Illustrated Physics
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Quantum

The ion that exists because its electrons keep apart

Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.
15 min read 5 figures Approximations that lieOnly some values fit

Assumes: Why an atom is the size it is · The order the shells fill

Where the electron probably is replaced the Bohr orbit with a cloud of probability. The order the shells fill explained why 4s fills before 3d by how much an s orbital penetrates the inner shells, why an atom is the size it is found the size of every atom from a balance of kinetic and potential energy with a screened nuclear charge, and the metal that is yellow because it is heavy added relativity. Each of those treated an atom’s electrons one at a time, each moving in the average field of the nucleus and the others, and each used screening — the others hiding part of the nuclear charge — as a number taken from a table.

The simplest atom where that approach can be tested exactly is helium, with two electrons, and the cleanest place it can fail is the negative hydrogen ion, H−\mathrm{H}^-, also with two electrons but a nuclear charge of one. The same calculation that gets helium’s energy within two per cent says H−\mathrm{H}^- does not exist. It does, and the reason is the part of the physics that a one-electron-at-a-time description leaves out by construction.

Helium without the answer given in advance

A helium atom has a nucleus of charge two and two electrons. With no repulsion between them, each would sit in the ground state of a hydrogen-like ion of charge two, with energy −Z2/2=−2-Z^2/2 = -2 hartrees, and the atom would have −4-4 hartrees, −108.8-108.8 electronvolts. The measured energy needed to strip both electrons is 79.0 electronvolts, so the repulsion matters by thirty electronvolts, and it has to be dealt with.

The energy of helium, one approximation at a time. The energy needed to strip both electrons from a helium atom, in electronvolts, as successive approximations give it, with the energy of the singly ionised atom He⁺ (dashed) for reference. In turn: no repulsion at all: −108.8 eV; repulsion added, orbitals unchanged: −74.8 eV; orbitals allowed to spread (ζ = 27/16): −77.5 eV; one electron in, one out: −78.3 eV; measured: −79.0 eV. Ignoring the electrons' repulsion overbinds the atom by 30 eV. Adding it without letting the orbitals respond underbinds by 4.2 eV. Letting each electron's orbital spread, as if it saw a nucleus of charge 27/16 rather than 2, comes within 1.53 eV — the second electron is screened by 5/16 of a charge, not by a whole one. The rest is correlation: the electrons avoiding each other.
Fig. 1 The energy of a helium atom, in electronvolts, as successive approximations give it, with the singly ionised atom plus a free electron at −54.4 eV (dashed). No repulsion: −108.8. Repulsion added, orbitals unchanged: −74.8. Orbitals allowed to spread, as for a charge of 27/16: −77.5. One electron held close and one further out: −78.3. Measured: −79.0.

The figure climbs down the approximations. Adding the repulsion between two electrons in the unchanged orbitals — its average is 5/85/8 of the nuclear charge, in hartrees — gives −74.8-74.8 electronvolts, now too little binding by four. The orbitals were shaped for a nucleus of charge two with nothing else present, and they should respond to the repulsion by spreading out. The method for letting them do so, with one adjustable number, is the variational principle.

One number, chosen by the atom

The variational principle says that the energy computed from any trial wavefunction is at least the true ground-state energy, with equality only for the true state. It turns the problem of solving an equation into the problem of minimising a number. The principle that fixes the energy instead of the clock found mechanics restated as a stationary principle over paths; this is quantum mechanics restated as a minimum principle over states, and it is the most-used approximation method in the subject.

The charge each electron behaves as if it sees. The energy of helium, in hartrees, with both electrons in the same hydrogen-like orbital shaped for a nucleus of charge ζ, against ζ: kinetic energy ζ², attraction to the real charge of 2, −4ζ, and the electrons' mutual repulsion, 5ζ/8. The least energy is at ζ = 27/16 = 1.6875, where it is −2.8477 hartree, −77.49 eV, against a measured −79.01 eV. Each electron behaves as if the other hid 5/16 of the nucleus's charge from it — not a whole charge, because the other electron is as often outside it as inside. Slater's empirical rule for two 1s electrons, a screening of 0.30, is this number rounded.
Fig. 2 The energy of helium with both electrons in one hydrogen-like orbital shaped for a nuclear charge ζ, against ζ: kinetic energy ζ2\zeta^2, attraction −4ζ-4\zeta and repulsion 5ζ/85\zeta/8, in hartrees. It is least at ζ = 27/16 = 1.6875, −2.848 hartree (−77.49 eV), against a measured −79.01 eV (dashed).

