A theoretical chemist's notebook

H2O

Three atoms, ten electrons, and one of the most studied substances in science. Follow water from the electrons that hold it together, through the bond that holds it to its neighbors, to the gas, the liquid, the many forms of ice, and the questions still open.

Behind this text: a live two dimensional water model, its hydrogen bonds forming and breaking as you read.

all scales

Why a theorist never finishes with water

On paper, water is a small problem: one oxygen, two hydrogens, ten electrons. For a single molecule, calculations from first principles now approach the accuracy of experiment[1]. Yet put two molecules together and the answer depends on how carefully you treat electron correlation. Put a few hundred together and you need statistical mechanics, nuclear quantum effects, and potentials that are correct to a fraction of a kilocalorie per mole across every environment the molecule might meet.

I spend my research time on the middle of that ladder: how water molecules interact, how those interactions add up in clusters, and how a molecule's vibrations report on its surroundings. This page is the wider view, organized by scale. Each chapter is one rung, and each rung has its own clock.

  1. attoseconds to femtosecondsAfter sudden ionization, charge can migrate across a molecule driven by electron correlation alone[2]
  2. 2.1 fsOne beat of a 1b₁/3a₁ hole wavepacket, from the orbital energies in chapter 01
  3. 8.9 fsOne O–H asymmetric stretch, from its 3756 cm⁻¹ frequency[3]
  4. 21 fsOne H–O–H bend, from its 1595 cm⁻¹ frequency[3]
  5. picosecondsA molecule in the liquid switches hydrogen-bond partners[4]

10⁻¹¹ m · attoseconds

Ten electrons, five orbitals

Everything water does starts with how its ten electrons arrange themselves around three nuclei. In the molecular orbital picture they fill five doubly occupied orbitals, each with its own energy and shape.

Hartree–Fock orbital energies (RHF/cc-pVDZ, computed for this site). By Koopmans' theorem, −ε approximates the energy needed to remove an electron from that orbital.
OrbitalCharacterε (eV)
1b₁out-of-plane lone pair, the HOMO−13.43
3a₁in-plane lone pair mixed with O–H bonding−15.43
1b₂O–H bonding, antisymmetric−19.04
2a₁mostly O 2s, bonding over both O–H bonds−36.38
1a₁oxygen 1s core−559.24

Experiment agrees with the ordering. The photoelectron spectrum puts the first ionization energy at 12.62 eV, from the 1b₁ orbital, with the next band about 2 eV higher[5]. The Hartree–Fock value, 13.43 eV, is close but not exact. Koopmans' theorem ignores how the remaining electrons relax and correlate once one is removed.

Interactive Watch these orbitals move Bohmian trajectories of one electron in a superposition of two of these orbitals, computed live in the browser. Open the electron-flow explainer →

10⁻¹⁰ m

Why water is bent

0.9572 Å

O–H bond length[7]

104.52°

H–O–H angle[7]

1.85 D

dipole moment[8]

A molecule of three atoms could be straight, like carbon dioxide. Water is not, and the reason sits in one orbital.

Follow the orbital energies as the H–O–H angle closes from 180°. One orbital, the one that becomes 3a₁, is a pure oxygen p orbital when the molecule is linear and does nothing for bonding. Bend the molecule and it can mix with the hydrogen 1s orbitals and with oxygen's 2s, and its energy drops steeply. Water has enough electrons to fill it, so bending wins. This is the argument of Walsh's 1953 correlation diagrams, still the quickest way to predict whether an AH₂ molecule is bent or linear[9].

The water molecule with its equilibrium geometry and dipole Oxygen at the top, two hydrogens below at 0.9572 angstroms and an angle of 104.52 degrees. Partial negative charge on oxygen, partial positive charges on the hydrogens. 104.52° 0.9572 Å O H H δ− δ+ δ+
The dipole lies along the symmetry axis (dashed), with the negative end at oxygen and the positive end between the hydrogens. At 1.85 D it is large for so small a molecule, and it is the first reason water dissolves salts and pulls on its neighbors.

