Density Functional Theory

Density Functional Theory (DFT) is one of the most popular methods of quantum mechanics. Today it is applied to calculate several molecular properties, for example, binding energies of molecules in chemistry and band structures of solids in physics, but also in other areas considered more distant to quantum mechanics such as biology and mineralogy.

## An Introduction

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DFT owes its versatility to the generality of its fundamental concepts and its flexibility to implement them. But, despite its flexibility and generality, DFT is completely based on a rigorous conceptual structure.

In quantum mechanics, all information on a system is contained in its wave function, Ψ. The usual method of quantum mechanics using Schrödinger's equation (SE) can be summarized by the following sequence

ν ( r ) ⇒ Ψ ( r 1 , r 2 , . . . , r N ) ⇒ o b s e r v a b l e s {\displaystyle \nu (r)\Rightarrow \Psi (r_{1},r_{2},...,r_{N})\Rightarrow observables}

First the system is specified by choosing the potential ν(r) and it is inserted into Schrödinger's equation. Then this equation is solved for the wave function and finally observables are calculated by taking the expected values of operators with this wave function. Electron density is an observable which is calculated in the following manner

ρ ( r ) = N ∫ d 3 r 2 ∫ d 3 r 3 . . . ∫ d 3 r N Ψ ∗ ( r 1 , r 2 , . . . , r N ) Ψ ( r 1 , r 2 , . . . , r N ) {\displaystyle \rho (r)=N\int d^{3}r_{2}\int d^{3}r_{3}...\int d^{3}r_{N}\Psi ^{*}(r_{1},r_{2},...,r_{N})\Psi (r_{1},r_{2},...,r_{N})}

Several powerful methods have been developed to solve Schrödinger's equation to treat problems of multiple bodies, such as methods of configuration interaction (CI) in chemistry. These methods require high computational resources and therefore, they are difficult to treat large and complex systems, such as the calculation of chemical properties of a 100-atoms molecule with full CI.

It is here where DFT provides a viable alternative, less accurate perhaps, but more versatile. DFT methods are cheap alternatives to include part of electron correlation. The best DFT methods achieve greater accuracy than Hartree Fock theory with only a modest increase in computational cost.

Electronic, magnetic and structural properties of materials have been calculated using DFT and the scope of DFT contributions to the science of molecules is reflected by the Nobel Prize in Chemistry 1998, which the Nobel Foundation awarded to Walter Kohn, one of the founding fathers of the DFT and to John Pople, who implemented DFT in computational chemistry.

Density Functional Theory aims to calculate the electronic ground-state energy of a system of N electrons only through its density, without prior knowledge of the wave function of the system. Thus, the 3N-dimensional problem is reduced to a three-dimensional problem and, at first, it should be easier to solve. DFT can be summarized by the sequence

ρ ( r ) ⇒ Ψ ( r 1 , r 2 , . . . , r N ) ⇒ ν ( r ) {\displaystyle \rho (r)\Rightarrow \Psi (r_{1},r_{2},...,r_{N})\Rightarrow \nu (r)}

In other words, knowledge of ρ(r) implies knowledge of the wave function and the potential, and hence of all other observables.

## Software

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There are several programs that implement DFT in their routines. Here is a list of the most common ones.

Firefly. http://classic.chem.msu.su/gran/gamess/
Gamess-UK. http://www.cse.scitech.ac.uk/ccg/software/gamess-uk/
Gamess-US. http://www.msg.chem.iastate.edu/gamess/
Gaussian. http://www.gaussian.com
ORCA. http://www.thch.uni-bonn.de/tc/orca/

## References

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W. Koch, M. C. Holthausen. A Chemist’s Guide to Density Functional Theory. Second Edition 2001 Wiley-VCH

K. A. Capelle. Bird’s-Eye View of Density-Functional Theory, Universidade de Sao Paulo, 2006.

F. Jensen. Introduction to Computational Chemistry. Second Edition 2007 John Wiley & Sons Ltd