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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:

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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 '20 at 6:32
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    $\begingroup$ @HitanshuSachania your question on the bounty has been asked and answered before: mattermodeling.stackexchange.com/questions/89/… $\endgroup$
    – ProfM
    Aug 26 '20 at 7:01
  • $\begingroup$ Could the self interaction correction DFT (SIC-DFT) be part of this list? $\endgroup$
    – NickZ
    Aug 31 '20 at 8:53
  • $\begingroup$ @ProfM I was utterly confused as to what DFPT as such was. I read your answer before coming here, but I somehow couldn't fit the pieces together then. Luckily, a bit more reading and clearing my mind of previous notions, I could finally grasp it. The main difference boils down to the difference in algorithm after all - finite difference vs. linear response. I wonder why they weren't given simpler names like phonon-by-fd and phonon-by-lr 😅. The bounty expires today; can you write a short answer with a link to the previous answer you shared so I can award it the bounty. $\endgroup$ Sep 1 '20 at 10:39
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    $\begingroup$ @HitanshuSachania I have posted an answer for DFPT adpating the one in the response above. DFPT in principle refers to linear response with some varying external parameter, so phonons are just a particular case of the methodology in which the perturbation is caused by atomic displacements. $\endgroup$
    – ProfM
    Sep 1 '20 at 12:25
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Real-time TDDFT (RT-TDDFT)

This is the straightforward non-perturbative solution of the TDDFT equations by means of direct propagation in time. Pioneered by Theilhaber and Yabana & Bertsch it has since found its way into several molecular or solid-state codes.

The TDDFT equations in the Kohn–Sham (KS) framework are $$ i \frac{\partial}{\partial t} \phi_i (t) = H^\mathrm{KS}(t)\phi_i (t), $$ where $\phi_i$ are the occupied KS orbitals and $H^\mathrm{KS}(t)$ is the time-dependent KS Hamiltonian.

In RT-TDDFT, the orbitals $\phi_i$ are evolved in time in a stepwise fashion. The propagation from time $t$ to time $t+\Delta t$ is performed by means of evolution operator $U(t+\Delta t, t)$: $$ \phi_i (t+\Delta t) = U(t+\Delta t, t)\phi_i (t), $$ where the evolution operator is constructed from the KS Hamiltonian. Because of the time dependence of the Hamiltonian, both explicit via a typically present time-dependent external field, and implicit via the time-dependent KS orbitals used to construct it, the TDKS equation is non-linear. Therefore, the time evolution is aided by some iterative or predictor-corrector scheme to ensure stable evolution. Examples of solvers include the Magnus, Crank–Nicholson or Runge–Kutta propagator.

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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? $\endgroup$
    – KFC
    Jul 24 '20 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 '20 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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  • $\begingroup$ 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! $\endgroup$ Jul 24 '20 at 17:11
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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:

$$ \frac{\partial E}{\partial\lambda_i}=\int\frac{\partial V(\mathbf{r})}{\partial\lambda_i}n(\mathbf{r})d\mathbf{r}, $$

$$ \frac{\partial^2 E}{\partial\lambda_i\lambda_j}=\int\frac{\partial^2V(\mathbf{r})}{\partial\lambda_i\lambda_j}n(\mathbf{r})d\mathbf{r}+\int\frac{\partial n(\mathbf{r})}{\partial\lambda_i}\frac{V(\mathbf{r})}{\partial\lambda_j}d\mathbf{r}. $$

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)
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    $\begingroup$ Once again, thank you, ProfM. I don't know why I couldn't understand $\lambda_{i}$ could be any parameter, which would then give a different meaning to the second derivative. Your answers in the forum are very helpful for people like me who are learning these topics and are relatively new to the field. Grateful for your participation. $\endgroup$ Sep 1 '20 at 20:07
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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.

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.

enter image description here

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.

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