the oceans the sea surface was encrusted with ore layers up to a kilometer thick. We use the scabs of the oxidised ferrous sea to build our steel reinforced cities today.

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…and their kin choked

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The oceans, now cleansed to a clear blue-green, could no longer hold back the corrosive gas. It escaped into the atmosphere and across the globe, wearing down anything that had not evolved chemical defenses.

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\"Oxidiser

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Diffusing across the skies and into ecological niches, •O–O• oxidised anything it contacted. Seeping through cell walls, oxygen triggered radical chain reactions that destroyed the membrane’s fatty layer, tore through DNA, and ripped apart protein machinery. Microbes unprepared for oxygen were suffused with this alien gas and erased from the genetic pool from the inside. Life had to develop new machinery to cope, or die. Survivors adapted [patchwork protection against this molecular cataclysm, imperfect oxygen disposal mechanisms that provided enough antioxidant shielding to outrun extinction. These emergency responses carried down in their descendants’ DNA.

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As atmospheric oxygen concentration slowly but steadily increased, more durable adaptations were required to transcend the chemical danger of this new reality. Some cells found seemingly providential protection under the aegis of oxygen-burning α-proteobacteria. This protection racket would go on to be one of the most successful chemical energetic exchanges in biology: they became mitochondrial powerplants of the incipient multicellular organism, forging previously untapped molecular stores of energy. They would come to have domain over this corroding new order.

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Most ancients did not survive, though pockets of anaerobe resilience are left in nooks unreached by this gas. The Purple Earth was bleached green by a corrosive spill. With most of the competition extinct, and chemical cycles sundered, the photosynthesisers began to remake the climate.

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…and the sky faded

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The cyanos had emerged first into a stifling atmosphere, a greenhouse gas layer which was more potent than ours due to high methane (CH4) levels. It was also chemically stultifying before the injection of oxygen, methane lifetimes were in the thousands of years as there were no oxidising radicals to scrub it. The prior bacteria were methanogenic, maintaining this warming blanket. Photosynthesis fundamentally changed this equation.

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\n \n CH4 + 2O2\n \n \n ⟶\n \n\n CO2 + 2H2O\n \n
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Within the same carbon budget, CH4 is 30–80x more potent a greenhouse gas than CO2. The millennia-long lifespan of methane was shortened by photosynthetic oxygen to just a decade, converted into carbon dioxide. The linear symmetry of O=C=O leaves it with fewer active infrared absorption modes than CH4: particular geometric distortions that can absorb longwave heat radiating back from the Earth’s surface. The 1:1 exchange of CH4 to CO2 by catalytic oxygen-linked radical reactions had made the greenhouse blanket more translucent to heat. While the oceans had buffered the cyanobacteria from the consequences of their waste gas for billions of years, their emissions destroyed the greenhouse layer in a thousandth of that . There was no human heavy industry emitting gigatonnes of CO2 into the atmosphere.

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The sun was dimmer back then. With nothing to compensate, a long winter was coming.

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…and the Earth froze

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\"Snowball

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An artist’s impression of “Snowball Earth” during the Huronian Glaciation. Oleg Kuznetsov [CC BY-SA 4.0]

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Tipping the Earth’s temperature balance, the cyanos’ solvent-fuel-transport wonder substance began to freeze. Glaciers crept from the poles towards the equator, dramatically envisioned as a “snowball Earth” where ice had a stranglehold on the globe. The full extent of the ice after the initial oxidation event is uncertain, likely not a complete encasement, but oxygen had set in motion an unstoppable feedback loop: as the surface reflectivity increased temperatures plunged. The lockpick that allowed cyanos to break down inexhaustible water now caused them to be encased in crystalline cages of it. It was an approach towards self-extinction. But a totality wasn’t reached, the climate oscillated in and out of frozen conditions, life barely clinging on in ice-free oases.

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The accumulation of greenhouse gases from the volcanic ruptures during tectonic shifts broke the globe free from this frozen end state. But this thawed world was a new one, the atmosphere had been injected with triplet oxygen, a potent electron acceptor for any metabolism aggressive enough to handle this incendiary substance.

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…and life rose up

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The oxygen apocalypse had largely swept aside the Purple Earth of the ancients, though archaea are still detectable in almost every habitat. The exact emergence timelines of various aerobic forms of life in oxic/anoxic and sulfur-dominated conditions are not completely clear-cut. But lifeforms would never be the same, eukaryotes —complex, and eventually multicellular, life— could emerge since there was more energy to work with. The trees in our parks were to be descended from the dominant cyanos, the oxygen evolving complex to be incorporated into the chloroplast of every leaf. While timepoints recording last and first common ancestors are still estimated in millions-of-years ranges, the outcome was inevitable: life would be oxygen-breathing, rocketed into fierce metabolic competition.

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\"Great

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Molecular, geological and fossil evidence for the arrival of complex life following the Great Oxidation Event (GOE). “Eukaryogenesis and oxygen in Earth history”, Mills, B.; Boyle, R.; Daines, S.; Sperling, E.; Pisani, D.; Donoghue, P.; Lenton, T.; Nature Ecology & Evolution, 6, 520–532, 2022

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…and aged away

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Far from being a rejuvenating substance or a breath of fresh air, the cellular combustion with oxygen is a corrosive bargain against the clock. Addition of a single electron to •O–O• during a biochemical process gone awry creates the superoxide radical (•O2–) which is not as strictly spin-forbidden against reacting with regular matter. This is the opening slippage in oxygen control that unshackles a cascade of other reactive oxygen species, which tear into biochemical structures and degrade cellular function. Oxidative stresses are one of the major contributors to cellular ageing in all aerobic organisms; the trade for a powerful metabolism is the shortening of cellular lifespan.

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The production of these reactive oxygen species is ironically also mediated through iron — the same buffering element precipitated out of the Archean-era ocean by the oxygen evolving complex, now facilitating the accumulation of cellular damage from the OEC’s waste.

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\n \n •O2− + H2O2\n \n \n Fe\n ⟶\n \n\n •OH + OH− + O2\n \n
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The net Haber–Willstätter(Weiss) reaction indicates how absorbed superoxide (•O2–) can create reactive oxygen species. It was co-proposed by Fritz Haber of Haber-Bosch fame in his final paper, published just after his death. The proposal was a breakthrough historical starting point (and a misnomer) but didn’t fully bear out in mechanistic studies. Fenton implicated iron species as the main pathway radical propagation. Direct reaction of superoxide with hydrogen peroxide would actually unleash singlet oxygen.

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Entire complex biological networks of antioxidants have been evolved to patch the damage of an oxygen metabolism: from superoxide converting enzymes, to radical scavenging vitamins, and phenolic aromatic compounds that can stabilise radical species into a less destructive form. Life’s chemical potential was turbocharged, but with it comes an accumulation of organic corrosions that has to be carefully managed by patchwork immolations until the cell fails.

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…and everything’s on fire.

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The atmosphere is now a great, slow, combustion engine. The steel girding our housing is smelt from the oxidised blood of the ocean. The plants are descendants of microbes which unpicked the water bond through a photocharged spin diode. They support our every breath, but their ancestors chemically obliterated the Earth. The ancients are all but wiped out wherever oxygen could reach. Our lungs fill with a corrosive gas, which only our mitochondria —internalised from an entirely different lineage— can burn with any modicum of control. A spin barrier ensures that this triplet oxygen, contributing the reactive fifth of our atmosphere that allows life to tick on, is not cause for immediate incineration. But now, living under an oxygen-rich sky, under the dictates of a rust-based order, still the law: all things decay and age.

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Oooooxxxyyygeeeennnn

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— To Be Kind, Swans

\n", "date_published": "2026-04-02T00:00:00+11:00" }, { "id": "https://keiran-rowell.github.io/sulfur/2026-01-18-fire-and-brimstone/", "url": "https://keiran-rowell.github.io/sulfur/2026-01-18-fire-and-brimstone/", "title": "Blue Fire and Yellow Brimstone: the world's waning appetite for hellish sulfur", "content_html": "

Photo Credit: Candra Firmansyah [CC BY-SA 4.0]

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Content notice: child labour, exploited labour / slavery, the Mafia, smog deaths.

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It’s just smoke. Floating over the volcano
\nIt’s just smoke. Go on, you know I can’t say no

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— “Smoke”, Caroline Polachek, 2023

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From hell, life: sulfur eating microbes

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In Earth’s hellish Hadean eon, the embers of ancient modes of life were kindled around geothermal vents, in the deep dark of the sea floor. Billions of years before photosynthesis arose as a way to nurture life from the sun, life found a way to turn the boiling chemical churn of these cauldrons into a stable energy source. The sulfurous nature of these volcano plumes was crucial to the chemotrophy of these Promethean life forms; they could fuel themselves through stripping electrons from this chemical soup.

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\"Underwater

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Archaea at the hydrothermal vents of the Giggenbach underwater volcano 800 km North-Northeast of New Zealand’s North Island. 2005, NOAA Vents Program

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4 billion years later, far removed from the sulfur-eating microbes that still cluster around deep-sea vents, humans toil at the base of volcanic craters to harvest the same element for their own purposes.

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Mountains of fire: Ijen’s sulfur volcano

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In the Indonesian archipelago a mining crew descends before dawn into the bowels of volcano Ijen to start their work. They are often surrounded by the hellfire of the element they are extracting: flames that burn not devilish red but ghostly blue.

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\"Ijen

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Blue fire spewing forth from the sulfur mine around the Ijen volcano complex. Trian Wida Charisma [CC BY-NC 2.0]

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These “soldiers of sulfur” are here for emas setan — Satan’s gold. Toxic hydrogen sulfide (H2S) and sulfur dioxide (SO2) gases risen hot from the caldera, crashing into the cooler atmospheric air mass. The triggered chemical reaction and phase change is so rapid that sulfur leaves its bonding partners and immediately solidifies. Yellow crystals of the volcano’s breath condense at its base. Most Sulfur mines are now museums for tourists, but were once valuable deposits of the element that fuelled the birth of the British chemical industry finishing the textile trade. Not too long ago, any sulfur deposits were once worth going to war over to guarantee cheap supply.

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An infernal gateway: sulfur and the Sicilian Mafia

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In 1901 a crew of boys gathers at one of the hundreds of entrances to the sulfur pit mines dotted across the Sicilian province. They are the carusi, the young lads, and their masters are enforcers there to protect the private affairs of the local famiglia, the nascent Sicilian Mafia. The feudal barons had been replaced by territorial landholders just a generation before, when the Sicilian island had been annexed in 1861 to form the unified Italian state.

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Six million years prior, the Messinian salinity crisis created the bedrock opportunity for industrial wealth in towns alongside the strait. The evaporating Mediterranean salt water left behind gypsum (CaSO4) crystals that dissolved in the rain to pool in crannies without much oxygen. There, sulfate-reducing bacteria stripped electrons from organic matter and donated them to the sulfur oxidised in sulfate to fuel their own metabolism, leaving behind limestone and native sulfur:

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\n \n 1.5 CaSO4·2H2O(s) + CH3COO-(aq) + H+(aq) → 1.5 CaCO3(s) + 0.5 CO2(g) + 5 H2O(l) + 1.5 S0(s)\n \n
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\"I_carusi\"

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‘I carusi’, Onfrio Tomaselli, 1905

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Techniques remained primitive. All mining had to be done by pick — explosives risked combustion of the brimstone they were gathering. Working without clothing was common, due to the heat and irritation. Over 400 deaths left records throughout the 1800s, due to flooding, collapse of shoddy mine shafts, or choking on sulfur dioxide fumes. The conditions were so bad that contemporary observers described the organisation to be as cruel as the chattel slavery occurring in the United States.

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I am not prepared just now to say to what extent I believe in a physical hell in the next world, but a sulfur mine in Sicily is about the nearest thing to hell that I expect to see in this life.

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— The Man Farthest Down, Booker T. Washington, 1911

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The conditions of the Sicilian mines are still inscribed in the memories of the Sicilian diaspora to the USA, and a bracing film depicting a child miner’s life in the Floristella mines

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The Mafia didn’t instigate these mines, organised sulfur mines had been built in Sicily by the Roman Republic. While sulfur was also widely used by other ancient civilisations (Greek, Egyptian, Chinese), Sicily held unique advantages with its tranquil golden vein of native sulfur. Other ancients had to visit active tectonic sites — like the Greek Island of Milos, or the Aoelian island of Vulcano in the Tyrrhenian sea. China, in contrast, by force of necessity developed complex alchemical ‘roasting’ processes to ‘sweat out’ sulfur from surface pyrites. Liberating the element from its ore and unleashing on the world another innovation of death — gunpowder.

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Hellfire: sulfur in gunpowder and Greek fire

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Sicilian sulfur was a coveted Medieval commodity: it provided the low temperature ignition of gunpowder, without which there was no thunderous crack of the cannon. A more closely guarded sulfur recipe was for Greek Fire. The Byzantine Empire’s naval flamethrower was devastating — it stuck to everything it was sprayed on but could not be extinguished by water.

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\"Greek

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“The Roman fleet setting fire to the enemy fleet [led by Thomas the Slav]”. Madrid Skylitzes, manuscript on the Byzantines for the court of Palermo in Sicily, 12th century CE

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The mixture for this exact hellflame is lost to history, but early sources for similar concoctions hint at sulfur’s inclusion in this demonic incendiary.

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[Automatic fire is composed of equal parts of:] sulfur, rock salt, ashes, thunder stone, and pyrite, and pound fine in black mortar, at midday sun. Also in equal amounts of each ingredient, mix together black mulberry sap and with Zakynthian asphalt, the latter in liquid form and free flowing, resulting in a product that is sooty coloured.

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— Kestoi [Fragment D25—Spontaneous Combustion], Sextus Julius Africanus [William Alder’s translation], c. 220 CE

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Yellow mammon: sulfur’s economic inversion by the petrodollar

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Today, oil refineries pay to have sulfur hauled away. The world can’t consume it faster than stockpiles mount as a waste product of the modern petrochemical industry. Even though sulfuric acid is the most produced chemical by volume, and vulcanisation transformed mechanical goods, nowdays sulfur often commands a negative spot price. From the ravenous fossil fuel gluttony of our combustion-driven world, sulfur falls out for free.

