We review theories of phosphorescence in cyclometalated complexes. We focus primarily on pseudooctahedrally coordinated metals (e.g., [Os(II)(bpy)3] 2+ , Ir(III)(ppy)3 and Ir(III)(ptz)3) as, for reasons that are explored in detail, these show particularly strong phosphorescence. We discuss both first principles approaches and semi-empirical models, e.g., ligand field theory. We show that together these provide a clear understanding of the photophysics and in particular the lowest energy triplet excitation, T1. In order to build a good model relativistic effects need to be included. The role of spin-orbit coupling is well-known, but scalar relativistic effects are also large -and are therefore also introduced and discussed. No expertise in special relativity or relativistic quantum mechanics is assumed and a pedagogical introduction to these subjects is given. It is shown that, once both scalar relativistic effects and spin-orbit coupling are included, time dependent density functional theory (TDDFT) provides quantitatively accurate predictions of the radiative decay rates of the substates of T1 in phosphorescent organotranstion-metal complexes. We describe the pseudo-angular momentum model, and show that it reproduces the key experimental findings. For example, this model provides a simple explanation of the relative radiative rates of the substates of T1, which differ by orders of magnitude. Special emphasis is placed on materials with potential applications as active materials in organic light-emitting diodes (OLEDs) and principles for the design of new complexes are identified on the basis of the insights provided by the theories reviewed. We discuss the remaining theoretical challenges, which include deepening our understanding of solvent effects and, vitally, understanding and predicting nonradiative decay rates.
KeywordsLigand field theory Relativistic density functional theory (DFT) Time dependent density functional theory (TDDFT) Organic light-emitting diodes (OLEDs) Ir(ppy)3 Spin-orbit coupling Scalar relativistic effects Triplet state Radiative decay rate Non-radiative decay rate Photoluminescent quantum yield (PLQY) Metal-to-ligand charge transfer (MLCT)Two major approaches have been taken to modelling the phosphorescence in organo-transition metal complexes: phenomenological or semi-empirical approaches, such as ligand field theory, and first principles approaches, primarily based on (relativistic) (time dependent) density functional theory. *