# Molecular orbital <!-- MICROSIMGEN:BEGIN v1.7 — generated by g08_place_microsims.py; three.js first (§15); do not hand-edit inside --> ## Microsims — three.js ### Molecular orbital (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Molecular_orbital.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Molecular orbital — three.js microsim"></iframe> </div> **Open it full-screen:** [Molecular_orbital.html](https://wikitube-3d-microsims.netlify.app/Molecular_orbital.html) · library `threejs` · route `microsim/threejs/` ### Related microsims Live sims on neighbouring articles: - [[Allotropes_of_oxygen]] - [[Atomic_orbital]] - [[Hemoglobin]] - [[Hydrogen_bond]] - [[Ozone_layer]] - [[Silicon_dioxide]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.* <!-- MICROSIMGEN:END --> ## Overview A molecular orbital is a one-electron wavefunction that belongs to the whole molecule rather than to any one atom. In the standard approximation it is built as a linear combination of atomic orbitals: two atomic orbitals combine to make two molecular orbitals, one bonding (in phase, electron density piled between the nuclei, lowered in energy) and one antibonding (out of phase, a node between the nuclei, raised in energy). The theory earns its keep on oxygen. A Lewis structure of dioxygen, O=O, pairs all twelve valence electrons and therefore predicts a diamagnetic molecule. Liquid oxygen visibly clings to the poles of a magnet: it is paramagnetic. Molecular orbital theory gets this right without adjustment, because filling the standard second-row diagram places the last two electrons in two degenerate pi* antibonding orbitals with parallel spins, following Hund's rule. Two unpaired electrons, hence paramagnetism. This is the textbook case where Lewis structures fail and molecular orbital theory succeeds. That argument is older than most textbooks admit. The molecular orbital picture was assembled between about 1927 and 1932 by Friedrich Hund, Robert Mulliken and John Lennard-Jones, and Lennard-Jones published the orbital account of the dioxygen ground state, unpaired electrons and all, in 1929 -- within a few years of the Lewis structure becoming the standard classroom object. Mulliken received the 1966 Nobel Prize in Chemistry for the work, and his Nobel lecture is still the clearest short statement of what the theory was trying to replace: not the idea of a bond, but the idea that electrons can be assigned to particular bonds at all. The conceptual move is to stop bookkeeping shared pairs and start counting occupancies. Once the atoms are close enough to interact, the atomic orbitals are no longer the right description of where the electrons are; the electrons occupy new, delocalised orbitals that belong to the molecule, and it is which of those orbitals are filled that fixes bond strength, bond length, spectroscopy and magnetism. Orbitals are conserved in the process: combine n atomic orbitals and you get exactly n molecular orbitals back, no more and no fewer. The two-orbital case above is simply the smallest instance of that rule. One asymmetry in that pairing matters more than its size suggests. The antibonding level is pushed up further than the bonding level is pushed down, so filling both members of a bonding and antibonding pair is not energetically neutral -- it is slightly worse than not forming the bond at all. That is the whole reason there is no He2 molecule: four electrons fill sigma(1s) and sigma*(1s) together, the bond order is zero, and the net effect is repulsive. Remove one of them and the arithmetic changes sign, which is why He2+, with bond order one half, is a real if fragile species. ## The physics For O2 and F2 the second-row ordering is sigma(2s) < sigma*(2s) < sigma(2p_z) < pi(2p_x) = pi(2p_y) < pi*(2p_x) = pi*(2p_y) < sigma*(2p_z) Note that B2, C2 and N2 swap sigma(2p_z) below the pi(2p) pair, because 2s-2p mixing pushes the sigma level up; the ordering above is specific to the O2/F2 end of the row. Spelled out, the swap runs like this. For B2, C2 and N2 the sigma(2p_z) level ends up above the pi(2p_x) = pi(2p_y) pair rather than below it, and the mechanism is a mixing between orbitals that the simple two-at-a-time construction ignores. sigma(2s) and sigma(2p_z) carry the same symmetry: both are cylindrically symmetric about the internuclear axis and both are even under inversion through the midpoint of the bond. Orbitals of the same symmetry are permitted to mix with one another, and when they do the lower of the two is pushed further down and the upper is pushed further up. The strength of that interaction