Variational Pair-Density Functional Theory: Dealing with Strong Correlation at the Protein Scale

Mikael Scott, Gabriel L. S. Rodrigues, Xin Li, and Mickael G. Delcey (2024)
Highlighted by Jan Jensen

As I've said before, one of the big problems in quantum chemistry is that we still can't routinely predict the reactivity of TM-containing compounds with the same degree of accuracy as we can for organic molecules. This paper might offer a solution by combining CASSCF with DFT in a variational way.

While such a combination has been done before, that implementation basically compute the DFT energy based on the CASSCF density. If you haven't heard of this approach, it's probably because it didn't work very well. 

This paper presents a variational implementation, where you minimise the energy if a CASSCF wavefunction subject to an exchange-correlation density functional, an the results are significantly better - in some cases approaching chemical accuracy! This is pretty impressive given that they used off-the-shelf GGA functionals (BLYP and PBE) so further improvements in accuracy with bespoke functionals is quite likely.

Oh, and one of the applications presented in the paper is multiconfigurational calculation on an entire metallo-protein!

This work is licensed under a Creative Commons Attribution 4.0 International License.

Pushing the frontiers of density functionals by solving the fractional electron problem

James Kirkpatrick, Brendan McMorrow, David H. P. Turban, Alexander L. Gaunt, James S. Spencer, Alexander G. D. G. Matthews, Annette Obika, Louis Thiry, Meire Fortunato, David Pfau, Lara Román Castellanos, Stig Petersen, Alexander W. R. Nelson, Pushmeet Kohli, Paula Mori-Sánchez, Demis Hassabis, Aron J. Cohen (2021) (OA version)
Highlighted by Jan Jensen

Part of Figure 1 from the paper. (c) 2021 The authors

This paper presents a new ML-exchange-correlation potential that gives improved results compared to state-of-the-art functionals, especially for barriers. Most importantly, it demonstrates the importance of including fractional charge and spin in the training set when developing new functionals. Fractional charge-systems help reduce the self-interaction error while fractional spin-systems supplies information about static correlation. For example, the current functional gives reasonable bond dissociation curves and future functionals of this kind may work considerably better on transition metal-containing systems with significant multi-reference character.   

This work is licensed under a Creative Commons Attribution 4.0 International License.

Empirical D3 dispersion as a replacement for ab initio dispersion terms in density functional theory-based symmetry-adapted perturbation theory

Robert Sedlak and Jan Řezáč (2017)

Highlighted by Amelia Fitzsimmons

Sedlak and Rezac presented an approximation to DFT-SAPT that replaces the ab initio dispersion terms in the popular but expensive SAPT calculation with a potential that is based on Grimme’s D3 dispersion. The D3 dispersion correction has become a popular way to improve the accuracy of DFT geometry optimizations and thermochemistry calculations for systems involving noncovalent interactions, and as applied to DFT-SAPT improves the efficiency of DFT-SAPT calculations. They demonstrated with the S66X8 and S66X6 test sets that this correction has root mean square errors of less than 1 kcal/mol for non-charge transfer species. I think this could be useful to anyone who is looking for a more efficient way to do energy decomposition analysis who has previously used DFT-SAPT, or anyone who is interested in noncovalent interactions and dispersion corrections to DFT. 