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\"Sulfur

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Unwanted mountains of sulfur piling up at the Syncrude Athabasca oil sands in Alberta, Canada. Jason Woodhead [CC BY 2.0]

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Sulfur can also be a liability to the petrochemical refining industry. Despite the Orinoco oil belt of Venezuela being the largest in the world, much of the crude is far too ‘sour’ (with sulfur content 6–10x the 0.5% threshold) to be economically viable. Yet this sulfur abundance-turned-burden in our fossil fuels has further unexpected uses.

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This economic inversion is what allows researchers to develop ‘no/low cost’ sulfur sponges cooked in –no joke– spent chip shop oil. The sponges sop up chemical waste, forming sulfur-gold bonds. The cleanup applies to both the highest and lowest tech industries in the world: gold can be recovered both from e-waste, and also the poisoned tailings of so-called “artisanal” gold mining. Artisanal gold mining is an industry of desperation; from alluvial deposits or small-scale ore holdings, rudimentary chemistry is used to extract pure gold. Gold readily forms an amalgam with its liquid metal neighbour mercury, which can then be collected by hand. Mercury’s boiling point is low (hence its use in thermometers), so it can be evaporated off from the raw gold on any stovetop. The consequences are harsh: artisanal mining alone releases roughly 40% of global mercury emissions into the atmosphere, the mercury vapours are potent neurotoxins inhaled by the miners, and the waterways are spoiled for all life.

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The overabundance of sulfur attests to the sheer scale of petroleum burnt each year. This glut is not a god-given law but an accident of chemistry. If humanity cleaned up its act and decarbonised, sulfur is likely to become a scarce resource again by 2040, with supply stream impacts on our fertiliser and lithium-ion battery production.

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Sulfurous sky: SOx and atmospheric pollution

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\"NYC

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Smog on the front page of The New York Times, photo by Neal Boenzi, November 25, 1966

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Sulfur’s calling card is its acrid fumes. Hydrogen sulfide (H2S) is noxious, notorious for its ‘rotten egg’ smell, and quickly fatal, but the sulfur oxides (SOx) have likely shortened more lifespans. We’ve turbocharged their concentration in the air; through our combustion engine and smelters humanity releases more than twice the suflur dioxide (SO2) emissions than volcanoes. These factories were often near population centres during the Industrial Revolution. Before emissions laws were in place, all it took was a pressure inversion in troposphere to press a smothering blanket of smog down on the cities, as seen in the 1948 Donora death smog, the 1952 Great Smog of London, and the 1966 New York City Smog (pictured above). The response to these hundreds of deaths prompted the legislation of atmospheric protection laws: the 1955 Air Pollution Control Act, the 1956 Clean Air Act, and the 1970 Clean Air Act, respectively.

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Such yellow sullen smokes make their own element. They will not rise, but trundle around the globe. Choking the aged and the meek, the weak…

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— Fever 103°, Sylvia Plath, 1963

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Our industries don’t always out-compete the Earth’s mantle for volume of fumes spewed into the air. Titanic volcanic eruptions can dwarf human activity and shock short-scale climate; in 1816 the eruption of Mount Tambora in Indonesia singlethroatedly caused “the year without summer”. The temperature plunged a couple of degrees, crops failed, and the summer skies in Europe were glum enough to compel Mary Shelley to write Frankenstein.

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Such global cooling effects are temporary; aerosolised sulfates increase the albedo of clouds, making them reflect more incoming heat back off earth. Unlike some of the worst “forever chemicals” of our own creation, sulfate emissions are short lived; lasting only weeks before plummeting to the surface as acid rain. Nevertheless, sulfate cooling is detectable: in 2020 the International Maritime Organisation reduced the sulfur limit of shipping’s dirtiest “bunker fuel” from 3.5% to 0.5%. Abrupt sea surface temperature rises and ocean cloud optical changes could be measured. This incidental inverse cloud seeding experiment does not however provide a tidy geoengineering case study to combat global warming: the respiratory and environmental consequences of runaway (SOx) emissions are catastrophic. The processes fuelling our modern life have us in a bargain with Mephistopheles: cleaning up our air exposes the full extent of our millennia-enduring green house gas emissions.

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An infernal engine: iron-sulfur clusters and bioelectricity

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Sulfur’s changeable electron count makes it useful: it is the non-metal element with the most versatile oxidation state. With a larger outer electron shell than its elemental bunkmate oxygen, sulfur’s third shell provides both readily emptiable or fillable sub-shells (\\(3p\\)). This gives sulfur bonding flexibility akin to an “expanded octet” that’s facilitated by higher-energy states lying closer to occupied orbitals than in most electron arrangements. By the fourth shell the elements are transitioning into metals, with electrons released into conductive valence band ‘sea’ of electrons characterising metallic bonding. Sulfur, therefore, finds use as a controllable electron exchange junction in redox processes. These junctions are made in the form of iron-sulfur clusters.

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\"Iron-Sulfur

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Iron-Sulfur clusters that act as waypoints in the electrical process of life, including the reactive ‘crucible’ of nitrogenase. Tracey A. Rouault & Wing-Hang Tong, “Iron-sulphur cluster biogenesis and mitochondrial iron homeostasis”, Nature Reviews Molecular Cell Biology, 6, 345–351, 2005.

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Electrons —whether generated from cracking an oxygen-phosphorus bond in the ‘battery molecule’ ATP, converted from sunlight by plant photosystems, ingested from Earth’s mineral crust, or swapped during redox in chemotrophy— must be channelled from junction to junction. Life’s electrons don’t typically get to their destination by traveling as free particles; they propagate as waves. This could be through free-space, but the wave is attenuated rapidly over short distances in that medium (around tenths of a nanometer, ~1.7 Å). Even travelling along a ‘non-conductive’ covalent bond is preferable to the vacuum. So, the amino acid linkages in proteins themselves act to wire up the way points (nucleic acids moreso due to delocalised electron sharing between stacked nucleobases).

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Along bonds, an electron’s attenuation distance increases, enough to tunnel from bond to bond in a process referred to by physicists as superexchange. Biological electron transfer is therefore a ‘through-bond’ process, and these quantum particles don’t take just one path: the coupling between donor and acceptor waypoints exhibits a wave’s phase, including constructive and destructive interference. The behaviour can’t be explained purely classically, much like the double-slit experiment couldn’t.

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The cell’s power is conducted by wavicles tunneling along the molecular bonds connecting life’s electrical circuits, reverberating across pathways and extending into dimensions inaccessible to our prosaic wires of extruded copper. Helping us breathe, helping plants turn photons into fuel. Doing their work to power all life through altering the charge density in the metal-sulfur core of these primordial clusters.

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Perhaps these clusters predate—and are a prerequisite for—life itself, states the iron-sulfur world hypothesis. The cell as a waterborne sack to harness electricity. Enzyme cores remodeled from, and still resembling, mineral clusters from the Earth’s crust. Both primates and our aquatic volcano microbial ancestors preserve highly coordinated machinery to build iron-sulfur clusters. That is how crucially guarded nature has kept the production of this molecular crucible.

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From the dribs of lava that pierced into the crust, a chemical soup contributed to the origins of us all.

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… but torture without end
\nStill urges, and a fiery deluge, fed
\nWith ever-burning sulfur unconsumed

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— Paradise Lost [Book I], John Milton, 1667

\n", "date_published": "2026-01-18T00:00:00+11:00" }, { "id": "https://keiran-rowell.github.io/dyes/2025-11-05-aniline-and-the-german-chemical-industry/", "url": "https://keiran-rowell.github.io/dyes/2025-11-05-aniline-and-the-german-chemical-industry/", "title": "From Muck to Mauve: the creation of modern dyes from coal tar", "content_html": "

Photo credit: James St. John. Flickr [CC BY 2.0]

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Content notice: this post discusses the early dye companies of industrialised pre-WWI Germany that were at one point put to horrific wartime use. World War II is alluded to in a literary work that was famously denied a Pulitzer prize in part because of its confronting content, including sections I find indefensible.

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In this post I’m aiming to give a broad-brush overview of how the near-infinite array of colours we see in textiles today were wrought from some of the most murky carbon substances known. This post will focus on purple (specifically, mauveine) but will meander through the history and chemistry of the blue hues.

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I’d like to write more on specific dyes in the future because they have so many fascinating chemical properties. For example, indigo forms from a dimer using the indole group common in biochemistry, placing it within striking distance of tryptophan (amino acid \\(W\\)) in the biosynthetic pathway.

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\"Indigo

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Indigo is made from joining two slightly modified indole molecules. Indole is such a common structure in biology that it is used to make tryptophan, one of the 20 standard amino acids, which can reveal protein folding properties through fluorescence. —Synthesis diagram from Plant Cell Reports, 2016, 35, 2449–2459.

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The development of dyes used to be covered in a (rarely taught) NSW syllabus option, “Chemistry of Art”, that touched on everything from indigenous ochres to transition metal pigments and atomic spectroscopy, but is sadly now gone.

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For those interested in colour more generally, I can recommend “The Secret Lives of Colour”. Its chapters are arranged by colour, making it addictively easy to pick up and dive into each hue. I also recommend “The History of Colour” for an academic history of the conception of colour — Chapter 2 starts with synthetic purple and details the explosion in commercially valuable tints that followed. If the development of industrial dyes interests you, BASF have an account of their history on their website including the “Aniline Women” who worked there.

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Some correspondence by scientists picturing this new cyclical chemical reality is contained in “Image and Reality: Kekulé, Kopp, and the Scientific Imagination” , providing primary sources around the murky question of whether he actually dreamed the snake. Image and Reality gives interesting rapid recount into the depictional development of atomic theory, including the radically clear-eyed early experimental proof by Alexander Williamson who provided, using etherification reactions, his pictorial theory of “an actual image of what we rationally suppose to be the arrangement of constituent atoms in a compound”.

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Williamson’s visual foresight gave a hazy glimpse at perceiving how constant molecular motion created reactivity, and many other scientists saw farther than the peers of their day by picturing the non-empirical.

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Pauvre mauve 🪻

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There’s an abundance of green in the natural world (likely because plant chlorophyll is tuned for reliable photosynthesis), but blues and purples stick out because they are so rare in nature. Maybe that’s why blue is humanity’s favourite colour. The rare examples of natural blue are unusual — bird and butterfly wings have to resort to optical trickery (metamaterials); blueberries get their name and antioxidant health craze because their skin is packed with the crop pigment anthocyanin; the blue warning signal of the poison dart frog doesn’t arise from its toxin (pumiliotoxin), but comes from skin iridophore cells containing stacks of reflective guanine platelets; and lobsters change spectacularly from blue to red upon boiling due to heat denaturing their crustacyanin.

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\"Anthocyanin

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Almost all natural pigments in the blue end of the spectrum (and others) are pigments built from the anthocyanidin skeleton shown above. —Wikimedia Commons, NEUROtiker, 5 August 2008.

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Similarly, there’s an almost complete absence of natural purple. Purple cabbage and eggplant contain the same anthocyanins as blueberries, similarly for violet’s flowers whose colour may be striking but is not colourfast when used to dye fabric. So, when nature was our only source of chemicals, people’s desire to wear the spectrum of ‘cool’ colours was met by crushing the leaves of the somewhat drab woad, and the perennially popular indigo — the name given to both the plant and one of Newton’s seven ‘ROYGBIV’ colours partitioning the rainbow.

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The only concentrated, colourfast sources of organic purple come from the seas. Aplysia sea slugs spray purple aplysioviolin molecules when threatened. Sea hares build this ink up in their glands by eating red algae; stripping the colourant molecule from the algae’s light harvesting proteins and concentrating an aplysioviolin derivative. A painstaking, but more reliable, way to obtain natural purple is to extract ink glands from the Murex sea snail. The ‘Tyrian purple’ dye harvested from these snail glands was, and is, outrageously expensive, fetching around $3,000 USD/g in the 21st century. Tyrian purple is in fact a brominated derivative of indigo, synthesised using a bromoperoxidase enzyme unique to Murex. Scientists still have not confirmed the active metal within the Murex’s bromoperoxidase enzyme (most bromoperoxidases use the rare vanadium), or which gene in the sea snail encodes for this purple producing enzyme.

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Lapis lazuli and other blues of value 🔵

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For making paints and dyes, even the colour diversity found within the minerals of the Earth was of little help; the naturally occurring mineral pigments are prone to dull hues, rapid fading, and discolouration into green (e.g. azurite). Synthetic mineral blue dyes were created in ancient times from copper silicates (mixed with calcium - “Egyptian Blue”, or barium - “Han Blue & Purple”), or otherwise just from indigo ‘baked into’ a mineral substrate, as detailed in a Chemical Society Reviews article on “The invention of blue and purple pigments in ancient times”. However, Han Purple did not provide a pure purple, its appearance was due to red copper oxide impurities. Similarly, pure, brilliant blue mineral pigment was only provided by eye-wateringly expensive lapis lazuli (ultramarine), mined only around the mountains on the North-Eastern tip of Afghanistan. Ultramarine would cost up to 100x other pigments, but its dazzling effect was valued in religious iconography across Asia and Europe, and has notable use in famous paintings (e.g. the blue headscarf in Girl with a Pearl Earring). Since mineral pigments and synthetics paled in comparison to “true” ultramarine, lapis lazuli was in demand well into the 19th century. The physical origin of the intense blue appearance of ultramarine is chemically interesting: rather than a metal at the centre of the mineral, the colour comes instead from a negatively charged trisulfur radical (S\\(_3^{\\bullet-}\\)).

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\"Sodalite\nThe negatively charged trisulfur radical caged in a zeolite with its surrounding sodium ions. —Chemical Society Reviews, 2013, 42, 5996-6005 adapted from Inorganic Chemistry 2002, 41, 11, 2848–2854.

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Now with modern chemistry, blue hues can be created that are so striking that an artist can make a multi-million dollar name for themselves with one famous blue dye, and scientific careers and reputations can be built upon chemists’ prowess to manufacture the bluest blue. The chemistry to produce commercially abundant purple dyes was however only cracked in the 19th century, and with it opened up new possibilities in organic synthesis. Even now, there is unplumbed depths to these cool hues: the colour terminology for purple and violet is murky, our visual perception strains near the UV wavelengths, and even the colour range of our computer monitors come up short.

