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For fixed electron-electron interaction and electron number, a nondegenerate ground-state electron density determines the external potential up to an additive constant, and hence determines the Hamiltonian and ground-state observables.

Suppose two potentials $v,v^{\prime }$ differing by more than a constant have different ground states $\Psi ,{\Psi }^{\prime }$ but the same density $\rho$. Strict Rayleigh-Ritz variational principle inequalities giveAdding produces the contradiction $E+E^{\prime }<E+E^{\prime }$. This proves the first Hohenberg-Kohn theorem under the stated nondegeneracy assumptions. A constant potential shift changes energy by $N$ times that constant without changing the density.^[ref]

Potassium is electropositive and transfers electron density toward the metal. The resulting surface dipole generally lowers the metal work function at such submonolayer coverages, reducing the tunnelling barrier and its inverse decay length $q\propto \sqrt{{\Phi }_{\mathrm{effective}}}$.

At the same specified $5\mathring{\mathrm{A}}$ separation, a smaller $q$ gives a larger $\exp (-2qs)$, so the expected tunnelling current is greater than $1\mathrm{nA}$ in the barrier-dominated comparison. No numerical value follows without the modified work function and electronic density of states. A real STM image also depends on the local density of states and adsorbate corrugation; equal geometric separation does not imply equal electronic barrier height.

Nitrogen-down adsorption places the positive hydrogen end farther from the metal, producing an outward molecular electric dipole moment and the observed decrease in work function. Its maximum normal component, however, is only $1.42$ D. Tilting alone can only reduce that component; it cannot explain an effective $2.20$ D.

The adsorbed molecule and metal therefore undergo appreciable electron density. Lone-pair donation into the metal, polarization of the substrate, image response and distortion of the adsorbate can change the effective interface dipole. The inferred increase relative to an upright isolated molecule is about $0.78$ D. For scale, transferring about $0.16$ elementary charges over $1$ Å would produce that dipole increment:This scale illustrates a plausible interfacial polarization; the work-function measurement alone does not uniquely determine a transferred charge or adsorption geometry.

The structure factor is the coherent scattering amplitude of the contents of a unit cell at a specified momentum transfer. Contributions from its atoms add with phases fixed by their positions. At a reciprocal lattice vector indexed by Miller indices, the crystal basis structure factor isChanging every phase sign conjugates the amplitude for real scattering factors and leaves the intensity unchanged. The positions here are fractions of the chosen cell edges. An atomic basis of a crystal should not be confused with a set of primitive translation vectors.

In kinematic elastic scattering the diffraction intensity of a Bragg reflection is proportional to $|S_{hkl}{|}^2$, after including geometrical, polarization, thermal-displacement and experimental factors. Powder integrated intensities also include multiplicity. The many-cell interference factor selects reciprocal-lattice vectors satisfying the diffraction geometry; the unit-cell factor determines their relative amplitudes and additional extinctions. Ring positions determine spacings through Bragg's law, while intensity patterns and systematic absences constrain lattice centering and atomic arrangements. Intensity loses the overall complex phase, so structure determination is not a direct measurement of every individual atomic phase.

X-rays scatter from electron density, giving an atomic factor that changes with momentum transfer and chemical charge distribution. Neutrons scatter coherently from nuclei with bound coherent scattering lengths, which need not track atomic number; magnetic neutron scattering is an additional channel in magnetic materials. Their different elemental/isotopic contrasts make the two probes complementary.

Choose longitudinal current along $x$, magnetic field along $z$, and define the oriented Hall coefficient by $R_H=E_y/(j_x{B}_z)$. For one carrier type with signed charge $q$, a stationary transverse force balance is $q(\mathbf{E}+\mathbf{v}\times \mathbf{B}{)}_y=0$. Since $(\mathbf{v}\times \mathbf{B}{)}_y=-v_x{B}_z$, this gives $E_y=v_x{B}_z$. Using $j_x=nq{v}_x$,This is the weak-field isotropic one-carrier approximation with Hall scattering factor one, not a general multiband identity.

Under the stipulated electron-dominated approximation, the given Hall magnitudes implyTake the ratio of the intrinsic activation law:Solving gives the model's activation-gap estimateThis result uses the supplied Hall-temperature ratio and the fixed-gap, fixed-prefactor assumptions. A temperature-dependent gap or changing Hall factors would modify its interpretation.

