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In the book Materials Modelling Using Density Functional Theory: Properties and Predictions by Feliciano Giustino, a timeline of milestones in DFT was given for achievements between 1964 and 1996:

$$\small\begin{array}{|c|c|c|} \hline \textbf{Year} & \textbf{Milestone} & \textbf{Researchers} \\ \hline 1964, 1965 & \text{HK Theorem/KS Formulation} & \text{Kohn, Hohenberg, Sham} \\ 1972, 1973 & \text{Relativistic DFT} & \text{von Barth/Hedin, Rajapol/Callway} \\ 1980, 1981 & \text{Local Density Approximiation(LDA)} &\text{Ceperley/Alder, Perdew/Zunger} \\ 1984 & \text{TDDFT} & \text{Runge, Gross} \\ 1985 & \text{First Principles MD} & \text{Carr, Parrinello} \\ 1986 & \text{Quasiparticle Corrections} & \text{Hybertsen, Louie} \\ 1987 & \text{Density Functional Perturbation Theory} & \text{Baroni, Giannozzi, Testa} \\ 1988, 1993 & \text{Toward Chemical Accuracy} & \text{Lee/Yang/Parr (1988), Becke (1993)} \\ 1991 & \text{Hubbard Correction} & \text{Anisimov, Zaanen, Andersen} \\ 1992, 1996 & \text{Generalized Gradient Approximation} & \text{Perdew/Burke/Ernzerhof} \\ \hline \end{array}$$

Has there been any milestone after 1996, or not included in the above list?

Please limit each answer to one milestone!


Since there's now so many answers that it takes a very long time to scroll down to a specific one, I Have created links for the milestones that have already been explained in an answer:

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    $\begingroup$ Very interesting question! What has Feliciano said about it? ;) $\endgroup$ – Nike Dattani May 5 at 4:04
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    $\begingroup$ There's definitely been new work, such as "density-corrected DFT" by Eunji Sim and Kieron Burke, work on dispersion by Matthias Scheffler, work on excited states by Tom Ziegler, better potentials by Becke-Johnson, and Staroverov, double-hybrids by Stefan Grimme, however the timeline in the book might have stopped because it's too early to tell how much these things should be considered "milestones". The papers might be getting thousands of citations, but who knows what it will be like 20 years from now? 1996 does seem to be a long time ago though. $\endgroup$ – Nike Dattani May 5 at 4:12
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    $\begingroup$ In the book the time frame was limited to 1965 to 2000 since post 2000 the field is evolving fast and it early to place recent developments in an historical perspective. $\endgroup$ – Thomas May 5 at 4:16
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    $\begingroup$ But I think we don't have to limit ourselves ;) $\endgroup$ – Thomas May 5 at 4:19
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    $\begingroup$ Exactly. So that's why I like this question :) $\endgroup$ – Nike Dattani May 5 at 4:46

13 Answers 13

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2006 (Grimme): Double hybrid functionals

The timeline of milestones you have given, includes a hybrid functional called B3LYP, which mixes a Hartree-Fock exchange functional with a GGA exchange-functional. In 2006, Stefan Grimme introduced what later became known as "double hybrid functionals", which not only mix the Hartree-Fock exchange functional with a GGA exchange-correlation functional, but also a many-body perturbation theory correction:

\begin{equation} E_{\textrm{xc}}=\left(1-a_\textrm{x}\right)E_\textrm{x}^{\textrm{GGA}} + a_\textrm{x}E_\textrm{x}^{\textrm{HF}} + bE_\textrm{c}^{\textrm{GGA}} + cE_\textrm{c}^{\textrm{MBPT}}. \tag{1} \end{equation}

For the MBPT correction, Grimme used MP2 with the Kohn-Sham orbitals and single-excitations neglected. He tried various double hybrids, and the one that he finally recommended was obtained after setting b=1-c, and then using the Becke-88 functional for $E_\textrm{x}^{\textrm{HF}}$ and the LYP functional for $E_\textrm{c}^{\textrm{GGA}}$. He called this new functional B2PLYP.

