By Assoc. Prof. Dr. M. Oluş Özbek

With rapid advancements in computer technologies and computational algorithms, the accuracy of computational methods has significantly improved, enabling researchers to gain unprecedented insight into complex chemical processes. This progress has not only deepened our understanding of fundamental reaction mechanisms but has also revolutionized catalysis research, providing powerful tools to design and optimize catalysts for industrial applications. To comprehend how Density Functional Theory (DFT) can be applied to catalysis research, it is essential to first delve into the history and development of DFT. Understanding its origins and fundamental principles provides a clearer picture of what DFT can achieve and the unique insights it offers into chemical processes.

DFT is a computational quantum mechanical method used to study the properties of atoms, molecules, and solids. In its most basic form, it solves the ground state electronic configuration of the given system. The starting point of DFT is the many-body Schrödinger’s Equation (ĤΨ = EΨ), formulated in 1926, and was awarded the Nobel Prize in Physics in 1933. Although this equation is exact, its solution is computationally impractical for large systems. The first practical approximate solution was introduced by Hartree in 1928 [1], who decomposed the total wave function as a product of single-electron wave functions (Ψ ≈ Π ψi). This approach enabled the transformation of the many-body (i.e., many-electron) Schrödinger equation into a set of one-electron equations. However, this significant simplification neglected the electron exchange and correlation effects. This approximation has limited accuracy because i. the physical state of a system must be invariant under the exchange of electrons (i.e., electrons are not distinguishable) and ii. the correlation arises from the fact that electrons interact via Coulomb forces, which are instantaneous and depend on their relative positions.

To address the shortcomings of the Hartree method, Fock and others introduced the Hartree-Fock (HF) method (1930) [2], which incorporated the Pauli exclusion principle. The HF method improved upon Hartree’s approximation by using a combination of single-electron wave functions (the Slater determinant) to construct the many-electron wave function. Although the HF method introduces an approximate exchange interaction through mean field approximation and can be solved iteratively (i.e., a self-consistent solution can be reached), it still neglects electron correlation. Another problem with the HF method is being computationally expensive due to high degrees of freedom (DOF). At this point, the DFT method used the electron density ρ(r), which depends only on three spatial variables, reducing the DOF and making a huge gain in the computation load.

One of the earliest density-based methods was the Thomas-Fermi model (1927–1928) [3,4], which approximated the energy of a system as a functional of the electron density. In 1964, Hohenberg and Kohn established the modern formalism of DFT by incorporating two theorems with their infamous ‘Inhomogeneous Electron Gas’ [5]: i. The ground-state properties of a many-electron system are uniquely determined by its electron density ρ(r), and ii. The energy of the system can be expressed as a functional E[ρ], which is minimized at the true ground-state density. Today the main form of DFT is based on the Kohn-Sham (KS) equations, developed by Kohn and Sham in 1965 [6], and was awarded the Nobel Prize in Chemistry in 1998. Their method introduced virtual orbitals (KS orbitals) that enabled the decomposition of energy functional E[ρ] into components such as kinetic energy, classical electrostatic energy, and exchange-correlation energy.

The success of DFT depends on two basic components. The first is the basis set used to describe the electronic structure of the atoms. A better representation would need a larger basis set that includes a larger number of parameters. However, since each new variable increases the DOF from 2n to 2n+1 (approx.), the selection of a suitable basis set is a trade-off between computational load and model correctness. Apart from the all-electron applications, most of the basis sets lighten the computational load using core (fixed) and valence (variable) electrons. 

Among these, pseudopotentials are commonly used and have a semi-empirical nature. The second component is the exchange correlation function that describes (or solves for) the interactions between the electrons of these atoms. It would not be totally wrong to say that the improvement in DFT applications stems from the improvements in the exchange functionals. After the first Local Density Approximation (LDA) [6] was introduced in 1965, the improvement came with Generalized Gradient Approximation (GGA) [7] in 1992. The GGA functional was further improved to overcome the shortcomings through PBE [8] in 1996 and RPBE [9] in 1999. To overcome the shortcomings of the pure GGA, hybrid functionals that include the HF into the exchange were developed, such as B3LYP [10] and PBE0 [11]. These hybrid functionals significantly improved the accuracy of the results; however, at a higher computational cost, rendering them impractical for larger systems. This chronological progression of exchange-correlation functionals highlights the continuous efforts to balance accuracy, computational efficiency, and versatility, enabling DFT to become a cornerstone of quantum chemistry and materials science. 

Currently Wikipedia lists more than 50 commonly used software [12]. These can be grouped into two categories with respect to their applications. First is the cluster models (Figure 1), which uses software such as Gaussian and ORCA and involves the approximation of a finite portion of a system, such as a molecule or a localized region of a material, to simulate its properties. These models are particularly useful for studying small-scale systems (isolated molecules) or specific reaction sites [13], allowing detailed study of localized interactions at a lower computational cost. While surface reactions can be modeled using a large enough cluster [14], this method suffers from edge effects and is not able to represent bulk properties.

Second group is the periodic models that use software like VASP or Quantum Espresso and utilize planewave basis sets, which are periodic themselves. As the name implies, these models use periodic boundary conditions to simulate infinite systems by repeating a unit cell. This approach is widely employed for studying crystalline solids, surfaces, and materials with translational symmetry. The major advantage over the cluster models is the ability to accurately represent materials with periodic structures, capturing long-range interactions and electronic band structures. These are especially well-suited for interpreting spectroscopic or crystallographic measurements. Practical applications cover simulations of surfaces and surface reactions, thin films, and interfaces under realistic conditions. Once again, the trade-off for these realistic models is the higher computational cost and complexity of setup. Not widely applied, but a combination of both approaches for comprehensive insights is also possible [15].