# Density Functional Theory

## Density Functional Theory

** Density Functional Theory (DFT**) is the most used electronic-structure computational method in quantum
chemistry and condensed matter physics, since it provides the highest accuracy/computational cost ratio. In
combination with powerful High-Performance Computing (HPC) hardware resources, DFT allows first
principles calculations of the electronic and optical properties of systems with thousands of atoms.

However, despite DFT is exact in principle, its practical accuracy depends on approximations concerning the exchange-correlation (XC) energy as a functional of the ground-state electronic density. In case of orbitalfree or subsystem-DFT calculations, approximations of the non-interacting kinetic energy (KE) functional are also required.

### Development of exchange-correlation functionals

The XC energy contains all the quantum many-body effects beyond the simple Hartree approximation, and it is the subject of heavy investigations in different research groups worldwide. In particular, meta-generalized gradient approximation (meta-GGA) and, more in general, orbital-dependent functionals, are attracting strong interest due to their accuracy. Meta-GGA functionals depend not only on the gradient of the electronic density, but also on the Laplacian of the density and/or on the kinetic energy density, which allows to distinguish one-electron region. Orbital-dependent functionals depend on all orbitals and thus include exchange exactly.

In the Computation group at CBN, we develop meta-GGA and orbital-dependent XC functionals using model systems and exact conditions to fix the functional form and the numerical parameters within it. This allows to obtain accurate XC functionals without recurring to a heavy empirical parametrization.

Key References:

F. Della Sala, E. Fabiano, L. A. Constantin, Int. J. Quant. Chem. 116, 1641 (2016)

E. Fabiano, P Gori-Giorgi, M. Seidl, F. Della Sala, J. Chem. Theory. Comput. 12, 4885 (2016)

### Development of kinetic energy functionals

While in the conventional Kohn-Sham (KS) DFT scheme KE is treated exactly via the one-particle orbitals, both orbital-free Density Functional Theory (OF-DFT) and subsystem DFT (Sub-DFT) require, in addition to the XC functional, the knowledge of the KE as well as of the corresponding kinetic potential as a explicit functionals of the electronic density (Ts[]). In addition, the KE kernel is a key quantity for the development of non-local KE functionals as well as in the non-local quantum hydrodynamical model (QHM) for the calculations of optical properties in plasmonics.

OF-DFT, Sub-DFT and QHM are attracting strong interest because these methods allow to treat very large systems with a linear-scaling computational costs, in contrast to KS-DFT, which cannot be exploited for large nanoscience applications. Thus, the development of reliable KE functionals is a fundamental topic in DFT. Nevertheless, it is also one of the hardest problems in the field. In fact, most of the modern kinetic functionals are not improving much with respect to the Thomas-Fermi functional, developed in 1927, which is also routinely used in modern QHM approaches; furthermore, the kinetic kernel has only been investigated for the infinite jellium model (i.e. the Lindhard function), whereas finite size effects have yet not been considered in literature. Thus, the development of accurate and efficient KE functionals is one of the biggest DFT challenges.

In the Computation group at CBN, we develop KE functionals which depend, other than on the density and its gradient, on the Laplacian of the electron density. In this way, they can univocally distinguish the nuclear and the bonding regions, thus being an important advance in the KE functional development.

Key References:

L. A. Constantin, E. Fabiano, S. Śmiga, F. Della Sala, Phys. Rev. B. 95,115153 (2017)

S. Śmiga, E. Fabiano, L. A. Constantin, F. Della Sala, J. Chem. Phys. 146, 064105 (2017)

### Ab-initio modeling of metal and heavily-doped semiconductors nanoparticles

Neither classical electromagnetism nor quantum hydrodynamic models take into account the exact core-valence coupling, which plays a key role in noble metals, and the atomic structure of nanoparticles, which is very important at small distances. Time-dependent density functional theory (TDDFT) can include both effects. However, first principles TDDFT is currently limited to systems below one thousand of atoms, while larger systems can be modeled using the Density Functional Tight Binding (DFTB) approximation.

In the Computation group at CBN, we model by TDDFT and TDDFTB the optical properties of metals and semiconductors clusters of different sizes, doping/charge and compositions. These studies have a strong impact for the understanding of the underlying physics, for the development of multiscale methods, and for the design of new plasmonics materials.

### Computing Center

At CBN we have a Linux Cluster (from ClusterVision) for High Performance Computing (HPC) with 24 nodes (in total 224 Intel cores and 994 GB di RAM), Infiniband interconnection network and 24TB cluster-file-system storage.