10⁻¹⁴ s · femtoseconds

A molecule that never stops moving

Three atoms have nine degrees of freedom: three for moving, three for rotating, and three for vibrating. Those three vibrations are water's fingerprints, and each one is a clock with a period between 9 and 21 femtoseconds.

Lab · normal modes

Idealized mode shapes, with the displacements exaggerated and the motion slowed by a factor of 10¹⁴. Frequencies are the measured fundamentals of the free molecule[3].

These numbers are where experiment and theory meet most precisely. A potential energy surface built from high-level calculations and refined against spectra reproduces thousands of measured water lines[10]. And because the two hydrogen nuclei are identical spin-½ particles, water comes in two nuclear-spin forms, ortho and para, in a 3:1 ratio at room temperature. They are hard to separate, but it has been done with a magnetically focused molecular beam[11].

vapor

Water on its own: the gas

In the vapor, a water molecule spends most of its time alone, flying between collisions. This is where the molecule of the first three chapters is measured most directly, and where its hydrogen bonds show their strength by their absence.

99.97 °C

water boils at 1 atm[13]

−60.3 °C

H₂S, a heavier molecule, boils at 1 atm[13]

2257 J/g

heat to vaporize water at its boiling point[13]

Those three numbers are the long-standing signature of hydrogen bonding. Hydrogen sulfide is heavier than water but forms only weak hydrogen bonds, so it boils 160 degrees lower. Boiling a gram of water takes 2257 joules, enough to heat that same gram from 0 to 100 °C more than five times over, because every hydrogen bond a molecule had in the liquid must be broken before it can leave[13].

Alone in the gas, water rotates as well as vibrates. All three of its moments of inertia are different, so it is an asymmetric rotor and its rotational levels form an irregular ladder. One rotational line, at 22.235 GHz, was the first emission from water ever detected in interstellar space, in 1969[14]. Closer to home, the same rotational and vibrational absorptions make water vapor the largest single contributor to Earth's present-day greenhouse effect, about half of it, ahead of clouds and carbon dioxide[15]. Because water vapor condenses and falls out as rain and snow, its amount follows temperature rather than setting it. It amplifies warming driven by gases that do not condense, such as CO₂[16].

3 × 10⁻¹⁰ m

The bond between molecules

A hydrogen bond forms when an O–H group on one molecule points at an oxygen on another. IUPAC defines it as an attractive interaction between a hydrogen attached to an electronegative atom and another atom or group, with evidence that a bond is forming[18]. Breaking the bond between two water molecules takes less than 3 percent of the energy needed to break a covalent O–H bond[19, 20], yet it decides almost everything about the liquid.

2.98 Å

O···O distance, vibrationally averaged[21]

≈ 5.0 kcal/mol

well depth Dₑ, computed[22]

3.16 kcal/mol

energy to separate, D₀, measured[19]

Lab · pull the dimer apart

An illustrative Morse curve anchored to the computed well depth and an equilibrium distance close to 2.9 Å. The real surface depends on six intermolecular coordinates, not one.

The gap between the two energies is quantum mechanics at work. Dₑ is the depth of the potential well. D₀, the energy actually needed to pull the pair apart, is smaller because the bound dimer can never sit still at the bottom. Its intermolecular vibrations carry zero-point energy. The difference, nearly 2 kcal/mol, is a large fraction of the bond.

Symmetry-adapted perturbation theory splits the interaction into physical pieces: electrostatics, exchange repulsion, induction, and dispersion[23]. All four matter for the water dimer, which is why simple point-charge pictures only get so far.

10⁻⁹ m

Clusters, and why the whole is not the sum of its pairs

Small water clusters are where the liquid's behavior first appears while the systems are still small enough for the most accurate methods. The trimer, tetramer, and pentamer form rings[28]. At six molecules, three dimensional cage, prism, and book structures compete. All three have been identified by rotational spectroscopy, with the cage lowest in energy[29].