Take as the trial state both electrons in the same hydrogen-like orbital, but shaped for a nucleus of charge ζ\zeta rather than two, and let ζ\zeta be chosen by the minimum. The energy is three terms: kinetic ζ2\zeta^2, attraction to the real nucleus −4ζ-4\zeta, repulsion 5ζ/85\zeta/8. The figure plots their sum. It is least at ζ=27/16=1.6875\zeta = 27/16 = 1.6875, and there the energy is −77.5-77.5 electronvolts, within two per cent of the measurement.

The number 27/1627/16 is the physically interesting part. It says each electron behaves as if the nucleus had charge 2−5/162 - 5/16: the other electron screens it by five-sixteenths of a charge, not by a whole one. A whole unit of screening would be right if each electron were always outside the other, as an outer electron is outside a closed inner shell. Two electrons in the same orbital are outside each other only about half the time, and the average they settle on is 5/16. Why an atom is the size it is used Slater’s empirical rules, fitted to atomic energies in 1930, which give two electrons in the same 1s shell a mutual screening of 0.30 — the variational 5/16, rounded.

Where five-eighths comes from

The repulsion term deserves a moment, because it is the only place the second electron enters. Each electron is a spherical cloud, the probability density of a hydrogen-like orbital, and the repulsion is the average of 1/r121/r_{12} over both clouds. Counting what comes out supplies the trick that makes the average easy: by Gauss’s law, the field of a spherical cloud at any point depends only on how much of the cloud lies inside that point’s radius. The second electron, at radius rr, feels the first as a point charge at the centre for the part of the first cloud inside rr, and feels nothing from the spherical shells of it outside rr except a constant potential.

So the second electron, wherever it is, is screened by exactly the fraction of the first electron that lies closer to the nucleus than it does. Averaged over where both electrons are, that fraction is the screening. It would be one if the first electron were always inside the second, and zero if always outside; for two electrons in the same orbital it comes out at a value that, weighted by where each electron’s energy is decided, gives the repulsion 5ζ/85\zeta/8 and the effective screening 5/165/16. The shape decides the falloff made the same point about a planet’s gravity: only what is inside counts. Screening in an atom is Gauss’s law applied to a cloud that overlaps the particle it acts on.

The same calculation, one charge lower

Apply exactly the same calculation to a nuclear charge of one — a proton with two electrons, H−\mathrm{H}^-. The minimum is at ζ=1−5/16=11/16\zeta = 1 - 5/16 = 11/16, with energy −(11/16)2=−0.473-(11/16)^2 = -0.473 hartree. A hydrogen atom plus a free electron at rest has −0.5-0.5 hartree. The trial state has more energy than the atom it is supposed to be binding an electron to, so by this calculation H−\mathrm{H}^- is unstable: it would shed an electron and gain 0.74 electronvolts.

How firmly the second electron is held, from H⁻ to C⁴⁺. For two electrons round a nucleus of charge Z, the energy needed to remove one of them, in electronvolts on a logarithmic axis, from the one-parameter orbital, from the in-and-out function, and measured. Z = 1: −0.74, 0.36, 0.76 eV; Z = 2: 23.07, 23.83, 24.59 eV; Z = 3: 74.09, 74.80, 75.65 eV; Z = 4: 152.32, 153.01, 153.90 eV; Z = 5: 257.76, 258.44, 259.35 eV; Z = 6: 390.42, 391.09, 392.01 eV. For every Z but one the simple guess is close. For Z = 1, the negative hydrogen ion, it gives a negative number — the ion falling apart — while the measured value is 0.754 eV: the ion exists only because the two electrons arrange themselves one near and one far.
Fig. 3 For two electrons round a nucleus of charge Z, the energy to remove one, in electronvolts on a logarithmic axis: one shared orbital, one electron in and one out, and measured. From Z = 2 to 6 the three agree closely (helium: 23.07, 23.83, 24.59 eV). At Z = 1 the shared orbital leaves the ion unbound, the in-and-out function binds it by 0.36 eV, and the measured value is 0.754 eV.