scales inversely with the energy gap separating the 2s and 2p levels of the free atom. That gap is the whole story, because it widens steadily from left to right across the period. The 2s orbital penetrates the core and feels the growing nuclear charge more directly than 2p does, so each step along the row stabilises 2s more than 2p and pulls the two apart. At the left-hand end -- lithium through nitrogen -- the gap is small, the mixing is strong, and sigma(2p_z) is driven clear above the pi pair. At the right-hand end -- oxygen and fluorine -- the gap is large, the mixing is slight, and the unmixed ordering printed above survives. The evidence is not merely theoretical. B2 has six valence electrons and is observed to be paramagnetic, which is only possible if its two highest electrons occupy a degenerate pi level singly and with parallel spins; had sigma(2p_z) been the highest occupied orbital, both electrons would have paired in it and B2 would be diamagnetic. C2, two electrons further along, fills that same pi pair and is duly diamagnetic. Photoelectron spectroscopy, which measures the ionisation energies of the individual orbitals, confirms the ordering directly in both regimes. Underneath every version of the diagram sits a single quantity, the overlap integral S = integral over all space of phi_A * phi_B dV where phi_A and phi_B are the two atomic orbitals being combined. S is a pure number that measures how much of the same region of space the two functions occupy, weighted by their amplitudes there. It is not a metaphor for closeness: when the nuclei are far apart the two functions simply do not reach one another, S goes to zero, the splitting collapses, and both molecular orbitals fall back onto the free-atom energy. No overlap, no bond. In the simplest two-orbital treatment the normalised combinations are (phi_A + phi_B) / sqrt(2(1 + S)) and (phi_A - phi_B) / sqrt(2(1 - S)), and their energies are E+ = (alpha + beta) / (1 + S) and E- = (alpha - beta) / (1 - S), with alpha the Coulomb integral -- roughly the energy of an electron left on its own atom -- and beta the negative resonance integral that does the actual bonding. The (1 + S) and (1 - S) denominators are where the He2 asymmetry noted above comes from: dividing by the smaller number pushes the antibonding level up harder than the bonding level is pushed down. The same integral fixes the vertical spacing of the ladder. Two 2p_z orbitals meet head-on, lobe against lobe along the bond axis, and overlap efficiently; two 2p_x orbitals meet side-on and can engage only through the flanks of their lobes. At the same internuclear distance the sigma-type overlap is therefore the larger, so the sigma(2p_z) and sigma*(2p_z) pair is split further apart than the pi and pi* pair. That single fact accounts for sigma*(2p_z) sitting alone at the top of the ladder in both orderings, and for the pi and pi* levels being crowded near the middle where a modest amount of s-p mixing can reshuffle them. The degeneracy of the two pi* orbitals, on which the entire paramagnetism argument rests, is not a convenience of the drawing and not an approximation that better calculation would remove. It follows from symmetry alone. A diatomic molecule is cylindrically symmetric about its bond axis, so the x and y directions perpendicular to that axis are physically indistinguishable -- there is no measurement that could tell them apart, and no reason for an electron to prefer one. pi*(2p_x) and pi*(2p_y) are consequently the same orbital rotated by ninety degrees about the axis, and must have exactly the same energy; the same holds for the bonding pi(2p_x) and pi(2p_y). Nothing short of destroying the cylindrical symmetry -- bending the molecule, applying a field across the axis, bonding one flank to something else -- can split them. That is what makes the prediction so hard to argue with: it does not depend on the accuracy of any computed energy, only on the molecule being linear. Bond order = (bonding electrons - antibonding electrons) / 2. Removing or adding electrons from the pi* level therefore changes the bond directly, and bond length tracks it: | Species | Valence e- | Bond order | Unpaired e- | Bond length | Magnetism | |---|---|---|---|---|---| | O2+ | 11 | 2.5 | 1 | 112 pm | paramagnetic | | O2 | 12 | 2.0 | 2 | 121 pm | paramagnetic | | O2- (superoxide) | 13 | 1.5 | 1 | 133 pm | paramagnetic | | O2 2- (peroxide) | 14 | 1.0 | 0 | 149 pm | diamagnetic | Every added electron goes into an antibonding orbital, so every step down the table weakens and lengthens the bond. Only peroxide, with