## Abstract

The past decades have seen density functional theory (DFT) evolve from a rising star in computational quantum chemistry to one of its major players. This Theme Issue, which comes half a century after the publication of the Hohenberg–Kohn theorems that laid the foundations of modern DFT, reviews progress and challenges in present-day DFT research. Rather than trying to be comprehensive, this Theme Issue attempts to give a flavour of selected aspects of DFT.

## 1. Introduction

Density functional theory (DFT) is a quantum-mechanical (QM) method used in chemistry and physics to calculate the electronic structure of atoms, molecules and solids. It has been very popular in computational solid-state physics since the 1970s. However, it was not until the 1990s that improvements to the method made it acceptably accurate for quantum-chemical applications, resulting in a surge of applications. The real forte of DFT is its favourable price/performance ratio compared with electron-correlated wave function-based methods such as Møller–Plesset perturbation theory or coupled cluster. Thus, larger (and often more relevant) molecular systems can be studied with sufficient accuracy, thereby expanding the predictive power inherent in electronic structure theory. As a result, DFT is now by far the most widely used electronic structure method. The huge importance of DFT in physics and chemistry is evidenced by the 1998 award of the Nobel Prize to Walter Kohn ‘for his development of the density-functional theory’ [1].

Nevertheless, even though DFT is an exact theory in principle, its approximate variants currently used are far from being fail-safe. Validation of these approximations is an important part of ongoing research in the field. New pitfalls are being discovered constantly, and there are still problems in using DFT for certain systems or interactions. One such fundamental problem that has become increasingly apparent as the systems that could be treated have become larger is the description of dispersion. A seemingly weak interaction *per se*, dispersion is omnipresent and can add up to a substantial force in large assemblies of atoms and molecules. It thus becomes very important in systems ranging from biomolecules to the areas of supramolecular chemistry and nanomaterials. However, in particular over the last decade, several new DFT approaches have been developed to overcome these problems. These range from highly parametrized density functionals to the addition of explicit, empirical dispersion terms. Research into this area (including the development of new functionals as well as assessment studies of these) continues to be very active.

The speed of DFT can also be exploited to perform very many energy and gradient calculations for a system to study its time evolution. DFT-based molecular dynamics (MD) methods, such as Car–Parrinello MD (CPMD [2]) and Born–Oppenheimer MD (sometimes collectively called *ab initio* MD, AIMD), have enjoyed a rapidly increasing popularity in the past decade, and are now ‘routinely’ applied in many areas of chemistry, physics, material and biomolecular sciences. Fully classical MD simulations based on inexpensive empirical force fields have long been a stronghold of biomolecular sciences. The DFT-based variants have the added benefit of higher predictive power beyond the validity of a bespoke force field and open the possibility to study chemical reactivity from first principles. DFT-based MD simulations allow a more realistic description of molecular systems and chemical processes, with a full description of dynamical ensembles at a given temperature, thus mimicking actual experimental conditions ever more closely. Large and complex condensed phase molecular systems can be investigated with AIMD, typically molecules immersed in solvents or infinite periodic surfaces in contact with solvents. To expand the length and time scales accessible with first-principles MD remains a challenge and a topical area of research in years to come.

During the preparation of this Theme Issue, the 2013 Nobel Prizes were announced. The Chemistry Prize went to computational chemists Martin Karplus, Michael Levitt and Arieh Warshel ‘for their development of multiscale methods for complex systems’. Techniques developed by the trio include hybrid methods that describe the central part of a large system using quantum-chemical methods (QM), whereas the surroundings are described by classical molecular mechanics methods (MM), i.e. QM/MM methods. Other techniques developed by the Nobel Prize winners include MM methods and MD. Whereas the foundations of the methods developed by the Nobel laureates were laid before DFT came of age, DFT has since seeped into these methods. Hybrid or QM/MM methods nowadays often use DFT to describe the QM part of the system and have become crucial especially in biomolecular modelling.