The figure applies the calculation across the two-electron ions from H−\mathrm{H}^- to C4+\mathrm{C}^{4+}. For every nuclear charge from two up, the simple guess is close — the energy to remove one electron is off by a few per cent, and less as the charge grows and the repulsion becomes a smaller fraction of everything. At a charge of one it gives the wrong sign. The measured binding is 0.754 electronvolts; H−\mathrm{H}^- is made routinely in ion sources, and its binding has been measured by detaching the electron with tuned laser light, to better than a part in a hundred thousand.

The failure is not the variational principle’s fault. The principle guarantees that the true energy is lower than any trial value, and the trial state here was simply too constrained to find it. What it could not do was let the two electrons be different.

One in, one out

In 1944 Subrahmanyan Chandrasekhar, working on the opacity of stellar atmospheres, tried a different trial state: one electron in an orbital of one size and the other in an orbital of another, symmetrised because electrons are indistinguishable, with both sizes chosen by the minimum.

The inner and outer orbitals of the negative hydrogen ion. The radial probability densities, against distance from the proton in Bohr radii, of the two orbitals in Chandrasekhar's function for H⁻, e^(−1.039r) and e^(−0.283r), with the one shared orbital of the simple guess, e^(−0.6875r), for comparison. The inner orbital peaks at 0.96 Bohr radii, almost exactly hydrogen's own ground state; the outer at 3.53, nearly four times further out. One electron makes an ordinary hydrogen atom, and the other orbits outside it, held by the atom's slight polarisation. Forcing both into one orbital puts each half the time too close to the other, and the repulsion that costs is more than the binding.
Fig. 4 The radial probability densities, against distance from the proton in Bohr radii, of the two orbitals in Chandrasekhar’s function for H−\mathrm{H}^-, e−1.039re^{-1.039r} and e−0.283re^{-0.283r}, and the single shared orbital of the simple guess, e−0.6875re^{-0.6875r} (dashed). The inner peaks at 0.96 Bohr radii, almost hydrogen’s own ground state; the outer at 3.53, nearly four times further out.

The minimum finds a strongly asymmetric arrangement, which the figure draws. One orbital is almost exactly hydrogen’s ground state; the other is diffuse, peaking at three and a half Bohr radii. One electron makes an ordinary hydrogen atom and the other orbits well outside it. The energy is −0.5133-0.5133 hartree, below the threshold: the ion is bound, by 0.36 electronvolts, about half the measured value, and the sign is right.

What changed is that the electrons are no longer required to be in the same place on average. When one is close to the proton, the other is likely to be far away, and vice versa: their positions are correlated in the radial direction. The simple guess allowed no correlation at all — each electron’s distribution was independent of where the other was — and so each spent half its time in the inner region with the other, paying a repulsion the real ion avoids. Holding the outer electron is the hydrogen atom’s polarisability: an electron approaching a neutral atom pushes the atom’s own electron cloud to the far side, leaving the proton partly exposed, and that small attraction, falling as the inverse fourth power of distance, is enough to hold one electron in a large, loose orbit and not enough to hold a third.

The rest of the missing binding comes from angular correlation — the two electrons preferring opposite sides of the atom — which requires a trial state depending on the angle between them, or directly on their separation. Egil Hylleraas introduced such functions in 1929 for helium, and they converge to the exact energy; for H−\mathrm{H}^- they reach the measured 0.754 electronvolts. The force with no force in it found that electrons with the same spin avoid one another through the exclusion principle alone; the two electrons here have opposite spins, the exclusion principle does not keep them apart, and the avoidance has to come entirely from the repulsion, which a single shared orbital cannot express.

What correlation is worth

The energy the one-orbital picture misses is called the correlation energy, and it is roughly the same in helium and H−\mathrm{H}^-.

What correlation is worth, to a helium atom and to H⁻. The energy needed to remove the second electron, by three descriptions, as a fraction of the measured value, for the negative hydrogen ion and for helium. Helium: one shared orbital 23.07 eV, one electron in and one out 23.83 eV, measured 24.59 eV — the simple picture is 94 per cent right. H⁻: −0.74, 0.362 and 0.755 eV — the simple picture is not merely inaccurate but has the wrong sign. The energy the simple picture misses — the electrons avoiding each other — is about 1.5 eV in both; in helium it is a small correction to 24.6 eV, and in H⁻ it is everything.
Fig. 5 The energy to remove the second electron by three descriptions, as a fraction of the measured value. Helium: one shared orbital 23.07 eV, one in and one out 23.83 eV, measured 24.59 eV. H−\mathrm{H}^-: −0.74 eV (unbound), 0.362 eV and 0.755 eV. The energy the simple picture misses is about 1.5 eV in both.