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Porphyrogeniture 👑

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Purple has long signified royalty and power in the ancient world, to the point that its use was restricted to high office; not just due to the expense of natural purple dye, but often enforced by social codes and law. Famously, Roman officials would colour their formal attire (togas) with a strip of purple along the border. This style, the toga praetexta (from Praetor, a magistrate position below Consul) wasn’t solely restricted to appointed officials, but likely had a religious origin and was recorded as being worn by priests and pre-adolescent children of fortunate birth. However, a toga fully clad in purple became restricted to Roman consuls and generals performing a Triumph, and later to the Emperor alone. Nero limited the use of purple to the imperial court, and the unsanctioned wearing of purple was even codified as an offence punishable by death by Theodosius II.

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The association of the purple hue with royalty extended beyond clothing, and Byzantine (Eastern Roman) queens would give birth to future rulers in rooms decorated with a deep-purple Imperial Porphyry stone (from Ancient Greek porphyra — purple). This birthright to distinction through colour was enforced from cradle to grave, as Emperors were buried in tombs made from porphyry, some survive today outside the Archaeological Museum in Istanbul. The purple name echoes throughout history, such as the sisters Zoe and Theodora Porphyrogenita who co-ruled the Byzantine Empire after a popular revolt.

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Born in the purple 🟣

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Ironically, despite the incredible rarity of steadfast purple in nature, purple was the first in a tidal wave of synthetic industrial dyes, bringing the colour to the masses and, as it goes with fads, becoming so commonplace as to be unfashionable quickly after reaching the mass market. The discovery of artificial purple dye (mauvine) was entirely accidental, but the chemical knowledge to synthesise it at massive scale required unravelling some of the fundamental mysteries in organic (carbon-based) chemistry; in the process clarifying how electrons create molecular structure and govern the interactions of dye molecules with light.

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The first artificial purple dye was synthesised by William Henry Perkin in 1856 from an accidental side discovery. Perkin’s motivation was not encolouration, but empire. What Perkin was searching for was a cheap synthetic route to quinine, an antimalarial substance isolated from cinchona tree bark, useful for stationing soldiers in colonies of the British Empire. Perkin’s synthesis attempts were misguided from the start, having only the empirical formula of quinine (C\\(_{20}\\)H\\(_{24}\\)N\\(_{2}\\)O\\(_{2}\\)) and no idea of its structure, he explored combining two equivalents of an organic starting material whose masses would sum to the total atom count of quinine once oxygenated. In reality quinine’s structure is quite complex and the quest to irrefutably create it from scratch kicked off 150 years of investigation and controversy into its difficult total synthesis. Incidentally, quinine is the source of tonic water’s ultraviolet glow under blacklight.

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What Perkin saw in his experiment was not emissions de-energised from the ultraviolet, but instead found that when diluting his failed synthesis product with alcohol during wash-up the solution turned purple. He opened up a dyeworks in Greenford the next year. The success of a synthetic dyes would lead to a complete collapse of the economic harvesting of natural dyes such as madder root.

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\"Sample

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Perkin’s mauve was a runaway success but through its overuse in fashion it came to be see as tawdry. The use of the gaudy colour fell out of favour for everyday wear. —Journal of the Society of Dyers and Colourists, Nov. 1906, via the Science History Institute .

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However, Perkin’s happy discovery at 18 years old started from being encouraged to experiment with the idea of synthesising quinine by August Wilhelm von Hofmann while working as his assistant. Perkin continued his experiments out of the lab, and in his own shed, in order to trial creating proper dyes and keep the profits. The raw organic feedstock to make these dyes became available at a commercial scale due to breakthroughs in organic synthesis and industrial processes to make flammable gas from coal tar. Though able to unlock amazing synthesis routes, Perkin, Hofmann and chemists were working on organic materials without knowing their true atomic connectivity. The substances they worked with were termed “aromatic” compounds due to a few representatives having a strong smell (e.g. benzoin, in incense), but the molecular structure of these aromatics was a puzzle from only their empirical formulae and reactivities.

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Muckraking ⚗️

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Coal tar’s shimmering opacity hides its chemical complexity. In the murky pitch black mixture are thousands of different hydrocarbon species, and until modern petroleum extraction they were the major feedstock to synthesise organic molecules. The stodgy goop’s appearance obscures a mosaic of complex carbon skeletons, ripe to transform into a rainbow of complex organics furnished from chemical reactions. August Wilhelm von Hofmann was one of the pioneers in these experiments, considered by science historians to be one of the “fathers of synthetic chemistry”. He and Jamaican born chemist John Blyth were the first to use the term “synthesis” in a publication.

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Hofmann’s interest in coal tar can be seen by the title of his first publication in 1843 “Chemische Untersuchung der organischen Basen im\nSteinkohlen-Theeroel” (“Chemical Investigation of the Organic Bases in Coal-Tar Oil”). Hofmann was able to isolate both aniline (the basis for dyes — mauvine was also called “aniline purple”) and quinoline (a degradation product of the quinine — Perkin was also interested in finding) from coal tar.

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Experiments with coal tar and other organic synthesis projects were a productive foundation for Hofmann to build a research career on. The structure of Hofmann’s lab is seen as the precusor to the modern industrial chemistry research lab and was a source of many discoveries, such as the isolation of benzene (then benzole) by his British student Charles Mansfield. Additionally, Hofmann was the first to build physical ball-and-stick “glyptic formulae” models to display an object to embody the atomic connections inside molecules. His croquet ball colour choices of black for carbon, white for hydrogen, and red for oxygen remain in wide use today.

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\"Hofmann's

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August Wilhelm Hofmann, “On the combining power of atoms”, Proceedings of the Royal Institution of Great Britain, 1865, 4, 401–430

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Despite Hofmann’s lab being in Imperial College London (he moved to the University of Berlin in 1865), and Perkin being the first to commercialise aniline dyes in Britain, this nascent dye and chemical industry was to come to be dominated by Germany, largely due to economic and technological factors. A key factor was the industrial production of synthetic liquid and gaseous fuels from coal tar, invented in Germany in 1920s. As Germany had vast coal deposits and almost no natural petroleum reserves, this Fischer-Tropsch synthesis of fuel from coal tar was crucial for the energy, transport, and military viability of Germany. By making huge quantities of syngas from coal tar deposits and building industrial chemical plants, the feedstocks and techniques for dye creation were linked to plentiful industrial production in the Rhineland.

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The economic advantage from industrial scale organic synthesis far outgrew the impact of only dyestuffs. A letter to Nature in 1934 puts it plainly:

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“WHILST the original discovery of a coal-tar dye was made by an Englishman… The consequent decline of the British coal-tar colour industry was already well marked in 1875, and in 1886 had proceeded so far that 90 per cent of the dyes then used in Britain were of foreign manufacture… It is not an overstatement to say that the development of this highly scientific and extremely profitable industry in Germany instead of in Great Britain had enormous, if not decisive, political and economic effects both before and during the War.”

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Enclosed with this submission to Nature is an additional letter where John Brunner (co-founder of a chemical firm that would later merge to become Imperial Chemical Industries) opines in 1915 that, if he and his chemist brother had money in their ‘teens’, their fascination with aniline dyed silk skeins would have driven them to keep the coal tar chemical industry in Britain.

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Coal tarred and feathered 🪨

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The importance of the profits from dyes in the founding of chemical companies can be seen in the names of industrial giants which are still around today, for example: BASF - Badische Anilin- und Sodafabrik (Baden Aniline and Sodium Carbonate Factory). BASF remains the largest producer of chemicals in the world: a multinational across 80+ countries with an annual revenue around €70 billion. Its history has touched on many chemical process famous in school curricula: BASF’s main trade was in dyestuffs until the 1900s; the rise of this company was meteoric after Haber and Bosch invented their method to turn nitrogen from the air into fertiliser; BASF invented polystyrene in the 1930s; and the sodafabrik part of BASF is infamous to New South Wales high school students who had to memorise the conditions of the Solvay process as part of their Industrial Chemistry module.

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Under Nazi rule before and during WWII, BASF would be united with other industrial titans into the chemical conglomerate I.G. Farben (loosely, “The Colour Cartel”) of notorious and horrific historical fame. This conglomerate was founded by Carl Bosch through merging BASF with Hoechst (now merged under Sanofi), Bayer (widely known for aspirin), and a few other smaller chemical companies. After WWII, IG Farben was seized by Allied forces — its forced labour operations inside Europe were shut down; many executives responsible for this deliberate and callous mass cruelty, industrially exploiting human lives, received light sentences at the Nuremberg trials that were not considered to match their crimes. The IG Farben conglomerate split up into its constituent components.

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In the reconstruction period, however, several of these companies such as BASF were refounded and allowed to rehire many of the same directorial staff who had served at the behest of depraved, horrific warmongers. These challenges to societal and base human conscience, the reshaping of entire industries, the discovery of new medical compounds (antifungals and antibiotics), and unfurling the spectrum of dyes for fabrics of all colours, resulted from an insight into the puzzling carbon skeleton structure of these newly synthesisable “aromatic” substances.

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Benzene: The daydream snake 🐍♻️♾️

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Benzene is a molecule, whose empirical formula defied the trends determined from other hydrocarbons. It took 40 years from the experimental isolation of benzene to solve its chemical structure. It clearly had a lower H:C ratio than “saturated” hydrocarbons, yet did not easily accept addition of new atoms like unsaturated hydrocarbons, and its boiling point just didn’t fit with other organics that matched its mass. This puzzled chemists even after they nailed down the true atomic weights of elements and were sure of the empirical composition of molecules. Any chemistry student will tell you it’s a hexagon. Benzene-as-a-hexagon is so commonplace to modern chemists that you can even buy hexagon stencils to help you breeze through your organic chemistry homework. But how was this shape arrived at?

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The story of the solution to puzzling structure of C\\(_{6}\\)H\\(_{6}\\) is one of the few times instructors can retell a founding myth; like Romulus and Remus being raised by wolves then raising Rome, or Mendeleev playing atomic solitaire until the cards of the Periodic Table just fell into place. (Though I’m favourable to alternative depictions, or spiralling out once in a while). Chemistry teachers get to share this dramatic origin story — a scientist called August Kekulé simply dreamed of benzene and woke up with the solution!

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In more detail, while dozing off on the last bus home from a friend’s place in London, Kekulé visualised chains of atoms in a whirling dance (at that time chemists were still figuring out how to apply recent studies by physicists on particle motion to their empirical chemical formulae). Suddenly, one chain of atoms seizes its own end, much like a snake biting its own tail. Kekulé awoke from his reverie and the structure of benzene was solved!

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Did Kekulé truly have a sudden prophetic dream solution? It’s impossible to verify someone’s recount of their internal thinking, and some have speculated on this story being embellished in order establish a legacy due to being retold only many years later by Kekulé at the “Benzolfest” in honour of his contributions to chemistry. It was unlikely a true bolt from the blue, as a more dramatic telling to first year chemistry students might go. These dreams often clarify conscious suspicions — organic reactions was Kekulé’s profession and solving structures a waking obsession. A defense of it truly being from a dream has been argued from studying contemporary letters: in his time nobody refuted either Kekulé’s primacy claim on clearly proposing a symmetrical hexagon carbon ring, or his disposition to inspiration from visual daydreams. It certainly didn’t stop Cormac McCarthy from using the story as a launching off point to argue that language is a virus (I find Laurie the most persuasive here).

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Of course, a snake grabbing its own tail is not an invention of Kekulé, the Ouroboros is an ancient and world-spanning icon symbolising eternity. Interestingly, the ouroboros has a direct throughline to chemical history through its use as a sygil in alchemy. It expressed the unity of all types of matter, with the ultimate project of course being the profitable demonstration of mastery over matter by gaining the ability to turn lead into gold.

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\"Ouroboros

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The ouroboros has a long history, including a connection to alchemy where it is illustrated here in relation to chrysopoeia (“gold-making”) through balancing “one in the everything”. —“Chrysopoeia of Cleopatra the Alchemist”, pseudonymous, 3rd or 4th century CE.

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The dreamsnake again rears its head (and heads its rear) in the novel Gravity’s Rainbow. The WWII IG Farben chemical colour cartel functions as a constant ominous industrial spectre stalking the book. It tragically mirrors how scientifically and commercially pioneering chemical companies were wrested into the control of war leaders to commit inhuman acts — the ‘Them’ of the military-industrial complex that haunts the book’s entire narrative. Kekulé’s serpentine solution to benzene’s structure is repurposed as a metaphor of the economic boon and destructive cycle unleashed by the WWII war economy:

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“Kekulé dreams the Great Serpent holding its own tail in its mouth, the dreaming Serpent which surrounds the World. But the meanness, the cynicism with which this dream is to be used. The Serpent that announces, “The World is a closed thing, cyclical, resonant, eternally-returning,” is to be delivered into a system whose only aim is to violate the Cycle. Taking and not giving back, demanding that “productivity” and “earnings” keep on increasing with time, the System removing from the rest of the World these vast quantities of energy to keep its own tiny desperate fraction showing a profit: and not only most of humanity—most of the World, animal, vegetable, and mineral, is laid waste in the process…

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No return, no salvation, no Cycle—that’s not what They, nor Their brilliant employee Kekulé, have taken the Serpent to mean. No: what the Serpent means is—how’s this—that the six carbon atoms of benzene are in fact curled around into a closed ring, just like that snake with its tail in its mouth, GET IT?”

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— Gravity’s Rainbow, Thomas Pynchon, pg. 412, Penguin 2nd Ed.

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The connection to colour and dyestuffs in the novel does not end there; the essay “Coloring Gravity’s Rainbow” applies a critical lens to these chromatic and chemical connections and charts how they shift throughout the book. The protagonist’s very name —Tyrone— could interpreted to be linked with Tyrian purple, his colour depiction alters as the narrative shifts towards an annihilating white, and the obsessive subjugation of nature through nomenclature and technology is linked explicitly to the terms of the organic chemistry industry.

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The connection between the dye industry and the ability to wage war was already well noted in the period between the world wars, as an American industrial pamphlet from the 1920s declares:

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“Who makes dyes today can tomorrow make high-explosives”

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—“World Disarmament and the Master Key Industry”, American Dyes Institute, 1921.

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How ring stability enables destruction and cures 💥💊

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The lack of reactivity of benzene was puzzling to chemists ever since it was first isolated. The high C:H ratio entails locations where carbons are double-bonded to themselves, leaving an opening for positions for where atoms could be attached if the double-bond could be broken during an addition reaction. It is now clear that the carbon rings in benzene and benzenoids are unreactive because their bonding arrangements are already stable. However, that does not mean benzene derived compounds are inherently inert. Quite the opposite, the core stability in the centre of benzenoids has led to the creation of molecules that either have phenomenal longevity or massive destructive power depending on what’s attached to the ring. These aromatic molecules have become notorious globally in their own right, with entire departments required to mitigate their danger to human life.