The essence of QM methods, DFT being no exception, is the computation of observables that can be directly linked to experiments. Besides the obvious quantities, energies and structures, spectroscopic properties are important targets for computation, because they mark areas where theory and experiment can team up fruitfully and complement each other. Vibrational, magnetic resonance and electronic spectroscopies are particularly important in this respect. In many cases, electronic structure theory is useful or even necessary for the interpretation of the experimental spectra. Owing to the ready applicability to ever larger and more realistic systems, DFT is constantly pushing the scope of questions, be they fundamental or applied, that can be addressed in such a concerted manner. In that respect, the calculation of nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) parameters by first-principles methods, including relativistic DFT, is currently a very active field.

Electronic excitation in molecular organic systems is another active field boosted by applications in photovoltaics or photochemistry. Even though DFT has been very successful in the prediction of ground-state properties, it is not well suited for the prediction of electronic excitation energies. The Kohn–Sham eigenvalues do not represent excitation energies and there is no DFT analogue of Koopmans' theorem. Fortunately, there are extensions to conventional DFT that allow the calculation of excited-state properties. One such method, time-dependent DFT (TDDFT) has become very popular for calculations of excited states. Going beyond the Born–Oppenheimer approximation is also required in certain cases where the decoupling between the nuclear and electronic motions does not hold any more, corresponding to non-adiabatic coupling(s). There are also theoretical challenges to be faced in this domain, and the DFT representation is at the forefront of new developments.

Fuelled by all this progress, the scientific impact of DFT on physics, chemistry and biology is huge. The computational efficiency of DFT means that larger (more realistic) systems can be treated, giving electronic structure theory much more predictive power and expanding its potential for applications. This trend is further boosted by continuing improvements in computer performance. As a result of these developments, many more joint experimental/computational studies are carried out today than 10–20 years ago. A very large portion of these use the DFT method as the workhorse, making it the most popular QM method in present use. Interestingly, simplified and approximate versions of DFT, for example the density-functional tight-binding (DFTB) method, are simultaneously being developed. With such semi-empirical methods based on the DFT framework, one would be able to access even faster electronic-based methods to reach much larger systems in size and time scale in MD simulations. Furthermore, as DFTB is built upon the DFT framework, it can easily benefit from any new improvement in DFT.

This apparent importance, if not dominance, of DFT in modern electronic structure theory spurred the idea to dedicate a Theme Issue of the *Philosophical Transactions* to this field. The 50th anniversary of the Hohenberg–Kohn theorems [3], published in 1964, widely considered as the birth of modern DFT, is a fitting occasion for such a Theme Issue. Comprehensive coverage of a vast topic such as DFT in this format is all but impossible. We rather chose a few selected contributions from leading researchers in the field, thereby providing representative snapshots from different facets of that diverse and vibrant discipline that DFT has become today.

## 2. Description of the Theme Issue

### (a) Beyond the Born–Oppenheimer approximation

The Born–Oppenheimer approximation [4] forms the basis of almost all electronic structure calculations. It greatly simplifies the Schrödinger equation by assuming that the electronic and nuclear motions can be separated. Although this assumption is valid in the majority of cases, corrections beyond the adiabatic (Born–Oppenheimer) level are sometimes required. Eberhard Gross (Max Planck Institute of Microstructure Physics, Halle, Germany) and Nikitas Gidopoulas (Durham University) discuss DFT beyond the Born–Oppenheimer approximation and introduce a non-adiabatic correction for the electronic equation in a new context, which couples the electronic and nuclear wave functions in a self-consistent way.

### (b) Development of density functionals

The development of new and more accurate density functionals is a very active research field. It is recognized that some functionals are better for some applications than others, though there are also efforts to develop more ‘general-use’ functionals. One group that is developing functionals aiming at broad applicability is Truhlar's group at the University of Minnesota. Naturally, new functionals need to be tested for their applicability and accuracy. The common way of doing this is by the comparison of results calculated by these functionals to corresponding results obtained with reliable high-level (usually *ab initio*) data. The article in this Theme Issue by the Minnesota group (authored by Roberto Peverati and Don Truhlar) exploits a set of databases of reference data to test and validate a wide range of density functionals, including their own Minnesota functionals. They conclude that the accuracy of DFT has increased notably over the last few decades, though there are still problem areas, for example multi-reference systems.