By 2011 there existed several different double hybrid functionals made by various other groups, and Goerigk and Grimme created an enormous dataset by combining 30 test sets, containing a grand total of 841 relative energies involving 1218 total energies, and compared the performance of 47 functionals (2 LDA, 14 GGA, 3 meta-GGA, 23 Hybrid, 5 Double Hybrid) on this test suite. Double hybrids were by far the most accurate family of functionals, with an estimated average error of only 1.8 kcal/mol:

                            

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  • $\begingroup$ I think double hybrids were originally proposed by Truhlar and coworkers in Zhao, Y., Lynch, B., Truhlar, D. J. Phys. Chem. A 2004, 108, 4786.. $\endgroup$ – Susi Lehtola Jun 10 at 12:30
  • $\begingroup$ @SusiLehtola I looked at Eqs 1 and 2 of that paper, and perhaps the $F^{\textrm{cor}}$ term resembles the last term of my Eq 1, but not quite. Grimme is using the KS orbitals to do essentially do MP2, which makes the calculation a lot more expensive, but can bring the total error down by quite a lot. The figure in my answer includes 5 double hybrids, but none of them are the 2004 one you're mentioning. I wonder why that is? It seemed this paper tried to be extremely comprehensive in including every relevant functional possible. $\endgroup$ – Nike Dattani Jun 11 at 1:03
  • $\begingroup$ sure but AFAIK Truhlar's 2004 paper was the first to propose mixing in MP2 correlation, even though they appear to have used HF orbitals for that contribution. (Using KS orbitals for both parts should be cheaper, not more expensive!) $\endgroup$ – Susi Lehtola Jun 11 at 7:47
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2015 (Sun et al.): SCAN functional

The SCAN meta-GGA functional is an extension of the popular PBE GGA [1] and the TPSS [2] and revTPSS [3] meta-GGAs, SCAN adheres to all 17 known exact XC constraints and is constructed to be almost exact for the noble gasses and jellium surfaces. Early evidence suggests that SCAN is more accurate than and of comparable efficiency to the aforementioned GGAs for crystal structure prediction [4-5], ab initio thermodynamics [6-7], and computational catalysis [8].

References:

[1] J. P. Perdew, K. Burke, and M. Ernzerhof,Phys. Rev. Lett.77, 3865 (1996).
[2] J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria,Phys. Rev. Lett.91, 146401 (2003).
[3] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin,and J. Sun,Phys. Rev. Lett.103, 026403 (2009).
[4] H. Peng, Z.H. Yang, J.P. Perdew, and J. Sun, Phys. Rev. X 6, 041005 (2016).
[5] J.H. Yang, D.A. Kitchaev, and G. Ceder, Phys. Rev. B 100, 035132 (2019).
[6] E.B. Isaacs and C. Wolverton, Phys. Rev. Mater. 2, 063801 (2018).
[7] Y. Zhang, D.A. Kitchaev, J. Yang, T. Chen, S.T. Dacek, R.A. Sarmiento-Pérez, M.A.L. Marques, H. Peng, G. Ceder, J.P. Perdew, and J. Sun, Npj Comput. Mater. 4, 9 (2018).
[8] G. Sai Gautam and E.A. Carter, Phys. Rev. Mater. 2, 1 (2018).

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    $\begingroup$ Robert: Thanks for this first answer of yours! Welcome to the site and we hope to see much more of you here!! I have edited it to make it look a bit more compact. Are you able to give a simple formula for this functional, like in my answer? Also I'm not sure how important it is to include all these references. Refs 1-3 are just for the non-SCAN functionals which are not the focus of your answer (to keep things compact, I didn't cite Becke-88, or LYP, or even B3LYP, in my answer, even though they were mentioned). As for Refs 4-9, they are just examples of applications where SCAN was a success. $\endgroup$ – Nike Dattani Jun 11 at 1:26
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Dispersion corrected methods (2007/2010)

Lots of answers already, I would say the main ones are covered. However, in the spirit of the question, I don't think anyone has done dispersion corrections yet. So,

There are many levels of Dispersion corrected methods, but I would say the most common is from Grimme et al. in 2010 (Grimme et. al. 2010 paper.)

The energy correction is calculated as (taken from Frank Jensen's textbook) \begin{equation} \Delta E_{\rm disp} = -\sum_{n=6} s_n \sum_{\rm AB} \frac{C_n^{\rm AB}}{R_{\rm AB}^n}f_{\rm damp}(R_{\rm AB}) \end{equation} Further variations can also account for higher order $R^n$ dependance.

Becke & Johnson have also done work on more physics based corrections, more can be found at a different question, here.