Hydrogen bonds in clusters are cooperative. A molecule that accepts a hydrogen bond becomes a better donor, so a ring of hydrogen bonds is stronger than the same number of isolated pairs. The tool for quantifying this is the many-body expansion:

E = Σi Ei + Σi<j ΔEij + Σi<j<k ΔEijk + …

One-body terms describe each molecule distorted into its cluster geometry, two-body terms the pair interactions, and three-body and higher terms the cooperativity. For ring clusters up to the hexamer, three-body terms reach about 30 percent of the interaction energy, while four-body and higher terms are negligible[28].

Lab · count the terms

Each k-body term is its own quantum chemistry calculation on k molecules, and there are C(n, k) of them. The cost line compares one CCSD(T) calculation on the full cluster, which scales as size⁷, with all of its dimer calculations, ignoring prefactors.

10⁻¹² s · picoseconds

A network that never holds still

In the liquid each molecule has roughly four nearest neighbors, with the first peak of the oxygen–oxygen distribution at 2.80 Å[31]. The network is mostly intact at any instant and completely different a few picoseconds later. Molecules change partners not by slow rotation but by sudden large-angle jumps[4].

Lab · cool the liquid

The two dimensional “Mercedes-Benz” model[32, 33]: each disk has three hydrogen-bonding arms at 120°, with the published parameters, simulated by Metropolis Monte Carlo at constant density. Cyan lines are hydrogen bonds. Lower the temperature and watch an ice-like hexagonal network try to form.

Real simulations of liquid water need a potential, and the choice is a trade. Rigid point-charge models such as SPC/E[34] and TIP4P/2005[35] are fitted to bulk properties and are fast. Data-driven many-body potentials such as MB-pol are built from correlated calculations on small clusters, using the many-body expansion of chapter 06[17]. Hydrogen is also light enough that its nuclei behave quantum mechanically. Zero-point motion and tunneling change the liquid's structure and dynamics measurably[36].

All of this adds up to a remarkable dielectric constant, about 78 at 25 °C[37]. The network's ability to reorient its dipoles is why water screens charges so well and dissolves salts. The same hydrogen bonds make liquid water hard to heat and hard to stretch. Its heat capacity at 25 °C, 4.18 joules per gram per kelvin, is unusually large for a liquid, and so is its surface tension, 72 millinewtons per meter[13].

bulk

The liquid that breaks the rules

Most liquids get steadily denser as they cool. Water gets denser only until 3.98 °C, then expands again on the way to freezing[38]. It is the most famous of water's anomalies, and the reason lakes freeze from the top down.

Data · density of liquid water at 1 atm

Computed from Kell's 1975 correlation for liquid water[38]. The inset magnifies 0–12 °C, where the maximum sits.

The density maximum has company. Water's compressibility is lowest near 46 °C[38], and its heat capacity at constant pressure is lowest near 36 °C[13]. In most liquids these quantities change steadily with temperature instead.

The molecular explanation is the competition you met in chapter 07. Hydrogen bonds favor an open, tetrahedral arrangement with lots of empty space. Thermal motion favors a denser, disordered packing. Cooling tips the balance toward the open structure. One proposed explanation for this and related anomalies is a second critical point, deep in the supercooled liquid, between two liquid forms of different density[39]. Long simulations of two realistic water models find such a point[40], and X-ray laser experiments that briefly melt compressed amorphous ice have watched the high-density liquid turn into the low-density one[41]. Whether real water has the critical point is still debated, because that region is so hard to reach before the liquid freezes.

pressure × temperature

One substance, many phases

Long before anyone could compute a hydrogen bond, water's phases had been measured with extraordinary care. Today they are summarized by international standard equations, fitted to the best experiments and maintained by the International Association for the Properties of Water and Steam[42].

273.16 K

triple point, at 611.657 Pa[42]

647.096 K

critical point, at 22.064 MPa[42]

−0.0075 °C/atm

shift of the melting point with pressure[43]

Lab · explore the phase diagram

Boundaries computed from the international equations for the vapor pressure, the sublimation pressure, and the melting pressures of ices Ih, III, V, VI, and VII[42, 44, 43]. Boundaries between the solid phases themselves are not drawn. Move across the chart, or tap it, to read the phase and the melting and boiling points at that pressure.

