This paper reviews a family of methods that compute electronic excited states by optimizing the electron orbitals for each state, a strategy called orbital‑optimized (OO) density functional theory (DFT). Unlike the widely used time‑dependent DFT (TDDFT), which follows how a system responds in time, OO methods are time‑independent and find excited states as stationary points of an energy surface. Because the orbitals are allowed to relax specifically for the excited state, OO calculations can give a more balanced description of states that differ a lot from the ground state.

The authors clarify the formal theory behind OO approaches and collect recent technical advances. They summarize new algorithms that locate excited states as saddle points on the electronic energy surface — points that are stationary in some directions and not minima in others — and that avoid a common failure called variational collapse, where the optimization falls back to the ground state. The review also brings together current recipes for treating open‑shell singlet excited states (a class of states with unpaired electrons), for computing transition properties and spectra, and for practical calculations of Rydberg, charge‑transfer, and core excitations.

At a high level, OO DFT works by solving Kohn–Sham (KS) style equations for a state that is not the ground state. The method replaces the usual variational minimum for the ground state with a stationarity condition for a labeled excited state. In practice this means using a state‑specific exchange‑correlation contribution (a key ingredient of DFT that accounts for electron interaction effects) and allowing non‑aufbau occupations, where higher‑energy orbitals can be occupied in the excited solution. These features let OO methods capture excitations that are difficult for standard, commonly used implementations of TDDFT, such as Rydberg states (highly excited diffuse electrons), long‑range charge‑transfer states, core‑level excitations, and states involving double excitations.