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Matrix mechanics represents an observable by an array of matrix elements between chosen quantum states. In an orthonormal basis of energy eigenstates, each a stationary state, the diagonal entries of the Hamiltonian operator are the energy levels; off-diagonal entries of other observables describe couplings between those states. The same array can therefore answer two different questions: which values can a measurement return, and which pairs of energy eigenstates can a specified interaction connect? The eigenvalues answer the first question; the relevant matrix elements, together with the energy differences, answer the second.

This makes linear algebra a practical language for atomic physics. One can calculate energy levels, electric-dipole selection rules and probability amplitudes for transitions without assigning an electron an unobserved classical orbit. The three-state example below separates a missing optical coupling from a missing energy level; the finite oscillator commutator boundary defect explains why a finite array requires care when approximating a quantum harmonic oscillator.

Cohesion in a metal involving delocalized valence electrons and positive ion cores. A dispersion-only sum between neutral molecular units omits this collective electronic contribution.

Heavy elements provide many ionic transitions at temperatures where hydrogen and helium have lost their most effective line emitters. Metal-line cooling can therefore increase the plasma cooling function substantially and broaden its temperature features. Low-energy fine-structure transitions also permit cooling below the hydrogen atomic threshold when the relevant ions and electrons are present.

A complex in which a metal is bound between two cyclopentadienyl-type rings, commonly with hapticity ${\eta }^5$ at each ring. In ionic electron counting, each unsubstituted cyclopentadienyl anion donates six electrons.

A metal-metal bond joins two metal centers directly through overlap of their atomic orbitals. A single bond contributes one electron to the local count at each metal in conventional electron counting. Bridging ligands can accompany direct bonding, but a short bridged separation alone does not establish its bond order.

Light initiates methane chlorination by photodissociation of $\mathrm{C}{\mathrm{l}}_2$ through homolysis. The elementary bookkeeping isHere $h\nu$ denotes the photon energy. The first propagation step replaces a chlorine radical with a methyl radical; the second regenerates a chlorine radical. Adding the two propagation equations cancels both free radicals and gives $\mathrm{C}{\mathrm{H}}_4+\mathrm{C}{\mathrm{l}}_2⟶\mathrm{C}{\mathrm{H}}_3\mathrm{Cl}+\mathrm{HCl}$: chloromethane and hydrogen chloride are made while the chain carrier is regenerated.^[ref]

For example, recombination $2{\mathrm{C}{\mathrm{H}}_3}^{\bullet }⟶{\mathrm{C}}_2{\mathrm{H}}_6$ of methyl radicals makes ethane and a covalent bond from two unpaired electrons, removing two carriers. Recombination of two ${\mathrm{Cl}}^{\bullet }$ also terminates carriers; in the gas phase, stabilization of the product requires disposal of released energy, for example to a third collision partner. The figure omits that partner. Termination competes with propagation, and further chlorination can consume chloromethane, so this cycle alone does not predict a pure product or a measured reaction rate.^[ref]

The methyl radical is the neutral radical ${\mathrm{C}{\mathrm{H}}_3}^{\bullet }$, with an unpaired electron. It carries the methane chlorination chain between its two propagation steps.

Robert Andrews Millikan's 1916 experiment used a magnetically turned wheel and rotary knife to expose sodium or lithium in vacuum. A surrounding copper-gauze/solid-copper collector caught electrons; an electrometer recorded accumulated electric charge over thirty seconds at each applied voltage. He located endpoints graphically from photocurrent curves. Mercury spectral lines, selected by a monochromator and filters, controlled frequency; rejecting shorter-wavelength contamination protected the endpoint. An oxidized collector suppressed reverse emission for selected lines. A copper test plate measured the contact potential difference by a displacement-null method; a certified voltmeter set the electrical scale. He compared stable-surface sweeps with rapid alternating-frequency measurements to control drift. The slope tested $h/e$; the threshold required the contact correction. His gauge did not measure mercury vapor. Confirmation of the equation coexisted with his continued skepticism about localized light quanta.^[ref]

For a modern repetition of the logic, first establish electrical zero with the illumination blocked, characterize leakage, and verify the voltmeter scale independently of the presumed Planck constant. Scan several voltages near each endpoint rather than calling one noisy zero a cutoff. Hold the illuminated area and surface state fixed when varying light intensity; vary frequency independently when testing the slope. Repeating a reference frequency between other readings reveals time-dependent offsets. Agreement of a slope with $h/e$ tests an energy relation; it does not alone identify every microscopic step of photon absorption and escape.

