Replace instantaneous acceleration and braking by legs of constant proper-acceleration magnitude $a$, retaining the same cruise rapidity $\eta$ and fixed endpoint separation. Each leg takes proper time $c\eta /a$ and displaces $(c^2/a)(\cosh \eta -1)$ in the endpoint rest frame. Subtracting the proper cruise time displaced by both legs gives the displayed correction. A cruise segment exists only if the total endpoint separation exceeds twice this acceleration distance.

In units $c=1$, constant signed proper acceleration $a$ givesPosition data determine $t_0,x_0$ but leave the initial rapidity ${\eta }_0$ free. Proper-acceleration magnitude alone also leaves its orientation unspecified. The familiar centred hyperbola $x^2-t^2=a^{-2}$ follows for the additional initial data $t_0=0$, $x_0=1/a$, ${\eta }_0=0$ and acceleration toward increasing $x$.

A solvent-mediated tendency of appropriately prepared water-repelling surfaces to associate. Surface chemistry, dissolved gas and the experimental geometry can strongly affect the measured force; it is not a universal replacement for every other colloidal interaction.

The Coulomb radial factors satisfy $[A_l,A_l^{†}]=a^2(l+1)/r^2=m{a}^2(H_{l+1}-H_l)/{\hslash }^2$. Commuting $A_l$ through its factorized Hamiltonian gives the displayed intertwining relation. A nonzero transformed radial eigenfunction has the same energy in the next angular-momentum sector; this changes the radial sector, rather than being the usual magnetic-quantum-number raising operator within a fixed $l$ multiplet.

An empirical measure of donor strength on a specified scale, including weak interactions when the scale defines them. A small nonzero value does not require a conventional O–H or N–H group.

Hydrogen-bond donor–acceptor matching compares possible hydrogen bond contacts by combining donor strength, acceptor strength, and geometry. A strong donor and acceptor can still fail to form a useful contact if their positions or orientations are incompatible.

In chemical solution, formation of a donor-acceptor contact also replaces interactions with the solvent. Matching therefore concerns the free-energy change for exchanging contacts, rather than the isolated strength of one hydrogen bond.

An empirical measure of hydrogen-bond accepting strength on a specified scale. It is not simply the count of lone pairs or heteroatoms.

An acyclic saturated hydrocarbon. The formula applies to a connected neutral alkane with $n\ge 1$ carbon atoms.

A conjugated hydrocarbon whose pi-site graph is bipartite: its sites can be divided into two sets with every retained resonance interaction joining opposite sets. The definition refers to the selected nearest-neighbour Hückel model, not simply to a visually alternating Lewis drawing.

The conjugated hydrocarbon with a five-membered ring bearing two endocyclic double bonds and one exocyclic methylene double bond. Its ideal planar structure has the C2v molecular point group.

The hydrocarbon with a three-membered carbon ring and an exocyclic methylene double bond at each vertex. The ideal planar structure belongs to the D3h molecular point group.

A cage isomer of benzene with one carbon–carbon double bond, conventionally described as tricyclo[3.1.0.0(2,6)]hex-3-ene. Its ideal cage geometry belongs to the C2v molecular point group. Molecular symmetry depends on the full three-dimensional cage rather than the apparent symmetry of a planar skeletal projection.

A fused pair of five-membered carbon rings with formula ${\mathrm{C}}_8{\mathrm{H}}_6$. Its neutral conjugated framework has eight pi electrons. Coordination and reduction alter its electron distribution and allow several metal-binding hapticities.

Replacement of a three-edge triangular linear resistance network by a three-arm star with the same relation between terminal pressures and flows. Each star arm equals the product of its two adjacent triangle edges divided by the sum of all three edges.

A network of two resistive rails joined by multiple cross-links. Finite rail resistance makes cross-link flows unequal; maintaining a three-terminal equivalent permits exact recursive reduction.

An effective local exchange coupling can favor parallel electron spins in different orbitals on the same atom. The displayed sign convention represents that preference. It underlies the high-spin tendency in Hund's rules and the alignment of itinerant and core spins used in double exchange. The coupling is an effective description of intra-atomic Coulomb and exchange energies, rather than a separate classical magnetic force.

In an orthonormal Hückel method basis, an occupied molecular orbital with occupancy $f_k$ contributes $f_k|c_{ik}{|}^2$ electrons to site $i$. Adding one electron to a fixed orbital changes the site population by $|c_i{|}^2$. A coefficient sign indicates orbital phase, not charge sign. A net atomic charge also needs a reference population and treatment of the electrons outside the model.

The change in energy on allowing a specified localized orbital reference to delocalize across a conjugated system. In a Hückel method calculation, the reference must use the same one-centre parameters and bond types as the delocalized molecule. With the signed convention $E_{\pi }-E_{\mathrm{localized}}$, stabilization is negative; its positive magnitude is often also called the delocalization energy, so the sign convention must be stated.