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The labels apply when the molecule has an inversion centre, as a homonuclear diatomic molecule does. The label $g$ means $\psi (-\mathbf{r})=\psi (\mathbf{r})$, and $u$ means $\psi (-\mathbf{r})=-\psi (\mathbf{r})$, with the origin at that centre. In a heteronuclear diatomic molecule, inversion exchanges different atomic nuclei and these are generally not valid symmetry labels.

For the p-derived combinations above,Inversion both exchanges the two centres and changes the sign of each individual p atomic orbital. Applying both operations to each sum or difference gives these labels. The star denotes antibonding molecular orbitals, independently of parity: a bonding molecular orbital need not always be $g$.

End-on overlap produces $\sigma$ and ${\sigma }^{\ast }$ molecular orbitals; side-on overlap produces $\pi$ and ${\pi }^{\ast }$ molecular orbitals. Matching signs of the facing atomic orbital lobes give constructive interference and a bonding molecular orbital. Opposite signs give destructive interference and an additional nodal plane between the atomic nuclei in the antibonding molecular orbital.

Take the internuclear direction as $x$, and define both atomic $p_x$ functions as positive on their own right-hand side. Up to wavefunction normalization, the combinations areThe minus sign in the bonding $p_x$ combination follows from this common-axis quantum phase convention: the facing lobes of the two unmodified $p_x$ functions have opposite signs. It does not make that combination antibonding. Every $\pi$ molecular orbital also has a nodal plane containing the internuclear axis; the additional midpoint node distinguishes ${\pi }^{\ast }$ from $\pi$.

A triangular prism has exactly the required nine metal–metal edges: six edges around two triangular faces and three edges joining corresponding vertices of those faces. It naturally offers two symmetry-distinct edge families, of sizes six and three.

With twelve $\mathrm{C}{\mathrm{l}}^{-}$ ligands, neutral $\mathrm{R}{\mathrm{e}}_6\mathrm{C}{\mathrm{l}}_{12}$ has total rhenium oxidation number $+12$, hence six formally Re(II), $d^5$ centres. After treating the Re–Cl interactions as ligand bonds, the available metal $d$ electrons areTwo localized bond order assignments use those fifteen pairs:In scheme I the three inter-triangle edges should be shorter; in scheme II the six triangle edges should be shorter. Both give each Re five electrons' worth of metal–metal bond-order contributions: two singles plus a triple, or two doubles plus a single. The electron count permits both schemes; identifying which family is shorter distinguishes them at this level. Actual cluster molecular orbitals are delocalized, so these integer bond orders summarize bonding rather than asserting isolated textbook multiple bonds on every edge.

The terms are ${}^3{\Sigma }_g^{-}$, ${}^1{\Delta }_g$ and ${}^1{\Sigma }_g^{+}$. Closed molecular orbital shells contribute no net spin angular momentum or axial orbital projection, so only the two equivalent ${\pi }_g$ electrons need be coupled. Their individual projections are $\lambda =\pm 1$.

Putting both electrons in the same projection gives $|\Lambda |=2$ and forces opposite spins by the Pauli exclusion principle, yielding ${}^1{\Delta }_g$. With one electron in each projection, the spin-adapted symmetric and antisymmetric spatial combinations give a singlet $\Sigma$ and a triplet $\Sigma$. Their reflection parities are $+$ and $-$, respectively. Both open-shell electrons are gerade, so their product is gerade. The dimensions $3+2+1=6$ exhaust the $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ microstates. By Hund's rules, the triplet is the lowest term, and ${}^1{\Delta }_g$ is the lower singlet.

Each site has a pi-electron population $4/3$, and each adjacent pi bond order is $2/3$. The two carbon sites supply one pi electron each and the nitrogen lone pair supplies two, giving four electrons in two occupied molecular orbitals. Orthogonality to the given Homo and LUMO fixes the remaining occupied normalized orbital, up to overall sign, asFor doubly occupied orbitals, the Hückel orbital charge density and Hückel pi bond order follow fromThe diagonal sums to four electrons. The negative $P_{13}$ is an off-diagonal density-matrix element between nonadjacent sites, not an additional directly connected chemical bond.