Reproducibility in density functional theory calculations of solids

Kurt Lejaeghere, Gustav Bihlmayer, Torbjörn Björkman, Peter Blaha, Stefan Blügel,Volker Blum, Damien Caliste, Ivano E. Castelli, Stewart J. Clark, Andrea Dal Corso,Stefano de Gironcoli, Thierry Deutsch, John Kay Dewhurst, Igor Di Marco, Claudia Draxl,Marcin Dułak, Olle Eriksson, José A. Flores-Livas, Kevin F. Garrity, Luigi Genovese,Paolo Giannozzi, Matteo Giantomassi, Stefan Goedecker, Xavier Gonze, Oscar Grånäs,E. K. U. Gross, Andris Gulans, François Gygi, D. R. Hamann, Phil J. Hasnip,N. A. W. Holzwarth, Diana Ius¸an, Dominik B. Jochym, François Jollet, Daniel Jones,Georg Kresse, Klaus Koepernik, Emine Küçükbenli, Yaroslav O. Kvashnin,Inka L. M. Locht, Sven Lubeck, Martijn Marsman, Nicola Marzari, Ulrike Nitzsche,Lars Nordström, Taisuke Ozaki, Lorenzo Paulatto, Chris J. Pickard, Ward Poelmans,Matt I. J. Probert, Keith Refson, Manuel Richter, Gian-Marco Rignanese, Santanu Saha,Matthias Scheffler, Martin Schlipf, Karlheinz Schwarz, Sangeeta Sharma,Francesca Tavazza, Patrik Thunström, Alexandre Tkatchenko, Marc Torrent,David Vanderbilt, Michiel J. van Setten, Veronique Van Speybroeck, John M. Wills,Jonathan R. Yates, Guo-Xu Zhang, Stefaan Cottenier Science 2016, 351, aad3000

Contributed by David Bowler
Reposted from Atomistic Computer Simulations with permission

A paper in Science (or equivalent journal) generally reports novel or ground-breaking research. At first sight, the paper I’ll discuss in this post[1] does not fit into that category: it reports an extensive set of tests on calculations for the equation of state (EOS) for 71 elemental solids using a variety of DFT codes, all using the PBE functional.

This paper is the product of a collaboration (you can find all the data, test suites etc on their web site[2]) that has been going for a while, and is both important and impressive. They have defined a single parameter, delta, which allows them to compare EOS calculated with different codes, giving a simple route to evaluating the reproducibility of DFT. This is immensely valuable, because different codes use different basis sets, different numerical solvers and different approaches to the external potential (full potential or a variety of pseudopotentials), and as a result will give different answers for the same simulation. The question is: how different are the answers ?

The key result from this paper is that modern DFT codes now achieve a precision[3] which is better than experimental; in terms of the paper, this means 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 summary table from the paper is reproduced below; the numbers given are the RMS value for delta across all 71 elements, while the colour indicates overall reliability.

Why is this work significant ? First, it gives a way to test new DFT codes and implementations, basis sets and approaches to the potential. So we now have an absolute reference against which codes can be compared. Second, it shows that there are now freely-available pseudopotential libraries which are precise in comparison to all-electron results (this is something that wasn’t true even five years ago - their Table 2, which shows the changing precision of different libraries over time, is fascinating). For both users and code developers, this is great news: there is no longer any question as to whether a particular pseudopotential is reliable, certainly within the context of single elements.

What could be added to the study ? Here are some ideas:

• More extensive tests. There are no tests of elements in different environments - and this can pose extreme challenges to pseudopotentials (think of the different oxidation states of transition metals, for instance).
• A comparison between the codes (e.g. speed, memory or parallelisation).
  This would be very challenging, but would be interesting data.
• More functionals and extensions of DFT will be important to include.

This paper is an immensely valuable contribution to the electronic structure community, as well as the wider scientific community, and it is good to see it published in a high-profile journal.

[1] DOI:10.1126/science.aad3000

[2] Delta value website from centre for molecular modelling

[3] Precision indicates the spread between different measured values, while accuracy indicates the deviation from the correct result (however “correct” is defined !) 

Small Atomic Orbital Basis Set First-Principles Quantum Chemical Methods for Large Molecular and Periodic Systems: A Critical Analysis of Error Sources

Sure, R.; Brandenburg, J. G.; Grimme, S. ChemistryOpen, EarlyView, DOI: 10.1002/open.201500192 (CC by-nc-nd)
Contributed by Grant Hill

The use of density functional theory (DFT) calculations to produce insights into the chemistry unveiled by experiment is widespread due to its relative ease-of-use and ability to explain many chemical phenomena of interest. In particular, the B3LYP hybrid functional [1] and the Pople-type 6-31G* basis set [2] are incredibly popular, with some referring to this as Default Favourite Theory (a play on DFT).[3] A recent review by Grimme and co-workers sets out a case against using this B3LYP/6-31G* model chemistry by careful examination of errors.

