Inversion Parity in Homonuclear Diatomics

Introduction

Homonuclear diatomic molecules have an inversion centre at the bond midpoint because the two identical nuclei exchange under inversion. Their molecular orbitals can be labelled gerade, g, if the wavefunction is unchanged, or ungerade, u, if its sign reverses. These labels add information beyond sigma or pi angular symmetry. They are essential for electric-dipole selection rules and are often confused with bonding versus antibonding, which is a different property.

Core explanation

Choose the origin at the midpoint between identical nuclei A and B. Inversion maps every point (x,y,z) to (−x,−y,−z), exchanging the two nuclear centres. For an orbital ψ, if iψ = +ψ, it is gerade; if iψ = −ψ, it is ungerade. Electron probability density ψ ² is unchanged in either case because the sign disappears on squaring. The g/u label refers to wavefunction phase, not positive or negative charge or a lopsided electron cloud.

For two equivalent s orbitals s A and s B, inversion exchanges them without an internal sign reversal. The sum s A+s B is unchanged and has g parity; the difference s A−s B changes sign and has u parity. In a simple H₂⁺ model, the symmetric s combination is bonding and the antisymmetric one is antibonding. It is tempting to infer that g always means bonding and u always means antibonding, but this inference fails for other orbital orientations.

Consider side-on p x functions on A and B with the bond along z, using the same global x-axis phase convention. An atomic p function changes sign under inversion about its own centre while the centres exchange. Thus inversion sends p {x,A} to −p {x,B}. The sum p {x,A}+p {x,B} is ungerade, while the difference is gerade. With the chosen phase convention, the same-sign side-on sum increases density in the bonding region and is a bonding πu orbital; the out-of-phase difference is antibonding πg . This explicit counterexample proves that inversion parity and bonding character must be assigned independently.

The electric-dipole operator has odd inversion parity: x, y and z each change sign under inversion. A transition moment integral between initial and final states can be nonzero only when the product of their parities with the odd operator is even. Thus g-to-u and u-to-g electric-dipole transitions pass the parity test, while g-to-g and u-to-u fail in an ideal centrosymmetric model. This is the Laporte parity rule. Passing it does not guarantee intensity; angular momentum, spin and overlap restrictions may still matter.

Only a molecule possessing inversion as an actual symmetry can have exact g/u state labels. A heteronuclear diatomic such as CO lacks inversion because C and O would exchange. A linear heteronuclear molecule can still use σ and π angular labels, but assigning g/u to its orbitals as exact symmetry species would be incorrect. Isotopic substitutions can also affect nuclear symmetry classification in a full molecular treatment.

The parity rule applies to the full electronic states involved in a transition, not merely to one orbital sketched in an MO diagram. In a simple one-electron excitation, orbital parity changes can guide intuition, but many-electron state symmetry and vibronic coupling can complicate the intensity picture. Vibrations that break inversion instantaneously can lend weak electric-dipole intensity to transitions that are forbidden at a static centrosymmetric geometry.

Step-by-step reasoning

First confirm that inversion maps every nucleus to an identical nucleus. Place the origin at the inversion centre and apply (x,y,z) → (−x,−y,−z) to the complete orbital wavefunction, including intrinsic p-orbital phase. Assign + as g and − as u. Determine bonding separately by phase and density between nuclei. For transitions, multiply initial parity, odd dipole parity and final parity, looking for an even product.

Visual explanation

Draw two nuclei equidistant from a centre mark. For s A+s B, shade both s lobes with the same phase and draw inversion arrows showing the sum returns unchanged. For side-on p x orbitals, mark plus lobes above and minus below each nucleus; inversion exchanges centres and lobes, giving a sign change for the bonding sum. Place g/u labels separately from bonding stars.

Real-world analogy

A symmetric drawing can look the same after turning it through its centre, while an antisymmetric signed pattern can return with every plus replaced by minus. The visible intensity of the signed pattern may still be unchanged because intensity depends on the square. This resembles g and u wavefunctions, but a drawing alone must include orbital phase to be meaningful.

Real-world example

Electronic spectra of homonuclear diatomics show strong parity restrictions for electric-dipole transitions. A transition between states of the same inversion parity may be absent or weak under the ideal rule, whereas a g-to-u transition can be allowed subject to other restrictions. Spectroscopists use these patterns to assign state symmetries alongside rotational and spin information.

Why?

Why does an odd dipole operator require opposite state parities? Multiplying g × u × u or u × u × g gives an even integrand capable of surviving inversion-symmetric integration. By contrast, g × u × g is odd and cancels. The rule is a direct-product selection rule expressed in simple ± parity language.

Common misconception

Gerade is not synonymous with bonding. A bonding π orbital can be ungerade, and an antibonding π orbital can be gerade. Nor does u mean a physically negative orbital or negative electron density. It means the wavefunction changes sign under inversion, while its probability density remains nonnegative.

Worked example

Let s A and s B be equivalent 1s orbitals of a homonuclear diatomic. Inversion exchanges them: i(s A+s B) = s B+s A = +(s A+s B), so the sum is g. Likewise i(s A−s B) = s B−s A = −(s A−s B), so the difference is u. If the in-phase sum increases density between nuclei, it is bonding σg; the out-of-phase difference is antibonding σu . This result is specific to the s-orbital combination and must not be generalised to every orbital pair.

Quick check

1. Can a bonding MO be ungerade? Answer: Yes. A bonding pi combination of suitable side-on p orbitals can be πu. 2. Which electric-dipole parity change is allowed in an ideal centrosymmetric molecule? Answer: A change between g and u, in either direction, passes the inversion-parity test.

Exam focus

Test inversion on the full wavefunction and state the inversion centre. Keep g/u separate from σ/π and bonding/antibonding labels. Apply the electric-dipole parity rule to electronic states, and qualify an allowed parity as only one of several conditions for intensity.

Advanced insight

Inversion parity is robust under any Hamiltonian term that preserves the inversion centre. If a perturbation breaks inversion, g and u cease to be exact labels and formerly forbidden state mixing can occur. This makes selection-rule violations informative: weak bands can signal vibronic coupling, environmental asymmetry or higher-order transition mechanisms rather than an invalid symmetry theorem.

Summary

Gerade and ungerade describe whether a homonuclear diatomic wavefunction is even or odd under inversion through the bond midpoint. They are independent of sigma/pi type and bonding character. Because the electric-dipole operator is odd, ideal g-to-u transitions can pass the parity test while same-parity transitions fail it. Heteronuclear diatomics generally cannot use exact g/u labels.

Practice questions

1. Why is the simple bonding side-on p x sum often πu rather than πg? Answer: Inversion exchanges the two p x centres and reverses each p orbital's internal sign, so the same-phase sum changes sign overall even though it increases bonding density. 2. Is a g-to-g electronic transition guaranteed absent from every measured spectrum? Answer: It is electric-dipole forbidden in the ideal centrosymmetric model, but vibronic coupling, symmetry breaking or other transition mechanisms can give weak intensity. 3. Why should CO not be assigned σg and σu orbitals as exact labels? Answer: Inversion through the bond midpoint exchanges carbon and oxygen, so inversion is not a symmetry of CO. Sigma labels remain valid, but g/u parity does not.