At one temperature and pressure, 273.16 K and 611.657 Pa, ice, liquid, and vapor coexist[42]. This triple point is so reproducible that, until 2019, it defined the kelvin itself[45]. Heat water in a sealed vessel and the liquid expands while the vapor above it is compressed, until at 647.096 K and 22.064 MPa the two become identical[42]. Beyond this critical point there is no boiling, only a supercritical fluid that changes continuously from gas-like to liquid-like.

The line between ice and liquid tilts the wrong way. Because ice is less dense than the liquid, squeezing favors the liquid, and the melting point falls by about 0.0075 °C per atmosphere[43]. James Thomson predicted this from thermodynamics in 1849[46], and his brother William, later Lord Kelvin, confirmed it by experiment the next year[47]. It was an early case of theory telling experiment where to look.

crystal

Ice: one solid, many lives

Ordinary ice, ice Ih, is a hexagonal crystal in which every oxygen sits at the center of a tetrahedron of four others. That open framework makes water expand by about 9 percent when it freezes, which is why ice floats[50, 38].

0.917 g/cm³

density of ice Ih near 0 °C[50]

3.37 J/(mol·K)

entropy left in ice at absolute zero, Pauling's estimate[51]

20+ phases

crystalline forms of ice known, including ice XXI[52, 53, 54]

Snow

The sixfold symmetry of every snowflake is that hexagonal lattice made visible. Whether a crystal grows as a flat plate, a column, or a branching star depends sharply on temperature and humidity, a dependence first mapped by Ukichiro Nakaya in the 1930s and still not fully explained[55].

Order in the oxygens, disorder in the protons

The oxygens in ice sit on a perfect lattice. The hydrogens do not. Bernal and Fowler proposed the ice rules: each oxygen has two hydrogens close to it, and each O···O link carries exactly one hydrogen[56]. Many arrangements obey these rules, so even a perfect ice crystal stays disordered at absolute zero. Pauling counted the arrangements, about (3/2)ᴺ for N molecules, and predicted a residual entropy of R ln(3/2), or 3.37 J/(mol·K)[51]. Calorimetry by Giauque and Stout found about 3.4 J/(mol·K) left over[57], one of the great successes of simple statistical reasoning.

Pauling's count ignores closed loops in the hydrogen-bond network, so it is not exact. Nagle's careful series calculation gives 3.41 J/(mol·K), about 1 percent higher and even closer to experiment[58]. For a flat, square version of ice, Elliott Lieb found the exact answer, (4/3)3/2 ≈ 1.540 arrangements per molecule, one of the landmark exact solutions of statistical mechanics[59].

Lab · shuffle the protons

Square ice, a two dimensional toy version of ice. Every oxygen keeps two hydrogens and every link holds exactly one. Each shuffle reverses closed loops of hydrogen bonds, sliding one proton along each bond in a loop, so every molecule stays intact. A large crystal of this kind has about 1.540ᴺ valid arrangements for N molecules[59], so even this patch of 36 has millions.

If the protons could rearrange freely, ice would order as it cools. In pure ice they are effectively frozen in place. Doping with a little potassium hydroxide lets them move, and below about 72 K the protons order into a new phase, ice XI[60].

Under pressure

  1. 1 atmIce Ih, the ice of snow, glaciers, and ice cubes[50]
  2. above ~2 GPaIce VII, two interpenetrating hydrogen-bonded networks[52]
  3. ~60 GPaIce X, with the proton midway between two oxygens[61]
  4. hundreds of GPaSuperionic ice XVIII, hot enough for protons to flow through a solid oxygen lattice[62]

Squeeze ice and its hydrogen-bond network folds into denser and denser frameworks. Seventeen polymorphs had been confirmed by early 2019[52]. Ice XIX followed[53], and then ice XXI, a metastable phase that forms at room temperature when water is compressed rapidly[54]. The list is still growing. Ice VII is more than a laboratory curiosity. Pieces of it have been found trapped inside diamonds from Earth's mantle, still held at pressures as high as 24 GPa, and it is now recognized as a mineral[63].