The figure sets the two systems side by side. In both, the one-orbital picture misses about one and a half electronvolts. For helium, whose second electron is held by 24.6 electronvolts, that is a six-per-cent correction, and the simple picture is a good first description. For H−\mathrm{H}^-, whose second electron is held by 0.754 electronvolts, the missing part is twice the answer, and the simple picture is qualitatively wrong. The lesson generalises. An approximation’s error is roughly a fixed amount of energy for similar systems, and whether it matters depends entirely on the size of the quantity being sought, which can differ by a factor of thirty between two systems that look alike.

This is the same structure the molecule that lives outside its own well found for a weakly bound state: when the binding is small compared with the scales of the system, the state lives mostly outside the region where the forces act, and it is decided by a small residual interaction that a coarse description gets wrong. H−\mathrm{H}^-'s outer electron is a weakly bound state of that kind, a few tenths of an electronvolt deep in a potential that is almost nothing — the residue of a neutral atom’s polarisation — and it lives far outside the atom it is bound to.

Why it matters that the negative ion exists

Chandrasekhar’s interest was not academic. In 1939 Rupert Wildt had proposed that the negative hydrogen ion was the main source of the opacity of the Sun’s visible surface — the reason the Sun has a sharp edge at all, rather than being transparent to its centre. The Sun’s outer layers are mostly neutral hydrogen, which cannot absorb visible light: its first excitation needs ultraviolet. A small fraction of the hydrogen atoms, about one in a hundred million, have picked up a spare electron supplied by easily ionised metals, and an H−\mathrm{H}^- ion absorbs any photon with more than its 0.754 electronvolts of binding — every visible and near-infrared photon, out to a wavelength of 1.64 micrometres — and releases the electron. That is enough, in the dense lower atmosphere, to make the gas opaque over a few hundred kilometres.

Chandrasekhar’s calculations of H−\mathrm{H}^-‘s binding and of its absorption, over several years in the 1940s, confirmed Wildt’s proposal and made the ion a central ingredient of the theory of stellar atmospheres. The number that decided it was the binding energy: if the ion were not bound, the Sun’s visible surface would be somewhere quite different, and the spectrum of sunlight would look different. A two-electron calculation, and whether it included the electrons’ avoidance of each other, decided what the surface of the Sun is made of.

Which atoms can hold an extra electron

H−\mathrm{H}^- is the simplest negative ion, and the same physics decides the others. An atom binds an extra electron — has a positive electron affinity — when the new electron can sit somewhere that the existing electrons screen incompletely. Chlorine, one electron short of a closed shell, binds an extra one by 3.6 electronvolts, because the vacancy in its outer shell lets the new electron in close, where the nuclear charge is poorly screened. Oxygen binds one by 1.46 electronvolts. The noble gases bind none: their closed shells screen the nucleus completely, and a new electron would have to go into the next shell out, where it sees almost no charge at all. Helium’s negative ion does not exist in its ground state. Nitrogen, whose half-filled outer shell would force the new electron to pair with one already there, binds none either, a direct consequence of the order the shells fill and of the extra repulsion of two electrons sharing an orbital.

Hydrogen sits between: nothing screens its proton when a second electron arrives, but the first electron occupies the only low orbital, and the second is held only by polarisation and correlation. That is why its affinity is small, and why an approximation too coarse to represent correlation gets it wrong while getting chlorine’s roughly right.

The ion’s importance depends on temperature in a way the binding energy explains. It needs free electrons to form and survives only where collisions are too gentle to detach its weakly held extra electron, so it dominates the opacity of stars somewhat cooler than about seven thousand kelvin, the Sun among them. In hotter stars the ions are knocked apart and hydrogen atoms themselves, absorbing from their excited levels, take over; in much cooler stars molecules do. The dividing temperatures are set by how 0.754 electronvolts compares with the thermal energy of a gas — about half an electronvolt at the Sun’s surface — the kind of ratio the exponential that decides everything found governing every such balance.

What the curves leave out

The calculations here are non-relativistic and treat the nucleus as infinitely heavy and point-like. For helium the corrections for the nucleus’s motion, for relativity and for quantum electrodynamics are a few parts in ten thousand of the energy, and are included in the measured values the figures compare against but not in the trial functions. The exact energies quoted for ions beyond helium are taken from an expansion in powers of one over the nuclear charge, accurate to about a thousandth of a hartree; for the conclusions drawn here that is ample.