The source's description of electrons as intrinsic silicon's majority carriers needs qualification. Intrinsic neutrality gives equal electron and hole concentrations, not an electron density majority. Electrons can nevertheless dominate conductivity and the Hall response because their mobility is larger. For two weak-field carriers,At $n=p=n_i$ this becomes $R_H=-({\mu }_e-{\mu }_h)/[e{n}_i({\mu }_e+{\mu }_h)]$. Thus ${\mu }_e>{\mu }_h$ explains the negative sign, while the one-electron formula is quantitatively accurate only if the hole contribution can be neglected. Equality $n=p$ itself is a concrete counterexample to a literal electron-number-majority claim. The numerical deductions above retain the approximation explicitly requested by the source.

The two body-centre/origin sites and the other six sites have different local environments. Their effective atomic scattering factors therefore need not be identical even in an elemental, chemically pure crystal. Write them as $f_A$ and $f_B$, respectively. The crystal basis structure factor becomesFor the low-angle families this givesThus $\{110\}$ can produce a weak extra powder diffraction ring below $\{200\}$. This is an accidental diffraction extinction from equal atomic scattering factors: its cancellation in part d is exact only in the identical-atom-scattering approximation. Small differences in the bonding electron density, or site-dependent Debye–Waller factors, can leave a small nonzero amplitude. For instance, taking $f_A=f+\delta f$ and $f_B=f$ gives $F_{110}=2\delta f$ while $F_{200}=4f+2\delta f$; the corresponding intensity suppression is quadratic in $|\delta f/f|$. No impurity is required. The $\{100\}$ and $\{111\}$ cancellations remain for arbitrary scalar $f_A,f_B$ in this two-site-class model.

Determining antiferromagnetic magnetic order is a good reason to use neutron scattering. The neutron has a magnetic dipole moment, so its scattering amplitude includes the spatial arrangement of electronic magnetic moments. Alternating magnetic moments can produce magnetic neutron diffraction peaks even when there is no change of the nuclear crystal structure. Ordinary nonresonant X-ray diffraction primarily measures the electron density, and is much less sensitive to this magnetic order.

Away from an absorption edge, the scalar atomic scattering factor is the Fourier transform of an approximately spherical, real atomic electron density. Opposite points contribute conjugate phases, so their sum is real. In Thomson scattering the electronic response is also effectively in phase with the driving field in this nonabsorbing approximation. Near an absorption edge, resonantly driven electronic excitations introduce a phase lag and absorption. The atomic scattering factor becomes $f_0+f^{\prime }+i{f}^{\prime \prime }$, with $f^{\prime \prime }$ describing the absorptive component and $f^{\prime }$ its dispersive companion. The sign of the imaginary part depends on the time and scattering-phase conventions; here it is the source's $+i{f}^{\prime \prime }$ convention. This discussion concerns scalar charge scattering, without magnetic or tensor resonant contributions.

Write $\mathbf{r}={\sum }_i{s}_i{\mathbf{a}}_i$. The electron density is periodic with period one in each $s_i$, so its three-dimensional Fourier series has integer harmonics:If $\Omega$ is a primitive unit cell of volume ${\Omega }_c$, Fourier orthogonality givesConversely, translation by ${\mathbf{a}}_i$ multiplies a general plane wave by $e^{i\mathbf{q}\cdot {\mathbf{a}}_i}$. Periodicity permits only $\mathbf{q}\cdot {\mathbf{a}}_i\in 2\pi \mathbb{Z}$, precisely the reciprocal lattice vectors. For a physical bounded periodic electron density, the expansion exists in the usual square-integrable sense.

To determine a crystal structure, scatter probes whose wavelength is comparable to atomic spacings, typically $0.1\mathrm{nm}$. X-ray diffraction, neutron diffraction and electron diffraction probe periodic order through coherent elastic scattering. With incident and outgoing wavevectors ${\mathbf{k}}_i,{\mathbf{k}}_f$, define $\mathbf{Q}={\mathbf{k}}_i-{\mathbf{k}}_f$. Elastic scattering has $|{\mathbf{k}}_i|=|{\mathbf{k}}_f|$ and constructive interference occurs when $\mathbf{Q}=\mathbf{G}$, a reciprocal lattice vector. The equivalent Bragg's law iswhere the detector scattering angle is $2\theta$, not $\theta$. Peak positions reveal plane spacings and the unit cell; systematic absent reflections constrain centring and the atomic basis.

