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Similar to: What are the types of charge analysis?, What are the types of bond orders?, and What are some recent developments in density functional theory?, I would like to ask: What are the different variations/flavors of DFT (density functional theory)?

I ask users to stick to one of the following, and explain it compactly as I did here:

  • DFTB: Density functional tight binding
  • DFPT: Density functional perturbation theory
  • SCC-DFTB: Self Consistent Charge DFTB
  • TD-DFT: time-dependent DFT
  • TD-DFRT: time-dependent density functional response theory [link to answer there]
  • BS-DFT: Broken-symmetry DFT
  • MDFT: Molecular DFT
  • MDFT-dev
  • DFT-D(EFP)
  • BDFT: Magnetic field DFT [link to answer there]
  • CDFT: Current DFT
  • KS-DFT: Kohn-Sham DFT
  • OF-DFT: orbital-free DFT [link to answer here]
  • TAO-DFT: Thermally-Assisted-Occupation DFT
  • DC-DFT: Density-corrected DFT [link to answer there]
  • Constrainted DFT
  • Conceptual DFT
  • vMSDFT (variational multi-state DFT)
  • ab initio DFT [link to answer there]
  • MCPDFT (Multiconfigurational Pair Density Functional Theory)
  • SCDFT (Superconducting DFT)
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  • $\begingroup$ The title makes me wonder how (often) a Discrete Fourier Transform could be used for analytical purposes in matter modelling. $\endgroup$ – Mast Jul 15 at 6:32
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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 $$j=j_p+\frac{e}{mc}n\mathbf{A}$$ 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
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  • $\begingroup$ Could you briefly define the paramagnetic current density and contrast it with the ordinary physical current density? $\endgroup$ – KF Gauss Jul 24 at 2:41
  • $\begingroup$ @KFGauss I have added what I know. Basically, the paramagnetic current density seems to just a computational trick to ensure you don't have gauge dependence. $\endgroup$ – Tyberius Jul 24 at 2:55
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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, $$ \tag{3} T_{TF}[n] = \frac{3}{10}(3\pi^2)^{2/3}\int \mathrm{d}\mathbf{r} \, n^{5/3}(\mathbf{r}) , $$ 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).
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    $\begingroup$ +1. It's a bit longer than ideal, but still a great contribution to the site! $\endgroup$ – Nike Dattani Jul 15 at 19:20
  • $\begingroup$ Well written. I made the bounty offer because I hoped we could get more answers here, but since none came, I re-read the existing answers carefully, and enjoyed your answer quite a lot. So the bounty is well deserved here :) It's also nice to get an answer from someone whose done a review paper on the subject! $\endgroup$ – Nike Dattani Jul 24 at 17:11

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