Density Functional Theory

Density functional theory (DFT) has a long story on the treatment of electronic problems. Mainly after the Hohenberg and Kohn theorems that state that the exact energy of the ground electronic state of a system can be obtained in a biunivocal relationship from the electron density $\rho$ expression [37]. This theorem shows that there exists an expression (a functional of the electron density) that connects electron density and the exact energy. The problem is that the theorem does not specify how this functional looks like.

The application of the DFT to the computational chemistry was not found useful until the late eighties. After that time, the DFT techniques have had an explosion of successful applications to chemical problems due to the usage of more appropriate functionals for these kind of systems [38,39].

The DFT applied to a molecular system may be solved in such a way that keeps many parallelism with the already explained Hartree-Fock wave mechanics. However, in this case the electron density distribution plays a central role rather than the many-electron wavefunction. In defense of the DFT it is said that while in some cases the wave mechanics must take into account thousands of determinants into the wavefunction, the DFT gives an easier understanding by its only three-space coordinates electron density.

Kohn-Sham equations:
The equivalent of the Hartree-Fock equations in DFT are the self consistent Kohn-Sham equations (KS)[40]. Although this is not the only way to consider the DFT equations, it is the most frequently used in computational chemistry, so we will focus on this particular aspect of the DFT, the rest of applications and theories may be found in the literature[41].

This strategy can be seen as modeling an independent particle model that is able to reproduce, in theory, the full interacting system. The energy is obtained through a sum of separated functionals of the electronic density $\rho$.

$$E_{DFT}[\rho] = T[\rho] + E_{Ne}[\rho] + V[\rho]$$ (2.42)

Both the kinetic term $T[\rho]$ and the electron-electron term $V[\rho]$ are unknown functionals. The $V[\rho]$ can be derived as a sum of the correlated coulombic interaction $J[\rho]$ and another functional that would account for the exchange found in HF.

$$V[\rho] = J[\rho] + K[\rho]$$ (2.43)

For the kinetic term $T[\rho]$ Kohn and Sham proposed to define an independent electron system described by a set of orbitals {$\Phi_i$} that give a density $\rho_0$

$$\rho_0(\vec r)=\sum_i^n \vert\Phi_i(\vec r)\vert^2$$ (2.44)

The central assumption of the Kohn-Sham strategy is to assume that the above fictitious system give the same electron density $\rho_0$ than the real system $\rho$

$$\rho_0(\vec r)=\rho(\vec r)$$ (2.45)

The kinetic term corresponding to this hypothetical system $T_0[\rho]$, although it is not the exact, is known and is the one that is used for HF.

$$T_0[\rho]=\sum_i^n \langle \Phi_i\vert-\frac{1}{2}\Delta_i\vert\Phi_i\rangle$$ (2.46)

$$E_{DFT}[\rho] = T_0[\rho] + E_{Ne}[\rho] + J'[\rho] + E_{xc}[\rho]$$ (2.47)

Where the uncorrelated coulombic functional $J'[\rho]$ has the known expression from the HF. And the functional $E_{xc}[\rho]$ must now include the correction to the Kinetic energy as well

$$E_{xc}[\rho] = (T[\rho]-T_0[\rho]) + (V[\rho] - J'[\rho])$$ (2.48)

This unknown exchange-correlation functional is where all the recent developments in DFT theory have achieved its major success. $E_{xc}[\rho]$ can be designed by local (LDA), gradient corrected or generalized gradient corrected (GGA) (e.g. BLYP[42,43]) and hybrid approximations (B3LYP [44]). The hybrid functional B3LYP has produced excellent results when it is compared with other strategies of similar computational cost. However this field is still in progress, and new functionals of the energy are being proposed to cover a wider variety of chemical systems [45].

Despite of the new nomenclature it is easy to see that equation 1.47 has similar functionality that the Hartree-Fock expression. So if the functional $E_{xc}[\rho]$ is only the exchange part $\hat K_j$ found in the HF scheme (equation 1.26) we recover the same HF expression. In this sense, the implementation of HF equations might be seen as a particular case of the DFT-KS equations. However, if the functional $E_{xc}[\rho]$ was known we would obtain the exact energy including all the electron correlation.

In the same manner the functional $E_{DFT}[\rho]$ is minimized with the constraint of the orthonormality of the Kohn-Sham orbitals. The Kohn-Sham orbitals can be spanned as a combination of basis functions centered on every atom leading to the Konh-Sham equations that have the same scheme of the Roothan-Hall procedure.

The computational cost of the KS equations is very similar to the Roothan-Hall-HF. However some differences exist, it could depend strongly on the expression used for the $E_{xc}[\rho]$ functional. Usually some integrals due to the exchange-correlation expression have to be evaluated numerically in a grid of points.

$$\langle \phi_r\vert V_{xc}[\rho (r), \nabla\rho(r)]\vert\phi_s\ra... ..._i^{points} V_{xc}[\rho (r_i), \nabla\rho(r_i)]\phi_r(r_i)\phi_s(r_i)\Delta v_i$$ (2.49)

The number of points, the size and the shape of the grid is crucial to reproduce a computational result.