Relative to neutral pi contributions $(1,2,1)$, the pi-system charges are $(-e/3,+2e/3,-e/3)$: electron density has moved from nitrogen toward the two carbon sites. Thus the physical electric dipole moment vector points from the midpoint of the carbon pair toward nitrogen along the molecular symmetry bisector. The conventional chemical dipole arrow, with a crossed positive tail, points oppositely: from nitrogen toward the carbon pair. This is only the pi contribution, not a prediction of the full sigma-plus-pi molecular dipole.

No: the lowest dissociation channel is $\mathrm{He}+{\mathrm{H}}^{+}$, which a doubly occupied helium-centred orbital can describe. In the heteronuclear molecular orbital diagram the helium $1s$ site lies below the hydrogen $1s$ site. As separation grows, the bonding orbital tends to helium $1s$, the antibonding orbital tends to hydrogen $1s$, and the two ground-state electrons remain in the lower orbital.

There is therefore no symmetry requirement to keep both sites equally occupied and no compulsory half-weight mixture of wrong ionic configurations at dissociation. The physical asymptote is already a closed-shell atom plus a bare proton. The alternative ${\mathrm{He}}^{+}+\mathrm{H}$ limit lies higher. A Restricted Hartree-Fock method curve approaches the Hartree–Fock helium energy rather than the exact correlated helium energy, so ordinary dynamic electron correlation is still missing, but this is not the homonuclear static-correlation catastrophe. In the preceding two-panel sketch, the helium hydride curves have the same qualitative dissociation channel with a small residual correlation offset, whereas the hydrogen restricted curve reaches a qualitatively wrong high limit.

The molecular orbital diagram has one level at $\alpha +2\beta$ labelled $A_{1g}$, two degenerate levels at $\alpha$ labelled $E_u$, and one level at $\alpha -2\beta$ labelled $B_{1g}$, in increasing energy for $\beta <0$. Each spatial molecular orbital admits two opposite-spin electrons. The diagram below also shows how one representative distortion lifts the middle degeneracy; the distorted scale is illustrative, not a fitted molecular spectrum.

Within each pair the in-phase and out-of-phase combinations have $A_1$ and $B_1$ symmetry respectively. One normalized choice isThe sign convention for $d$ is chosen to make the off-diagonal interaction positive in the corresponding two-by-two block. It does not change the physical predictions. These are pair symmetry-adapted linear combinations; the actual molecular orbitals are linear combinations within each of the two equal-symmetry pairs, found by diagonalizing the blocks.

Put $D=\sqrt{n^2-2n+5}=\sqrt{(n-1{)}^2+4}$. For $0<n<1$, $D>n+1$, since $D^2-(n+1{)}^2=4(1-n)>0$. Remembering $\beta <0$, the two lowest trapezium molecular orbital energies areThe remaining levels lie at $2\alpha -{ϵ}_2$ and $2\alpha -{ϵ}_1$. Filling the lowest spatial orbitals with opposite-spin pairs givesFor the square four-electron case, two electrons occupy the lowest orbital and two occupy the degenerate $E_u$ set. Their summed one-electron energy is fixed here, but this Hückel method calculation alone does not decide the spin or correlation within that open shell.

A zero-dimensional example is buckminsterfullerene, ${\mathrm{C}}_{60}$: a closed cage with 60 vertices, 90 edges, 12 pentagons and 20 hexagons. Euler's relation is $60-90+32=2$. Each carbon has three neighbors in a curved, approximately $s{p}^2$-bonded network; bonds are not all equivalent. Its cage diameter is of order $0.7\mathrm{nm}$.

Confinement in all directions gives discrete molecular orbitals, unlike the extended dispersion of graphene. It is an electron acceptor with characteristic electronic and vibrational spectra. Its pentagons supply the positive curvature needed to close the carbon network. A molecular solid of cages is held together predominantly by intermolecular forces, so its mechanical and electrical properties differ from those of an individual strongly bonded cage. Undoped fullerene is not a high-conductivity graphene sheet; doping, packing and charge transfer can strongly change transport.

