Variational Calculations of the Excited States of the Charged NV-center in Diamond Using a Hybrid Functional

TL;DR

This work addresses the challenge of accurately describing excited states of the NV- center in diamond within a scalable first-principles framework. It employs time-independent variational density-functional theory (DO-MOM) with the HSE06 hybrid functional to compute both vertical and adiabatic excitations, including relaxed structures, for the NV- defect. The results show a vertical triplet excitation at 2.29 eV and a zero-phonon line at 2.01 eV for the triplet manifold, with singlet states at 0.72 eV (vertical) and 0.60 eV (relaxed) relative to the ground state, all in good agreement with experimental data, and reveal Jahn-Teller–like distortions in the excited-state relaxations. The study demonstrates that variational DFT with a hybrid functional can provide accurate, experimentally consistent excited-state energetics and offers a practical screening tool for identifying defect systems suitable for quantum technologies. This approach thus bridges gap between high-level correlation methods and scalable solid-state calculations, enabling more reliable predictions for defect-based quantum devices.

Abstract

The excited electronic states involved in the optical cycle preparation of a pure spin state of the negatively charged NV-defect in diamond are calculated using the HSE06 hybrid density functional and variational optimization of the orbitals. This includes the energy of the excited triplet as well as the two lowest singlet states with respect to the ground triplet state. In addition to the vertical excitation, the effect of structural relaxation is also estimated using analytical atomic forces. The lowering of the energy in the triplet excited state and the resulting zero-phonon line triplet excitation energy are both within 0.1 eV of the experimental estimates. An analogous relaxation in the lower energy singlet state using spin purified atomic forces is estimated to be 0.06 eV. These results, obtained with a hybrid density functional, improve on previously published results using local and semi-local functionals, which are known to underestimate the band gap. The good agreement with experimental estimates demonstrates how time-independent variational calculations of excited states using density functionals can give accurate results and, thereby, provide a powerful screening tool for identifying other defect systems as candidates for quantum technologies.

Figures (3)

• Figure 1: (a) Atomic structure of the NV$^-$ center and the threefold symmetry axis [111]. (b) The energy levels of spin orbitals within the band gap. A bar over the orbital symbol indicates spin down. (c) The orbitals $e_x$ and $e_y$. (d) Schematic illustration of the optical cycle used for preparing the spin-pure ground state. Measured values of the zero phonon lines (ZPL) of triplets davies_optical_1976 and singlets rogers2008infrared are indicated. Intersystem crossings (ISC) are shown as dashed lines. (e) The highest occupied orbitals of mixed states $^m\Phi_2$ and $^1\Phi_3$. Right hand side illustrate the occupied spin orbitals of $^m\Phi_2$ and $^1\Phi_3$.
• Figure 2: Energy levels of NV$^-$ center. The solid lines denote vertical excitations and the dashed lines denote relaxed energy levels. The colored region indicate the experimental estimates of $^1A_1$ and $^1E$ energy levels. $^a$ Ref. chen2023multiconfigurational, $^b$ Ref. ma_quantum_2020, $^c$ Ref. ivanov_electronic_2023, $^d$ Ref. davies_optical_1976wolf_nitrogen-vacancy_2023. Ref. davies_optical_1976 reports a vertical excitation energy of 2.180 eV and the ZPL of 1.945 eV. Ref. wolf_nitrogen-vacancy_2023 gives the experimental estimates of the singlet states.
• Figure 3: (a-b) Atomic displacements from $^3A_2$ to $^3E$ shown from two different viewpoints. The opaque and transparent atoms are from $^3A_2$ and $^3E$, respectively. (c-d) The atomic displacements from $^3A_2$ to $^1E$. The opaque and transparent atoms are from $^3A_2$ and $^1E$, respectively. Structural relaxation is highly localized around the NV defect. Apart from the displaced atoms adjacent to the NV center, the atomic displacements of all other carbon atoms in the supercell are approximately one order of magnitude smaller and are omitted from the figure for clarity. The displacements are doubled for clearer visualization.