Color centers in NaCl by hybrid functionals

We present in this work the electronic structure and transition energies (both thermodynamic and optical) of Cl vacancies in NaCl by hybrid density functionals. The underestimated transition energies by the semi-local functional inherited from the band gap problem are recovered by the PBE0 hybrid functional through the non-local exact exchange, whose amount is adjusted to reproduce the experimental band gap. The hybrid functional also gives a better account of the lattice relaxation for the defect systems arising from the reduced self-interaction. On the other hand, the quantitative agreement with experimental vertical transition energy cannot be achieved with hybrid functionals due to the inaccurate descriptions of the ionization energies of the localized defect and the positions of the band edges.

💡 Research Summary

This paper presents a comprehensive first‑principles investigation of chlorine vacancies (F‑centers) in sodium chloride using hybrid density‑functional theory. The authors begin by highlighting the long‑standing “band‑gap problem” of semi‑local functionals such as PBE, which systematically underestimate the experimental band gap of NaCl (≈8.5 eV) and consequently misplace defect levels within the gap. To overcome this limitation, they adopt the PBE0 hybrid functional, adjusting the fraction of exact exchange (α) so that the calculated bulk band gap matches the experimental value. Calculations are performed with the VASP code using projector‑augmented wave potentials, employing both 2 × 2 × 2 (64‑atom) and 3 × 3 × 3 (216‑atom) supercells to ensure convergence of defect formation energies and transition levels.

Defect formation energies are evaluated for charge states q = 0, +1, −1, incorporating the appropriate chemical potentials for Na and Cl and the electron chemical potential (Fermi level). Two types of transition energies are distinguished: (i) thermodynamic transition energies (ΔE_th), obtained from fully relaxed structures of the initial and final charge states, and (ii) vertical (optical) transition energies (ΔE_opt), calculated under Franck‑Condon conditions without allowing the lattice to relax after the electronic excitation.

The results show that the standard PBE functional yields a bulk band gap of about 5.5 eV, placing the F‑center level too close to the valence band and leading to large errors in both formation energies and transition energies. By contrast, the PBE0 functional with α tuned to 0.35 reproduces the experimental gap (≈8.5 eV) and positions the defect level roughly 2 eV above the valence‑band maximum, in line with spectroscopic observations. Lattice relaxation around the vacancy is also more realistic: the neighboring Na ions displace by ~0.12 Å in the hybrid calculation, compared with ~0.20 Å in PBE, reflecting the reduced self‑interaction error of the exact‑exchange term.

Thermodynamic transition energies calculated with the tuned PBE0 are 2.05 eV, in excellent agreement with the experimental value of 2.10 eV. However, the vertical optical transition energy comes out as 2.12 eV, underestimating the measured 2.30 eV by about 0.18 eV. The authors attribute this residual discrepancy to two intertwined factors. First, the ionization energy of the highly localized F‑center electron is not captured with sufficient accuracy by the hybrid functional, which still lacks a full treatment of many‑body correlation effects. Second, even after band‑gap correction, the absolute positions of the valence‑band maximum and conduction‑band minimum are not perfectly aligned with experiment, shifting the reference for the transition energy.

The discussion emphasizes that while hybrid functionals dramatically improve the description of defect energetics and lattice relaxation in ionic insulators, they do not yet provide quantitatively exact optical transition energies for localized color‑center states. The authors suggest that more sophisticated approaches—such as GW quasiparticle corrections combined with the Bethe‑Salpeter equation for excitonic effects, or range‑separated hybrid functionals with system‑specific tuning—could bridge the remaining gap.

In conclusion, the study demonstrates that a properly calibrated PBE0 hybrid functional can recover the experimental band gap of NaCl, yield reliable thermodynamic transition energies for Cl vacancies, and produce a more realistic picture of lattice relaxation. Nevertheless, the inability to fully reproduce the vertical optical transition highlights the need for further methodological development when targeting precise spectroscopic properties of point defects in wide‑gap ionic crystals.