Choose one atomic orbital ${\varphi }_A$ and ${\varphi }_B$ on the two atoms and approximate a molecular orbital by the linear combination of atomic orbitalsProjecting the Schrödinger equation onto the two orbitals gives a generalized eigenvalue problem $H\mathbf{c}=ES\mathbf{c}$, where $H_{ij}=⟨{\varphi }_i|\hat{H}|{\varphi }_j⟩$ and $S_{ij}=⟨{\varphi }_i|{\varphi }_j⟩$. In an orthonormal basis the matrix isThe coupling mixes the atomic states and produces a lower bonding molecular orbital and upper antibonding molecular orbital. For equivalent real orbitals with overlap $s$ the normalized combinations are $({\varphi }_A\pm {\varphi }_B)/\sqrt{2(1\pm s)}$; the precise energy ordering follows from the Hamiltonian matrix elements. This restricted orbital basis is an approximation to the molecule's wavefunction, rather than a claim that exact isolated-atom orbitals remain unchanged in the molecule.

An electron occupying a pi-type molecular orbital. In a conjugated planar molecule, the pi subspace is commonly built from p orbitals perpendicular to the molecular plane; its electron population excludes the sigma framework.

A radical has one or more unpaired electrons. In a simple atomic orbital or molecular orbital occupancy picture, an unpaired electron occupies a spatial orbital without an opposite-spin partner in that orbital. A dot such as ${\mathrm{H}}^{\bullet }$ marks this electronic property; it does not mean a missing electron or positive electric charge. Free radicals can be neutral or charged. Terminology also has a boundary: metal ions exhibiting paramagnetism are not normally called free radicals just because they have unpaired electrons.^[ref]

Compare three species containing the same single proton. The hydrogen atom ${\mathrm{H}}^{\bullet }$ has one electron in its $1s$ atomic orbital, and is a neutral radical. The hydrogen cation ${\mathrm{H}}^{+}$ has no electrons, so no unpaired electron. The hydrogen anion ${\mathrm{H}}^{-}$ has two electrons paired in its $1s$ atomic orbital. It is charged but is not a radical in its ground state. With $N_p$ protons, $N_e$ electrons, and positive elementary charge $e$, the independent electric charge count isThe electron pair and electric charge questions require different information. An odd $N_e$ ensures that not all electrons can be paired; an even $N_e$ alone does not ensure pairing, as the molecular orbital description of molecular oxygen illustrates.

The distinction matters whenever a chemical reaction mechanism uses dots and charge signs together. Homolysis gives one bonding electron to each fragment, while heterolysis assigns both to one fragment. A radical chain reaction can pass an unpaired electron from one species to another without generating net electric charge. Detection then exploits the unpaired electron's spin and magnetic moment, rather than merely measuring whether the sample carries an electric charge.

A single nitrate Lewis structure singles out one N=O connection, yet the three N–O chemical bonds of the isolated symmetric ion are equivalent. Chemical resonance reconciles that symmetry with the integer electron pair allocations of a Lewis structure. The equal chemical bond lengths in nitrate are discussed in The Open University, Discovering chemistry: resonance structures.

Chemical resonance uses several Lewis structures with the same chemical atom connections and total electric charge to describe electronic structure that one allocation cannot fully express. The chemical atoms stay in place; the positions of bond pairs, lone pairs, and formal charges differ between drawings. A two-headed arrow $\leftrightarrow$ connects the drawings and is distinct from a chemical equilibrium arrow $\rightleftharpoons$. See the IUPAC Gold Book, resonance form.

In each nitrate resonance structure, use the existing formal charge convention $\mathrm{FC}=v-n-b/2$:Here $v$ counts neutral-chemical atom valence electrons, $n$ counts nonbonding electrons, and $b$ counts incident bonding electrons. The sum is $+1-1-1+0=-1$, agreeing with the nitrate ion's net electric charge. The formal charge rules and their distinction from actual atomic electric charge are given in Flowers et al., Chemistry 2e, 7.4: formal charges and resonance.

