What are some recent developments in density functional theory?

Question

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!

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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:

• 1993 (Becke) Hybrid Functionals
• 1995 (Casida) TD-DFRT
• 2004 (Yanai et al.) Range separation
• 2005 (Bartlett) ab initio DFT
• 2006 (Grimme) Double Hybrids
• 2007 (Becke-Johnson) Dispersion correction
• 2013 (Kim et al.) Density-corrected DFT
• 2015 (Sun et al.) SCAN functional

Answer 1

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:

                            

Answer 2

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).

Answer 3

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)

Answer 4

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.

Answer 5

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)

Answer 6

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.

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

Answer 7

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.

Answer 8

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?"

Answer 9

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

Answer 10

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...

Answer 11

1997 (Marzari & Vanderbilt): MLWF

• These methods enable a more qualitative view of the electron density by projecting the Bloch wavefunctions into localized Wannier functions [1], which is especially useful when it comes to transition metal systems, but not limited to these.
• This description enables DFT practitioners to "talk" to the modelling community, since now we can approximate the physics and chemistry of the system in terms of orbitals.
• Two major milestones are the description of the maximally localized Wannier functions (MLWF) method by Marzari and Vanderbilt [2] and its community open-source implementation as Wannier90 [3].
• There are still ongoing work to perfect the method (such as alternative projection criteria), and to apply it to study other (more exotic) material systems. For instance, recently it has been used to describe the effective physics of the spin liquid candidate $\alpha$-$\ce{RuCl_3}$ [4].

References:

[1] G.H. Wannier, Phys. Rev. 52, 191 (1937), doi:10.1103/PhysRev.52.191
[2] N. Marzari & D. Vanderbilt, Phys. Rev. B 56, 12847 (1997), doi:10.1103/PhysRevB.56.12847; N. Marzari et al., Rev. Mod. Phys. 84, 1419 (2002), doi:10.1103/RevModPhys.84.1419
[3] A.A. Mostofi et al., Comput. Phys. Commun. 178, 685 (2008), doi:10.1016/j.cpc.2007.11.016; A.A. Mostofi et al., Comput. Phys. Commun. 185, 2309 (2014), doi:10.1016/j.cpc.2014.05.003; G. Pizzi et al., J. Phys. Cond. Mat. 32(16), 165902 (2020), doi:10.1088/1361-648X/ab51ff
[4] C. Eichstaedt et al., Phys. Rev. B 100, 075110 (2019), doi:10.1103/PhysRevB.100.075110

Answer 12

2020 Furness et al: r$^2$SCAN functional

The SCAN functional is the most recent meta-GGA functional constructed from first principles, which satisfies all known bounds. However, SCAN is also numerically pathological: getting converged energies requires huge quadrature grids, and constructing pseudopotentials is difficult. This motivated the construction of the regularized SCAN (rSCAN) functional of Bartók and Yates in J. Chem. Phys. 150, 161101 (2019), which eliminated the numerical instabilities in SCAN.

The r$^2$SCAN functional, published in J. Phys. Chem. Lett. 11, 8208 (2020) builds upon rSCAN, but restores some of the exact constraints of SCAN. In addition to being numerically stable and thereby faster than SCAN, r$^2$SCAN has been found to yield spectacular performance, see e.g. recent work by Grimme and coworkers

Answer 13

2001 (Taylor et al.): DFT+NEGF

• Combining density functional theory (NEGF) with 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

This paper has been chosen as one of Physical Review B 50th Anniversary Milestones:

The following is the comments:

Over the last few decades, there has been an explosion in the realm of nanotechnology, nanodevices, and nanomaterials, where anything ‘nano’ has become part of the condensed matter and materials physics lexicon. As has often been restated, this is due to the great potential for technological applications. However, this potential produced the need for a better understanding of the fundamental physics at the atomic scale not just for molecular modeling, but for device and application purposes as well. In the early 2000s, the use of density functional theory (DFT) and ab initio modeling continued to revolutionize the way we understand materials. Two outstanding PRB papers reported on the incursion of DFT into the arena of quantum electron transport properties by means of nonequilibrium Green’s functions. They helped pave the way for the advancement of device modeling at the atomistic level.

Answer 14

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).

Answer 15

2016: Reproducibility of DFT calculations (Lejaeghere et al)

Lejaeghere et al.$^1$ compared the calculated values for the equation of states for 71 elemental crystals from 15 different widely used DFT codes employing 40 different potentials. They defined a single parameter, Δ, which allowed the comparison of EOS calculated with different codes, giving a simple route to evaluating the reproducibility of DFT. Δ was defined as the root-mean-square energy difference between the equations of state of the two codes, averaged over all crystals in a purely elemental benchmark set.

The key result from this paper is that modern DFT codes now achieve a precision which is comparable to high-precision experiments; a delta value which is better than 1 meV/atom. This precision applies across various basis sets: plane waves, augmented plane waves, and numerical orbitals. It also applies to all-electron, PAW, and both ultra-soft and norm-conserving pseudopotential calculations. The work demonstrates that the precision of DFT implementations can be determined and also shows that the pseudopotential approach using recent libraries are precise in comparison with all-electron results.

The summary table from the paper is shown below; the numbers given are the RMS value for Δ across all 71 elements and colour indicates overall reliability.

References:

1. Lejaeghere, Kurt, et al. "Reproducibility in density functional theory calculations of solids." Science 351.6280 (2016).
2. https://molmod.ugent.be/deltacodesdft
3. https://davidbowler.github.io/AtomisticSimulations/blog/dft-reliability

Answer 16

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.