What are the types of DFT?

KS-DFT: Kohn-Sham DFT

The KS-DFT is proposed to deal with the problems of orbital-free DFT (OFDFT), which has been explained by @wcw. OFDFT attempts to compute the energy of interacting electrons, as the functional of the density. While this brute force approach is in principle correct, in practice it is not very accurate. This is due to the lack of accurate approximations for the kinetic energy functional.

Then Kohn and Sham proposed an alternative approach in order to overcome this problem.

• Phys. Rev. 140, A1133 (1965)

In the Kohn-Sham approach, a fictitious non-interacting system is constructed, in such a way, that its density is the same as that of the interacting electrons. The core ideas can be summarized in the following figure.

By introducing a fictitious KS non-interacting electron system, the problem OFDFT encountered is well solved.

You can find more details on this lecture note written by C.-K.Skylaris.

Density functional perturbation theory (DFPT)

This method refers to the calculation of the linear response of the system under some external perturbation.

Consider some set of parameters ${lambda_i}$. The first and second derivatives of the total energy with respect to these parameters in DFT read:

$$ frac{partial E}{partiallambda_i}=intfrac{partial V(mathbf{r})}{partiallambda_i}n(mathbf{r})dmathbf{r}, $$

$$ frac{partial^2 E}{partiallambda_ilambda_j}=intfrac{partial^2V(mathbf{r})}{partiallambda_ilambda_j}n(mathbf{r})dmathbf{r}+intfrac{partial n(mathbf{r})}{partiallambda_i}frac{V(mathbf{r})}{partiallambda_j}dmathbf{r}. $$

The trickier part of the calculation is the derivative of the density $n(mathbf{r})$, which can be written in terms of the first order change of the Kohn-Sham orbitals in the presence of the perturbation, which in turn can be determined using standarnd first order pertubation theory.

This general formalism can be applied to study a variety of physical properties depending on the choice of parameters ${lambda_i}$. For example, if the parameters are chosen to be atomic coordinates $lambda_i=mathbf{R}_i$, then the second derivative above corresponds to the matrix of force constants that can be used to calculate phonons. Another commonly used perturbation is that caused by an external electric field.

References:

1. Baroni, Gironcoli, Dal Corso, Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001)

OF-DFT: Orbital-free density functional theory

Hohenberg and Kohn established that the ground state energy, $E$, of interacting electrons in a potential, $v(mathbf{r})$, is a functional of the electron density, $n(mathbf{r})$:

$$ tag{1} E[n] = F[n] + int mathrm{d}mathbf{r} , v(mathbf{r}) n(mathbf{r}) . $$

While this statement is formally true, we do yet not have a convenient way to compute the "universal functional" $F[n]$ exactly for most cases. To carry out OF-DFT, one chooses an explicit form for $F[n]$, likely an approximation, and varies the electron density to find the particular $n(mathbf{r})$ that minimizes $E[n]$, yielding the ground state. Quantities like forces and stresses then follow from Hellmann–Feynman-type formulas.

Both the advantages and challenges of OF-DFT stem from its simplicity; wave functions and density matrices are eschewed altogether. For cases when OF-DFT is suitably accurate, it is extremely attractive from a computational standpoint, in significant part because $n(mathbf{r})$, the sole working variable, grows only linearly with system size. However, for much of the periodic table, OF-DFT remains less accurate than other electronic structure methods.

Modern implementations of OF-DFT often build on the insights of Kohn and Sham, who considered $F[n]$ in the form $$ tag{2} F[n] = T_s[n] + E_{Hxc}[n] , $$ where $T_s[n]$ is the kinetic energy of an auxiliary system of noninteracting electrons (with the same electron density as the interacting system) and $E_{Hxc}[n]$ subsumes electrostatic, exchange, and correlation contributions. The full Kohn-Sham scheme determines $T_s[n]$ implicitly, but exactly, following the introduction of single-electron orbitals. The corresponding orbital-free approach, in contrast, approximates $T_s[n]$ with an explicit density functional, while utilizing the same approximations for $E_{Hxc}[n]$. Simple approximations to $T_s[n]$ include the Thomas-Fermi functional, $$ tag{3} T_{TF}[n] = frac{3}{10}(3pi^2)^{2/3}int mathrm{d}mathbf{r} , n^{5/3}(mathbf{r}) , $$ and the Weizsäcker functional,

begin{align} T_W[n] & = -frac{1}{2} int mathrm{d}mathbf{r} , n^{1/2}(mathbf{r}) nabla^2 n^{1/2}(mathbf{r}) tag{4}\ & = int mathrm{d}mathbf{r} , left[ frac{1}{8} frac{|nabla n(mathbf{r})|^2}{n(mathbf{r})} - frac{1}{4} nabla^2 n(mathbf{r}) right],tag{5} end{align}

both of which are exact for certain limiting cases and predate the Hohenberg-Kohn theorems by decades.

For more (disclaimer: from my perspective), here is a recent review of successful OF-DFT applications in materials science:

• W.C. Witt, B.G. del Rio, J.M. Dieterich, and E.A. Carter, Orbital-free density functional theory for materials research, Journal of Materials Research 33 (2018) (DOI: 10.1557/jmr.2017.462).

CDFT: Current DFT

Current DFT is defined via the generalized Hohenberg-Kohn theorem (HKT), which extends the traditional HKT to account for the effect of magnetic fields. The generalized HKT says that the scalar potential $mathbf{V}$, the (nondegenerate) ground state wavefunction $Psi$, and the vector potential $mathbf{A}$ are uniquely determined by the ground state density $n$ and the paramagnetic current density $j_p$. From [1], the physical and paramagnetic current densities are related by $$j=j_p+frac{e}{mc}nmathbf{A}$$ Note, the total/physical current density is not used, as the factor involving the vector potential leads to gauge dependence and thus wouldn't uniquely determine the ground state.

Similar to standard DFT, this results in a variational principle, where the true $n$ and $j_p$ minimize a functional for the ground state energy. This can in turn be shown to be equivalent to solving a set of one-electron equations, e.g. Kohn-Sham CDFT. One of main challenges in the development of this area is formulating new functionals that incorporate $j_p$ into the exchange-correlation functional while maintaining gauge invariance.

References:

1. G. Vignale and Mark Rasolt "Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields" Phys. Rev. B 37, 10685 DOI: 10.1103/PhysRevB.37.10685
2. James W. Furness, et al "Current Density Functional Theory Using Meta-Generalized Gradient Exchange-Correlation Functionals" J. Chem. Theory Comput. 2015, 11, 4169−4181 DOI: 10.1021/acs.jctc.5b00535