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\"TNT,

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(In)famous aromatic structures that led to: (TNT, left) a profitable explosive and arms company that funded the creation of the Nobel Prize, with Alfred Nobel believing in a 20th century version of mutually assured destruction, earning himself a mistaken obituary claiming that he “could hardly pass as a benefactor for humanity”; (DDT, centre) the founding of the US Environmental Protection Agency under pressure from an exposé on pesticides; (PCB, right) industrially valuable capacitor materials, that nevertheless are carcinogens that have persisted after their global ban by the 2001 Stockholm Convention.

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The reason for the proliferation of organics containing these aromatic rings is the central hexagon (or more rarely pentagons and other polygons) supplies a scaffold that provides more lateral directions to build out the bonds of a molecule. Small aromatics are also readily bioavailable, and hence frequently absorbed as toxins or cures (indeed many of your own key biomolecules contain aromatic rings). Ring systems are everywhere in pharmaceuticals. Small organic molecules containing ring scaffolds describes many types of drugs including: opioids, painkillers, psychotropics, antibiotics, anti-inflammatories, etc, etc.

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\"A

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An example of how often different rings occur within a 3566 compound dataset (they can occur multiple times in a structure). — Roughley, S.; Jordan, A., “The Medicinal Chemist’s Toolbox: An Analysis of Reactions Used in the Pursuit of Drug Candidates”, Journal of Medicinal Chemistry, 2011, 54, 10, 3451–3479

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The birth of “resonance” and fictitious intermediates ⌬ ↔ ⏣

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So, what did Kekulé’s benzene look like? While it’s commonplace to call the drawing of a hexagon with static alternating single and double bond lines the “Kekulé structure” to distinguish it from the “Dewar benzene” that is bifolded by a single-bond through the middle (which can exist, and Dewar also didn’t back the structure that carries his name), these chemists were too empirically-minded to be drawn on a fixed depiction. What is clear instead from the drawings in Kekulé’s papers is an unsatisfied valence in a C\\(_{6}\\)H\\(_{6}\\) chain which is only satisfied when the chain is closed by attaching back to its beginning.

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\"Kekulé's

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Kekulé’s schematic of benzene illustrates a sense of overlap, but doesn’t accord with current ways of representing organic molecules. —“Sur la constitution des substances aromatiques”, Kekulé, M., Bulletin mensuel de la Société Chimique de Paris, 1865, 98–111.

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The oblongs were not meant to show that the molecule literally lay in a straight line, but what the sausage atoms (literally wurst in his articles) were meant to represent was contemporary chemical thought which held the valency property of an element was due to multiple subparts each capable of forming a connection. This is clearer in other diagrams where these particles of ‘atomicity’ are drawn like they’re wrapped in a sock to form the overall element.

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While purely a trick to be able to depict the multibond connections available to these elongated multivalent elements, the dimensional limitation of these sketches seems to have led Kekulé to some mistaken structural assignments, such as initially assigning a triangular form of benzene to achieve chain closure. However, other organic chemists immediately picked up on this based on experimental evidence of 6-fold positional symmetry (additions to the benzene ring creating isomer patterns that are unique to a regular hexagon). Regardless, Kekulé derived the need for complete symmetry in the ring from carbon’s tetrahedral tetravalent bonding. His supposition was not of alternate structures in equilibrium, but that relationship of each carbon to its ring neighbour was held together in equal spacing, mediated by oscillating collisions with two neighbouring carbons and one hydrogen. Kekulé’s daydreamed timescale was likely intramolecular vibrational, not reactive, nor electronic (since the electron was only discovered the year after he died).

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The importance of the number of carbon atoms (often six) was recognised by early chemists as well. Armit and Robison popularised the term “aromatic sextet” through many persuasive molecular examples that differed greatly from benzene but still maintain rings of six delocalised electrons. However it was Ernest Crocker, a scientist who never pursued the “Doctor” title, who got there first. He did this by applying the valence electron “octect rule” to all atoms in benzene with Lewis-Langmuir bonding and accounting for the aromatic stability from the remaining six electrons. He places the electrons clearly “outside” the ring due to charge considerations, but states that: “electrons are continually vibrating in the plane of the ring… The net physical effect… serving to bind together all carbon atoms present into virtually a single atom.”

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These days, it is common to introduce students to benzene’s cycle through a smearing of two “resonance” structures. We come to a resonance structure due to a dogged adherence to Lewis bonding and “formal charges” as a pedagogical accounting tool for building up (“aufbau”) bonding lines from atoms. Two valid Lewis structures, with alternating double bonds, are drawn with “resonance equilibrium arrow” between them (though the reality being “1.5” bonds is generally acknowledged). So, how did we arrive at two so-called ““Kekulé”” structures of benzene if the early pioneers and experimental evidence evinced a complete symmetry in the nature of benzene?\n\"Benzene

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The two “resonance contributors” of benzene, where the single and double bonds are shown to alternate in viable positions.

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Likely the instructional ease of drawing Lewis bonding structures helped the misnomer “resonance” stick. I’ve labelled it a misnomer as there is not a “resonation” allowing for two nuclear or bonding position extrema to interconvert in equilibrium; benzene in an unaltered state is a purely symmetrical (\\(D_{6h}\\)) flat hexagon. What the resonance concept does is bring the concept of energetic degeneracy (i.e. multiple quantum states that have identical value upon measuring a specific observable) into the chemical drawing system learned by practising chemists. The adoption of this useful fiction was turbocharged by populariser extraordinaire: Linus Pauling.

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Linus Pauling ran an exposition campaign to bring the insights of the new quantum mechanical theories of bonding to chemistry departments. While Pauling’s explanations focused on chemical conceptual accessibility rather than the rigours of strict physical chemistry research, this approach was widely popular. It was couched in a visual process that unlocked the conceptual basis of structural chemistry in a way that could be latched on to by scientists designing organic synthesis reactions. While Pauling may be responsible for popularising electron delocalisation concepts in language and rules palatable to the synthetic chemists of his day, Fritz Arndt may be the source for the double-headed “resonance equilibrium” arrow that can mislead some chemistry undergraduates. Arndt published papers expounding on a “Zwischenstufe” (intermediate state) in resonance stabilised structures, though to Arndt’s credit he never committed to an oscillating model, or timescale, and emphasised purely electronic bonding rearrangement in a pre-quantum mechanical explanation. Decades later Pauling’s prestige, accumulation of credits, and renowned teaching skills would generate a best seller, popularise a dubious concept, that led to a lot of wasted money and time despite a bongo accompanied ode to orange juice from an equally lionised scientist going through his final days.

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The Soviets, however, were having none of it. Resonance structures were decried as “anti-realist” and antithetical to the true teaching of dialectical materialism. The fictitious mathematical idealisation of real atomic positions was viewed with suspicion, and the Soviet chemists adopted a terminological compromise, with the mathematical constructs themselves considered more palatable to a realist philosophy than these bourgeois anti-realist “resonance hybrid” notions. To object to such notions is not surprising, as they clash with the official Marxist-Leninist views of a state that held Engels as an authoritative expert on geometric cosmology.

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“A motionless state of matter is therefore one of the most empty and nonsensical of ideas – a “delirious fantasy” of the purest water.”

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— Anti-Dühring: Philosophy of Nature. Cosmogony, Physics, Chemistry, Friedrich Engels, 1877, Part I, Section IV.

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Resonance joins a pantheon of scientific concepts which at one time or another were suppressed by the Bolshevik state, including: Darwinian evolution; Ernst Mach’s philosophical moves from Netwon’s clockwork universe to one that permitted relativity in motion but also sensational experience; early cybernetic thought (which appears to have been shucked down into systems theory in STEM fields, while cybernetics remains a domain of HASS departments); non-Pavolvian physiology; and interpretations of wave-particle duality.

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It should be conceded that, while formally fictitious, the concept of resonance hybridised orbitals played an outstanding explanatory role in revealing bonding behaviour in typical aromatic organic molecules. It has been argued they can still be considered a “correct” representation of aromatic molecules (particularly of charged delocalised tautomers) since dominant resonance contributor representations can be constructed from wavefunction-based molecular orbitals by applying natural bonding orbital analysis.

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The electronics of dyes ⚛️

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Planar, hexagonal benzene geometry was key to advancing the quantum mechanical explanation of its bonding pattern. The six “remainder” electrons after accounting for co-valent bonding do not lie “outside” the carbons. Instead, contrary to the Crocker proposal, the electrons ‘circulate’ both above and below the ring at once. In fact, in the plane of the ring is the one area the “remainder” carbon electron orbitals cannot be due to a node of density absence caused by the angular momentum of the orbitals. Angular momentum is one of the few properties that an electron can have, being particles otherwise indistinguishable from one another. These remainder electron orbitals are commonly described as hybridised — another concept Pauling intuited and popularised from the mathematical constructs created by the likes of Slater. The outer orbitals of carbon each have a one angular momentum “slice” through them, making it a \\(p\\)-orbital (from now-nonsensical vestigial spectroscopic notation). In the hybridised view, all 4 carbon valence electron arrange into the tetrahedrally splayed \\(sp^{3}\\) hybrid orbitals to join with 4 neighbours. The cyclic chains in aromatic systems instead permit for 3 \\(sp^{2}\\) hybridised carbon covalent bonding orbitals, with the “remainder” \\(p\\)-orbitals supplying ‘free’ electron density whose adjacent densities can join in what is called a \\(\\pi\\)-bond. Because electrons do display wave-particle duality, these ‘wavicles’ can add their amplitudes because their wavefunctions are in phase. \\(\\pi\\)-bonds provide typical double bonds, but what makes aromatic bonding special is continuous, cyclical, \\(\\pi\\)-bonds tracing a ring an electron can flow around.

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Chemical aromaticity arises due to a planar cyclic \\(p\\)-orbital system that can delocalise to form a π-electron system around a closed ring. —LibreTexts Chemistry 15.9: “What Is the Basis of Hückel’s Rule?”, Tim Soderberg (University of Minnesota, Morris)

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Hückel solved the quantum mechanical bonding rules of benzene, even as “resonance” was being bandied around, using two constants: \\(\\alpha\\) – the energy of a \\(p\\)-orbital, and \\(\\beta\\) – the stabilisation gained by allowing adjacent \\(p\\)-orbitals to delocalise into a \\(\\pi\\)-orbital. This is because electron position and momentum observables follow the Uncertainty Principle; they multiply together to some (Planck) constant. A highly constrained electron has very low position uncertainty, meaning broad uncertainty in its momentum, allowing a high kinetic energy value. Conversely, the delocalisation stabilisation energy \\(\\beta\\) occurs because the electron density is spread over a large position space in an extended \\(\\pi\\)-orbital, lowering the kinetic energy of the system (\\(\\nabla^{2}\\)).

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It turns out six isn’t the sole magic number allowing an “aromatic” ring of delocalised electrons, nor must every atom strictly be carbon (if \\(p\\)-orbitals are still available after covalent hybridisation). So “aromatic” rings can be created whenever \\(\\pi\\)-orbital electrons can loop back on themselves. This leads to the common rule-of-thumb in chemistry.

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Delocalised “aromatic” systems arise in cyclic arrangements of atoms, where \\(4n+2\\) \\(p\\)-electrons are “shared” between all atoms in the ring; \\(n=1,2,3...\\)

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In fact the overlapping orbitals can be obtained from unusual geometric configurations, even highly exotic twists like Möbius aromaticity. More commonly, several smaller rings are joined together to make one large delocalised electron system. A famous such aromatic scaffold is the porphyrins: four pentagonal nitrogen-containing heterocycles in conjugated linkage, and so named because of their red-purple hues. Research into the physical basis of “aromaticity” is still a hot field in chemistry, just this year scientists made a massive 18-porphyrin ring that displays current across 242 electrons running through the entire system.

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Understanding the electronic basis of benzene’s bonding explained its unreactivity through aromatic stabilisation. But it also explained something more visible: colour.

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Tuning the resonant hue 🎨

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The origin of the colour of aromatic dyes can be studied when two aromatic rings are strung together with a conjugated carbon chain. This chain provides the length along which the electron wavefunction can form standing waves. This behaviour is effectively a complicated particle-in-a-box constraint. Consequently, lengthening the conjugated chain lowers the frequency of the electronic wavefunction. But for colour what matters is how this electron wavicle interacts with light. A photon can excite the electronic wavefunction from the highest lying resting state, to next available (‘virtual’) standing wave, provided the energies match. In the process the photon is absorbed, leading to the observation of colour due to a spectral gap in the white light reflected from an object. Longer chains will have a smaller occupied-virtual gap, shifting the absorption to redder light and making the dye appear the complementary colour (bluer). Thus, for calculating the colour of dyes, what matters is the gap between the highest occupied electronic state and the next available lowest energy ‘virtual’ state — the excitation energy.

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Seth Olsen (vale) was an Australian theoretical chemist, and like many physical chemists the excitation energies of dyes fell inside his many interests. Relevant here is his work using modern computational modelling with molecular orbital theory to provide predictive explanation of Leslie Brooker’s rules of resonant dye colour absorption discovered in the 1940s. Brooker studied dyes with two aromatic ring systems linked by a simple conjugated carbon chain of different lengths. In the process, Brooker discovered a relationship when he contrasted dyes that were symmetrical (both ends the same), versus asymmetrical dyes. The relationship was this:

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Dyes with asymmetric ends absorb at a photon wavelength at the mean value of the two symmetric parents, but deviating by a blueshift. This deviation is related to the different abilities of the end rings to pull electrons from the central chain (their “basicity”). The larger the difference in basicities, the greater the blueshift deviation from the simple average.

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\"Brooker's

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The asymmetrical dye deviates from its “resonant” isoexcitation point — the average of the two symmetric parents. This deviation is directly linked to the electron attractive nature of the end ring systems. — Olsen, S.; McKenzie, R., “Bond alternation, polarizability, and resonance detuning in methine dyes”, The Journal of Chemical Physics, 2011, 134, 114520

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Brooker’s Deviation Rule was worked out experimentally within Kodak labs, decades before computational chemistry methods could be used to find explanations. Seth Olsen and Ross McKenzie showed this empirical deviation rule can be explained by molecular orbital theory, though the orbital active spaces involved go well beyond the simple Hückel picture given here, even for classic green and blue dyes. One key result is that the one-electron Brooker basicities can be treated as a constant for each aromatic ring end, regardless of what other ring it is conjugated to, because the localised frontier orbitals remain unchanged. This greatly simplifies the prediction of the excitation energy, meaning the colour of any such dye can be predicted if the constants of each ring system is known — without the need to actually construct them and measure physical values.