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    $\begingroup$ +1. Great to see a new user here! We hope to see much more of you! Just to point out: This formula for the dispersion energy goes back to at least 1930 (see papers by Fritz London, and the reason why the C6/r6 dispersion potential is often called the "London interaction"). Damping functions date back to decades before Grimme (see work of Scoles, Tang/Toonies, and others). Also, Grimme's 2016 review paper shows that M06 (not Grimme's DFT-D) is the most popular DFT method accounting for dispersion (according to number of citations, but this can sometimes be misleading). $\endgroup$ – Nike Dattani Jun 8 at 23:43
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    $\begingroup$ You might find this interesting too: materials.stackexchange.com/q/63/5 $\endgroup$ – Nike Dattani Jun 8 at 23:45
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    $\begingroup$ Thanks. Hopefully I can be the second or third person to work on something, but get 17K citations for it :) $\endgroup$ – HipperThanHop Jun 8 at 23:52
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    $\begingroup$ I added a link to the other question, since they included the physics based approach of Becke $\endgroup$ – HipperThanHop Jun 8 at 23:54
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    $\begingroup$ Well, the Becke-Johnson methods refer to specific ways of calculating the C6 and f_damp in the equation that you presented. Grimme's paper calculates C6 and f_damp in a different way. The formula you presented was used since several decades ago (almost a century ago!), but the methods to calculate C6 and f_damp are what is different in the Becke-Johnson approach, vs the Grimme approach, vs the TS approach, etc. $\endgroup$ – Nike Dattani Jun 8 at 23:57
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2013: Density-Corrected DFT (DC-DFT)

The goal of Density-Corrected DFT (DC-DFT) is not only to get better accuracy but also to understand and correct the true error in the functional approximation.[1,2] In any approximate density functional, the DFT error is $\Delta E = \tilde E[\tilde n] - E[n]$ where $E$ and $n$ are exact functional and density while $\tilde {}$ represents approximate counterpart. Therefore, any density functional calculation has errors due to two causes: approximate functional and approximate density. The true functional error is

$$\Delta E_F = \tilde E[n] - E[n] = \Delta E_{XC}[n]$$

and the remaining error is

$$\Delta E_D = \Delta E - \Delta E_F = \tilde E[\tilde n] - \tilde E[n]$$

called the density-driven error. In most cases, $\Delta E_F$ dominates $\Delta E$ but, if DFT energy is density-sensitive, i.e., largely affected by density, $\Delta E_D$ is non-negligible and worth examining.

There is no guarantee that the HF density is closer to the exact density than DFT self-consistent densities, but HF-DFT[5,6] (DFT energy evaluated on HF density) is probably one of the simplest and most practical ways to perform DC-DFT. Moreover, HF-DFT has shown to provide better results than standard approximations for various density-sensitive cases that are not spin contaminated including electron affinities, potential energy curves, spin gaps for coordination compound, and noncovalent interactions.[3,4]

  1. M.-C. Kim, E. Sim, K. Burke, Phys. Rev. Lett., 111, 073003 (2013)
  2. A. Wasserman, J. Nafziger, K. Jiang, M.-C. Kim, E. Sim, K. Burke, Annu. Rev. Phys. Chem., 68, 555 (2017)
  3. Y. Kim, S. Song, E. Sim, K. Burke, J. Phys. Chem. Lett., 10, 295 (2019)
  4. S. Vuckovic, S. Song, J. Kozlowski, E. Sim, K. Burke, J. Chem. Theo. Comp., 15, 6636 (2019)
  5. P. Verma, A. Perera, R. J. Bartlett CPL, 524, 10-15, 2012
  6. P.M.W. Gill, B.G. Johnson, J.A. Pople, M.J. Frisch, Int. J. Quant. Chem., 44,319 (1992)
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    $\begingroup$ +10. Welcome to the site Eunji! 감사합니다. $\endgroup$ – Nike Dattani Jun 11 at 1:45
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2004 (Yanai et al.): Range separation

Often, the source of DFT improvement comes from Hartree-Fock as is also obvious from the answer involving double hybrid functionals. So too it is with range-separation. The electron-electron Coulomb operator for the exchange contribution is separated into a short and long range contribution.

\begin{equation} \frac{1}{r} = \frac{1-\text{erf}(\omega r)}{r} + \frac{\text{erf}(\omega r)}{r} \end{equation}

where $\text{erf}$ is the standard error function. The $\omega$ parameter determines when to switch from using the short range part to the long range part. In this strategy the short range contribution is calculated using a density functional, and the long range contribution is calculated using HF. HF is good at this. What HF is not good at is correlation, but, there is a different functional for that. Right now, we are improving the electron-electron calculation.