the pi* level filled and paired, is diamagnetic -- the one case where Lewis and molecular orbital theory agree. Bond order is worth taking seriously as a physical quantity rather than as bookkeeping, because it tracks the two things about a bond that can actually be measured. More net bonding density between the nuclei pulls them closer together and makes them harder to separate, so bond length falls and dissociation energy rises together as bond order climbs. Across the right-hand end of the second period, where the ordering above applies throughout, the trend is unmistakable: | Molecule | Bond order | Bond length | Dissociation energy | |---|---|---|---| | N2 | 3 | 110 pm | 945 kJ per mole | | O2 | 2 | 121 pm | 498 kJ per mole | | F2 | 1 | 142 pm | 158 kJ per mole | The two measured columns move in opposite directions, but neither is linear in bond order, and the deviation is itself informative: N2 is nearly six times harder to break than F2 for only three times the bond order, because the F2 bond is additionally destabilised by repulsion between the lone pairs crowded onto two small, electron-rich atoms -- an effect the bond order does not see. The relationship is a strong monotonic trend rather than a formula. It predicts the direction of a change confidently and its magnitude only roughly. Within the dioxygen family in the table above, each electron added to the pi* level costs half a bond order, lengthens the bond by of order ten picometres, and lowers the energy needed to break it; the dioxygenyl cation O2+, with an electron removed from pi* instead, has the shortest and strongest bond of the four. ## Term symbols and the low-lying states of dioxygen Written out in full, with the g and u labels that record whether an orbital is even or odd under inversion through the midpoint of the bond, the valence configuration of ground-state O2 is (sigma_g 2s)^2 (sigma_u* 2s)^2 (sigma_g 2p)^2 (pi_u 2p)^4 (pi_g* 2p)^2 which counts out as twelve electrons, six from each atom's 2s2 2p4. Eight of them are in bonding orbitals and four in antibonding ones, giving the bond order of two that the table above records. The state that results is labelled X 3Sigma_g-, where the leading superscript three is the spin multiplicity 2S + 1: with two unpaired electrons the total spin quantum number S is 1 and the ground state is a triplet. The physical reason Hund's rule holds here is exchange. Electrons of the same spin are forbidden by the Pauli principle from occupying the same point, so a parallel-spin pair keeps further apart on average and pays less electrostatic repulsion, and putting them in different spatial orbitals reduces that repulsion further still. Two unpaired electrons give a spin-only magnetic moment of sqrt(n(n+2)) = sqrt(8) = 2.83 Bohr magnetons, and gaseous dioxygen accordingly shows a molar magnetic susceptibility near +3.4 x 10^-3 cm^3 per mole at room temperature -- positive, and orders of magnitude larger than the small negative values typical of ordinary closed-shell molecules. Below 90 K the liquid concentrates that susceptibility enough to make the classroom demonstration dramatic: poured between the poles of a strong magnet, liquid oxygen does not fall through but bridges the gap and hangs there until it boils away. Liquid nitrogen, every valence electron paired, pours straight through. The same two electrons in the same two orbitals also generate dioxygen's excited states, and this is where the molecular orbital picture supplies something the Lewis structure cannot even frame. If the two pi* electrons are paired into a single pi* orbital instead of spread across both, the result is the singlet state a 1Delta_g, about 94 kJ per mole (0.98 eV) above the ground state; a second singlet arrangement, b 1Sigma_g+, lies about 157 kJ per mole (1.63 eV) up. The first of these is "singlet oxygen", a chemically distinct and far more aggressive species that matters in photodynamic therapy, polymer degradation and atmospheric photochemistry -- and it exists as a separate, long-lived state precisely because the ground state was a triplet. That triplet ground state also resolves something the Lewis picture makes puzzling: why an atmosphere holding twenty-one per cent of a powerful oxidant does not simply ignite. A direct reaction between triplet O2 and an ordinary singlet organic molecule to give singlet products would have to change the total spin, which is forbidden to first order, and the resulting kinetic barrier is most of what keeps the biosphere from burning. Oxygen's magnetism and oxygen's sluggishness are the same fact