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By providing a representationally simple two-state model of dyes, these studies yield interesting results for the oxonol dye chromophore housed inside Green Fluorescent Protein (GFP), the most widely used bioluminescent probe in molecular biology experiments. These include that the prevalent form of the dye is the anionic one (since the resonant colour is more easily tuned to fit evolutionary pressures), and the existence of an unobserved ‘dark’ state in GFP absorption.

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The molecular orbital picture of dyes can get stranger still. Take azulene, a vivid dark blue isomer of naphthalene. While azulene has been isolated as a dye for hundreds of years by steaming chamomille, research in 2023 reveals that the molecule’s excited state has unique anti-aromatic ring currents which makes it disobey a long standing principle in photochemistry — azulene’s photon emission is not from its lowest lying excited state (Kasha’s rule) but from the 2nd excited state. More widespread but no less entrancing is the class of compounds that are a byword purple itself: the porphyrins. They often function as the molecular holster for a metal atom: the combinations of the transition metal \\(d\\)-orbitals with the delocalised orbitals extended over the aromatic ring system have beautiful molecular orbital representations, as illustrated below with an iron atom.

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\"CAS(14e,14o)

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Ten active orbitals in the electronic structure of an iron-porphyrin complex typical of haem proteins. Calculated at the CAS(14e,14o) level of theory with \\(A_1\\) symmetry. —“Methodological CASPT2 study of the valence excited states of an iron-porphyrin complex”, Journal of Molecular Modelling, 2017, 23, 53.

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Porphyrins everywhere!

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The porphyrin with greatest recognition is chlorophyll: a modified porphyrin ring holds the central magnesium atom and a poly-terpene hydrocarbon chain extends off as a tail. Chlorophyll is of course the key photon antenna for photosynthesis, and the ultimate input channel of energy into all biological ecosystems, captured from radiation cast off during nuclear fusion. The high school formula 6CO\\(_{2}\\) + 6H\\(_{2}\\)O + light \\(\\rightarrow\\) C\\(_{6}\\)H\\(_{12}\\)O\\(_{6}\\) + 6O\\(_{2}\\) completely hides the spooky resonant action-at-a-distance of absorbing stray photons, channelling their energy to a photosynthetic reaction system, and charging electrons to then drive formation of carbohydrates. Words fail me and I’ll let Drew Berry’s gobsmacking animation of the photosynthesis process for the Walter & Eliza Hall Institute of Medical Research speak for itself.

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We started this colour journey with the abundant green from the chlorophyll of plants, but porphyrins are responsible for far greater share of the rainbow displayed by living organisms. Natural colours from porphyrins include:

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🌈 Bilins, which are “unwrapped” porphyrins. They display interesting light harvesting properties, e.g. in pychobilin (algae). Their colour spans the rainbow range upon chemical degradation.
\n🔴 Haem (as in haemoglobin) the source of unmistakable red of oxygenated blood, which is broken down within 3-4 months by your liver to produce…
\n🟢 Bile, a disconcerting colour to ever see.
\n🟡 Urine, and other bodily fluids. Unless you are unlucky enough to be suffering from…
\n🟣 Porphyria disease where badly formed enzymes causes porphyrin build-up, turning your urine purple.
\n🟤 The brown of egg shells, unless transformed into…
\n🔵 The blue of a robin’s egg from oocyanin.
\n⚫ The petrochemical darkness of crude oil and shale, which are full of geoporphyrins, some of the first clues that petrol has biological origin.
\n🌟 The ‘glow-in-the-dark’ of the Sierra luminous (Motyxia) millipede comes from a porphyrin containing photoprotein whose structure is unresolved (a candidate for AlphaFold prediction?).

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Retinal: in microbes 🦠 and in our eye 👁️

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Life has an even more ancient connection to purple, harking back to rise of the microbes. Bacteriorhodopsin occurs in the “purple membrane” of phototrophic archaea. Similar rhodopsins are found across diverse microbes, including proteobacteria, enabling a protean metabolism — traditional chemical food can be supplemented with a light-driven energy source. The bacteriorhodopsin purple protein operates as a light driven proton pump as animated in this video where the absorption of a photon triggers proton transfers and subsequent isomerisation, causing the protein to change shape.

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The structure of bacteriorhodopsin was solved to atomic resolution using cryogenic electron microscopy by Richard Henderson and co-workers in 1990, who would later go on to win a Nobel prize in part due to that work. This partnership of a photosensitive retinal contained inside a rhodopsin protein is found across all domains of light sensing organisms, including the structures solved this year for the Asgard archaea domain of life.

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\"HemdallR1

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Structures of HeimdallR1 rhodopsin with the retinal chromophore, energy minimised using a QM/MM method. —“Structural insights into light harvesting by antenna-containing rhodopsins in marine Asgard archaea”, Nature Microbiology, 2025, 10, 1484–-1500.

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The purple membrane shares a remarkable similarity with vision — not through direct ancestry but through convergent evolution. Microbial proton-pumps (Type I) and visual opsins (Type II) both adapted to use retinal as the light harvester to conduct their work. Type II opsin molecules are directly evolutionarily related to each other, while Type I pumps show strong structural conservation, indicative of shared ancestry even without much sequence similarity. Amazingly, both use variants of the same photochemistry shown above: rotation around the double-bonds upon light absorption.

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Which brings us full circle. Purple gave nature rhodopsin, allowing microbes to feed off light. Evolution rediscovered this molecular trick for the opsins that now mediate our vision, (watch “Phototransduction: How we see photons” for the best explanation I’ve seen), while our cones allow us to gaze on purple in all its power and write a post of purple prose about it.

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Purple is my favourite colour :)

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Deep Purple is shrouded in mystique, while there’s something spectrally special about High Violet hinting at energies just beyond human perception, pushing into the Ultraviolet.

\n", "date_published": "2025-11-05T00:00:00+11:00" }, { "id": "https://keiran-rowell.github.io/nitrogen-fixation/2023-06-14-nitrogen-fixation/", "url": "https://keiran-rowell.github.io/nitrogen-fixation/2023-06-14-nitrogen-fixation/", "title": "Nitrogen fixation: artificial lightning, bird poop, and how kudzu grew out of control", "content_html": "

Photo credit: Clinton Steeds. Flickr [CC BY 2.0]

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Content notice: this post discusses the history and chemistry that was occurring within the context of the invention of some of the worst chemical agents of World Wars I & II, but doesn’t go into detail. It also mentions a suicide, deaths in an industrial accident, bombing campaigns, and indefinite detention of migrants.

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Nitrogen fixation is a subject I was taught about ever since high school chemistry, where the Haber process was part of the curriculum on equilibria. The book that most influenced this current post is The Alchemy of Air, which details the life of Fritz Haber and Carl Bosch.

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Carl Bosch was an industrialist unafraid to apply the highest pressures and the wildest materials in the search for a new catalyst. Bosch guided the German chemical conglomerates in 1930s that provided Germany chemical supplies (BASF and Bayer were denazified after WWII and thrive to this day). When the Nazis began to dictate to industry, Bosch was removed from his positions for speaking up against anti-Semitism. Fritz Haber was zealously loyal to imperial Germany, developing many horrific chemical warfare gases; a loyalty that horrified his pacifist wife who took her life following Haber’s gas attacks on the battlefield, and ended up meaning nothing to the ascendant Nazi regime which attempted to erase his legacy because he was Jewish. Bosch and Haber were also involved in the early pushes for a synthetic dye industry — a subject of a later post.

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N≡N: the strongest bond in the air

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Nitrogen gas (N2) is the most common molecule in the atmosphere (~80%), yet it sits there inert, each nitrogen atom strongly bound to the other. Because each nitrogen atom requires 3 electrons, the surrounding oxygen atoms can’t easily fulfil nitrogen’s bonding requirements since they only have 2 unpaired electrons, so the N≡N triple bond dominates the air. When nitrogen-oxygen bonds are made (mostly in car engines), the molecules that are formed (NOx) are at the top of the list of atmospheric pollutants.

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Despite N2 being plentiful in the atmosphere, most of the biosphere is nitrogen starved. Nitrogen, in a form that is less strongly bound than N≡N, and hence bioaccessible, is crucial to provide organisms the feedstock for manufacturing the DNA and amino acids that build the complex machinery of cells. This is what fertilisers are for. Despite making up 4/5ths of the air, nitrogen is found in the soil at ~50-100 mg/kg, and this is because only a few bacteria and one industrial reaction generates bioaccessible nitrogen from the air (termed nitrogen fixation). The industrial reaction to break the N≡N bond and form useful chemicals is one of the most energy-intensive reactions invented by humanity, and essential to our survival.

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N≡N + ⚡: the Promethean way to unbind nitrogen

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It was at the beginning of the 20th century that humanity snapped the N≡N bond open through sheer energy, when the Norwegian hydro-electric nitrogen stock company (Hydro) built the Vemork power plant. Vemork’s world-leading 108 MW output was used to generate arcs of artificial lightning that split the surrounding N2 and O2 gases, overcoming the N≡N triple bond strength of 942 kJ/mol (twice that of oxygen gas). The split nitrogen and oxygen gases formed NO gas, which was shepherded along to produce feedstock nitric acid. This arc lightning process was so energy intensive that it was unable to compete economically when the alternative Haber-Bosch process, which requires only pressure and heat, was brought to industrial scale.

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Vemork began producing other chemicals, including separating heavy water, D2O, from water by electrolysis. D2O is H2O whose hydrogens are now deuterium (D) because they are ‘heavier’ by one neutron. D2O was a potential route making nuclear energy and weapons if enough could be stockpiled (neutrons slow down when passing through heavy water, allowing to them to be absorbed into atoms which then become nuclearly unstable). Vemork was the only site that produced heavy water industrially, and while D2O stockpiles were exported to France from then-neutral Norway, Hydro management caved in and agreed to produce new heavy water after the Quisling government surrendered to the Nazis. This launched a sabotage mission by Norwegian resistance ski troops, an operation portrayed in the TV series The Heavy Water War.

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Hydro continues as Norway’s renewable energy company and one of the largest global suppliers of aluminium. Norway’s abundant hydro power is still leveraged by the Norwegian state as the largest generator of renewable energy in Europe.

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Removed a reference to FREYR since the company abandoned plans for battery manufacturing from hydro power.

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N≡N + 🗜️: the brute force way to make NH3

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Fixation of atmospheric nitrogen through electrical arcs was not viable almost anywhere else in the world. Bioaccessible nitrogen was needed to meet the fertiliser demands of mechanised agriculture, and mined supplies of nitrates in the form of niter (a vague historical term for various mineral deposits) or guano (bird droppings, detailed below) were not going to be able to meet demand. Research had been underway for an alternative source of nitrogen, particularly in Germany which lacked niter deposits (a vulnerability that was exposed in WWI when the German navy was blockaded from colonies and foreign trade). Commercially viable artificial nitrogen fixation was developed by Haber, Bosch, and others at BASF in 1909–1910. The development of the Haber-Bosch process for nitrogen fixation was largely down to brute force searching driven by need: a wide array of catalysts were tested (originally Os, later to be replaced by affordable Fe-based surfaces), and the industrial capabilities of the Rhine were used to make the sturdiest metal walls to allow the reaction vessels to withstand the intense pressures (hundreds of times Earth’s atmosphere) necessary to force nitrogen and hydrogen gases into NH3 (ammonia).

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BASF built an industrial plant at Oppau to generate Germany’s fertiliser and munitions, but the ammonium nitrate manufactured is inherently dangerous and in peacetime between WWI & WWII more than 500 people died when ammonium nitrate plastered on the storage silo walls was detonated during the, at that time routine, use of small explosive charges. Nevertheless, Oppau was built up as an industrial powerhouse until the end years of WWII when city was virtually levelled in a series of continuous bombing campaigns by the Allies.

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The Haber-Bosch process is used unchanged in its fundamentals today, consumes 1–2% of all the world’s energy, and creates the fertilisers that provide nitrogen to crops that sustain half the world’s population.

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🐦 + 💩: white gold of the seas

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Nitrogen-containing mineral deposits (historically known by the vague term ‘niter’) crystalise on cave walls, of which potassium nitrate (‘saltpeter’) was extensively mined for gunpowder and fertiliser. In these caves, it wasn’t only the mineral deposits that were a source of nitrates, but also the bat droppings. The volume of bat poo available pales in comparison to islands covered in bird poo, which was harvested by people of the Andes mountain region who called it guano. Since nitrates dissolve in water, dry conditions were needed for rich guano deposits, so it only piled up on certain islands. In the 19th century, Europeans used their vast shipping infrastructure to run a guano industry that fervently extracted hundreds of thousands of tonnes of the fertiliser for intensive agricultural use. This made intensive agriculture reliant on a few islands. This led to war between Spain and its former colonies for the Chincha guano islands.

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Hard labour is still performed to extract guano from the droppings of 4 million birds on Peruvian islands, where it is stacked in giant sacks among the throng of cormorants. Guano was also vital to American farming; causing the U.S. to pass the Guano Islands Act, allowing any U.S. citizen who stumbles upon an uninhabited guano island to seize it on behalf of federal government. The U.S. acquired many islands this way (often transiently, just to extract guano), including the Midway Atoll of the Battle of Midway fame. The Guano Islands Act is still in force, for any Americans brave enough to venture in search of new islands to claim.

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Australia has its own involvement in the administration of a guano island, Nauru, a 21 km\\(^2\\) country whose economy collapsed once its guano and calcium phosphate deposits were depleted. The Australian government uses the now guano-free land of Nauru for an offshore immigration processing centre, representing the most extreme outcome of Australia’s controversial indefinite detention laws.

