LDA+U Method for Correlated Electron Systems

• LDA+U is an extension of DFT that introduces explicit on-site Hubbard corrections to accurately capture electron correlation in localized d and f orbitals.
• The method augments the DFT energy functional with a Hartree–Fock–like U term and subtracts a double-counting correction to mitigate self-interaction errors.
• It is widely implemented across various codes and parameterized empirically or ab initio, providing improved predictions of gaps, magnetic moments, and orbital order in correlated materials.

The LDA+U method is an extension of density functional theory (DFT) that introduces explicit on-site Coulomb corrections for localized electron states—most often transition-metal 3d and 4f/5f electrons—enabling first-principles calculations to capture Mott insulators, orbital polarization, and other physics beyond reach of conventional local (LDA) or semilocal (GGA) functionals. By augmenting the Kohn–Sham DFT total energy with a Hartree–Fock–like Hubbard U correction and subtracting a double-counting term, LDA+U corrects the self-interaction and delocalization errors that plague DFT in correlated systems. The method has been systematically developed and benchmarked across a wide range of compounds, providing a computationally tractable route to improved gaps, magnetic moments, orbital order, multiplet splitting, and excitation spectra in correlated materials.

1. Theoretical Principles and Formalism

LDA+U modifies the total-energy functional of DFT to explicitly penalize partial occupancy of localized atomic orbitals (typically $d$ or $f$ shells) via an on-site Hubbard correction. The general LDA+U functional is

$$E_{\rm LDA+U}[\{\psi\},\{\phi_m^I\}] = E_{\rm LDA}[\rho] + E_{\rm Hub}[\{n_{mm'}^{I\sigma}\}] - E_{\rm dc}[\{n^I\}]$$

where $n_{mm'}^{I\sigma}$ is the occupation matrix of correlated orbitals on site $I$. Multiple rigorous forms are in use:

• Rotationally invariant (Liechtenstein et al.):

$$E_{\rm Hub} = \frac{1}{2} \sum_{I\sigma mm'k} \langle m, m''|V_{ee}| m', m'''\rangle\ n_{mm'}^{I\sigma} n_{m''m'''}^{I-\sigma} + (\mathrm{exchange\ term})$$

The interaction tensor $V_{ee}$ is parameterized by Slater integrals $F^k$.

• Simplified (Dudarev et al.):

$$E_U = \sum_{I,\sigma} \frac{U_{\rm eff}}{2}\, \mathrm{Tr}\left[ n^{I\sigma}(1-n^{I\sigma}) \right]$$

where $U_{\rm eff} = U - J$ is the difference of averaged Hubbard and Hund couplings.

The double-counting term is necessary to subtract the contribution of on-site interactions already present in the LDA or GGA functional. The Fully Localized Limit (“FLL”) expression is

$$E_{\rm dc}^{\rm FLL} = \sum_I \left[ \frac{U}{2} n^I(n^I-1) - \frac{J}{2} \sum_\sigma n^{I\sigma}(n^{I\sigma}-1) \right]$$

with $n^I = \sum_{m,\sigma} n_{mm}^{I\sigma}$. Variants such as the Around Mean Field (AMF) scheme exist but are less commonly applied for Mott insulators (Himmetoglu et al., 2013, Lohani et al., 2023, Lee et al., 2010, Keshavarz et al., 2018).

2. Implementation and Parameterization

LDA+U is implemented in various electronic structure codes, including VASP, Elk, OpenMX, and TB-LMTO-ASA, often using either atomic-like, pseudo-atomic, or Wannier-type projectors for defining the correlated subspace (Nakamura et al., 2010, Lohani et al., 2023, Lee et al., 2019). Parameters $U$ and $J$ can be:

• Empirical, chosen to reproduce experimental observables (gap, moment, XMLD, etc.) (Sterling et al., 2021, Meinert et al., 2012, Griffin et al., 2014).
• Ab-initio derived, using methods such as constrained DFT, linear-response supercell techniques, or constrained Random Phase Approximation (cRPA) (Karlsson et al., 2010, Aras et al., 2013, Himmetoglu et al., 2013).