For X-ray diffraction, a laboratory tube or a synchrotron supplies a collimated beam. In a monochromatic single-crystal experiment a goniometer rotates the specimen and an area detector records reflection positions and integrated wave intensities. A powder experiment uses many randomly oriented crystallites; the reflections form cones recorded as rings or peaks versus $2\theta$, permitting indexing and determination of lattice constants without selecting one crystallite. For a cubic cell, for example, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The white-beam Laue method is especially useful for crystal orientation.

The crystal basis structure factor isafter geometric and thermal corrections, including the Debye–Waller factor. For X-rays, $f_j$ describes scattering from the atomic electron density. Thus positions provide lattice geometry while relative wave intensities constrain atomic positions and species. Wave intensities do not directly give the phases of $F_{\mathbf{G}}$: structure refinement or other constraints must address the phase problem.

Neutron diffraction uses a reactor or spallation source, wavelength selection and a detector bank or time-of-flight detection. Neutrons interact with nuclei through nuclear scattering lengths that vary between elements and isotopes rather than simply with atomic number. This makes them useful for light atoms, isotope contrast and distinguishing some neighbours in the periodic table; their magnetic interaction also probes magnetic order. Electron diffraction in a transmission electron microscope gives strong signals from very small thin crystallites, although strong multiple scattering often complicates quantitative wave intensity analysis. These techniques complement one another.

A phonon dispersion relation requires both momentum and energy transfer, so elastic diffraction alone is insufficient. In coherent inelastic neutron scattering, the measured transfers for phonon creation areHere $\mathbf{q}$ is the phonon wavevector reduced into a Brillouin zone, and $s$ labels an acoustic phonon branch or optical phonon branch. Negative energy transfer corresponds to annihilating an already occupied phonon. For neutrons $E={\hslash }^2{k}^2/(2m_n)$, so wavelength and energy selection are linked. Typical thermal-neutron energies of a few to tens of millielectronvolts ($1\mathrm{meV}=10^{-3}$ electronvolts) are well matched to many lattice vibrations.

A triple-axis spectrometer uses a monochromator crystal to select ${\mathbf{k}}_i$, the sample as the second scattering axis, and an analyser crystal to select ${\mathbf{k}}_f$ before detection. Rotate these elements and the specimen to hold $\mathbf{Q}$ fixed while scanning $E_i-E_f$; inelastic peaks locate ${\omega }_s(\mathbf{q})$. Repeat at different $\mathbf{q}$ along symmetry directions to map the phonon dispersion relation. The dynamic structure factor governs peak weights, including the selection of phonon polarization by the scattering geometry. A time-of-flight spectrometer instead infers outgoing neutron energy from travel time over a known flight path and records many directions simultaneously.

Inelastic X-ray scattering can also resolve phonons when very high energy resolution is available. Optical Raman scattering and Brillouin scattering measure vibrations near the zone centre because photon momentum transfers are small compared with a typical reciprocal lattice spacing; they complement, rather than generally map the entire zone like, momentum-resolved neutron measurements. Ultrasound measurements give the long-wavelength sound speed, the initial slope $d\omega /dq$ of an acoustic phonon branch.

A positive electric potential difference between gate and substrate attracts electrons towards the insulating interface and repels the mobile electron holes of the p-type semiconductor substrate. The exposed, negatively charged acceptor impurities form a depletion region; the thin mobile electron layer at its surface is the semiconductor inversion layer. The conduction band and valence band bend downwards towards the positively biased gate, because the electron potential energy is $-e\psi$.

To specify the conventional MOS inversion threshold, let $N_A$ be the ionized acceptor density, $n_i$ the intrinsic carrier concentration, andIn the neutral bulk the intrinsic level lies $e{\varphi }_F$ above the semiconductor chemical potential. A positive surface potential ${\psi }_s$ lowers both band edges by $e{\psi }_s$, giving, in the nondegenerate approximation,At ${\psi }_s={\varphi }_F$, the surface is intrinsic: $n_s=p_s=n_i$. The usual onset of strong inversion instead meansThus the surface electron density has grown to the bulk majority-carrier density. The chemical potential need not actually enter the conduction band. The phrase “just forms” can also refer to the earlier weak-inversion crossover; the diagram uses the conventional strong-inversion threshold and labels it explicitly.

