Parity and Selection Rule Checker

Parity describes how a wavefunction behaves under spatial inversion (x → -x). A state is called even parity if it is unchanged by inversion, and odd parity if it flips sign. For a hydrogen-like orbital, the parity is exactly (-1)^l, so s and d orbitals (l=0, 2) are even, while p and f orbitals (l=1, 3) are odd.

The Laporte rule states that electric dipole (E1) transitions can only connect states of opposite parity, equivalently, |Δl| must be odd (in practice, exactly 1 for a single-electron transition). It follows directly from the electric dipole operator (proportional to position x) having odd parity itself.

A transition is 'forbidden' relative to a specific radiation mechanism when the corresponding transition matrix element is exactly zero by symmetry. E1-forbidden transitions can often still occur through weaker mechanisms, magnetic dipole (M1) or electric quadrupole (E2) radiation, which have different parity and Δl requirements and are typically thousands to millions of times slower.

M1 transitions arise from the interaction of the atom's magnetic moment with the oscillating magnetic field of light, and require Δl = 0 with no parity change. E2 transitions arise from a higher-order term in the light-matter interaction and require Δl = 0 or ±2, also with no parity change. Both are much weaker than E1 because they involve smaller physical couplings, but they are not literally forbidden.

In a dense lab gas or solid, an excited atom is knocked into a different state by a collision long before it has a chance to decay via a slow M1 or E2 transition. In the extremely tenuous gas of a nebula, densities are so low that collisions are rare, giving atoms enough time (seconds to hours) to eventually emit the E1-forbidden line, which then becomes a bright, diagnostic feature of the nebula's spectrum.

The green [O III] line at 500.7 nm, denoted with square brackets to indicate it is forbidden, comes from an M1/E2 transition between fine-structure levels of doubly-ionized oxygen with the same orbital configuration (Δl=0). It is one of the strongest lines in many planetary nebulae and was originally mistaken for a new element, 'nebulium', before its true origin was understood.

The electric quadrupole operator contains terms proportional to both x² (which connects states differing by 0 or 2 units of angular momentum) unlike the dipole operator's single power of x. This is why E2 transitions can connect states of the same l (as long as other quantum numbers like m or J differ) in addition to states two units apart.

No, it is deliberately scoped to the parity and Δl part of the selection rules for a single active electron's orbital quantum number, the first level of detail taught in most courses. Real multi-electron atoms have additional ΔJ selection rules (0, ±1 for E1/M1, 0/±1/±2 for E2, always excluding J=0 to J=0) that this calculator does not check.

No. Since parity is (-1)^l, Δl = 0 always means the parity stays the same, and E1 transitions strictly require a parity change (the Laporte rule). A Δl = 0 transition can only proceed via M1 or E2 radiation, never E1, regardless of any other quantum numbers.

A transition with |Δl| ≥ 3 is forbidden to electric dipole, magnetic dipole, and electric quadrupole radiation alike, since none of their selection rules permit such a large jump in orbital angular momentum for a single active electron. Higher multipole orders (M3, E4, and so on) could in principle connect such states, but these are vanishingly weak and rarely relevant.