ungerade

Ungerade is a German adjective meaning not even, commonly used to describe odd numbers. In mathematics, it denotes an odd integer: a number that is not divisible by 2. An integer n is ungerade if n = 2k + 1 for some integer k. Examples include 1, 3, 5, 7. Parity rules for ungerade numbers include: the sum of two ungerade numbers is gerade (even); the product of two ungerade numbers is ungerade; the sum of an ungerade and a gerade number is ungerade; the product of an ungerade and a gerade number is gerade.

In physics and chemistry, ungerade describes parity under spatial inversion. For a wavefunction ψ in a centrosymmetric system, inversion maps r to -r. If ψ(-r) = -ψ(r), the state has ungerade parity; if ψ(-r) = ψ(r), it has gerade parity. In molecular spectroscopy and group theory, g (gerade) and u (ungerade) labels indicate electronic and vibrational states with gerade or ungerade symmetry. The Laporte selection rule states that electric dipole transitions in centrosymmetric molecules occur only between states of opposite parity (g ↔ u), making many transitions forbidden if the parity does not change. The terms are common in the description of coordination chemistry and polyatomic molecules with an inversion center.

Etymology: derived from the German prefix un- meaning "not" and gerade meaning "even." The concept is used across mathematics, physics, and chemistry.