The long range contribution from HF helps ensure that DFT overlocalization of charge separation is removed - notably yielding much improved excitation energies for charge-transfer states. Overall: Range-separation helps solve self-interaction errors and improves excitation energies among other benefits.

[1] T. Yanai, D.P. Tew and N.C. Handy, Chemical Physics Letters, 393, (1-3), 51-57 (2004)

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    $\begingroup$ I appreciate the homage to $\omega$B97-XD, but I wonder if you would mind shortening it perhaps to just one or the other and keeping it to somewhere between 1-3 paragraphs like the dispersion corrections section? It would be nice if we could have about 20 compact, short summaries of recent milestones in DFT. It's also what both the asker and the bounty granter have asked for. Currently we are also having a discussion about whether or not to allow double-answers like this on our site. I fully appreciate all the time you took to write this full answer: Perhaps you give a link to longer version? $\endgroup$ – Nike Dattani Jun 4 at 3:42
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    $\begingroup$ sure, but OP should use singular language too :) $\endgroup$ – Charlie Crown Jun 4 at 3:49
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    $\begingroup$ It is very appreciated Charlie. As for the question asking for "Milestones" instead of a single "Milestone", I did not catch that, because I figured that after 20 people each give 1 milestone, there will be several milestones listed. However I will edit the question to make it even more clear and unambiguous! $\endgroup$ – Nike Dattani Jun 4 at 3:55
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    $\begingroup$ The Heyd-Scuseria-Ernzerhof functional, which you link, is an inverse separation. It used HF for the short range where as my post is about using it as the long range. HSE is used for solid state calculations $\endgroup$ – Charlie Crown Jun 4 at 4:47
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    $\begingroup$ Thanks for noting that difference. As for the origin of the idea to use HF for the long-range: the CAM-B3LYP paper you cited says that they got the idea from Tsuneda, but the Tsuneda paper was cited as "in press". Maybe it took from 2004-2014 for it to get pressed! $\endgroup$ – Nike Dattani Jun 4 at 4:55
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1995 (Casida): TD-DFRT

Time-Dependent Density Functional Response Theory is a linear response formulation of TDDFT for the calculation of excitation energies and corresponding transition amplitudes, that in turn allows to evaluate electronic spectra of molecular and condensed matter systems.

The time-dependent density functional theory (TDDFT) in the Kohn–Sham formalism is a set of differential equations for the time evolution of orbitals under the influence of an external field. However, in perturbation theory, TDDFT can be recast as an algebraic equation that describes the response in the frequency domain. Furthermore, instead of directly solving the response to an external field of particular frequency, one can calculate the resonant frequencies that correspond to the system's excitation energies (and to the poles of response functions).

This is achieved by the Casida equation, whose most general form is $$ \pmatrix{A & B \\ -B^* & -A^* } \pmatrix{X_N \\ Y_N} = \omega_N \pmatrix{X_N \\ Y_N} , $$ where terms $A$, $B$ are the Hessians (also called coupling matrices), $\omega_N$ is the $N$-th excitation energy, and vectors $X_N$, $Y_N$ contain the corresponding transition amplitudes. The equation can be further simplified by considering the Tamm–Dancoff approximation (neglecting $B$) that can be added as a part of this direction of DFT development (Hirata & Head-Gordon 1999).

For real orbitals and frequency-independent DFT kernel (part of the Hessian), the equation can alternatively be simplified by defining $$ CZ_N = \omega_N^2 Z_N , $$ where $C=(A-B)^{1/2}(A+B)(A-B)^{1/2}$ and $Z_N=(A-B)^{1/2}(X_N-Y_N)$. The Casida equation has the form of an eigenvalue equation with excitation energies being the eigenvalues.

The equation was introduced by Casida in 1995 and is now part of all major DFT codes and the primary DFT way of calculating excitation energies and excited state properties of wide range of systems.