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N≡N + 🦠: the organic way to fertilise soil

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Pre-existing nitrogen in the soil has been put there by nitrogen fixing bacteria roughly 2.5 billion years ago, around the time the atmosphere became oxygen rich and well before plants learned to breathe. Nitrogen fixing bacteria contain nitrogenase: an enzyme that funnels electrons to a molecular-scale crucible of metal clusters where the N≡N bond is broken. While the structure of this ‘crucible’ has been determined in exquisite detail, to my knowledge exactly where and how famously unreactive N≡N binds inside nitrogenase is still an unsolved, subtle, complex molecular puzzle actively researched with modern techniques. The efficiency of nitrogenase leaves contemporary molecular science a little embarrassed; what bacteria can do in the open air humans must still force through using electric arcs or hundreds of atmospheres of pressure.

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Some nitrogen fixing bacteria, known as rhizobia, require a plant host (specifically legumes: beans, clover, peas, etc) in which rhizobia form root nodules. Rhizobia feed legumes fixed nitrogen until their host dies, at which point rhizobia infect a new legume or return to the soil, providing bioaccessible nitrogen. This is the answer to the children’s song, “Do you, or I, or anyone know how oats, peas, beans, and barley grow?” — legumes are included in the crop rotation to ensure the soil remains nitrogen-rich. Several other bacteria can fix nitrogen, including some blue-green algae which have a symbiotic relationship with many plants, including giant Rhubarb (Gunnera manicata); an example of which can be found in the Tasmanian Botanic Gardens.

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🌿 + 🦠: kudzu and the overrun of the American south

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Kudzu is a vine native to the Asia-Pacific and is a legume with a phenomenally successful symbiotic relationship with nitrogen fixing bacteria. Kudzu’s hardiness and scrabbling tendrils make access to plentiful nitrogen supply via rhizobia a dangerous combination. Kudzu took root in the U.S. after being introduced both for its fashionable purple flowers and as fast growing fodder for livestock. Kudzu is capable of growing at a rate of a foot a day, and was declared a weed as it spread over the landscape, blocking the sun and smothering all below it. Kudzu’s growth has been so efficient in the American South that kudzu engulfed entire settlements, as captured in eerie quiescence of R.E.M’s debut album cover. Kudzu’s onward march appears unstoppable, and the Midwest is set to be next.

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N≡N + 🧑‍🔬 + 🔎: research to improve nitrogen fixation

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As it stands, nitrogen fixation remains critical to global food supply, uses 1–2% of the world’s energy (concomitantly producing 1–2% of human CO2 emissions), and underpins a roughly $60 billion ammonia industry. A fundamental change in how the world gets its nitrogen would be transformative.

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Some alternatives to the Haber-Bosch process could be:

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I have no expertise in this field, but there’s plenty of literature on such approaches for those interested.

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For the approach of developing inorganic nitrogen fixing catalysts, a recent review surveys the staggering diversity of metals (Mo, Re, V, Zr, Ti, …) and structures investigated for potential catalysts, of which only Mo-based catalysts work under ambient conditions. Breakthroughs in the use of inorganic nitrogen fixing catalysts will come when there is a combination of: an inexpensive catalytic metal, such as Fe which is the key metal inside nitrogenase; use of H2O as a hydrogen source; and function under ambient conditions. If these developments can be achieved, commercial scale nitrogen fixation might be done without huge industrial plants.

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For ‘green ammonia’, the Haber-Bosch process itself is already efficient at combining hydrogen and nitrogen gas, once reactors could contain high pressures. Renewable energy might best be directed towards powering gas compressors, and to electrolyse H2O for a source of hydrogen, since the use of fossil fuels as a hydrogen source in the Haber-Bosch process is the largest contributor to its CO2 emissions. It has been estimated that an electricity driven Haber-Bosch process will be cheaper and involve less energy loss than the fossil fuel based status quo.

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It could be possible to selectively mutate sections of pre-existing nitrogenases (in a process known as ‘directed mutagenesis’) to increase the efficiency of the enzyme. I can’t easily find any literature on improvements to nitrogenases through mutagenesis, which surprises me since the technique has been around for a while, so I suspect the structure of nitrogenase is so finely tuned to N2 that most site mutations kill the enzyme’s efficiency. Mutagenesis was, however, used on nitrogenase in the 1980s to show that upon replacement of crucial components (specific amino acids) nitrogenase ceases to fix nitrogen, and of the two ‘crucible’ sections (Fe-containing and MoFe-containing) activity continues even when only the Fe-containing ‘crucible’ is present. In fact, nitrogen fixing bacteria have a range of nitrogenases using alternative metals for when Mo is not available in the environment.

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As for entirely new artificial enzymes, I don’t know if this has been achieved by anyone, but one approach has been to combine parts of the nitrogenase system from different bacteria by exchanging gene segments. I don’t have enough working molecular biology knowledge to fruitfully read the nitty-gritty of these papers, but a key result is that hooking up the electron transport infrastructure from one type of bacteria to the nitrogenase of another can lead to greater efficiency. Gains can also be made by doing things like turning off the regulatory networks that monitor nitrogen production in a cell.

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Despite N2 being abundant and simple in structure, it has underpinned entire ecosystems, determined wars, and continues to escape the complete control of chemists. It may be worth reflecting on N2, the particle you collide with the most throughout your life, the next time you eat your (likely fertiliser grown) rice, toast, or other grain in the morning.

\n", "date_published": "2023-06-14T00:00:00+10:00" }, { "id": "https://keiran-rowell.github.io/photobiology/2023-04-15-photobiology/", "url": "https://keiran-rowell.github.io/photobiology/2023-04-15-photobiology/", "title": "The chemistry behind bioluminescence", "content_html": "

Photo credit: Adam Foster. Flickr [CC BY-NC-SA 2.0]

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This post is adapted from a talk I gave at one of the University of Sydney’s Theory Group meetings, but is ultimately a refashioned book report of Bioluminescene by Thérèse Wilson & J. Woodland Hastings.

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The Theory Group talks were an fascinating initiative, where other theoretical chemists were able to present on any topics in science that interested them. I hope my talk was interesting, but I don’t think anything could top the talk given by some postdocs on the optimal physical chemistry for cooking pasta and making espresso.

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Glow worms, fireflies, and glowsticks

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Glow worms, beyond inspiring a calming ballad by Vashti Bunyan, are bewitching organisms, especially if you are lucky enough to be in the Southern Hemisphere and see them up close in the Waitomo Caves or the Blue Mountains. Glow worms are the larvae of gnats, and glow to attract insects to their sticky thread, entrap them, and digest them. When desperate or territorial they will resort to cannibalism. The larvae metamorphose into adults that have no mouth, live for 2-6 days, mate, then die. Hardly seems worth it. Nevertheless they put on a spectacular show, and carry the Māori name titiwai meaning ‘lights reflected in water’.

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The North Americans get fireflies, which are associated with a popular hit, and encompass a far broader church of glowing beetles. Adult fireflies have mouths. Beyond that I can’t comment, I’m not a biologist. But fireflies have captivated people throughout history.

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Below is a rough chemical overview of where the magic ‘glow’ comes from: ultimately it is the energy released upon breaking a strained O-O bond and forming two C=O bonds that affords the generation of an electronically excited molecule, which proceeds to emit light when the electrons relax to the ground state. There is some full-on physical chemistry behind this process.

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In glowsticks a simple dioxetane is used, energy is released when the glowstick is cracked, and a dye is electronically excited and begins glowing. If the dioxetane is used by itself, an electronically excited carbonyl is created, but triplet carbonyls are poor photon emitters – so instead nature devises a concurrent charge transfer process, creating a singlet excited state and ensuring a photon is emitted from the O-O bond breakage. A chemical schematic of the process looks like this.

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Bioluminescence as oxygen disposal

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Bioluminescence is found in creatures from sea to sky, but is not evenly distributed amongst organisms on the phylogenetic tree. The central argument in “Bioluminescence: living lights, lights for living” is that bioluminescence was actually an evolutionary accident: a side reaction for disposing toxic oxygen, back when oxygen was plentiful in Earth’s atmosphere and organisms had not yet adapted the chemistry to cope. This “oxygen detoxification” cause for bioluminescence is a compelling argument, but I don’t believe it is proven conclusively. Regardless, here’s a summary of the molecules used to create bioluminescence in different organisms.

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Luciferin + luciferase = 💡

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The reactions that cause bioluminescence involve: a luciferin (‘light-bearer’), a molecule that is electronically excited upon addition of O2; and a luciferase, a protein that catalyses the addition of O2 to luciferin and alters the colour of the emitted photon based on the amino acids surrounding the luciferin.

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Every class of organism seems to have independently evolved a different luciferin and luciferase, lending credence that it was an oxygen disposal mechanism. Here are the different luciferins used by different organisms.

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Crustaceans

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Crustaceans use the simplest known bioluminescence process: they release two secretions, one containing the luciferin and the other contains the luciferase. The glow begins (see below) when the secretions come into contact.

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\"Crustacean

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The bioluminescence of the Vargula hilgendorfii crustacean.

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The luciferin is composed mainly of three amino acids and a central imidazopyrazinone, whose N-C=O bond is ultimately broken, releasing CO2 and light.

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\"Crustacean

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The luciferin of the Vargula hilgendorfii crustacean. The imidazolepyrazinone (highlighted in red) reacts with oxygen, generating an excited state similar to the excited state carbonyl in dioxetanes that was outlined above.

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Jellyfish

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Sea creatures use many different luciferins, and the luciferin for the sea pansy is shown below, it is again is based on three amino acids and a central imidazolepyrazinone.

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\"Sea

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The luciferin of the Renilla reniformis sea pansy closely resembles that for Vargula shown above, but is surrounded with different amino acids.

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However, the bioluminescence most well-known in the sea is that of jellyfish. Jellyfish bioluminescence is also the most widely used in the lab, supplying the ubiquitous Green Fluorescent Protein (GFP). GFP is used in countless assay and biological experiments: it’s easier to image tissue if it glows.

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In jellyfish, the luciferin is covalently bound to the luciferase, surrounded by a beta barrel structure (see below) which admits only water and provides control of the colour of bioluminescence.

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\"Jellyfish

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The Green Fluorescent Protein in Aequorea victoria, where an enveloping beta barrel is seen around the luciferin. The amino acids in the barrel both prevent the luciferin reacting with external substances, and alter the colour of the light emitted (otherwise the luciferin would glow blue).

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Algae

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Several algae can also bioluminesce, leading to the beautiful appearance of sea sparkles in certain shorelines of the world.

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Here, a new type of luciferin is found where a modified porphyrin-like molecule is used. The chemistry here is slightly different, adding oxygen to a cyclopentanone group and yielding H2O, but light is generated just the same.

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\"Algae

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The luciferin of Lingulodinium polyedrum dinoflagellate algae species.

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The bioluminescence of these algae only activate at night, but glow brightly (see below).

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\"Algae

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Glowing Lingulodinium polyedrum dinoflagellate algae, where the light-emitting organelles (scintillons) can be clearly seen.

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Fireflies

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The glow of the lantern on a firefly’s bum is a well-known sight, and looks stunning en masse.

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\"Firefly

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The impressive light organ located on abdomen of Photinus pyralis.

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The chemisty of firefly luciferin is again different (see below).

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\"Firefly

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The chemistry of firefly luciferin is driven by ATP (the ‘battery molecule’ of biology). Ultimately a dioxetane is again formed, and breaking the O-O bond yields an electronically excited carbonyl that emits light.

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Quantum chemical calculations on firefly luciferin

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In the 20th century several biological and chemical experiments gave us an understanding of photobiology. In the 21st century, new methods in computational chemistry and increases in compute power allows us to look into the physical chemistry that drives the generation of an excited state luciferin through O-O bond breakage.

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These calculations can get very complicated, so much so that people like Isabelle Navizet and Roland Lindh have made careers out of this problem.

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Some diagrams follow for the theoretical chemists amongst us.

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For those without training: suffice to say the character of several excited states are involved, the presence of an anionic group on the luciferin is required to generate a photon-emitting singlet excited state, and curiously a conical intersection exists along the reaction co-ordinate that ejects the CO2. The conical intersection was, at least to me, unexpected because it should lower the quantum yield of the luciferin since the conical intersection allows relaxation to the electronic ground state without emitting a photon. A very good review of bioluminescence using the insights gleaned from computational chemistry is provided in The Chemistry of Bioluminescence: An Analysis of Chemical Functionalities.

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\"Luciferin

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Density Functional Theory calculations indicate that the luciferin is in a (π,σ∗) ground state, and the anionic C-O- is crucial since the extra electron undergoes an electron transfer process to the site of the broken O-O bond to generate a singlet excited state (\\(S_1\\)) upon ejection of the CO2.

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\"Luciferin

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Multiconfigurational quantum chemistry calculations show that the first singlet excited state (\\(S_1\\)) is a (π,π∗) excitation, and the transition state (TS) ejecting the CO2 corresponds to a very small energy gap between the \\(S_1\\) and \\(S_0\\) electronic states.

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\"Luciferin

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The transition state (TS) ejecting the CO2 is generated by an avoided crossing between the (π,σ∗) ground and excited states. As the CO2 separates from the luciferin, the luciferin can either proceed along the (π,π∗) excited state surface and eventually emit a photon to relax back down to the ground state, or follow the seam of a sloped conical intersection to access a closed-shell singlet product.

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QM/MM calculations on luciferin

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Finally, a hybrid quantum mechnical/molecular mechanical (QM/MM) simulation can be used, where the luciferin is treated with a quantum mechanical method to describe the electronic excitation, while the luciferase protein is simulated with standard classical molecular mechanics to allow its shape to move.

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This allows a full mechanistic movie of oxygen entering the luciferase protein, binding to the luciferin, and generating an excited state. I’ve summarised the key findings of such QM/MM calculations in the image below. For those interested, it is worth reading the original paper QM/MM Study of the Formation of the Dioxetanone Ring in Fireflies through a Superoxide Ion.

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\"Luciferin

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I first learnt about this topic when I picked up a secondhand copy of Computational Methods for Large Systems, in which Chapter 12 details “Modelling Photobiology Using QM and QM/MM Calculations”.

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\"Fireflies

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Thank you for reading!

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I think it’s incredible how nature manages to generate light by combining chemicals inside the bodies of small organisms, and put on an entrancing show.

\n", "date_published": "2023-04-15T00:00:00+10:00" }, { "id": "https://keiran-rowell.github.io/guide/2023-04-12-picking-an-active-space/", "url": "https://keiran-rowell.github.io/guide/2023-04-12-picking-an-active-space/", "title": "Selecting an active space", "content_html": "

This post is adapted straight from my PhD Thesis, and deals with the struggle of creating a workable active space for multiconfigurational calculations.