### (c) Fast density functional theory approximations

With some judicious further approximations, DFT calculations can be made significantly faster. One such approach, which can achieve a speed-up of two to three orders of magnitude, is the DFTB method. Two leading developers and users of this approach, Markus Elstner and Gotthard Seiffert, from the Karlsruhe Institute of Technology and TU Dresden in Germany, respectively, describe its foundation, current and foreseeable extensions and the scope of application. Although the gain in speed comes at the expense of somewhat reduced accuracy, this method is useful for the study of truly large and complex systems, from biomolecules to nanomaterials, and is thus a showcase example for how DFT can be applied at and across the boundaries between physics, chemistry and biology.

### (d) Relativistic density functional theory for nuclear magnetic resonance and electron paramagnetic resonance parameters

NMR and EPR are among the most useful spectroscopic techniques available to the chemist; NMR because of its immense potential as an analytical and structural tool, and EPR because of its high sensitivity towards paramagnetic systems. When the first *ab initio* methods for calculating NMR properties (at the Hartree–Fock level) became available in the 1980s, they were hailed by some as a new dimension of quantum chemistry. When DFT entered the field of computational NMR in the 1990s, it quickly became the leading player, with computational EPR following suit. The current cutting edge of method development in this area is the extension of the commonly used non-relativistic approaches to their relativistic generalizations, in order to be able to describe systems containing heavy elements. A pioneer in the field, Jochen Autschbach from the University of Buffalo, describes the state of the art of such relativistic DFT computations for molecular NMR and EPR parameters.

### (e) Density functional theory in liquid water: *ab initio* molecular dynamics

Since the dawn of AIMD simulations, liquid water has been a popular target in order to further our understanding of its unique properties that make it the element of life. Together with his co-workers, one of the founders of the popular CPMD method that bears his name, Michele Parrinello (ETH Zürich), presents a review of AIMD simulations, focusing on applications on liquid water and its constituents, the hydronium and hydroxide ions, as well as aqueous solutions of ions and molecules. The authors illustrate how these simulations can provide insights into the microstructure of the solvent, both pristine and around a solute, and how, on the other hand, subtle inaccuracies in the simulated structure and dynamics of water can inform on shortcomings of the underlying DFT model.

### (f) Density functional theory for excited states

The study of the electronic and optical properties of organic systems is crucial to a large variety of fields, including photovoltaics, photochemistry and photosynthesis. Extensions to conventional DFT that allow the calculation of excited-state properties have been developed, for example TDDFT. The formal foundation of TDDFT is a theorem that has been introduced by Erich Runge and one of the contributors to this Theme Issue, Eberhard Gross. The Runge–Gross theorem is the time-dependent analogue of the Hohenberg–Kohn theorem [5]. In this Theme Issue, Xavier Blase and co-workers (from the Joseph Fourier University, Grenoble in France) review standard DFT and its time-dependent variations, as well as methods based on many-body Green's function perturbation theory, for the calculation of excited-state properties.

### (g) Density functional theory in the solid state

From its roots in solid-state physics, DFT has expanded into solid-state chemistry, surface science, materials science, mineralogy and other fields that are investigating materials in the solid state. Modern DFT simulation codes for solid-state calculations can calculate a vast range of structural, chemical, optical, spectroscopic, elastic, vibrational and thermodynamic properties, and it is nowadays common practice to include computational results in experimental studies on materials and surfaces. A large number of programs for calculations on periodic systems using either plane-wave or atom-centred basis sets are available and under active development. In this Theme Issue, the use of DFT in the solid state, with emphasis on calculations done with the CASTEP code, is reviewed by Phil Hasnip and colleagues from the UK's CASTEP Development Group.

## Footnotes

One contribution of 8 to a Theme Issue ‘Density functional theory across chemistry, physics and biology’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.