observed twice. ## Reading the phase of an orbital Every picture of an orbital, including the one in the simulation above, is a choice. An orbital has no boundary -- the wavefunction decays smoothly to zero and never quite arrives -- so what gets drawn is an isosurface, the locus of points where the wavefunction takes some chosen value. Raising or lowering that value inflates or shrinks the shape without changing the physics, which is why the isosurface control matters: set it low and the diffuse tails dominate, so lobes appear to merge generously across the midplane; set it high and only the dense core near each nucleus survives. Neither picture is the orbital, and comparing them is more instructive than trusting either. Two surfaces are drawn for each orbital, one at a positive value of the wavefunction and one at the negative of it, and the contrasting colours record the sign of the wavefunction and nothing else. The sign is not itself observable -- the probability density is the square, and multiplying an entire molecular orbital by minus one changes nothing -- but the relative sign of the two atomic contributions is exactly what distinguishes bonding from antibonding. Where two same-signed lobes merge into a single body of amplitude between the nuclei, the atomic wavefunctions have added constructively and an electron there is attracted by both nuclei at once. Where a positive lobe faces a negative one across an empty gap, they have cancelled, and the surface on which the wavefunction is exactly zero is a nodal plane. Counting nodes is the fastest way to read a diagram. A node perpendicular to the bond axis, bisecting the internuclear region, is the signature of an antibonding orbital and appears in sigma*(2s), sigma*(2p_z) and both pi* orbitals. A node containing the bond axis is the signature of pi character and appears in pi(2p_x) and pi(2p_y) as well as their starred partners; a pi* orbital therefore carries both kinds at once, which is why it looks like four separated lobes. The sigma bonding orbitals have neither and are the only ones with uninterrupted amplitude along the whole axis. One trap is worth naming, because the phase bookkeeping for orbitals with angular nodes is easy to get backwards. Take two 2p_z orbitals on atoms lying along the z axis, each drawn in the usual way with its positive lobe pointing toward +z. The lobes that actually face one another across the gap then carry opposite signs, so the constructive, bonding combination is the difference of the two atomic functions rather than their sum. For 2p_x orbitals meeting side-on, the facing lobes already match and the bonding combination is the sum. The reliable criterion is never the plus or minus sign in the formula but the physical question the isosurface answers directly: does amplitude build up between the nuclei, or is it scooped out. ## Controls -> what each maps to | Control | Maps to | Range / values | Physical meaning | |---|---|---|---| | Oxygen species | valence electron count | O2+, O2, O2-, O2 2- | Adds or removes pi* electrons; drives bond order, length and magnetism | | Molecular orbital to display | selected MO | the 8 valence MOs | Which one-electron wavefunction is drawn in 3D | | Isosurface level | percentage of peak amplitude | low - high | Where the surface is cut; low values show lobes merging across the midplane | | Show nodal planes | -- | on / off | Reveals the node between the nuclei that defines an antibonding orbital | | Cut away the near half | -- | on / off | Exposes the interior phase structure | | Auto-rotate | -- | on / off | Rotation only; disabled under prefers-reduced-motion | ## Learning objective After playing, a learner can compute a bond order from a filled molecular orbital diagram, explain why dioxygen is paramagnetic when its Lewis structure says otherwise, and identify a bonding versus an antibonding orbital from the presence of a node between the nuclei. A learner who has worked through the sections above should be able to go further: to say why the diagram is drawn one way for B2, C2 and N2 and another for O2 and F2, and to name the 2s-2p energy gap as the reason; to predict the direction in which bond length and dissociation energy move when an electron is added to or removed from the pi* level, before looking up either; and to state what would have to be done to the molecule for the two pi* orbitals to stop being degenerate, which is the same as saying what the paramagnetism argument actually depends on. ## Limits and connections This is the linear-combination-of-atomic-orbitals approximation at its simplest: