Structural Chemistry pp 221-289 | Cite as

# Density Functional Theory: From Conceptual Level Toward Practical Functionality

## Abstract

Density Functional Theory (DFT), the most productive and attractive method of computational chemistry in the last decades, a beacon expected to endure for a long time, is discussed in this chapter in terms of its conceptual and practical sides. After introducing the historical roots of DFT, the proofs of the founding Hohenberg–Kohn theorems are exposed. The issue of the exchange-correlation hole is introduced and illustrated by a strategy of heuristic artifices, describing the promises of DFT and its current limitations and compromises. Several technical issues are approached, making clear, for instance, the analytic operations leading to the celebrated *ρ* ^{4/3} pattern for the approximated local density exchange functional. The various flavors of DFT calculations (species and acronyms of consecrated functionals) are treated briefly. A special emphasis is put on features specific to DFT, not attainable in the techniques pertaining to the wave function theories (WFT). These are: the conceptual possibility of fractional occupation numbers and an orbital formalism (Kohn–Sham) with eigenvalues substantiated as derivatives of the total energy with respect of level populations. The first derivative of energy as a function of occupation numbers (taken with changed sign) is invested with an important meaning: absolute electronegativity. The second derivatives are interpreted as measures for the strength of acids and bases in the Lewis definition (accepting or donating electronic density), namely the so-called chemical hardness. The chapter proposes the term of *electrorigidity*, instead of chemical hardness, underlining the meaning of this second derivative as a “force constant” resistant to the deformation of the electronic cloud. Although not extensively, the chemical significance of the lemmas enabled by DFT is emphasized, mentioning the principles of electronegativity equalization, maximum hardness, and the mutual affinity trends in the hard and soft acids and bases (HSAB) taxonomy. One issue is the so-called DFT+U technique, using plane wave methods to alleviate non-physical trends in the account of metal ions in compounds. The explanation brings to the level of chemists’ intuition technicalities from physicists’ language. A very suggestive illustration of exposed issues is done with the help of an original development: the energies of atomic bodies as continuous functions of shell populations. With the
*ad hoc* proposed model, one concretizes several aspects of conceptual DFT, discussing comparisons between calculation and experiment, verifying theorems and approximations related to absolute electronegativity and chemical hardness (*electrorigidity*). After giving a general overview of the DFT realm and a compressed briefing of its constitutional rules, several technical aspects are revisited in more detail in the final part of the chapter. This gives an analytical survey of the main workable kinetic, exchange, and correlation density functionals, within local and gradient density approximations. They generally fulfill the *N*-contingency, assure the total energy minimization, influence the different levels of approximation, i.e. local density or gradient density frameworks, control the bonding through electronic localization functions, and decide upon reactivity through the electronic exchange relating the electronegativity and chemical hardness indices.

## Keywords

Electronic density Exchange-correlation functional Exchange hole Correlation hole Long range behavior Kohn–Sham orbitals Janak theorem Fractional occupations DFT+U methods Electronegativity Chemical hardness (electrorigidity) Electronegativity equalization Maximum hardness principle Hard and soft acids and bases (HSAB) scales Continuous energy function## References

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