2.1.5. Quantum Density Functional Theory (DFT)

Among the quantum mechanical methods applied to the study of the optical properties of CDs, DFT methods are by far the most used. Compared to pure wave function theory methods such as HF, DFT methods has a greater capability of coupling reasonable accuracy with relatively low computational cost. DFT has the benefit to incorporate some of the correlation among electrons, with a much lower computational cost than correlated post HF wave function methods.

DFT methods derive from the 1964 Hohenberg and Kohn theorems,¹³⁶ stating that all the ground-state properties of an N-electron system are uniquely determined by the total electron density. In the later Kohn–Sham DFT formulation (by far the most used among those existing), the total energy of the ground state is expressed as a sum of exact terms, and an important (although small) contribution to the energy is given by the exchange-correlation term E_XC:

where T is the noninteracting electron kinetic energy, J the Coulomb energy, and V is the energy due to the external field generated by the nuclei. The first three terms can be computed exactly, while the exact E_XC functional form is not yet known except that for a uniform electron gas, and only approximate forms can be used. An increasing number of functionals, each one with its strength and limitations and differing in the way they approximate the exact XC term, is available in the most common quantum mechanics codes.

The simplest approximation of the XC term is the local density approximation¹³⁷ (LDA) and its generalization including electron spin LSDA (local spin density approximation). The energy is typically separated to an exchange and in a correlation part:

While LDA has been widely used for studying bulk properties in solid state physics, it is not appropriate to study surfaces or molecules, since it overestimates the bond energies and produces too short bond lengths.

A more sophisticated approximation emerged in the 1980s, making use of both the spin densities and of their gradients (GGA, generalized gradient approximation); typically, but not necessarily, the exchange and correlation terms are separated. The two main approaches to GGA make use of parameters obtained either by a fitting to some data sets, as in the B86 Becke approach,¹³⁸ or derived using theoretical conditions, as in the Perdew and co-workers approach.^139,140 The development of these functionals was fundamental for DFT to enter into chemistry.

A broadly used variant of the GGA, largely applied to the study of the structure and properties of CDs, is constituted by the hybrid-GGA methods, which typically combine the HF exchange integrals with a GGA exchange functional. These types of functionals are commonly referred as “global hybrids” since they are applied on the whole spatial domain without any truncation in short- and long-range domains as in the case of range-separated (RS) hybrids. The popular B3LYP, developed by Stephens and co-workers,¹⁴¹ is an example of global hybrid functional, and it was derived from the three-parameter hybrid GGA functional B3PW91,¹⁴² obtained in 1993 by Becke, by replacing in the PW91¹⁴³ the correlation terms with the LYP GGA.

There is a large and increasing number of hybrid functionals, and popular quantum mechanical software packages, like Gaussian^144,145 and NWChem,¹⁴⁶ allow one to choose among dozens of them or even to tune them through a flexible combination of their HF and GGA components.

The most used functionals in chemistry and material science are those based on the B3LYP and PBE functionals,¹⁴⁷ respectively. However, there are several limitations in using these functionals, such as the reproduction of dispersion forces and the incorrect behavior of the XC functional at long-range, which have relevant impact on the charge-transfer excitation. A promising approach is represented by the RS hybrids functionals in which, differently from the above-described functionals, the XC functional is divided into short-range and long-range contributions.^148,149 The popular CAM-B3LYP and ωB97XD belong to this class.

Another DFT related approach which has been recently improved is the so-called multireference configuration interaction (DFT/MRCI) method.^150,151 In this approach, the total electronic correlation is described by properly adding to the truncated multireference expansion, a DFT contribution in order to take into account the dynamic correlation. However, as shown in Table 3, the applications of DFT/MRCI methods to CDs systems are still very limited and mainly focused on benchmarking. In particular, the DFT/MRCI can outperform DFT in the case of (i) emission energies calculations, (ii) correct ordering of the excited states, and (iii) open-shell or double excited electronic configurations (see section 2.1.11 for a detailed description). However, this approach lead to an overall increase in the computational cost, due to its intrinsic multireference nature.¹⁵²

Focusing on the specific challenge of including the dispersion in DFT calculations, it must be noted that an important source of error is due to the inaccurate description of van der Waals interactions.¹⁵³ In particular, LDA or GGA functionals and the majority of hybrids originating from a simple linear combination of them (as B3LYP) lack in the description of the long-range van der Waals interactions, therefore missing the accurate description of the typical attractive 1/R⁶ (where R is the internuclear distance) contribution.¹⁵⁴ A straightforward solution to this problem is to add an empirical pairwise term proportional to C₆/R⁶, which depends only on the internuclear distance. The main issue related to this approach is the actual choice of the tabulated C₆ coefficients that are usually dependent on the chemical environment. This approach has been fully extended by Grimme, and the whole periodic table has been covered in order to describe different types of interactions.¹⁵⁵ These kind of functionals are usually denoted as DFT-D. Finally, a fundamental source of error in standard DFT is the so-called “self-interaction error” (SIE) arising from the impossibility to distinguish two-body electrostatic interactions from spurious self-interaction contributions.¹⁵⁶ Even if suitable procedures have been developed in order to partially reduce SIE, pure exchange correlation functionals cannot completely remove SIE, which is still an open problem in the DFT field.

The inclusion of a significative portion of HF exchange into the hybrid exchange-correlation functional is considered as the best solution to deal with SIE even if a complete SIE cancellation cannot be obtained.¹⁵⁶ Alternative solutions involve the extension of the Perdew–Zunger self-interaction correction (SIC) to DFT,¹⁵⁷ the use of localized orbital scaling corrections,¹⁵⁸ or multiconfiguration pair-density functional theory (MC-PDFT).¹⁵⁹

Concerning the applications of DFT methods in CNDs studies, from the data reported in Table 1 and Table 2, where the most relevant computational details on the calculations performed with the QM method on CDs are summarized, it can be clearly seen that most investigations have focused on understanding CNDs structure and properties using small molecular models make use of DFT methods and that B3LYP is the most used functional. The target feature is in general the optical absorption since the computations of the emission properties are not always straightforward. Sometimes the density of states (DoS) or the vibrational spectra are also calculated and compared to the experimental results. It is important to note that the fact that the B3LYP functional is the most used does not imply that it is always to be recommended, and concerning this point, we refer the reader to the studies performed to compare the performance of different functionals in the system of interest for CNDs.^113,160 A detailed description on benchmarking applied to CNDs can be found in section 2.1.11.