Within the depletion approximation the negative space charge is approximately $-e{N}_A$ over a widthwhere ${ϵ}_s$ is the absolute semiconductor permittivity. Its outer edge is where the bands become flat and mobile electron holes recover their bulk density. The electrical insulator blocks steady carrier transport while sustaining an electric field; the metal gate supplies positive surface charge balancing depletion and inversion charge. Under gate bias, the gate and substrate electrochemical potentials are different, so a single equilibrium Fermi level should not be drawn through the whole biased stack.

For one isotropic carrier species in the Drude model, the longitudinal electrical resistivity at zero transverse electric current is ${\rho }_{xx}=1/(ne\mu )$, and the Hall electrical resistivity has magnitude $|{\rho }_{yx}|=B/(ne)$. Equivalently, the electric field balances the magnetic deflection through $E_y=v_{x}B$, while $|v_x|=\mu |E_x|$. Therefore the magnitude of the Hall angle obeysThe required ratio givesThe electron density cancels. With electric current along $+x$ and field along $+z$, electrons have negative Hall coefficient and hence $E_y<0$; the requested one-percent condition concerns the magnitude. This calculation assumes electrons dominate transport and the transverse contacts draw no electric current, so a Hall field can develop.

A covalent bond is stabilized by shared electronic states extending across neighboring atomic nuclei. In a simple two-centre description, combining comparable atomic orbitals produces a bonding state with enhanced electron density between the nuclei and an antibonding state with a node there. For equivalent normalized orbitals ${\varphi }_A,{\varphi }_B$ with real overlap $S$,Their energy ordering follows from the molecular Hamiltonian; constructive interference and reduced gradients help explain why the bonding combination can be lower. Opposite-spin electrons can occupy the bonding orbital together. The familiar molecular-hydrogen bond is a basic example, while extended covalent networks form solids such as diamond. Orbital shape and hybridization make covalent bonding strongly directional and constrain bond angles. Occupying antibonding states reduces the net bond strength.

Ionic bonding instead emphasizes electric charge transfer and the electrostatic attraction of the resulting oppositely charged ions. Removing an electron from one atom costs ionization energy and adding it to another gains electron affinity; these isolated-atom terms alone need not favor transfer. In a crystal, the electrostatic lattice energy can stabilize the ionic arrangement. For a schematic lattice of singly charged ions at nearest-neighbor distance $r$,where the geometrical Madelung constant $A_M$ sums attractions and repulsions throughout the lattice. Short-range overlap repulsion prevents collapse and gives an equilibrium separation. Sodium chloride is an example of an extended ionic crystal, rather than isolated NaCl molecules in the solid.

Ideal ionic bonding is comparatively nondirectional, so packing, ion sizes and electric charge balance strongly influence coordination. ions cannot freely migrate in the ordinary solid; melting or dissolution can make ionic electric charge transport possible. Covalent materials do not obey one universal electrical conductivity rule: their electronic bands and filling determine whether mobile electronic carriers are available.

The two descriptions are limiting models, not a sharp classification of every bond. Unequal electronegativities can produce polar covalent bonds with partial electric charge transfer, and a material may contain both covalent groups and ionic interactions between those groups. Chemical bonding reflects a balance of electronic kinetic energy, electron-nucleus attraction, electron-electron repulsion and nuclear repulsion; it cannot be reduced to a force between two assigned point charges in every material.

Put $x=E_I/(k_{B}T)$. With $\zeta =1/2$, the heat capacity formula becomesTaking $E_I=13.6\mathrm{eV}$ gives $\boxed{T=1.02\times 10^4\mathrm{K}}$. At half ionization, $n_e{n}_p/n_H=n_e$, so the Saha ionization equation directly givesThis uses $2g_p/g_H=1$ as specified in part (a); a different internal-state convention must be carried consistently into the numerical electron density. The original nuclei density is $n_0=2n_e\simeq 10^{21}{\mathrm{m}}^{-3}$.

Use the preceding electron density as the line-of-sight mean, $L/c=2000\mathrm{yr}$, $\omega =2\pi (1.4\times 10^9){\mathrm{s}}^{-1}$ and $\Delta \omega =2\pi (4.0\times 10^7){\mathrm{s}}^{-1}$. ThenThis estimate corresponds to dispersion measure about $69.5\mathrm{pc}\mathrm{c}{\mathrm{m}}^{-3}$. A cloud density inferred from absorption need not equal the actual path average, so a real pulsar may differ even at the same distance.

