Category:Strongly correlated electrons

Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity. To model such systems, several extensions of DFT have been developed.

DFT+U

DFT+U is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.

The total energy within DFT+U can be written as \begin{equation} E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right], \end{equation} where the first term is the standard DFT energy, the second term is the Hubbard on-site interaction and the third term accounts for the double counting. The on-site interaction is described by $U^I$ and $n_m^{I\sigma}$ are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.

VASP provides the following approaches to include the Hubbard corrections:

• LDAUTYPE=1: The rotationally invariant formulation of the Hubbard correction that eliminates the dependence on the specific choice of the localized basis set.

• LDAUTYPE=4: The same approach as LDAUTYPE=1 but uses spin-averaged expression that's simpler and assumes an average spin configuration.

• LDAUTYPE=2: The simplified approach which is also rotationally invariant but uses isotropic effective interaction $U^I_{eff}=U^I-J^I$ ^[1]. This approach neglects the anisotropy of the orbitals and thus the on-site interaction depends on the occupations but not the orbitals themselves.

A common approach is to treat the effective on-site interaction $U$ as an adjustable parameter, tuning it to reproduce selected experimental observables such as the band gap, magnetic moments, or lattice parameters. Alternatively, fully ab initio schemes exist that determine the Hubbard interaction directly from first principles, avoiding empirical fitting.

• LDAUTYPE=3: Linear-response calculation of $U$. Within this approach the effective interaction $U$ can be determined via the linear response approach ^[2]

\begin{equation} U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}. \end{equation} The shortcoming of this method is that the effective interaction accounts for the response due to all electrons including the target states (localized $d$ or $f$ orbitals), thus leading to double counting in the derived effective potential U.

Constrained Random Phase Approximation (cRPA)

cRPA is a first-principles method used to compute the effective interaction parameters for the DFT+U or DMFT calculations. cRPA allows to separate the screening originating from the target states from the rest of the system and to determine the effective interaction $U$ that is free of double counting.

The response function without the contribution of the target states or constrained polarizability $\chi_c$ is calculated by explicitly removing the response in the target space $\chi_d$ from the total response function: $\chi_c(\omega) = \chi(\omega) - \chi_d(\omega)$.

VASP provides several approaches for calculating $\chi_d$

Dynamical Mean-Field Theory (DMFT)

DMFT is an dynamical extension of DFT+U, where U is considered to be frequency dependent (DFT+DMFT). DMFT provides an accurate treatment of strongly correlated materials, which are fully neglected in DFT+U ^[3]. Within DMFT a lattice problem is mapped onto a self-consistent quantum impurity model by embedding a single correlated site in an effective bath that represents the rest of the system, and the key approximation is that the self-energy is local (frequency-dependent but momentum-independent). DMFT constitutes a state-of-the-art approach for the accurate description of strongly correlated systems, capturing essential many-body effects such as quasiparticle renormalization, Hubbard bands, and Mott metal–insulator transitions. DMFT can be used in combination with cRPA, where cRPA is used as a preliminary step to determine the effective screening $U(\omega)$ without the contribution from the target space to avoid double-counting in the subsequent calculation of the self-energy within DMFT.

Other methods

There are other methods that, while not specifically designed for strongly correlated systems, have nonetheless been demonstrated to improve their description and electronic structure.

Hybrid functionals

By incorporating a fraction of exact exchange, hybrid functionals partially mitigate the self-interaction error inherent to standard DFT. This reduction of the self-interaction error has been shown to yield an improved description of strongly correlated systems ^[4][5].

Hybrid functionals + U

A well-known limitation of hybrid functionals is their uniform treatment of all electronic states, which can result in markedly different levels of accuracy for states with varying degrees of localization. The inclusion of the Hubbard on-site interaction term within the hybrid functional framework has been shown to address inaccuracies arising from the overscreening of localized states ^[6].

QPGW

The GW approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW, which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons ^[7].

Tutorials

• Tutorial for NiO LSDA+U calculations.
• Tutorial for NiO DFT+U calculations.
• Tutorial for NiO Calculate U for LSDA+U calculations.
• Tutorial for CRPA of SrVO3 calculations.
• Tutorial for Bandstructure and CRPA of SrVO3 calculations.
• Tutorial for NiO DFT+DMFT calculations.
• Tutorial for NiO GGA+U calculations.

References