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Multiconfigurational methods are where the training wheels come off, you can no longer use a “black-box” combination of method and basis set – you have to think about the chemical problem. There is therefore no right answer, but chemical intuition comes into play (what bonds break/form, which orbitals are populated?). Start with simple orbitals and build up. So, small basis sets, make sure it converges, then enlarge the basis set if needs must. Try to treat the “which orbitals?” and “how small a basis set can I get away with?” problems separately. My advisor also gave me guidance that calculating the cation (thus giving you “pulled in” orbitals) is another trick to get started.

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The field is developing fast, to the point where there are methods to automatically select an active space, but I believe “bespoke” active space selection will still have its uses, and is an art as much as a science.

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Complete Active Space (CASSCF) calculations and Minimum Energy Conical Intersection (MECI) searches

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In order to search for an optimised minimum energy conical intersection (MECI), a configuration where two electronic states are degenerate in energy, a multiconfigurational quantum chemistry method must be used. This involves optimising both the orbital coefficients but, unlike single-reference methods, also optimising the combinations of multiple Slater determinants. In multiconfigurational methods choice of an appropriate set of electron configurations in configuration state functions (CSFs) are crucial for a qualitatively correct description of the wavefunction.

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While individual CSFs could be manually selected, one of the most common approaches is the complete active space (CAS) method where all possible configurations inside a selected orbital active space are treated. The self-consistent field (SCF) is used, and so the method at at this level of theory is known as CASSCF. The use of an active space removes the burden of choosing particular CSFs to include in the SCF optimisation, but requires the selection of appropriate active space orbitals which capture the chemical process of interest. Usually this includes any orbitals near the frontier orbitals, those involved in making and breaking bonds, the inclusion of correlated bonding and anti-bonding pairs, and any orbitals which are calculated to have incomplete occupation according to orbital population schemes.

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Generation of Natural Bond Orbitals

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Selection of orbitals to include in the active space is more easily done if natural bond orbitals (NBOs) are used since they correspond to chemical intuition and are localised to the reactive space, while canonical orbitals are often too delocalised to be interpretable.

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Minimal basis sets are easier to converge in a CASSCF calculation than large basis sets, and also aid in interpretability. The starting point for all calculations was generation of an initial orbital population at the \\(S_1\\) TS configuration, using NBOs at the HF/STO-3G level of theory. An example Gaussian input file to generate these NBOs is given below. This process was also repeated if a large basis set, such as 6-31+G(d), was used.

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%chk=[file_path]/save_NBOs.chk\n# HF/STO-3G Pop=(Full,SaveNBOs)\n\nTitle Card Required\n\n0 1\n[Molecule input]\n
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The generated NBOs can also be examined in an external program (Chemcraft in this example). The Gaussian input file below shows the extra input at the bottom required to call on the NBO program to print the calculated natural bond orbitals to file. These will be labelled FILE.\\(X\\) where \\(X\\) is 31–37. Chemcraft can open up FILE.31 directly, and should recognise the FILE.\\(X\\) files for import. Chemcraft can then render NBOs from these files using the drop-down options: Tools → Orbitals → Render molecular orbitals → NBOs. Be aware: there is a reordering from the .log file and the .chk file, so only rely on the numbering in GaussView when identifying orbital indexes.

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%chk=[file_path]/print_NBOs.chk\n#P HF/STO-3G Pop=(Full,NBORead,SaveNBO) gfoldprint\n\nTitle Card Required\n\n0 1\n[Molecule input]\n\n$NBO BNDIDX PLOT $END\n
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Selection of the active space

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It is useful to check both the character of these NBOs from their density distribution, as well as their occupation value according to the NBO scheme. Any molecular orbitals which have occupations that differ significantly from the 0/2 value for virtual/occupied orbitals are likely candidates for inclusion in the active space.

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The conical intersection of interest is between the \\(S_1\\)/\\(S_0\\) states of the carbonyl. This conical intersection will be present near the vicinity of the transition state which involves a 1,5–H-shift from the carbonyl oxygen to the \\(\\gamma\\)-hydrogen. Correlated pairs of bonding and anti-bonding orbitals should be included, for example if a C-H σ bonding orbital is included the corresponding C-H σ∗ antibonding orbital should also be included. The oxygen \\(n\\) orbital which interacts with the \\(\\gamma\\)-hydrogen and also changes occupation in the excited state is crucial to include in active space, as well as the C=O π bonding and antibonding orbitals.

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Additionally, since the photoexcited C=O moiety abstracts a hydrogen atom to form a C-O-H bond, the C-O orbitals were found to be important for inclusion in the active space. This is particularly true when a STO-3G minimal basis set was used, which would deliver optimised C-O bond lengths approximately 0.3 Å larger than in the transition state structure. With large basis sets the C-O bond length varied little from the transition state structure and hence the inclusion of corresponding orbitals was of less importance.

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A reasonable (8,7) active space for typical saturated carbonyls includes:

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The NBOs which correspond to this active space are shown in the example of butanal below at an isosuraface value of 0.1. While literature calculations use a (8,7) active space, a (10,8) active space that includes the alternate O \\(n\\) oxygen-centred NBO was used as the largest active space. The literature active space is provided in Kletskii et al. Competing Mechanisms of Norrish and Norrish-Like Reactions in a Wide Range of Systems — from Carbonyl Compounds to Nitrogen Oxide Donators

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\"Butanal

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Natural bond orbitals for butanal, showing those typical for the active space of a CAS(8,7) calculation on a saturated carbonyl. CAS(10,8) active spaces include the other oxygen-centred NBO.

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For unsaturated species, it was found to be beneficial to also include the π and π∗ orbitals of the point of unsaturation. A (12,10) active space is very computationally demanding, so for unsaturated species the (10,8) active space sized is preserved by removing the C-O σ and σ∗ NBOs, and replacing them with the π and π∗ orbitals from the point of unsaturation, as illustrated below.

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\"2-Oxobutanal

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Natural bond orbitals for 2-oxobutanal, showing those typical for the active space of a CAS(10,8) calculation on an unsaturated carbonyl. In these carbonyls with a point of unsaturation the other π and π∗ NBOs are included instead of the C-O σ and σ∗ NBOs to keep the active space manageable.

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Once identified, these orbitals of interest need to be rotated into the active space. The orbitals which are considered in the active space are those nearest the HOMO/LUMO frontier according to the (\\(n\\),\\(m\\)) active space chosen. The (\\(n\\),\\(m\\)) nomenclature means enough occupied molecular orbitals to host \\(n\\) electrons are treated as active. These are taken from the HOMO index and those sequentially below. The number of virtual orbitals in the active space is \\(m\\) less the number of active occupied orbitals, and they are indexed from the LUMO and those sequentially above it.

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Performing CASSCF and MECI calculations

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The easiest way to perform this active space selection with the orbitals in the correct index is to interchange orbitals read in from the NBO checkpoint file. This process is illustrated in the example Gaussian input file below — interchanging as an example orbitals 9 and 25, as well as 40 and 31.

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Note: before running the input file below, copy the checkpoint file containing the saved NBOs to have same name as the checkpoint filename used here, so the CASSCF calculation can read in the correct NBOs. The molecular geometry does not need to be supplied since it is read from the checkpoint file.

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The iop(5/7=N) keyword sets the amount of CASSCF convergence cycles used. While this can be increased, a slow or difficult to converge CASSCF calculation is often indicative of a poor active space. In some large molecules convergence can be slow, but if the energy is seen to be monotonically decreasing with each cycle then simply increasing the amount of available cycles may be all that is needed.

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%chk=[file_path]/[molecule]_S0_8-7.chk\n# CASSCF(8,7)/STO-3G iop(5/7=200) Guess=(Read,Alter) Geom=Checkpoint \n\n\nTitle Card Required\n\n0 1\n\n9,25\n40,31\n
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If this CAS(\\(n\\),\\(m\\)) calculation converges on the \\(S_0\\) state at configuration of the \\(S_1\\) NTII TS geometry calculated by TD-DFT, then this wavefunction is taken as a good initial guess for a beginning a conical intersection optimisation.

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An example input file of running a minimum energy conical intersection (MECI) search with Gaussian is given below. Again, the checkpoint file from a previous job must be copied to this checkpoint filename specified in the current job to read the orbitals, in this case from the converged CAS \\(S_0\\) calculations. Note: in Gaussian 16 the state average weights must be included at the bottom of the input file, whereas Gaussian 09 does not need this extra input and defaults to the 0.5, 0.5 weighting between the upper and lower state of the same spin.

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%chk=[file_path]/[molecule]_CI_search.chk\n# CASSCF(8,7)/STO-3G iop(1/8=5) Guess=Read Geom=Checkpoint Opt=Conical  \n\nTitle Card Required\n\n0 1\n\n0.5 0.5\n
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The iop(1/8=N) keyword sets the maximum step size during the optimisation, and generally needs to be decreased from its default value of \\(N = 30\\) which corresponds to 0.3 Bohr. Since the \\(S_1\\) Norrish Type II TS structure is taken as being close MECI a small step size is appropriate and avoids issues where the optimiser can overshoot the MECI several times if large step sizes are used. The maximum number of convergence cycles may again need to be increased through the use of the iop(5/7=N) keyword, however since an already converged \\(S_0\\) CAS wavefunction is used as the initial guess the convergence during MECI searches tended to be well behaved, and convergence issues were often an indication that the geometry optimiser has strayed into a bad part of configuration space and a new guess geometry must be used.

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The \\(S_1\\)/\\(S_0\\) state energy difference is reported in the log file as Energy difference= X and this difference should be monitored, as well as the usual geometry convergence criteria of force and displacement values. If the energy difference, forces, or geometry begin to oscillate around a central zero value then decreasing the step size at this stage may improve the MECI search.

\n", "date_published": "2023-04-12T00:00:00+10:00" }, { "id": "https://keiran-rowell.github.io/guide/2023-04-12-compchem-methods-basics/", "url": "https://keiran-rowell.github.io/guide/2023-04-12-compchem-methods-basics/", "title": "An Ersatz Ansatz", "content_html": "

This post is adapted straight from my PhD Thesis, and is intended as a primer for beginner computational chemists. Thanks goes out to Dr Laura McKemmish, whose notes on compchem for undergraduates is the urtext for this guide.

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Many computational chemistry talks begin with the famous Dirac quote:

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“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It there fore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.” Dirac, P.A.M. Quantum Mechanics of Many-Electron Systems, Proceedings of the Royal Society A, 1929, 123, 714–733.

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The practice of computational chemistry is then left to picking the appropriate approximate method to answer the scientific question that interests you. Below is a rough primer of the basic quantum chemistry methods.

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Electronic Structure Theory

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All chemistry is governed by the behaviour of electrons and nuclei in molecules, which are described by the Schrödinger equation \\(\\hat{H}\\Psi = E\\Psi\\), where the molecular energy (\\(E\\)) is an eigenvalue found by applying a Hamiltonian operator (\\(\\hat{H}\\)) to the wavefunction (\\(\\Psi\\)) of the molecule. Given the correct Hamiltonian, solution of the full Schrödinger equation in principle yields exact properties, but is in practice insoluble for molecular systems. All quantum chemistry methods therefore rely on computational techniques for solving approximations to the full Schrödinger equation.

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In the full Schrödinger equation the Hamiltonian operator \\(\\hat{H}\\) contains all potential and kinetic energy terms from all electrons and nuclei and the interactions between them. The non-relativistic Hamiltonian is commonly simplified by invoking the Born-Oppenheimer approximation: since electrons have smaller masses and faster timescales of motion compared to nuclei, the electronic wavefunction can be solved in a field of nuclei that are considered fixed. This approximation removes the nuclear kinetic energy term from the Hamiltonian and makes the nuclear-nuclear interaction term constant, resulting in the simpler electronic Hamiltonian:

\n\n\\[\\hat{H}_{\\mathrm{electronic}} = \\underbrace{-\\sum \\frac{1}{2} \\nabla^{2}_{i} -\\sum_{i}^{elec.} \\sum_{s}^{nuc.} \\frac{Z_{s}}{\\vec{r}_{is}}}_{\\hat{h}_{i}} +\\sum_{i \\lt j}^{elec.} \\frac{1}{\\vec{r}_{ij}}\\]\n\n

Atomic units are used here to simplify equations. The first term in \\(\\hat{H}_{\\mathrm{electronic}}\\) is the kinetic energy of the electrons, the second term the electron-nuclear attraction, and both are encompassed in the one-electron operator (\\(\\hat{h}_{i}\\)) which is summed over all electrons in the molecule. The third term represents the electron-electron repulsion and can not be solved exactly for multi-electron systems. Wavefunction-based quantum chemistry methods differ in the approximations they introduce to solve the electron-electron repulsion term.

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In order to solve the electronic Schrödinger equation, spin orbitals (\\(\\chi_{n}\\)) which depend on position of one electron (\\(\\mathbf{x}_{n}\\)) are used. Since electrons are fermions, the wavefunction must obey the Pauli principle and be antisymmetric with respect to the exchange of two electrons. This is achieved by representing the wavefunction of a many electron system (\\(\\Psi\\)) as a Slater determinant of one-electron spin orbitals:

\n\n\\[\\Psi_{0}(\\mathbf{x}_{1},\\mathbf{x}_{2},\\cdots,\\mathbf{x}_{n}) = \\frac{1}{\\sqrt{N!}}\n\\begin{vmatrix}\n\\chi_{1}(\\mathbf{x}_{1}) & \\chi_{2}(\\mathbf{x}_{1}) & \\chi_{3}(\\mathbf{x}_{1}) & \\cdots & \\chi_{N}(\\mathbf{x}_{1}) \\\\\n\\chi_{1}(\\mathbf{x}_{2}) & \\chi_{2}(\\mathbf{x}_{2}) & \\chi_{3}(\\mathbf{x}_{2}) & \\cdots & \\chi_{N}(\\mathbf{x}_{2}) \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\chi_{1}(\\mathbf{x}_{N}) & \\chi_{2}(\\mathbf{x}_{N}) & \\chi_{3}(\\mathbf{x}_{N}) & \\cdots & \\chi_{N}(\\mathbf{x}_{N}) \\\\\n\\end{vmatrix}\\]\n\n

Quantum chemistry is typically performed according to molecular orbital (MO) theory, where the wavefunction can be represented as a product of MOs (\\(\\phi_{i}\\)). These MOs are constructed from a linear combination of atomic orbitals (LCAO): \\(\\phi_{i} = \\sum\\limits_{n}C_{ni}\\chi_{n}\\), where each coefficient (\\(C_{ni}\\)) determines the contribution of each atomic orbital to a MO. The optimal value of these \\(C_{ni}\\) coefficients for a set of atomic orbitals can be determined according to the variational principle, which states that energy of the approximate wavefunction will always be higher than the true wavefunction. Therefore, when using a variational method, the coefficients which yield the lowest total energy are the best estimate of energy of the true molecular wavefunction.