The trial functions also say nothing directly about excited states. H−\mathrm{H}^- has no bound excited state at all — the one bound state is the only one — and helium’s excited states, where one electron is promoted to a higher orbital, are well described by the one-electron picture because the promoted electron is far outside the other and hardly correlated with it. It is precisely the ground states with two electrons in the same region where correlation matters most.

Still open: how correlation should be computed for many electrons

For two electrons the exact energy can be computed to twenty digits. For many electrons, correlation is the central difficulty of quantum chemistry. Methods that expand the wavefunction in many configurations converge but grow in cost as a high power of the number of electrons; density-functional theory, which is used for most practical calculations, includes correlation through an approximate functional whose exact form is unknown, and its failures are concentrated exactly where correlation is strong — in stretched bonds, in some transition-metal compounds, and in weakly bound negative ions like H−\mathrm{H}^-, whose extra electron many common functionals fail to bind. Finding functionals that treat such cases correctly, and understanding why the successful ones succeed, is one of the most active problems in the field.

The habit worth carrying away is to compare an approximation’s error with the size of what is being computed, not with the size of the whole. A picture in which each electron moves in the average field of the others misses the energy of their avoiding one another — about 1.5 eV in both helium and H−\mathrm{H}^- — which is a small correction to helium’s 24.6 eV and larger than H−\mathrm{H}^-'s entire 0.754 eV, so the same calculation that describes helium well says H−\mathrm{H}^- does not exist. An ion that exists only because its electrons keep apart is what gives the Sun a surface.

Part 5 of 5

This essay is one argument about Atomic structure. The others:

The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely. The metal that is yellow because it is heavy Part 4 — Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

Where this goes next

Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more. Why an atom is the size it is A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude. The hole that holds exactly one particle. How likely a second particle is to be found a distance away from a first, relative to a gas with no correlation at all, for three cases that differ in nothing but the symmetry of the state under swapping the two labels. There is no interaction anywhere in this calculation: no Coulomb term, no potential, no force. Distinguishable particles give a flat line, which is what no interaction ought to give. Identical fermions dig a hole that reaches exactly zero at zero separation and fills back in over about a wavelength. Identical bosons do the opposite and pile up to twice the density. Integrating the fermion hole gives 0.9992 particles missing from around each one — exactly one, and the sum rule holds at any density, because raising the density narrows the hole in exact proportion. That is what makes the effect worth a name of its own. It is often called an exchange force and it is not a force: nothing carries momentum between the particles, and no term in the energy is proportional to a distance. It is a statement about which states exist. What follows from it is most of chemistry — the reason two atoms with filled shells repel, the reason a metal's electrons cost so much less Coulomb energy than a random arrangement would, and the reason matter takes up room. The force with no force in it Two identical fermions keep apart and two identical bosons crowd together, and neither is being pushed. The Hamiltonian contains no interaction at all: what produces the hole and the pile is which many-particle states exist, and the hole it digs around each electron holds exactly one particle at any density whatever. A bound state that lives outside its well. The radial probability density of the one bound state of a square well of radius R, against distance in units of R, for four depths just past the one that first binds a state. In units of ε = ħ²/2μR², a state bound by 0.00589 ε has 93 per cent of its probability outside the well and a mean radius of 7.0 R; a state bound by 0.0397 ε has 82 per cent of its probability outside the well and a mean radius of 3.0 R; a state bound by 0.245 ε has 62 per cent of its probability outside the well and a mean radius of 1.5 R; a state bound by 1.4 ε has 35 per cent of its probability outside the well and a mean radius of 0.9 R. The more weakly the state is bound, the further its exponential tail reaches, and the less of it is where the force is. The molecule that lives outside its own well The harmonic picture describes the bottom of a well, where a molecule vibrates near the length of its bond. The last bound level lives at the top, and when it is bound by almost nothing it is not near the bond at all. Two helium atoms whose potential well bottoms out at three ångströms form a molecule whose atoms sit on average about fifty ångströms apart, most of the time where the force between them is negligible. Such a state forgets the shape of the well that made it. Its size, its binding and the way it scatters are all set by one number, and nothing inside the well can be read from them.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Atomic structureEffective nuclear chargeElectron correlationHelium atomIonisation energyNegative ionScreeningVariational principle