A representative computational workflow proceeds as:

1. Perform LDA/GGA calculation.
2. Project occupations onto correlated orbitals.
3. Add the +U correction and subtract double-counting in the Kohn–Sham equations.
4. Iterate self-consistently to convergence for charge and spin densities.
5. Optionally, adjust $U, J$ to match experiment or by self-consistency (cf. cRPA-LDA+U) (Karlsson et al., 2010).

For systems with strong spin–orbit coupling or multipolar order, explicit inclusion of SOC and multipolar tensor expansion of the local density matrix are essential (Zhou et al., 2011, Pi et al., 2013).

3. Physical Interpretations and Effects

LDA+U shifts partially occupied correlated bands away from the Fermi level, favoring integer occupancy, and thus stabilizes insulating states in Mott and charge-transfer systems. For sufficiently strong $U$:

• The correlated subspace splits into lower and upper Hubbard bands, opening a gap (Nakamura et al., 2010, Sterling et al., 2021).
• Magnetic moments and crystal-field splittings are more accurately captured, as shown for NiO, Gd, and 4f/5f oxides (Karlsson et al., 2010, Zhou et al., 2011).
• Orbital selectivity emerges in materials with multiple correlated orbitals, as interorbital hybridization and local crystal fields cause $U$ and $J$ to act differentially (see FeSe vs FeTe) (Lohani et al., 2023).
• In hybrid-functional and DFT+U combinations, the $U$ parameter can be tuned to restore experimentally measured gaps and d-level binding energies, yielding improved energetics and ground-state structure (Aras et al., 2013).

4. Applications Across Correlated Materials

LDA+U has enabled:

• Restoring correct insulating Fermiology and anisotropy: In Sr$_2$VFeAsO$_3$, large $U$ applied on V $d$ drives the blocking layer Mott insulating, yielding conventional Fe-based superconductor Fermi surfaces and strong transport anisotropy (Nakamura et al., 2010).
• Spectroscopic accuracy and limitations: For Co$_2$FeSi, LDA+U opens a minority gap and recovers total moment, but the crystal-field splitting and minority-d bandwidth are overestimated, degrading agreement with x-ray magnetic linear dichroism. Fixed-spin-moment GGA provides better XMLD agreement (Meinert et al., 2012).
• Phonon spectra and electron–lattice coupling: In La$_2$CuO$_4$, LDA+U with optimal $U$ (set by optical gap and local moment) reproduces Cu–O half- and full-breathing phonon frequencies in line with neutron data—standard LDA/GGA severely underestimates mode energies (Sterling et al., 2021).
• Multipolar exchange and spin–orbit-driven order: Extensions allow explicit calculation of dipole– and quadrupole–quadrupole interactions, e.g. in UO$_2$ via a pair-flip mapping over the LDA+U+SOC energy landscape (Pi et al., 2013).
• Self-interaction and crystal-field accuracy in 4f/5f dioxides: Self-interaction–corrected LDA+U combined with on-site multibody fits recovers experimental crystal-field splittings within ~10–20 meV (Zhou et al., 2011).
• Exploration of metastable states: Random density matrix control provides a systematic way to traverse the local minima in the multi-dimensional LDA+U energy landscape, ensuring comprehensive magnetic and orbital state identification (Keshavarz et al., 2018).
• Cluster Mott and $J_{\mathrm{eff}}$ physics: For lacunar spinels, charge-only LDA+U formalism (vs. spin-polarized) stabilizes well-defined $J_{\mathrm{eff}}$=1/2 or 3/2 bands, with quantitative metrics for band purity (Lee et al., 2019).