Keep one labelled oxygen fixed while reading the three nitrate resonance structures. Its connection is double in one drawing and single in two. Equal weighting within this three-contributor Lewis structure model gives the average bond orderThe same calculation applies to all three N–O connections. Averaging the drawn formal charges similarly gives $-2/3$ per oxygen, while nitrogen retains $+1$; $+1+3(-2/3)=-1$. These fractions summarize the chosen contributors. They are not a measurement of partial atomic charge, a literal fraction of an electron sitting on an oxygen, or time spent in one drawing.

The resonance hybrid has one spatially continuous electron density extending over the ion. None of its N–O chemical bonds is permanently selected as the double connection of one contributor. The dashed marks above are symbols; electrons do not travel along them or make the marks float between chemical atoms. For isolated symmetric nitrate, symmetry makes the three N–O chemical bonds equivalent; an environment that distinguishes the oxygen sites can remove that equivalence. The fraction $4/3$ alone predicts neither a chemical bond length nor a dissociation energy. The general relation between multiple chemical bonds, lengths, and strengths is discussed in 7.5 Strengths of Ionic and Covalent Bonds; its tabulated averages are not a nitrate-specific calibration of $4/3$.

Ozone provides a useful two-contributor comparison. Its $3\times 6=18$ valence electrons give three bond pairs and six lone pairs. One resonance structure has a double O–O connection on the left and a single one on the right; the other reverses these allocations while retaining the same chemical atom positions and connections. The central oxygen has one lone pair and formal charge $+1$; the singly connected terminal oxygen has three lone pairs and formal charge $-1$; the doubly connected terminal oxygen has two lone pairs and formal charge zero. Thus $6+12=18$ electrons and zero net electric charge are conserved. The equal-contributor bond order count is $(2+1)/2=3/2$ for either connection. See MIT 5.111, lecture 10, pp. 4–5: ozone resonance structures.

For carbon dioxide, $\mathrm{O}=\mathrm{C}=\mathrm{O}$ and ${\mathrm{O}}^{-}-\mathrm{C}\equiv {\mathrm{O}}^{+}$ share sixteen valence electrons and the same O–C–O connections. The mirror drawing ${\mathrm{O}}^{+}\equiv \mathrm{C}-{\mathrm{O}}^{-}$ is also possible. The two charge-separated drawings are equivalent to one another for an isolated symmetric carbon dioxide molecule, but neither is equally dominant with $\mathrm{O}=\mathrm{C}=\mathrm{O}$ in the elementary formal charge comparison.

Chemical resonance does not mean that a molecule rapidly flips between drawings, nor that separate molecules at chemical equilibrium carry the individual drawings. The actual electron density belongs to one combined electronic description. In a quantum mechanics treatment, contributors enter a linear combination of many-electron wavefunctions; a molecular orbital treatment can describe the extended electronic arrangement through molecular orbitals. The drawings alone provide neither a measured density nor numerical weights. Their usefulness is to expose the limitations of one localized allocation without pretending that every algebraically allowed Lewis structure contributes equally.

A spatial molecular orbital occupied by one electron. In an open-shell molecule, its empty opposite-spin slot can accept another electron even though the spatial orbital is occupied.

A spatial orbital is a normalized spatial wavefunction used in a one-particle description. An atomic orbital is the corresponding notion for a chemical atom; a molecular orbital belongs to a molecule. A spatial orbital is not a trajectory and does not by itself specify spin.

Electron correlation caused by the need for multiple important configurations, often when several molecular orbitals are nearly degenerate. Stretching a homonuclear two-electron bond makes a single doubly occupied orbital retain unphysical ionic contributions; a suitable combination of configurations cancels them.

A linear combination of atomic orbitals chosen to transform under an irreducible representation of the molecular point group. Relative orbital signs must be transformed as well as the positions of their centres. Such combinations organize a Hamiltonian operator into symmetry blocks; combinations of the same symmetry generally still mix and need not themselves be molecular orbitals.