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Hartee-Fock theory

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The Hartee–Fock (HF) method is a variational approach to solving an approximation to the electronic Schrödinger equation. In HF theory the electron-electron repulsion term is solved iteratively, using a “mean-field” of the other electrons, until electron distributions and energies converge within a self-consistent field (SCF). The HF equations are cast as an eigenvalue problem: \\(\\mathbf{FC} = \\epsilon \\mathbf{SC}\\), where a Fock operator (\\(\\mathbf{F}\\)) acts on a matrix of \\(C_{ni}\\) coefficients (\\(\\mathbf{C}\\)) to yield a vector containing the molecular orbital energies (\\(\\epsilon\\)). The Fock operator (\\(\\hat{F}_{i}\\)) contains the one-electron operator (\\(\\hat{h}_{i}\\)), and operators for the exchange (\\(\\hat{K}\\)) and Coulomb repulsion (\\(\\hat{J}\\)) energy between electrons:

\n\n\\[\\hat{F}_{i} = \\hat{h}_{i} + \\sum_{j \\neq i}^{N/2} (2\\hat{J}-\\hat{K})\\]\n\n

This HF solution is often used as the initial trial wavefunction for calculations that use higher accuracy quantum chemical methods.

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Correlated methods

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Since the HF equations are solved in a “mean-field” of electrons, this method lacks the dynamic electron correlation energy necessary to make quantitative predictions of molecular properties. A number of wavefunction formalisms are available to recover this correlation energy using a correction to the HF wavefunction. These post-HF methods are often referred to as “correlated” methods. The energetic predictions of post-HF methods are systematically improved as higher-order terms of electron correlation are included, but at the cost of increased scaling of computational cost with system size.

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An alternative quantum chemical approach is to bypass the wavefunction and calculate molecular properties based solely on the electron density. These density functional theory (DFT) methods have reduced computational cost, since a \\(3N\\) dimensional wavefunction does not have to be computed, but are not systematically improvable since the exact form the exchange-correlation functional should take is not known.

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Multiconfigurational methods

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Both the DFT and post-HF methods outlined below are “single-reference” methods based upon a single reference Slater determinant. In cases such as electronic degeneracies, a single Slater determinant may be qualitatively inadequate to describe the wavefunction, and single-reference DFT and wavefunction methods will fail. In such cases, multiple Slater determinants must be included through a multiconfigurational (MC) approach. The MC form of HF is known as the multiconfigurational self-consistent field (MCSCF) method, and a common variant of MCSCF is to include all Slater determinants describing a chemically relevant active space (CASSCF).

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Formalisms for different spin states

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For closed-shell molecular spin states, where all electrons are paired, a “restricted” formalism of the quantum chemical method (e.g. RHF) can be used, where the orbitals optimised contain both spin-up and spin-down (\\(\\alpha\\) and \\(\\beta\\)) electrons. Restricted formalisms are used for singlet electronic states (\\(S_0, S_1, S_2\\)). Photoexcitation can produce open-shell molecules containing unpaired electrons. In these cases “unrestricted” (e.g. UHF) or “restricted open-shell” (e.g. ROHF) formalisms can used. The former uses two independently optimised sets of orbitals to treat the \\(\\alpha\\) and \\(\\beta\\) electrons separately, while the latter uses both singly and doubly occupied orbitals. Unrestricted methods can be used for open-shell states (\\(T_1, T_2, D_0\\)), since unrestricted methods are computationally simpler and more amenable to post-HF correction than restricted open-shell methods. However, UHF calculations can encounter “spin-contamination”, where the contributions of higher state solutions contaminate the UHF wavefunction and hence the spin operator (\\(\\mathbf{S}^{2}\\)) of the UHF wavefunction is no longer a valid spin eigenfunction. The deviation from the \\(\\mathbf{s(s+1)}\\) expectation value should be monitored as a diagnostic in all calculations of open-shell species. Unrestricted DFT (UDFT) appears to suffer less from spin-contamination due to the use of Kohn-Sham orbitals.

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Post–Hartee-Fock wavefunction methods

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Møller-Plesset perturbation (MPn)

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A Møller-Plesset (MPn) perturbative correction (\\(\\lambda \\hat{V}\\)) can be applied to the HF Hamiltonian (\\(\\hat{H}_{0}\\)) to yield a more exact Hamiltonian (\\(\\hat{H}\\)): \\(\\hat{H} = \\hat{H}_{0} + \\lambda \\hat{V}\\). Typically, the perturbation applied is a second order correction (MP2), which takes into account electron repulsion integrals (ERIs) between occupied (\\(i,j,\\ldots\\)) and virtual (\\(a,b,\\ldots\\)) orbitals in the Slater determinant:

\n\n\\[E^{\\mathrm{MP2}} = \\sum_{ij}^{occupied} \\sum_{ab}^{virtual} \\frac{(ia|bj)(jb|ia)}{\\epsilon_{i} + \\epsilon{j} - \\epsilon_{a} - \\epsilon_{b}}\\]\n\n

This MP2 correction recovers \\(\\sim\\)80–90% of the total dynamic electron correlation which was lacking in the HF solution, allowing better chemical predictions at the increased computational cost of evaluating ERIs.

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Resolution of the Identity (RI)

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The resolution of the identity (RI) approximation replaces computationally expensive four-centre ERIs with three-centre or two-centre integrals through the use of a larger auxiliary basis set. This leads to dramatic gains in computational efficiency with little loss in accuracy, particularly with larger basis sets containing higher angular momentum orbitals. The use of the RI approximation for correlated methods, such as RI-MP2, allows correlated calculations on far larger systems than otherwise possible. The Coulomb and exchange integrals can also be treated with a form of RI, known as RIJK. RIJK can be used for many methods, while the “chain-of-spheres” algorithm for exchange integrals (RIJCOSX) can be used for excited state calculations.

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Coupled cluster (CC)

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Coupled cluster (CC) theory uses a cluster operator (\\(\\hat{T}\\)) that contains excitations that promote occupied orbitals in the Slater determinant to virtual orbitals to obtain a more exact wavefunction: |{\\(\\Psi_{cc}\\)}\\(\\rangle\\) = \\(e^{\\hat{T}}\\)|{\\(\\Psi_{0}\\)}\\(\\rangle\\). The cluster operator can be truncated to include only double excitations (CCD), single and double excitations (CCSD), single, double and triple excitations (CCSDT), etc.:

\n\n\\[|{\\Psi_{\\mathrm{CC}}}\\rangle = \\left(1 + \\hat{T}_{1} + (\\hat{T}_{2} + \\frac{1}{2}\\hat{T}_{1}^{2}) + (\\hat{T}_{3}+\\hat{T}_{1}\\hat{T}_{2}+\\frac{1}{3!}\\hat{T}_{1}^{3}) + \\cdots \\right)|{\\Psi_{0}}\\rangle\\]\n\n

CC calculations deliver some of the best energetic predictions of any quantum chemical method, but the inclusion of high order excitation results in large computational scaling exponents. For example, CCSD formally scales as \\(\\mathcal{O}(N^{6})\\) with system size, while CCSDT scales as \\(\\mathcal{O}(N^{8})\\) and is therefore impractical for modestly sized molecules (\\(\\sim\\)6 non-hydrogen atoms). The use of CCSD with a perturbative triple excitation correction, CCSD(T), scales as \\(\\mathcal{O}(N^{7})\\). CCSD(T) energies are often the “gold standard” that other quantum chemical methods are benchmarked against.

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Density Functional Theory (DFT)

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Density functional theory (DFT) calculates the total energy through the use of functionals which depend on the electron density (\\(\\rho\\)):

\n\n\\[E[\\rho] = T_{s}[\\rho]+ E_{eN}[\\rho] + J[\\rho] + E_{xc}[\\rho]\\]\n\n

The first term (\\(T_{s}[\\rho]\\)) is the electron kinetic energy which can be calculated from a Slater determinant similarly to HF theory, though in DFT Kohn-Sham orbitals are used. The Coulombic electron-nuclear attraction (\\(E_{eN}[\\rho]\\)) and electron-electron repulsion (\\(J[\\rho]\\)) terms can be calculated classically. The final term, \\(E_{xc}[\\rho]\\), contains the electron-electron exchange-correlation functional whose exact form is not known. There are myriad DFT methods using different functional approximations for \\(E_{xc}[\\rho]\\), however the accuracy of their approximations cannot be known a priori, unlike with wavefunction methods. Though some functionals are non-empirical, many DFT methods are semi-empirical as their functional forms are parameterised with reference experimental data in an attempt to improve their accuracy.

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Functionals which incorporate more physical parameters of the electron density have accordingly higher computational costs, but tend to show increased accuracy. This provides a convenient classification of DFT methods into rungs on the conceptual “Jacob’s ladder” of accuracy, which is illustrated below. The representative accuracy of each rung in Jacob’s ladder is taken from the DFT review by Goerigk and Grimme, A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions. A popular DFT method is listed as an example for each rung.

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\"Jacob's

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A schematic of the “rungs” of accuracy in the conceptual “Jacob’s ladder” organisation system of density functional methods. The qouted accuracy of each rung comes from a review by Goerigk & Grimme on density functional methods. Schematic of Jacob’s ladder modified from: Robert Fludd, Utriusque Cosmi, 1671, illustrated by Johann Theodor de Bry, Oppenheim and Frankfort.

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Local density approximation (LDA) functionals use only the local electron density (\\(\\rho\\)), generalised gradient approximation (GGA) functionals incorporate the electron density gradient (\\(\\nabla\\rho\\)), and meta-GGA functionals incorporate the second derivative or kinetic energy of the electron density (\\(\\nabla^2\\rho\\)). Improved accuracy relative to pure DFT functionals can be achieved by mixing in a component of a wavefunction method. When HF exchange is incorporated into DFT it is known as a “hybrid” method, and addition of a perturbative MP2-like correction yields a “double-hybrid” method. The RI approximation can be used for the MP2-like correction in double-hybrid methods. DFT methods, being formulated upon local electron densities, do not properly capture dispersion interactions and so are frequently combined with empirical dispersion corrections.

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Global hybrid functionals

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Pure DFT functionals suffer from self-interaction error (SIE), where the Coulomb repulsion of an electron with itself is non-zero due to the form of the exchange functional. The Slater determinant adopted for the HF exchange energy (\\(E_{x}^{\\mathrm{HF}}\\)) exactly cancels SIE, and so SIE is lessened in hybrid functionals. Global hybrid (GH) functionals use a set fraction of HF exchange and have the form:

\n\n\\[E_{xc}^{\\mathrm{GH}} = (1 - a_{x})E_{x}^{\\mathrm{DFT}} + a_{x}E_{x}^{\\mathrm{HF}} + E_{c}^{\\mathrm{DFT}}\\]\n\n

As an example, in the popular B3LYP method \\(E_{c}^{\\mathrm{DFT}}\\) is calculated with the Lee, Yang, and Par (LYP) correlation functional, \\(E_{x}^{\\mathrm{DFT}}\\) with the Becke88 (B88) exchange functional, and the HF exchange mixing coefficient, \\(a_{x}\\), is 0.2 at all distances.

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Double-hybrid functionals

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Double-hybrid density functionals (DHDFs) supplement \\(E_{xc}[\\rho]\\) with a mixture of both non-local exchange and correlation energy from wavefunction methods, allowing them to recover more dynamic correlation and dispersion energy. DHDFs have the form:

\n\n\\[E_{xc}^{\\mathrm{DHDF}} = (1 - a_{x})E_{x}^{\\mathrm{DFT}} + a_{x}E_{x}^{\\mathrm{HF}} + (1 - a_{c})E_{c}^{\\mathrm{DFT}} + a_{c}E_{c}^{\\mathrm{PT2}}\\]\n\n

One of the first DHDFs was B2-PLYP which uses the same B88 and LYP functionals as B3LYP, and wavefunction mixing coefficients \\(a_{x} = 0.47\\) and \\(a_{c} = 0.27\\). The wavefunction mixing coefficients were re-optimised for specific chemical applications for the B2\\(X\\)-PLYP set of DHDFs. In a systematic survey by Karton et al. the “general-purpose” B2GP-PLYP functional, with \\(a_{x} = 0.65\\) and \\(a_{c} = 0.36\\), was found to be the most robust of the B2\\(X\\)-PLYP set methods, with maximum errors below 8 kJ/mol across all benchmark datasets surveyed.

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Range separated functionals

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The use of a single \\(a_{x}\\) mixing coefficient in GH functionals leads to incorrect behaviour at long-range electron-electron distances, resulting in poor accuracy for charge-transfer excitations. In range separated hybrids (RSHs) \\(\\alpha\\) and \\(\\beta\\) parameters are introduced to the exchange functional form, to smoothly connect between GH exchange at short range and exact HF exchange at long range:

\n\n\\[E_{x}^{\\mathrm{RSH}} = (1 - \\alpha)E_{x}^{\\mathrm{DFT}} + \\alpha E_{x}^{\\mathrm{HF}} + \\beta E_{x,\\mu}^{\\mathrm{HF}} - \\beta E_{x,\\mu}^{\\mathrm{DFT}}\\]\n\n

The value of \\(\\mu\\) in a RSH functional determines how rapid the switch is between short range and long range exchange behaviour. For example, the CAM-B3LYP functional uses 0.19 \\(E_{x}^{\\mathrm{HF}}\\) + 0.81 \\(E_{x}^{\\mathrm{B88}}\\) at short-range but transitions to 0.65 \\(E_{x}^{\\mathrm{HF}}\\) + 0.35 \\(E_{x}^{\\mathrm{B88}}\\) at long-range.

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I can provide a post on basis sets later, but the nomenclature in the field is messy. Best to go straight to the Basis Set Exchange

\n", "date_published": "2023-04-12T00:00:00+10:00" } ] }