The less abundant mobile charge carrier species: electron holes in an n-type semiconductor, or electrons in a p-type semiconductor. A small equilibrium population does not imply an unimportant role in an operating transistor.

A mitochondrion is an organelle with an outer mitochondrial membrane and an inner mitochondrial membrane. The latter encloses the mitochondrial matrix and folds into cristae; the intermembrane space lies on its opposite side. This architecture separates a chemical workspace from a membrane that couples electron transport to ATP formation.^[ref]

For an original location exercise, place soluble label A in the mitochondrial matrix and soluble label B in the interior of a crista. In the depicted geometry, B can reach the peripheral intermembrane space through the fold's opening without crossing a lipid bilayer. A must cross the inner mitochondrial membrane to reach either location. A fold projecting into the mitochondrial matrix therefore does not make its enclosed fluid part of the mitochondrial matrix. Following a continuous path through the colored regions proves the distinction; merely asking whether a label is “inside the mitochondrion” loses it.

The functional connection is equally spatial. The mitochondrial respiratory chain pumps protons from the mitochondrial matrix side across the inner mitochondrial membrane. Their return through ATP synthase can drive ATP formation. Oxidative phosphorylation develops the mechanism and chemiosmosis gives its free energy bookkeeping. Compartmentalization is useful because it lets a shared electrochemical gradient couple different proteins without direct transfer of a chemical intermediate between each pair.

Under aerobic conditions, mitochondria supply much of the ATP in many eukaryotic biological cells; the contribution varies with biological tissue, demand and oxygen supply. Glycolysis also makes ATP in the cytosol. Mature mammalian red blood cells lack mitochondria, and bacteria have no mitochondria at all. Neither all ATP production nor all biological cells fit the familiar mitochondrial picture.^[ref]

Mitochondria descend from a bacterial endosymbiosis and retain a genome and ribosomes, although many required proteins are encoded by nuclear genes. Endosymbiotic theory develops the ancestry argument; organelle protein import explains delivery of nuclear-encoded proteins. A functioning isolated mitochondrion demonstrates biochemical autonomy under supplied conditions, rather than independently establishing that ancestry.

NADH supplies two electrons to respiratory complex I, then ubiquinone, respiratory complex III, cytochrome c and cytochrome c oxidase; terminal oxygen reduction forms water. Succinate dehydrogenase, or respiratory complex II, supplies another entry to ubiquinone. Respiratory complex I, respiratory complex III and cytochrome c oxidase move protons outward; respiratory complex II does not.^[ref]

For an original electron balance, supply exactly $6\mu \mathrm{mol}$ of NADH to an ideal preparation with no other electron donor or competing electron acceptor. The donor supplies $12\mu \mathrm{mol}$ of electrons. Complete reduction of one oxygen molecule needs four electrons, so the corresponding oxygen consumption is $3\mu \mathrm{mol}$. Adding the two formal half-reactions makes the result explicit:The cancelling electrons determine the donor-to-oxygen ratio. These chemical equations do not count transmembrane protons or determine an ATP yield. Chemical protons consumed in water formation and transported protons have different roles in the ledger.

A mobility edge separates localized from extended energy eigenstates on the energy axis. Three-dimensional disordered models can have both. In the infinite one-dimensional tight-binding model with independent continuous random site energies, all states are localized for any nonzero disorder, with probability one. This statement excludes specially correlated disorder.^[ref]

A mobility edge is not a band gap: states can exist on both sides of $E_c$, but differ in spatial extent. For electrons, the location of the Fermi energy relative to a mobility edge matters for low-temperature electrical conductivity. A three-dimensional sample need not have an extended interval at every disorder strength. In particular, the finite one-dimensional calculations below neither locate a three-dimensional mobility edge nor establish a nonzero universal critical disorder.