5. Parameter Choice, Double Counting, and Best Practices

Key choices in LDA+U calculations include:

• Projectors: Choice determines the correlated subspace and directly affects occupation matrices, $U$, and physical predictions (Himmetoglu et al., 2013, Karlsson et al., 2010).
• U and J estimation: Empirical procedures may target the gap or moment, while self-consistent cRPA or linear-response DFT provide first-principles $U$ values (Karlsson et al., 2010, Aras et al., 2013, Lee et al., 2019, Lohani et al., 2023).
• Double counting: FLL is standard for Mott and charge-transfer insulators, AMF for itinerant metals. Uncertainty in the form is a pervasive open issue (Himmetoglu et al., 2013).
• Spin treatment: For systems where Hund's $J$ and exchange dominate, charge-only DFT+U is recommended to avoid unphysical enhancement of Stoner exchange (Lee et al., 2019, Keshavarz et al., 2018).
• SOC and multipolarity: Manifest necessity in systems with strong spin–orbit coupling or multipolar order (especially 4f/5f compounds), requiring generalization to tensorial occupation matrices (Zhou et al., 2011, Pi et al., 2013).
• Hybrid and combined approaches: LDA+U can be combined with hybrid functionals (e.g. HSE+U) to further alleviate delocalization and self-interaction errors, often improving structure and gap predictions (Aras et al., 2013).

6. Strengths, Limitations, and Future Directions

LDA+U:

• Strengths: Delivers computational efficiency, systematic correction for DFT errors in correlated subspaces, orbital selectivity, and has a well-understood effect on Fermiology and gapping. LDA+U is essential for ground-state structural, magnetic, and spectroscopic calculations of Mott and charge-transfer insulators, multipolar ordered systems, and $J_{\mathrm{eff}}$ magnets (Himmetoglu et al., 2013).
• Limitations: Remains static (frequency-independent $U$), neglects dynamical correlation and fluctuations (important for quantitative ARPES, low-energy collective modes), and is sensitive to projector and double-counting ambiguity. LDA+U+SOC fails to describe broad multiplet spectra without explicit energy-dependent kernels (Lee et al., 2010). Overcorrection and poor orbital selectivity can arise if $U$ is applied indiscriminately, as evidenced by XMLD failure in Co$_2$FeSi (Meinert et al., 2012).
• Development directions: Self-consistent, frequency-dependent $U$ (dynamical mean field theory, cRPA-LDA+U), generalization to LDA+U+V (inter-site), tensorial forms with full orbital-resolved $U_{mm'm''m'''}$, and integration with TDDFT or GW for excited states (Karlsson et al., 2010, Himmetoglu et al., 2013).

7. Summary Table: Representative LDA+U Applications

Material/System	LDA+U Role	Key Outcome(s)	Reference
Sr$_2$VFeAsO$_3$	Layer selectivity, $U$ on V	Mott insulating perovskite, textbook Fe-pnictide FS	(Nakamura et al., 2010)
Co$_2$FeSi	FLL scheme, $U$ on Co/Fe, XMLD comparison	Over-broadened XMLD, FSM GGA preferable	(Meinert et al., 2012)
La$_2$CuO$_4$	Dudarev $U$ on Cu	Gap and breathing phonons matched to experiment	(Sterling et al., 2021)
FeSe, FeTe	TB-LMTO-ASA, $U,J$ explicit, FLL DC	Orbital/nematic correlations, orbital-selective effects	(Lohani et al., 2023)
UO$_2$	LDA+U+SOC, tensor expansion, pair-flip	Antiferro-dipolar, ferro-quadrupolar exchange	(Pi et al., 2013)
II-VI chalcogenides	HSE+U hybrid	Optical gaps, formation energies improved	(Aras et al., 2013)
4f/5f dioxides	LDA+U+SIC, many-body mapping	Crystal field splittings within 10–20 meV of experiment	(Zhou et al., 2011)
GaM$_4$Se$_8$	Charge-only LDA+U (FLL), $J_{\mathrm{eff}}$	Molecular $J_{\mathrm{eff}}$ bands, controlled Hund's J	(Lee et al., 2019)
FeAs, TM oxides	Dudarev LDA/GGA+U, empirical $U$ range	Limited moment correction, structure vs. magnetism trade-off	(Griffin et al., 2014, Keshavarz et al., 2018)

LDA+U stands as a foundational, physically motivated, and widely adopted augmentation to standard density functional approximations, providing robust access to the rich phenomenology of correlated electron systems within an efficient mean-field framework. Continued methodological innovation addresses its key limitations through self-consistent $U$, dynamical extensions, and integration with many-body and hybrid functionals.