Ultraviolet is electromagnetic radiation with wavelengths shorter than the violet end of ordinarily visible light. The CIE convention uses $100$–$400\mathrm{nm}$, subdivided into UV-A at $315$–$400\mathrm{nm}$, UV-B at $280$–$315\mathrm{nm}$ and UV-C at $100$–$280\mathrm{nm}$. These divisions name intervals; they do not mark universal changes in every material's response. The boundary with visible light is not sharp, and other fields use near-, far-, vacuum- and extreme-ultraviolet divisions. ^[ref]. For example, NASA's Solar Dynamics Observatory observes extreme-ultraviolet bands below $100\mathrm{nm}$; the CIE interval is therefore not the full extent of astronomical ultraviolet usage. ^[ref].

The CIE ultraviolet interval corresponds to frequencies about $7.49\times 10^{14}$–$3.00\times 10^{15}\mathrm{Hz}$ and photon energies $4.97\times 10^{-19}$–$1.99\times 10^{-18}\mathrm{J}$, using vacuum wavelength and $E=hc/\lambda$ with Planck constant $h$ and speed of light $c$. Shorter wavelength means larger energy per photon; larger irradiance means more incident power per area. These are different comparisons. At the same incident power, a $254\mathrm{nm}$ ultraviolet beam has fewer photons per second than a $940\mathrm{nm}$ infrared beam, even though each ultraviolet photon has more energy.

A common ultraviolet mechanism is photon absorption promoting an electron from a ground state to an excited state of a chemical atom or molecule. By contrast, many infrared absorption bands involve stretching or bending chemical bonds. For either mechanism, both the energy difference and the coupling to the incident electromagnetic field matter. One concrete example is the $217\mathrm{nm}$ absorption discussed for butadiene: an electron moves between molecular orbitals, and the position of the absorption reveals their energy separation. Ultraviolet absorption can therefore probe molecular orbitals, while infrared spectroscopy probes many molecular motions. ^[ref].

Subsequent chemical reactions depend on the absorbing chemical substance and its environment; the label ultraviolet alone does not predict a reaction rate. At sufficiently short wavelength, a photon can cause photoionization if its energy exceeds the relevant ionization energy. There is no single wavelength that separates all ionizing from all nonionizing behavior for every chemical substance. Likewise, eventual heating after photon absorption is possible in both ultraviolet and infrared. The distinction is useful because different optical spectra expose different processes, not because the two ranges obey different laws of electromagnetism.

Ritter's 1801 observation used a chemical readout rather than a thermometer. In Caltech/IPAC's account, sunlight passed through a glass optical prism and silver chloride was placed in different parts of the optical spectrum. Darkening was weak toward red, stronger toward violet, and pronounced beyond the visible violet edge. The measured evidence was a change in the chemical substance's appearance at a spatial location with no visible color, establishing radiation beyond the eye's ordinary range. It was not a calibrated measurement of a universal ultraviolet optical spectrum. ^[ref].

A controlled modern interpretation of such a chemical test would compare equal exposure durations and matched fresh samples, keep an unexposed sample in darkness, monitor temperature, and distinguish scattered visible light from the intended short-wavelength beam with characterized filtering. These are proposed controls, not an assertion that Ritter performed them. Unequal sample preparation or heating can change a reaction rate. The optical prism, terrestrial atmosphere and sample response restrict which wavelengths the test detects. Darkening is a detector-weighted response, so its spatial maximum is not automatically the maximum of incoming radiant flux or of photon energy.

Thomas Larason and Jeanne Houston's NIST report documents a quantitative ultraviolet detector comparison. An argon source feeds a double-diffraction grating spectrometer; a shutter and mirrors deliver the selected band to a working-standard or test photodiode, successively at the same position. A beam splitter supplies a monitor photodiode. Amplifiers and voltmeters record detector and monitor simultaneously; shutter-closed readings provide dark baselines. The $200$–$500\mathrm{nm}$ scan, typically with $4\mathrm{nm}$ bandpass, measures spectral responsivity. Monitor normalization suppresses source drift; known spectral lines calibrate wavelength. The report assesses temperature effects and unwanted light outside the band, weighted by detector response. The latter need not cancel between different spectral responsivities. ^[ref].