The molecular count $N$ is a dimensionless integer; the amount of substance $n$ has unit mole, symbol $\mathrm{mol}$. They satisfyThe Avogadro constant is exact by the modern definition of the mole: see SI base unit: mole by Bureau International des Poids et Mesures. Thus two water molecules correspond to $2/N_{\mathrm{A}}$ moles, whereas two moles of water contain $1.204428152\times 10^{24}$ molecules. A mole is a unit for counting specified entities, which may also be chemical atoms, ions or electrons.

In the International Union of Pure and Applied Chemistry (IUPAC) mass spectrometry convention, a molecular ion is formed by removing or adding one or more electrons to a molecule without fragmentation of its molecular structure.^[ref] For example, removing one electron from intact water gives ${\mathrm{H}}_2{\mathrm{O}}^{\bullet +}$: two hydrogen and one oxygen chemical atom, total electric charge $+e$, and an unpaired electron, hence a radical cation. Atom inventory, net electric charge, and electronic state are separate pieces of information. Hydronium, ${\mathrm{H}}_3{\mathrm{O}}^{+}$, has a different inventory; it is a polyatomic ion, rather than the molecular ion obtained by electron removal from water.

A molecular orbital is a one-electron spatial wavefunction for the combined atomic nuclei of a molecule. For a normalized molecular orbital, the Born rule gives the position probability density $|\psi (\mathbf{r}){|}^2$, with ${\int }_{{\mathbb{R}}^3}|\psi {|}^2{d}^3\mathbf{r}=1$. The molecular orbital is the amplitude, rather than its squared modulus or a classical trajectory.

In the independent-orbital approximation, a doubly occupied molecular orbital contributes $2|\psi {|}^2$ to the electron number density, whose integral is two. An empty molecular orbital contributes zero, even though its shape is still part of the model. The molecular hydrogen example therefore uses one occupied spatial molecular orbital for two electrons, distinguished by their spins. A negative real probability amplitude means a relative quantum phase of $\pi$, not positive electric charge; multiplying the whole wavefunction by $-1$ changes only its global phase.

Molecular orbital theory describes the electronic arrangement of a molecule using molecular orbitals: one-electron spatial wavefunctions extending over the combined atomic nuclei. Filling bonding molecular orbitals can stabilize a chemical bond; filling antibonding molecular orbitals opposes that stabilization. A molecular orbital is a building block of a many-electron description, rather than the entire many-electron wavefunction. See Robert W. Field, MIT 5.61, lecture 24, pp. 1–4: molecular orbitals and the LCAO approximation.

First make the interference visible with an exactly normalized one-dimensional toy. Replace the three-dimensional $1s$ atomic orbitals of the hydrogen atom by two Gaussian functions centred at $x=\pm a$, with dimensionless $x$ and $a>0$:The normalized sum and difference are ${\psi }_{\pm }=({\varphi }_A\pm {\varphi }_B)/\sqrt{2(1\pm S)}$, following the same linear combination of atomic orbitals algebra. At $a=1$, the orbital overlap is $S=e^{-1}\approx 0.367879$. At the midpoint, ${\varphi }_A(0)={\varphi }_B(0)\approx 0.455581$, givingBoth probability densities have total integral one: interference redistributes probability, rather than creating an extra electron. These are toy wavefunctions, not actual hydrogen atom orbitals or predictions of molecular hydrogen energies.

Now fill the two $1s$-derived molecular orbitals in the minimal model. The lower bonding molecular orbital is labelled ${\sigma }_g$; the upper antibonding molecular orbital is ${\sigma }_u^{\ast }$. The Pauli exclusion principle permits at most two electrons in each spatial molecular orbital, with opposite spin projections along a chosen axis. For the lowest electron configuration in this model, fill ${\sigma }_g$ first. The subscripts $g$ and $u$ denote even and odd behaviour under inversion through the midpoint; the star marks an antibonding molecular orbital. See Flowers et al., Chemistry 2e, 8.4: occupancy and bond order and Field, MIT 5.61, lecture 25, pp. 1–3: bonding, antibonding, and symmetry labels.