Pauling electronegativity is a relative, dimensionless scale motivated by bond-dissociation energies. The interesting observation is that a chemical bond between unlike chemical atoms can be stronger than a simple baseline inferred from the corresponding like-atom chemical bonds. An empirical excess energy can then be associated with an electronegativity difference. The 1932 paper The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms built such a map; its original coordinates used hydrogen as zero, so those coordinates should not be mistaken for the modern tabulated Pauling electronegativities.^[ref]

One arithmetic-baseline version of the Pauling scale relation isHere $D_{AB}$, $D_{AA}$ and $D_{BB}$ are consistently defined bond-dissociation energies per molecule, and the arithmetic mean is the chosen baseline. With molar energies instead, write $\sqrt{{\Delta }_D/(96.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1})}$, approximately $0.102\sqrt{{\Delta }_D/(1\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1})}$. The denominator makes the result dimensionless; the factor $0.102$ is not a unit-independent magic constant. This is an empirical relation with a specified baseline, not a definition valid for every chemical bond.^[ref]

For an explicitly synthetic example, suppose $D_{AA}=D_{BB}=300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and $D_{AB}=396.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ with the same temperature and dissociation convention. Then ${\Delta }_D=96.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and the model gives an electronegativity difference of $1$. Replacing $D_{AB}$ by $300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ gives zero; a negative excess would make this real square-root model inapplicable, not produce a negative electronegativity difference. These invented inputs illustrate the formula and are not measurements of named chemical substances.

Only differences appear, so adding the same constant to every Pauling electronegativity leaves this relation unchanged. A reference value must fix the origin. The bond-dissociation energies alone also do not tell which endpoint attracts electron density more strongly, because interchanging $A$ and $B$ leaves the formula unchanged. Chemical comparisons fix that direction. Thus Pauling electronegativity is useful as a compact ordering, while the underlying measured energies and approximations remain essential.

The phase problem is the loss of the wave phases of the crystal basis structure factors when an experiment records their squared magnitudes. Write $F_{hkl}=|F_{hkl}|e^{i{\varphi }_{hkl}}$. A Fourier series reconstruction of a periodic electron density uses both $|F_{hkl}|$ and ${\varphi }_{hkl}$, whereas an ordinary wave intensity measurement supplies only $|F_{hkl}{|}^2$. Estimating the missing phases therefore needs additional constraints or measurements.^[ref]

There is also an unavoidable choice of origin. Translating all motif positions by the same fractional crystal coordinate vector $t$ multiplies $F_{hkl}$ by $e^{2\pi i(h,k,l)\cdot t}$ without changing $|F_{hkl}{|}^2$. The physical arrangement is unchanged. A reconstructed coordinate list must therefore distinguish a coordinate convention from information actually recovered from the measurements.

A polar covalent bond shares electrons unequally, as in hydrogen chloride.

The word “polar” qualifies the sharing within a covalent bond; it does not require complete transfer of an electron or an isolated cation–anion pair. In $A^{\delta +}-B^{\delta -}$, the signs identify the relatively electron-poor and electron-rich ends, while $\delta$ leaves the amount unspecified. A neutral Lewis structure can give both chemical atoms zero formal charge and still correspond to a polar covalent bond: equal division of the bond pair is a formal allocation, whereas bond polarization concerns the actual electron density.

Local and molecular descriptions answer different questions. Calling one chemical bond a polar covalent bond does not establish that the whole molecule is a polar molecule. Combine the bond dipoles with the molecular geometry, then check for molecular dipole cancellation. This lets a short electronegativity comparison explain a local tendency without confusing it with the observable electric dipole moment of the whole molecule.

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 resonance hybrid is the combined electronic description expressed using several resonance structures. The word does not describe a mixture of separate molecules or ions: a nitrate ion has one electron density, rather than three species in chemical equilibrium. Equal weighting is appropriate for the equivalent nitrate contributors in the elementary model; the unequal carbon dioxide contributors illustrate why it is not a general rule. See Chemistry 2e, 7.4: resonance hybrids do not fluctuate between forms.