Compress further and the proton stops choosing a side. In ice X it sits midway between two oxygens and the hydrogen bond becomes symmetric[61]. Simulations that treat the protons quantum mechanically show that tunneling and zero-point motion are central to this change[64]. At the pressures and temperatures inside Uranus and Neptune, simulations predicted superionic ice, a solid oxygen lattice through which protons flow like a liquid[65]. Twenty years later, laser-driven shock experiments observed it and measured its crystal structure, now called ice XVIII[62].

Glassy ice, layered ice, and ice with guests

Squeeze ordinary ice at 77 K and it does not find a new crystal. It collapses into a glass, high-density amorphous ice[66]. Decompress and warm it, and it switches abruptly to a low-density amorphous form, much like a first-order transition[67]. These two glasses are part of what motivated the two-liquid picture of chapter 08[39].

Even ordinary ice is not always hexagonal. Ice that freezes from deeply supercooled water is a random stack of hexagonal and cubic layers, called stacking-disordered ice[68]. Ice surfaces are never quite dry, carrying the liquid-like layer of chapter 09[49]. And water can freeze around guests. Clathrate hydrates are ice-like cages that trap gas molecules such as methane, and enormous amounts of methane are stored this way in seafloor sediments and permafrost[69].

network

How a proton travels without moving far

An extra proton in water moves through the liquid far faster than an ion of its size should. The idea behind the explanation is more than two centuries old[70]. The proton does not travel, the charge does.

Schematic · the Grotthuss relay

Each hydrogen moves only a fraction of an ångström, from one oxygen to the next along a hydrogen bond. The positive charge, shown by the halo, travels the length of the chain.

One proton hops onto a neighboring molecule while that molecule passes one of its own protons further down the chain[71]. Simulations treating electrons and nuclei quantum mechanically show the excess proton fluctuating between two limiting forms, a hydronium-like ion and a proton shared evenly between two waters, with the transfer driven by rearrangements of the surrounding solvent[72].

meters

Water is blue for a vibrational reason

Pure water is faintly blue, visible through a few meters of it. Almost every colored substance owes its color to electronic transitions. Water does not. Its electronic absorption starts far in the ultraviolet.

The color comes from vibrations. Overtones and combinations of the O–H stretch, the same motion from chapter 03 excited several quanta at once, reach into the red end of the visible spectrum. Water absorbs a little red, so the light that survives looks blue[73]. It is one of very few natural colors that come from molecular vibration.

frontier

What we still don't know

  • A potential that works everywhere. The same molecule sits in a cluster in the gas phase, at an interface, around an ion, and in the bulk. A single model that is accurate across all of these, with the molecule allowed to respond to its surroundings, is still a target. It is the long-term aim of my work on environment-dependent water potentials.
  • The supercooled liquid. Simulations and experiments now point toward two distinct liquids[40, 41], but whether real water has a liquid–liquid critical point, and where, is still argued over[39].
  • Quantum nuclei at scale. Treating hydrogen nuclei quantum mechanically in long simulations of large systems remains expensive, and their effect on properties is subtle[36].
  • Water next to other things. Around small solutes water rearranges its hydrogen bonds. Around large ones it cannot, and hydrophobic assembly changes character at about a nanometer[74].
  • Ice under extreme conditions. New ice phases keep appearing at high pressure, and the superionic ices that may fill the interiors of ice giants are only beginning to be mapped[62].
  • Convergence of the many-body expansion for higher levels of theory and for larger, more varied clusters, which decides how far accurate methods can scale.

Every one of these questions comes back to the scales above: electrons that set the shape, a shape that sets the hydrogen bond, and hydrogen bonds that, added up with all their cooperativity and quantum blur, make the gas, the liquid, and every form of ice.

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