Molecular orbitals and electronic symmetry

For a linear molecule, a sigma orbital has zero projection magnitude of orbital angular momentum along the internuclear axis, while a pi orbital has projection magnitude one in units of hbar. Higher projections give delta and further labels. For a homonuclear diatomic, inversion through the molecular center is also a symmetry: gerade functions retain their sign and ungerade functions change it. A heteronuclear diatomic generally lacks that inversion symmetry, so g/u is not an exact label there.

An asterisk in a qualitative orbital diagram denotes antibonding character, not an additional symmetry eigenvalue. Bonding or antibonding concerns how occupation influences an energy comparison and often correlates with internuclear nodes in a simple basis. Inversion parity and axial projection instead describe transformations of a function. They should not be interchanged: an orbital can be ungerade and bonding, or gerade and antibonding, depending on the atomic functions and their combinations.

Apply inversion to a matrix element between opposite-parity functions. The even Hamiltonian is unchanged, the bra keeps its sign, and the ket reverses sign. The element therefore equals its own negative and must be zero. Functions of the same symmetry may mix, but symmetry alone does not guarantee large mixing; their energy separation and coupling strength still matter. An external field or distortion that breaks inversion can enable a previously forbidden matrix element.

In a finite orbital basis, solving the secular problem produces approximate orbitals adapted to symmetry. For multiple electrons, a single determinant automatically enforces electronic antisymmetry but usually omits some correlation. Mixing determinants can improve the electronic state, especially near bond dissociation or configurations close in energy. The Born–Oppenheimer approximation and the single-determinant approximation are logically separate: nuclei can be clamped while the electronic state still needs several configurations.

A spatial orbital can hold two electrons of opposite spin because its two spin orbitals are distinct. Occupation of degenerate orbitals requires attention to electron interactions and spin symmetry; filling a sketch by energy alone is not a substitute for solving the many-electron problem. In common qualitative models, antibonding occupation reduces bond order, while removing such an electron increases it. This trend does not specify an equilibrium length, a dissociation threshold, or the ordering of all molecular electronic states.

Likewise, summing one-electron orbital energies generally double-counts some interactions and omits required terms in the total energy. A spectroscopy assignment ultimately concerns many-electron level differences and transition matrix elements. Symmetry is especially valuable because it can remain exact within a stated Hamiltonian even when computed energies are approximate. The practical sequence is to label transformations, identify permitted couplings and occupancies, then state which electronic approximation supplies quantitative numbers.