In 1964, P. Hohenberg and W. Kohn (In 1998, Walter Kohn received -shared with John A. Pople- the Nobel prize in Chemistry for his work on DFT) published a pair of theorems constituting the basis for Density-Functional Theory (DFT). Only one year later, the development of the Kohn-Sham (KS) scheme allowed to make DFT a practical theory for all kind of (intended) calculations, as it is known today (KS-DFT).
These authors showed that there always exists a one-to-one relation (correspondence) between the energy and the electron density of a system, i.e. it is in principle possible to obtain directly the exact energy from this density through an universal functional. However, the mathematical formulation that delivers this energy is still unknown. This approach completely circumvents the paths classically forming the core of Quantum Chemistry: the wavefunction is no longer needed and the associated Schrödinger equation does not need to be correspondingly solved.
The key is thus to model or mimic the subtle effects dominating matter at the quantum scale by means of a functional of the electronic density. The machinery should accurately include exchange and correlation effects, in order to address structure and bonding of molecules, and it should be more advantageous than the ab initio methods, either by reducing the computational cost associated to any calculation or by introducing theoretical models able to rationalise chemical reactivity or physical concepts.
It was not until the 1980s that modern approximations to that universal functional were proposed. That means to dispose of expressions able to deliver the stabilising effects of matter arising from a purely quantum-mechanical (non-classical) origin after inserting the density of any system into the specific chosen mathematical form. The development of these expressions (the density functionals) is normally a hard work, needing extensive calibrations and applications before its wide adoption by the community. Apart from the Local Density Approximation (LDA), the extensions coined as Generalized Gradient Approximation (GGA), the HYBRID functionals containing a portion of exact-like exchange, the meta-GGA functionals, the DOUBLE-HYBRID functionals also containing a portion of perturbative-like correlation, local hybrid functionals, range-separated hybrid functionals, and other orbital-dependent functionals (i.e. RPA) are available today for running calculations.
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