Foundations

A molecular symmetry operation leaves the specified nuclear arrangement indistinguishable from its original arrangement.

In an octahedral crystal-field model, d orbitals separate into a lower t2g set and an upper eg set.

Electronic excitation between metal-centred or ligand-involving states can absorb visible light and contribute to complex colour.

Valence occupation and electronic stabilization organize main-group chemistry: alkali metals commonly form +1 ions and alkaline-earth metals +2 ions, while p-block elements support varied covalent structures and oxidation states.

In a common mononuclear two-electron oxidative addition, a substrate bond is replaced by two metal–ligand bonds and the metal's formal oxidation state increases by two.

Migratory insertion combines ligands within a metal coordination sphere, such as an alkyl and CO forming an acyl group.

Matrices representing symmetry operations on a chosen basis must reproduce group composition.

A transition matrix element ⟨initial|O|final⟩ can be nonzero only when Γinitial*⊗Γoperator⊗Γfinal contains the totally symmetric representation.

Coordination compounds can share composition while differing in ligand connectivity or spatial arrangement.

Neutral-fragment and oxidation-state counting consistently allocate ligand and metal electrons when their conventions are used throughout.

Dissociative ligand substitution loses a ligand before addition, while an associative route adds an incoming ligand before loss.

Ligand-field MO models combine metal and ligand orbitals with compatible symmetry, adding covalent interactions beyond a point-charge crystal field.

Hydrogen in an ionic hydride is formally H− and can act as a strong base or reducing species; proton transfer with water produces hydrogen gas and hydroxide.

Boron trihalides have trigonal-planar molecular reference structures and can accept donor electron pairs; BF3 and fluoride form BF4−.

Nitrogen forms oxides with several formal oxidation states.

Phosphorus(III) and phosphorus(V) oxides have different oxygen content and formal oxidation states.

Oxide O²−, peroxide O₂²− and superoxide O₂− contain different numbers and arrangements of oxygen atoms and electrons.

Halogens and halides have distinct oxidizing and reducing roles; for example, chlorine can oxidize bromide under suitable solution conditions.

Same-excess reaction-progress experiments compare rates at matched reactant concentrations but different catalyst histories; differences can reveal deactivation, product effects or activation.

An abundant observed catalyst state can be a resting state or off-cycle reservoir, while a low-population intermediate may carry productive flux.

Deuterium isotope effects can constrain C–H functionalization pathways, but their interpretation depends on whether rates, competition or product distributions were measured.

Carrow and Hartwig compared isolated palladium hydroxo/halide and boronic-acid/boronate species and their rates.

The 2025 copper study combines atomistic surface/kinetic calculations with STM comparison to examine CO-induced reconstruction during CO2 reduction.

A working electrocatalyst can change structure and coverage during operation.

The 2026 device study reports separating sorbent regeneration and CO2 liberation through redox-decoupled electrolysis.