This determines electrical response without visual perception. Bandpass and spatial nonuniformity limit the calibration; aging and changed temperature can affect later use. The scan includes visible light and ultraviolet; no response below $200\mathrm{nm}$ follows from it.

For a narrow beam, the idealized substitution calculation isHere $R$ is spectral responsivity; $I$ and $M$ are dark-subtracted detector and monitor electric currents. This follows by writing $I=RP$ for each detector and using the monitor to track changes in radiant flux. It assumes linear response, unchanged beam splitting and collection geometry, and a sufficiently narrow band. For an illustrative standard with $R_{\mathrm{std}}=0.15\mathrm{A}{\mathrm{W}}^{-1}$, ratios $I_{\mathrm{std}}/M_{\mathrm{std}}=2.0$ and $I_{\mathrm{test}}/M_{\mathrm{test}}=1.6$ give $R_{\mathrm{test}}=0.12\mathrm{A}{\mathrm{W}}^{-1}$. These invented arithmetic inputs illustrate the method; they are not NIST measurements.

An independent photon-counting model connects spectral responsivity to quantum efficiency. If $\eta (\lambda )$ is the number of collected electrons per incident photon, there is no internal multiplication, and $q$ is the elementary charge, thenAt illustrative constant $\eta =0.60$ and $P=1\mu \mathrm{W}$, this gives about $0.123\mu \mathrm{A}$ at $254\mathrm{nm}$ and $0.455\mu \mathrm{A}$ at $940\mathrm{nm}$. The ultraviolet readout is smaller in this model because there are fewer incident photons, not because their individual energies are smaller. Real quantum efficiency depends on wavelength, windows, absorption and carrier collection; a detector's band gap and construction determine its usable range. Constant $\eta$ across these two wavelengths is an analytical comparison, not a claimed specification of a real photodiode.

The curve is the calculated relation $E=hc/\lambda$, and the shaded ranges use the stated CIE conventions. The lower diagram abstracts the detector comparison rather than reproducing an instrument drawing. Together with the infrared heat-balance illustration, it shows why a calibrated detector response, rather than what looks bright to an observer, is the bridge between invisible radiation and a quantitative measurement.

The VSEPR model predicts approximate molecular geometry by arranging the electron domains around a central chemical atom to reduce their mutual repulsion. Start with a Lewis structure, count the domains, and then distinguish the resulting electron pair geometry from the positions of the chemical atoms themselves. A lone pair counts as a domain even though it is not another chemical atom.^[ref]

Three worked counts have the same total of eight valence electrons. In methane, carbon's four plus the contributions of four hydrogen chemical atoms give $4+4=8$: four bond pairs, no lone pairs, and tetrahedral molecular geometry. In Ammonia, nitrogen's five plus three from hydrogen give $5+3=8$: three bond pairs and one lone pair, producing trigonal pyramidal molecular geometry. In water, oxygen's six plus two from hydrogen give $6+2=8$: two bond pairs and two lone pairs, producing bent molecular geometry. All three have four electron domains; their molecular geometries differ.

The regular tetrahedral directions can be represented by the unit vectorsEach pair has dot product $-1/3$, so the ideal bond angle is $\theta =\arccos (-1/3)\approx 109.47^{\circ }$. This is a geometric calculation, rather than a prediction that every four-domain molecule has that exact bond angle. Lone pairs and different kinds of chemical bonds distort the ideal arrangement; water's H–O–H angle is about $104.5^{\circ }$.^[ref]

The model is attractive because small-number bookkeeping produces a spatial prediction without first solving for every molecular orbital. Shape then helps explain chemical polarity and which hydrogen bond donor or hydrogen bond acceptor sites are accessible. It is a qualitative model, not a full calculation of the electronic energy or a universal solution for every chemical compound. Gillespie and Nyholm's 1957 review ^[ref] is a historical anchor for its development.