fixed atomic basis, no configuration interaction, no correlation, and orbital energies drawn schematically rather than computed. Real calculations solve for the coefficients self-consistently and the picture stays qualitatively intact, which is why the diagram survives in every textbook. The shapes being combined here are the subject of [[Atomic_orbital]]; what happens when the same atoms bond into an extended solid instead of a molecule is [[Silicon_dioxide]]. It is also worth being clear that a molecular orbital is not an observable. Only the total electronic wavefunction is, and the one-electron orbitals are an artefact of the approximation that lets each electron move in the averaged field of the others. What rescues the picture from being merely a mnemonic is that its orbital energies map approximately onto measurable ionisation energies, so photoelectron spectroscopy can test the ordering band by band -- which is how the s-p mixing story above stopped being an argument and became a measurement. Where the simple diagram is quantitatively poor, the failures are instructive rather than fatal: the ground state of C2 and the weakness of the F2 bond both need electron correlation beyond a single configuration to come out right, and adding it changes the numbers without disturbing the qualitative picture. The same orbitals reappear whenever dioxygen does chemistry. When O2 binds to the iron of a haem group, it is the pi* orbitals that accept electron density from the metal, and the resulting complex is diamagnetic: the magnetism that identifies free dioxygen disappears on binding, which is a large part of the story in [[Hemoglobin]]. Stretch the same construction over three centres instead of two and the pi system delocalises across the molecule, which is what makes ozone a different substance from dioxygen rather than merely a bigger one -- the subject of [[Allotropes_of_oxygen]] and, for the atmospheric consequences, [[Ozone_layer]]. ## References - Atkins, P. W.; Friedman, R. S. *Molecular Quantum Mechanics*, 5th edition. Oxford University Press, Oxford, 2011. Chapters 8 and 9 cover the variation principle, the LCAO construction and diatomic molecules. - Atkins, P. W.; de Paula, J.; Keeler, J. *Atkins' Physical Chemistry*, 12th edition. Oxford University Press, Oxford, 2022. Focus 9, molecular structure, including the dioxygen paramagnetism argument. - Housecroft, C. E.; Sharpe, A. G. *Inorganic Chemistry*, 4th edition. Pearson Education, Harlow, 2012. Chapter 2 treats homonuclear diatomics, s-p mixing, and the B2-N2 versus O2-F2 orderings. - Levine, I. N. *Quantum Chemistry*, 7th edition. Pearson, Boston, 2014. Chapter 13, the electronic structure of diatomic molecules and their term symbols. - Mulliken, R. S. "Spectroscopy, Molecular Orbitals, and Chemical Bonding." Nobel Lecture in Chemistry, The Nobel Foundation, Stockholm, 12 December 1966; reprinted in *Science*, volume 157, number 3784, pages 13-24, 1967. - Lennard-Jones, J. E. "The Electronic Structure of Some Diatomic Molecules." *Transactions of the Faraday Society*, volume 25, pages 668-686, 1929. The original molecular orbital account of the dioxygen ground state and its paramagnetism. - Hund, F. "Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel." *Zeitschrift fuer Physik*, volume 33, pages 345-371, 1925. Origin of the rule of maximum multiplicity. - Herzberg, G. *Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules*, 2nd edition. Van Nostrand, New York, 1950. Symmetry classification and term symbols for diatomics. - Huber, K. P.; Herzberg, G. *Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules*. Van Nostrand Reinhold, New York, 1979. Reference source for diatomic bond lengths, dissociation energies and the excited states of O2. - Coulson, C. A. *Valence*, 2nd edition. Oxford University Press, Oxford, 1961. Classic exposition of the sigma and pi classification and of bond order. - Pauling, L. *The Nature of the Chemical Bond*, 3rd edition. Cornell University Press, Ithaca, New York, 1960. The valence bond alternative, for contrast. - Shakhashiri, B. Z. *Chemical Demonstrations: A Handbook for Teachers of Chemistry*, volume 2. University of Wisconsin Press, Madison, 1985. Procedure for the liquid-oxygen-and-magnet demonstration. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Molecular_orbital) : [Wikitube](https://en.wikitube.io/wiki/Molecular_orbital) ## Previous hub tags Tree parent: [[Oxygen]]. Legacy hubs: `REACTION`. --- *Created 2026-08-05 - append-only - hand-authored to WIKI_REPOPULATION_PROTOCOL v1.0 section 5 - 0 deletions*