The diagonal one-centre energy ${\alpha }_i$ assigned to a pi-basis orbital. This one-electron parameter is distinct from a two-electron Coulomb repulsion integral in Hartree-Fock theory.

For real orthonormal site functions, $p_{ij}={\sum }_k{f}_k{c}_{ik}{c}_{jk}$, where $f_k$ is orbital occupation. It measures the off-diagonal pi density matrix; a complete C–C bond also contains a sigma contribution.

For a bipartite pi system with a common diagonal energy $\alpha$, the orbital spectrum is paired about $\alpha$. If $H=\alpha I+K$ and $P$ changes the sign on one sublattice, then $PKP=-K$; an eigenvector of energy $\alpha +ϵ$ is mapped to one of energy $\alpha -ϵ$. Their site-probability magnitudes are equal.

For a four-site cycle with site energy $\alpha$, three resonance integrals $\beta <0$ and one weakened integral $n\beta$, $0<n<1$, put $D=\sqrt{(n-1{)}^2+4}$. In the independent-electron Hückel method, filling the two or four lowest spin orbitals gives total electronic energies $E_2=2\alpha +\beta (n+1+D)$ and $E_4=4\alpha +2\beta D$. At the square limit $n=1$, these become $2\alpha +4\beta$ and $4\alpha +4\beta$. The weakened-edge distortion raises the two-electron sum but lowers the four-electron sum. Near $n=1$ the latter change is $\beta (1-n{)}^2/2+O((1-n{)}^4)$, so its first-order term cancels. This electronic comparison omits structural repulsion and other restoring forces, and does not determine the open-shell spin at the square limit.

With one orthonormal site orbital per cube vertex, common site energy $\alpha$, and coupling $\beta$ along every edge, the one-electron Hamiltonian is $\alpha I+\beta A$, where $A$ is the cube adjacency matrix. Its eigenvalues are $\alpha +3\beta ,\alpha +\beta ,\alpha -\beta ,\alpha -3\beta$ with multiplicities $1,3,3,1$.

One proof labels each vertex by three signs. The adjacency operator flips one sign at a time, so a product of $k$ coordinate signs is an eigenfunction of $A$ with eigenvalue $3-2k$. There are $\left(\genfrac{}{}{0ex}{}{3}{k}\right)$ such products. For $\beta <0$, eight electrons fill the lowest four spatial molecular orbitals, yielding total one-electron energy $8\alpha +12\beta$.

A genomic grouping of related Hox genes whose organization can support shared and sequential developmental regulation. Clustering is widespread but is not a universal requirement for positional Hox expression.

A grouping of related Hox genes corresponding to an ancestral member retained in duplicated clusters. Sequence relationships, synteny and gene history jointly support the assignment.

A context-dependent tendency for posterior Hox functions to dominate more anterior Hox functions where their domains overlap. It is not a universal rule for all tissues.

Consolidation under approximately equal applied pressure in all directions at elevated temperature. In waste treatment it can produce a dense ceramic inside a containment can.

Recombination between related chromosomes derived from different ancestral genomes, distinguished from recombination between true homologues.

The centromere-associated histone-H3 variant supporting kinetochore assembly. Its genomic occupancy helps mark centromeric identity, which can persist without one universal underlying repeat sequence.

Histone methylation adds methyl groups to selected residues of histones. The position and number of methyl groups affect which binding partners recognize the mark. H3K4 methylation, H3K9 methylation and H3K27 trimethylation therefore cannot be treated as one interchangeable signal. Their distributions can be associated with different transcriptional states, and direct causation requires more than correlation between a mark and gene expression. ^[ref]

Reingestion of selected caecal material, allowing a later digestive passage to recover nutrients including microbial products. It is more specific than indiscriminate faeces consumption.

The guanine-nucleotide-binding subunit of a heterotrimeric G protein. Nucleotide state changes its interactions with receptors, Gβγ and effectors.

The associated β and γ subunits of a heterotrimeric G protein. Their interactions with downstream effectors transmit signals as well as support heterotrimer assembly.

A staggered alignment in which the lowest electron states and most favorable hole states occur in different materials. Unlike type-I band alignment, it can reduce electron–hole spatial overlap. Comparing the two band gaps alone does not determine whether this alignment occurs.