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1993 (Becke): Hybrid Functionals

Axel D. Becke introduced the adiabatic-connection model, which allows for mixing of DFT exchange and Fock-like exchange via the formula $$ E_{\text{x}} = a \cdot E^{\text{HF}}_x + b \cdot E^{\text{GGA}}_x $$ to obtain the exchange part of the exchange-correlation energy. Typically, one imposes $a+b = 1$, but some authors have sometimes abandoned the summation to $1$ or introduced exchange from the local density approximation (LDA a.k.a. Slater-Exchange) to the mix (B3LYP falls into this category).

Hybrids show improved performance over GGAs and meta-GGAs for the HOMO-LUMO gap, thermochemistry and excited states via time-dependent DFT. Especially for the latter, mixing parameters that are dependent on the electron-electron distance have yielded good results, leading to "long-range corrected" or "range-separated" hybrid functionals such as LC-PBE.


Technically, this is not more recent than 1996. Given the impact of hybrid functionals, I think it is proper to list this milestone regardless.

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  • $\begingroup$ Becke's 1993 paper is already listed in the question. $\endgroup$ – Nike Dattani Jun 5 at 15:24
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2005 (Bartlett): ab initio DFT

Basically one takes the xc functional from a wave function approach such as MBPT(2), CC, etc. and constructs an xc potential from them using density conditions or a functional derivatives approach. Summary of developments is best captured in the following article: "Adventures in DFT by a wave function theorist."

Details on how to construct a local exchange potential from the HF exchange energy can found in the following article: "Exact exchange treatment for molecules in finite-basis-set Kohn-Sham theory," while details about constructing the correlation potential from MBPT(2) energy can be found in the article title "Ab initio density functional theory: The best of both worlds?"

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I would add some developments in TDDFT that came around 1996 and resonated only later such as:

  • the Casida equation (Casida 1995) that allows to calculate excitation energies and electronic spectra

  • the real-time TDDFT (Yabana & Bertsch 1996) a non-perturbative TDDFT technique in which the time-dependent Kohn–Sham equation is solved by direct propagation in time

  • the van Leeuwen theorem (van Leeuwen 1999) a generalization of the Runge–Gross theorem that also underpins the Kohn–Sham construction in TDDFT

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    $\begingroup$ Hi @LukasK, this is very useful information indeed! And welcome to the site! We hope to see much more of you here. Would you be able to pick one of them and explain it in the format that (most of) the other answers are in? $\endgroup$ – Nike Dattani Jun 11 at 20:18
  • $\begingroup$ Hi, thanks. Yes, I can do it, and not just for one of them. $\endgroup$ – LukasK Jun 12 at 8:26
  • $\begingroup$ Preferrably you can pick one and focus on it. That's what I would prefer anyway. That's also what I did (I could have written several answers here but chose to focus on one). $\endgroup$ – Nike Dattani Jun 12 at 10:33
  • $\begingroup$ OK, so the Casida equation it is (most citations, most widespread, before 1996 but not on the original list). $\endgroup$ – LukasK Jun 12 at 12:41
  • $\begingroup$ I think it's ok to add something from before 1996, if it's not already on the list. $\endgroup$ – Nike Dattani Jun 12 at 15:05
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Most chemists' point of view could be condensed as follows:

Implementation of DFT in Gaussian (Pople et al, 1992) LDAs and GGAs were implemented in Gaussian 92/DFT by Pople, Gill and Johnson [Chem Phys Lett 199, 557 (1992)].

DFT better than ab initio (Johnson et al, 1993) BLYP was found to yield more accurate equilibrium geometries, dipole moments, harmonic vibrational frequencies and atomization energies than ab initio methods, using the 6-31G* basis set [J Chem Phys 98, 5612 (1993)].

B3LYP (Stephens et al, 1994) The B3LYP functional was proposed based on Becke's earlier suggestion, where the correlation functional was just changed from PW91 to LYP.

This culminated in the infamous B3LYP/6-31G* model chemistry...

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    $\begingroup$ +1. Especially for the reference to the under-appreciated paper by Stephens. As for DFT being better than ab initio methods, maybe it would be help to list the specific methods that DFT was tested to be better than, because I'm sure it's not better than ab initio methods such as FCI and CCSDTQ! $\endgroup$ – Nike Dattani Jun 11 at 1:38
  • $\begingroup$ @NikeDattani even FCI is not that good in 6-31G* which was the basis they used.... $\endgroup$ – Susi Lehtola Jun 11 at 7:48
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  • DFT+NEGF (2001):

  • Combining density functional theory (NEGF) and nonequilibrium Green's function method (NEGF), a self-consistent first-principles technique for modeling quantum transport properties of atomic and molecular scale nanoelectronic devices under external bias potentials is reported.

