In a centrosymmetric molecule, a fundamental vibration cannot be both electric-dipole infrared active and ordinary Raman spectroscopy active. Dipole components are odd under inversion; polarizability components are even. This selection rule concerns modes, not accidental numerical equality of unrelated frequencies.

The 24-operation molecular point group of an ideal regular tetrahedron, including improper operations. Its irreducible representations are $A_1,A_2,E,T_1,T_2$, with dimensions $1,1,2,3,3$. The polar-vector representation is $T_2$, the axial-vector representation is $T_1$, and symmetric quadratic functions transform as $A_1+E+T_2$.

The 24-operation molecular point group of a regular planar hexagon with equivalent sites. Its irreducible representations have inversion labels $g,u$; polar-vector components transform as $A_{2u}$ for $z$ and $E_{1u}$ for $(x,y)$.

The six-operation molecular point group with a threefold proper rotation axis and three vertical mirror planes. Its rotational symmetry number is three because mirror operations do not count as proper spatial rotations.

The molecular point group containing identity, one twofold rotation, an inversion centre and a mirror plane perpendicular to the twofold axis. Its four operations are $E,C_2,i,{\sigma }_h$. An extended anti conformation of hexa-1,5-diyne provides an example; changing the conformation can change the symmetry.

The point group of orthogonal spatial transformations that preserve a fixed molecular geometry, including the positions and chemical identities of all nuclei. Changing a conformation can change the molecular point group even when the molecular connectivity is unchanged. Its group representations on atomic orbitals depend on orbital signs and rotations as well as the permutation of centres.

A transformation that preserves a specified structure, including the chemical identity of sites in a molecule. Rotations, reflections, inversion and improper rotations can occur in a molecular point group. The associated action on atomic orbitals also changes their orientations and signs, which must be included when computing a character of a representation.

The eight-operation molecular point group of an ideal square pyramid. Its polar-vector components transform as $A_1$ for the principal-axis coordinate and $E$ for the transverse pair. Symmetric quadratic functions also contain $B_1$ and $B_2$, allowing Raman spectroscopy activity beyond the infrared modes.

The molecular point group of a centrosymmetric linear molecule. Longitudinal coordinates have $\Sigma$ symmetry; their behaviour under inversion about the inversion centre supplies the $g$ or $u$ label, and a vertical reflection supplies the sign on a $\Sigma$ representation.

A linear combination of atomic displacements adapted to an irreducible representation of a molecular point group. Symmetry coordinates reduce the dynamical matrix into independent symmetry blocks; several coordinates in the same block can still mix.

For a free molecule in a uniform ideal gas, $q_i$ is proportional to volume. The ratio $f_i=q_i/V$ is independent of the container volume at fixed temperature and has units of inverse volume.

If energies are measured upward from a species reference ${\epsilon }_0$, its absolute single-particle sum is $q_{\mathrm{abs}}=e^{-\beta {\epsilon }_0}q$. The ideal-gas chemical potential is then ${\epsilon }_0-k_{B}T\ln (q/N)$. A zero-point factor can be included either in $q$ or in the reference energy, but not both.

A slow spatial modulation produced by overlapping periodic structures with different projected spatial frequencies or orientations. Its spacing need not equal an atomic lattice spacing.

Let $c>0$, $0\le p\le 1$ and $\lambda >0$. With probability $p$, take $Y=-c$; otherwise draw an exponential distribution of rate $\lambda$. This mixed probability distribution has a negative atom and a continuous nonnegative component. Its moments are $\mathbb{E}[Y]=-pc+(1-p)/\lambda$ and $\mathbb{E}[Y^2]=p{c}^2+2(1-p)/{\lambda }^2$. The law of total variance equivalently gives $\mathrm{Var}(Y)=(1-p)/{\lambda }^2+p(1-p)(c+1/\lambda {)}^2$.

When $p<1$, conditioning on $Y\ge 0$ selects exactly the exponential component, so $P(Y\ge t\mid Y\ge 0)=e^{-\lambda t}$ for $t\ge 0$. An ordinary probability density function for the continuous component integrates to $1-p$, not one; the missing mass is the atom, not a normalization error.

Spatial changes in actinide composition during service. Temperature gradients, chemical potentials, oxygen chemistry and transport through the solid or across pores can contribute. A temperature gradient alone does not determine a unique plutonium migration direction.

Alpha-beta pruning evaluates a minimax search tree while omitting branches that cannot change an already available decision. The parameter $\alpha$ bounds what Max can already secure, and $\beta$ bounds what Min can already enforce along the current search path. Once $\alpha \ge \beta$, further children of that node cannot improve the relevant ancestor's choice.

If Max already has an alternative worth five, and a new Min node has a child worth three, that node's value is at most three. Its remaining children can be skipped. Full-depth pruning preserves the root minimax value; applying it to a depth-limited evaluation preserves that truncated value, not necessarily the true game result. Good move ordering makes cutoffs occur sooner.^[ref]

For a clean isotropic three-dimensional weak-coupling s-wave model, use the energy-valued gap $\Delta$, a density of states $N(0)$ per spin, and free-energy density ${\alpha }_{\Delta }|\Delta {|}^2+{\beta }_{\Delta }|\Delta {|}^4/2+K_{\Delta }|D\Delta {|}^2$. Near $T_c$, ${\alpha }_{\Delta }\simeq N(0)(T-T_c)/T_c$. The quartic coefficient follows fromThe integral is $\pi /(2|{\omega }_n{|}^3)$ and the positive odd-integer sum is $7\zeta (3)/8$, yielding the displayed coefficient. The isotropic Fermi-velocity average gives $K_{\Delta }={\beta }_{\Delta }{\hslash }^2{v}_F^2/6$. The pair charge is $q=-2e$ in $D=\nabla -iq\mathbf{A}/\hslash$. These values change under a rescaling of $\Delta$, different density-of-states convention or different material model; the local expansion is a near-transition approximation.^[ref]

A pairing-channel Hubbard–Stratonovich transformation replaces an attractive fermion interaction by a complex Cooper pair field coupled to fermion pairs. A Grassmann Gaussian integral then gives a trace-log effective action, with its trace convention fixed to avoid doubling independent spin-pair blocks. The uniform stationary equation is the BCS gap equation. A small-field, slow-gradient expansion near the transition yields Ginzburg-Landau theory of superconductivity; the normalization of the phenomenological field need not equal that of the microscopic gap.

A liposomal miR-34a microRNA mimic studied in a phase 1 trial. Serious immune-related toxicity illustrates limits of inferring therapeutic safety from endogenous regulatory function.^[ref]

A metallacarborane containing a cobalt vertex. CpCo can replace BH in a suitable isolobal skeletal-electron count.