In the modern world, we find ourselves facing a wide range of scientific and technological problems ranging from creating more efficient energy storage devices, to the design of better computer processors, and the development of new pharmaceuticals to tackle ever-evolving diseases. At their core, these problems can all be reframed as a material problem, that is the development of some novel “functional” material or chemical substance that is designed to solve the specific problem.

Despite a plethora of particles in nature, as described by the Standard Model of Particle Physics, it turns out only three are required to yield the incredibly versatile set of building blocks of the chemical elements: electrons, protons and neutrons, with the latter two forming the nucleus of the atom. Chemistry and electronic structure are concerned with the behaviour of electrons and nuclei. Despite the simplicity of the constituents, the behaviour of the whole turns out to be immensely complex and rich. It leads to a wide range of phenomena in the so-called “quantum materials” whose behaviour cannot be explained using classical physics. In the face of such complexity, the ability to screen a wide range of candidate materials accurately requires the accurate computational simulation, at the level of first-principles or ab initio techniques. 

At the heart of condensed matter physics and chemistry lies quantum mechanics and Schrödinger’s equation, laying the theoretical groundwork for the behaviour of all microscopic objects, and playing an analogous role to Newton’s law of motion for classical objects. This equation allows us to predict how electrons and nuclei behave under the influence of each other. However, even after disentangling the electronic and nuclear motions, the direct solution of Schrödinger’s equation for electrons alone is intractable for all but the simplest systems. This is because the complexity of the solution scales exponentially fast with electron number.  

The challenge lies with the many-body wavefunction, which describes all the possible positions of all of the particles in a system, and is so information-dense that storing it (never mind actually attempting to calculate it) poses unsurmountable difficulty: storing the wavefunction for one quantum state of a single carbon atom (six electrons, a central element in organic chemistry) on a coarse grid of just 10×10×10 grid points per electron, requires more than a million terabytes of storage! To store the wavefunction for one quantum state of the nitrogen atom (the next element), would require a thousand times larger capacity. 

Figure 1: A 3D function on a discrete grid, with the function's value coloured according to transparency and colour of the pixels. Red and opaque means a high value, blue and transparent is a small value.

Fortunately, a sleight of hand in the form of density functional theory (DFT) allows us to predict properties of the real (physical) many-electron system of interest, without solving Schrödinger’s equation for the many-body wavefunction. As the name implies, DFT shifts the focus, from the many-body wavefunction (which is a function on a 3N-dimensional grid) to the electronic density of the physical system (which is a function on a 3D grid) where N is the number of particles (see figure 1). The word ‘functional’ indicates a fundamental theorem of DFT, namely that once the electronic density is found, all properties of the physical system are in turn determined by it (or are functionals of it). Then, solving an equation for the electron density, rather than Schrodinger’s equation for the wavefunction, allows us to predict efficiently and accurately the behaviour of matter (see figure 2). 

Figure 2: DFT turns an interacting system (left) into a non-interacting system (right). where the electrons interact with each other via a background effective potential vₛ (r). This potential is a functional of the density (hence the name DFT) that may be partitioned into three terms. We know how to write two of the three potential/ energy terms vₑₓₜ (r) and Hartree/ classical electrostatic vₕ(r) terms. However, we know there must be some portion of interaction felt by electrons that goes beyond classical electrostatic which contributes to vₛ (r) as well as the total energy - this is encapsulated by the exchange-correlation potential vₓc (r).

CASTEP is a leading code to simulate the behaviour of materials using DFT. It is developed by a team of UK researchers, including those at Durham, and used worldwide both in academia and industry. As we enter the exascale era of supercomputing, with wider access to more computational resources and increasingly efficient numerical algorithms and robust simulation codes, the ubiquitousness of DFT in the toolbox of material scientists, physicists and chemists only continues to increase.  

At Durham, the research into DFT and algorithmic development (CASTEP) can be classed into two broad categories. The first involves the development and implementation of the theoretical framework. Although DFT is a formally exact theory, its practical implementation requires the development of increasingly accurate, widely applicable approximations, and the understanding of their limitations and why they work.  

The second area of research is the application and utilisation of DFT. DFT is an ab initio computational approach, meaning ‘from the beginning’ and computes properties of materials by solving the fundamental quantum mechanical equations, rather than by fitting a model with empirical parameters, which may bias the model to a specific class of materials. This enables transferability across a wide range of materials, from those with novel forms of magnetism to new energy storage devices and even simulating chemical reactions, for instance the action of a new drug on the body or investigating the feasibility of synthesising new materials that have yet to be discovered experimentally. 

Find out more:

• Discover more about CASTEP.
• Read more about DFT in Walter Kohn's Nobel Prize lecture.
• Find out more about DFT by reading Physics Magazine's Nobel Focus
• Learn more about self-consistent equations including exchange and correlation effects.