With $N_b$ electrons in bonding molecular orbitals and $N_a$ in antibonding molecular orbitals, the model's bond order is $b=(N_b-N_a)/2$. The three elementary counts are:

The zero bond order says this minimal model gives no net covalent bond for ${\mathrm{He}}_2$. It does not rule out every bound pair of helium chemical atoms: a very weak Van der Waals force can bind a neutral helium dimer, which has been observed experimentally. See Zeller et al., Imaging the He2 quantum halo state using a free electron laser, PNAS 113, 14651–14655 (2016), p. 14651: van der Waals binding and Luo et al., The weakest bond: Experimental observation of helium dimer, Journal of Chemical Physics 98, 3564–3567 (1993). The bond order count does not include that weak binding mechanism.

The useful surprise is that sharing alone does not determine a chemical bond: the relative quantum phase and the occupancy matter together. The same two shapes explain why adding an electron to a bonding molecular orbital increases the model's bond order, while adding one to an antibonding molecular orbital decreases it. Molecular orbital theory also accounts for the unpaired electrons and paramagnetism of molecular oxygen, which its all-paired Lewis structure misses. See Chemistry 2e, 8.4: the molecular orbital description of oxygen.

A molecule, using the International Union of Pure and Applied Chemistry (IUPAC) convention, is an electrically neutral entity containing two or more chemical atoms. Its constituents form a bound unit through chemical bonds; merely drawing a boundary around neighboring particles does not make a molecule. The neutrality requirement concerns total electric charge, not the absence of internal charge separation: water is a polar molecule with zero total electric charge.^[ref]

One water molecule, ${\mathrm{H}}_2\mathrm{O}$, contains two hydrogen and one oxygen chemical atom. One molecular oxygen molecule, ${\mathrm{O}}_2$, contains two oxygen chemical atoms; a molecule need not contain different chemical elements. One carbon dioxide molecule, $\mathrm{C}{\mathrm{O}}_2$, contains one carbon and two oxygen chemical atoms. The molecular formula records these inventories, while molecular geometry describes their arrangement. Water and carbon dioxide are chemical compounds; molecular oxygen is a form of one chemical element.

Ordinary crystalline sodium chloride, $\mathrm{NaCl}$, contains an extended arrangement of sodium cations $\mathrm{N}{\mathrm{a}}^{+}$ and chlorine anions $\mathrm{C}{\mathrm{l}}^{-}$. Its formula unit specifies one of each for compositional accounting, without selecting a unique isolated pair. Bulk neutrality therefore does not imply a material consists of neutral molecules. This statement concerns the ionic crystal; the physical state and conditions matter when deciding what particles a chemical formula represents.^[ref]

A polyatomic ion is a charged bound group of chemical atoms, such as hydronium, ${\mathrm{H}}_3{\mathrm{O}}^{+}$. A molecular ion formed by removing an electron from intact water, ${\mathrm{H}}_2{\mathrm{O}}^{\bullet +}$, retains the same atom inventory but has net electric charge $+e$. Neither satisfies the neutral-molecule definition. Distinguishing these entities makes molecular count and amount of substance and mass spectrometry interpretable: say whether the count is of chemical atoms, neutral molecules, ions, or formula units.

The useful surprise is that a molecule can be treated as one particle for motion and counting while retaining internal structure. Its molar mass connects a particle inventory to bulk mass; molecular rotation and molecular vibration reveal internal organization. The continuum approximation uses bulk fields in place of individual molecular positions and motions.

In the alpha form, MoCl3 has edge-sharing octahedral layers distorted by Mo–Mo dimer formation. Metal-centred electron pairing contrasts with the more localized d electrons of CrCl3.

For counts $N(p)dp$ per electron momentum interval, the allowed spectrum is proportional to $p^{2}F(p)(Q-T{)}^{2}dp$ when recoil and neutrino mass are neglected. Thus the displayed ordinate is linear in kinetic energy under a constant-shape assumption. Counts per kinetic-energy interval instead have the factor $p{E}_e$ in natural units; the Jacobian $dT/dp=p/E_e$ accounts for the difference. Mixing those two binning conventions changes the inferred ordinate.

An atom is monovalent in a specified bonding or conduction model when its relevant valence is one. In the simple metallic free electron model, one monovalent atom per primitive unit cell supplies one conduction electron per cell. Counting the two spin states then fills half of a single energy band at zero temperature.