The review contends that the use of small basis sets such as 6-31G* leads to relatively large errors due to both basis set superposition error (BSSE) and basis set incompleteness error (BSIE). It's not entirely clear how to separate these two terms and as a result the review mostly focuses on the BSSE element, with some emphasis on intramolecular BSSE in addition to the more familiar intermolecular BSSE. The question of why B3LYP/6-31G* still performs well in a number of cases is then examined in terms of a fortuitous cancellation of errors between BSSE and London forces (dispersion energy). It is demonstrated that this cancellation cannot be relied upon in all cases and a convincing case is made for choosing different basis sets and methods. While the review mostly focuses on intermolecular interactions, there is some generalisation to other problems of interest.

A number of alternative methods for including dispersion in DFT calculations are reviewed, and final recommendations are made that can easily be incorporated into the workflow of a non-specialist, without a significant increase in computational cost. This includes the use of the def2-SVP basis set of Weigend, Ahlrichs and co-workers.[4] This review should make for interesting reading for anyone routinely using DFT methods in conjunction with double-zeta basis sets.

References:
[1]a) Stephens, P. J.; Devlin, F. J.; Chablowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. b) Becke, A. D. J. Chem. Phys. 1993, 98, 5648.
[2] Hehre, W. J.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257.
[3] I first heard this at the Computational Molecular Science conference in 2008, but I can't recall the originator. The late Nick Handy responded by suggesting that DFT could instead be "Damn Fine Theory".
[4] See Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297 and references therein. These basis sets are available to download from the EMSL basis set exchange in formats suitable for most electronic structure packages.

The accuracy of semi-local functionals

Contributed by David Bowler
Reposted from Atomistic Computer Simulations with permission

The great strength of density functional theory (DFT) in its purest form is that the energy only depends on the charge density, whether its magnitude alone (LDA), or with its gradient (GGA) or beyond. This simplicity is, of course, also the source of some of its largest errors: poor band gap predictions, excessive delocalisation of charge density, poor description of charge transfer, and problems with band widths. Many of these issues can be solved by the addition of a judicious amount of exact exchange, giving hybrid functionals. But exchange is very expensive to calculate, and requires the use of orbitals in the place of a charge density. And there is the associated question of what fraction of exchange to mix (there are many hybrid functionals available, each with a different fraction of exchange) and even whether the exchange interactions should be screened (and if so, by how much…). An interesting recent review and extended opinion piece from one of the key practitioners in the field[1] gives more details on some of these problems, and is well worth a read.

There have been recent attempts to construct semi-local (GGA) functionals which have at least some of the features of exact exchange, in the hope of improving the accuracy. The first of these[2] constructed the potential as a functional of the charge density which matched the OEP (optimised effective potential - an approach to solving the exact exchange problem) very well for atoms. It has been modified[3] to create a potential that can be used in solids, and which gives good band gaps (known as the mBJ or TB-mBJ potential). However, this is only a potential, and because of its form, it is not possible to create a corresponding energy functional. This limits the applicability to post-hoc corrections of the band structure. (The potential can also diverge where there are large areas of vacuum, such as surfaces.)

A similar attempt has been made to create consistent potential and energy functionals which also reproduce key features of the exact exchange[4,5]. One of these, the derivative discontinuity (DD), describes a change in the gradient of the energy with respect to electron number at integer fillings; this behaviour is missing from most semi-local functionals. The AK13 functional[4] restores this by considering the asymptotic behaviour of the functional, giving reasonable accuracy for band gaps, without the need for fitting (the mBJ functional has parameters that are fit to improve the accuracy).

A very recent paper, which presents a comparison of results and potentials from these functionals[6], shows, as ever, that there is no single functional that gives good results in all situations (though in principle the exact density functional should do this). As DFT practitioners we must choose whether we want to use highly fitted functionals which give excellent accuracy within certain limits, or to use functionals which respect certain important limits and behaviours for the electron gas. There is no universal function, just as there is no universal basis set, and part of the practice of DFT simulations is to quantify carefully the limitations and effects of the approaches we choose.