  • Implementation packages: QuantumATK, Nanodcal and Questaal ...

  • Reference: Phys.Rev.B 63, 245407

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    $\begingroup$ Can you please follow the format of the other answers? $\endgroup$ – Nike Dattani Sep 18 at 14:48
  • $\begingroup$ This a good start to a couple different answers, but ideally we would prefer each answer to address one topic. $\endgroup$ – Tyberius Sep 18 at 14:49
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    $\begingroup$ I would recommend keeping this answer and removing the other two. We highly prefer that a user doesn't answer several times on the same question, and instead of spending time on 3 topics, spends 3x the amount of time on 1 topic. Since the other two answers are just copying and pasting abstracts from other people's papers, I would recommend to keep this one :) $\endgroup$ – Nike Dattani Sep 18 at 15:35
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    $\begingroup$ @Jack You say that you doubt that it's reasonable for us to have each answer addressing just one topic: You are free to express that opinion here: mattermodeling.meta.stackexchange.com/a/126/5 (there have been some arguments for following the one-topic-per-answer format (for example if 3 topics were in one answer, are we upvoting because we like OFDFT, or because we like DFT+NEGF or because we like the answer about band-gaps? What if someone likes OFDFT but does NOT like the other two, how do they split the vote)? We also don't 3 different comment chains on one answer. $\endgroup$ – Nike Dattani Sep 19 at 0:06
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    $\begingroup$ I gave you +1 on this answer now, but I see that other people gave you -1 on all three, and a moderator deleted your other two answers. I did not tell anyone to do either of those things (I was not even at my computer for the last 7 hours), it was just that others did not like what you did. Also the mod that deleted your answers didn't downvote you (Tyberius has 1 total downvote). In general if you're just going to be copying and pasting an abstract, it's probably better to just write a link to the paper in a comment. You can't earn rep for posting an abstract of someone else's paper :) $\endgroup$ – Nike Dattani Sep 19 at 0:10
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2014 (Gagliardi): MPCDFT

Multiconfiguration Pair-Density Functional Theory (MC-PDFT) is a theoretical framework that combines multiconfigurational wave functions with a generalization of density functional theory. As the reference wavefunction is multiconfigurational rather than being a single Slater determinant, it has the advantage that it can describe strongly correlated systems, bond dissociations, and electronic excitations. This requires a new type of density functionals (functionals of the total density, its gradient, and the on-top pair density) that can be obtained by translating conventional density functionals of the spin densities. As the on-top pair density is an element of the two-particle density matrix, this goes beyond the Hohenberg−Kohn theorem that refers only to the one-particle density.

References:

  1. G. Li Manni, R. K. Carlson, S. Luo, D. Ma, J. Olsen, D. G. Truhlar, L. Gagliardi, Multiconfiguration pair-density functional theory. J. Chem. Theory Comput. 10, 3669–3680 (2014).

  2. L. Gagliardi, D. G. Truhlar, G. L. Manni, R. K. Carlson, C. E. Hoyer, J. L. Bao, Multiconfiguration pair-density functional theory: A new way to treat strongly correlated systems. Acc. Chem. Res. 50, 66–73 (2017).

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  • $\begingroup$ +1. Welcome to our community, and thank you for contributing your excellent answer here! We hope to see much more of you !!! I had actually asked Laura Gagliardi and Giovanni Li Manni to write something about MPCDFT here, but they were both very busy during our COVID-filled summer! $\endgroup$ – Nike Dattani Sep 20 at 18:46
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  • Solving the band-gap problem at DFT level (2008):

  • To obtain the correct band-gap in semiconductor physics and materials science is very important for device applications, such as charge transport and optical absorption. It is well known that DFT with PBE exchange-correlation functional will underestimate the band-gap of semiconducting materials. Currently, some methods based on DFT have been proposed, such as hybrid functional and GW approximation. Unfortunately, these schemes are computationally prohibitive for large systems, especially when the spin-orbit coupling becomes important. How to solve the band-gap problem at the DFT level cost is the core problem of this field. This following cited paper tries to explain the failure of conventional exchange-correlation functional from the viewpoint of the fractional charge, which can be considered as one of the most important developments in recent years.

  • Ref: Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction.

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