[[1] Rev. Mod. Phys. 87, 897 (2015) DOI:10.1103/RevModPhys.87.897

[[2] J. Chem. Phys. 124, 221101 (2006) DOI:10.1063/1.2213970

[[3] Phys. Rev. Lett. 102, 226401 (2009) DOI:10.1103/PhysRevLett.102.226401

[[4] Phys. Rev. Lett. 111, 036402 (2013) DOI:10.1103/PhysRevLett.111.036502

[[5] Phys. Rev. B 91, 035107 (2015) DOI:10.1103/PhysRevB.91.035107

[[6] Phys. Rev. B 91, 165121 (2015) DOI:10.1103/PhysRevB.91.165121

An isomeric reaction benchmark set to test if the performance of state-of-the-art density functionals can be regarded as independent of the external potential

T. Schwabe, Phys. Chem. Chem. Phys., 2014,16, 14559.

Contributed by Martin Korth.

It's an open question: Can more parameters efficiently replace knowledge about the underlying physical behaviour? Some examples:

Classical potentials usually have a fixed functional form, which is chosen to capture the basics of (inter)molecular interactions. Neural networks[1], Gaussian approximation potentials[2] and other machine-learning approaches do not rely on fixed potentials, but it's unclear whether systems with complex chemistry (for instance involving more than a handful of different elements) can be efficiently treated this way in general.

One step up on the theory ladder, semi-empirical quantum mechanical (SQM) methods relying more heavily on parametrization (like PM6, PM7) seem to perform overall as good as SQM methods trying to capture more of the correct physics by introducing orthogonalization corrections (OMx), but if one looks at complicated chemistry, the latter ones turn out to be more robust.[3]

Another step up, the highlighted paper by Schwabe comes into play: Also in the field of density functional theory (DFT) exchange-correlation (XC) functional development, people have tried both pathways; more physics (like including dispersion in a DFT-D type fashion[4]) and more parameters (how some Minnesota type functionals by Truhlar and co-workers capture -- at least mid-range -- dispersion[5]).

Schwabe now assesses a number of functionals for isomerization reactions in which heteroatoms are systematically replaced with heavier atoms of the same group. With this setup he has found a clever way to find out whether the performance of a method depends on the elements involved and thus the external potential - which should not be the case at least for the 'true' functional. Schwabe indeed finds no such dependence except for one case, M11-L, a functional on the more-parameter side of things, thereby suggesting that like for SQM methods, the more-physics pathway might be the safer road to follow.

It would be interesting to see how SQM methods (including SCC-DFTB) perform on Schwabe's set, though in this case not a dependence on the external potential but element-specific parametrization effects would be investigated (in the past we did not find element-specific trends for PM6[3] and also no such trends for WFT/DFT methods -- including some Minnesota functionals -- when we benchmarked 'mindless'[6]).

References:

[1] J. Behler,  Representing Potential-Energy Surfaces by High-Dimensional Neural Network Potentials, J. Phys.: Condens. Matter 26 (2014) 183001.

[2] A. P. Bartok, M. C. Payne, R. Kondor, G. Csanyi, Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons, Physical Review Letters 104 136403 (2010)

[3] M. Korth, W. Thiel, Benchmarking Semiempirical Methods for Thermochemistry, Kinetics and Noncovalent Interactions: OMx Methods are Almost as Accurate and Robust as DFT-GGA Methods for Organic Molecules. J. Chem. Theory Comput., 2011, 7, 2929.

[4] S. Grimme, Density functional theory with London dispersion corrections, WIREs: Comp. Mol. Sci. 2011, 1, 211.

[5] R. Peverati, D. G. Truhlar, The Quest for a Universal Density Functional: The Accuracy of Density Functionals Across a Broad Spectrum of Databases in Chemistry and Physics, Philosophical Transactions of the Royal Society A 372, 20120476/1-51 (2014).

[6] M. Korth, S. Grimme, 'Mindless' DFT Benchmarking, J. Chem. Theory Comput., 2009, 5, 993.