Calculation of the energies of the multideterminant states of the nitrogen vacancy center in diamond with quantum Monte Carlo

Certain point defects in solids can efficiently be used as qubits for applications in quantum technology. They have spin states that are initializable, readable, robust, and can be manipulated optically. New theoretical methods are needed to find the best host materials and defect configurations. Most methods proposed so far rely either on cluster models or restrict the many-body treatment of the defects to a subspace of single-particle orbitals. We explore best practices and theory for the use of quantum Monte Carlo to predict the excitation spectra for spin defects, by using the negatively charged nitrogen vacancy (${\mathrm{NV}}^{-}$) center in diamond as a test system. Quantum Monte Carlo can be used to explicitly simulate electronic correlations with larger systems and sets of orbitals than previous methods due to favorable scaling with respect to system size and computing power. We consider different trial wave functions for variational and diffusion Monte Carlo methods, explore the nodal surface errors of the ground and excited state wave functions, and study whether the variational principle holds for the excited states. We compute the vertical excitation energies in different simulation cell sizes and extrapolate to infinite system size, and include backflow corrections to the extrapolated energies. The final results for vertical excitation energies are found to overestimate the experimental estimates, but the triplet-to-triplet and singlet-to-singlet transitions are accurate against experiment. Finally, we list further developments for QMC needed to address the problem of accurately predicting structural and spin properties of the solid-state defects.

Figure 1

Sketch of energy structure of the single-particle orbitals (left) and the many-body states (right) of the ${\mathrm{NV}}^{-}$ center. A split between the spin channels of the single-particle orbitals is shown, so that the spin-majority (spin-up) channel in the left has orbitals with lower energies than the spin-minority (spin-down) channel. The electronic occupation of the single-particle orbitals in the $3A_2$ ground state is denoted for spin-up (upward arrow) and spin-down (downward arrow) electrons. The many-body states in energetic order from lowest to highest energy are $3A_2$ (blue), $1E$ (pink), $1A_1$ (green), and $3E$ (red). The double arrows denote the triplet-to-triplet and singlet-to-singlet transitions, with experimental values shown in eV. The energies of the doubly degenerate orbitals $e_{x,y}$ and the many-body states $3E$ and $1E$ are represented with two lines, displaced vertically for easier depiction.

Figure 2

The single-particle orbitals $a_1$ (left), $e_x$ (middle), and $e_y$ (right), shown from the direction of the high-symmetry axis of the ${\mathrm{NV}}^{-}$ center. The black spheres represent atoms nearest to the vacancy; the atom in the middle is N, the others are C, and N is behind the C atoms in the figure. The negative (blue) and positive (red) parts of the orbitals are plotted. The orbitals are obtained from a spin unpolarized DFT simulation with a PBE correlation functional.

Figure 3

Energies of the S.U. (top row) and S.R. (bottom) single-particle orbitals in 63- and $215\text{−}\mathrm{atom}$ simulation cells, computed with PBE and HSE06 correlation functionals. The green line represents the energy of the orbital with $a$ symmetry, and red the energies of orbitals with $e$ symmetry. The red lines representing degenerate states are vertically displaced for presentative purposes.

Figure 4

Total energies (top) and vertical excitation energies (bottom) of the Slater-Jastrow many-body states of the ${\mathrm{NV}}^{-}$ center in a $64\text{−}\mathrm{atom}$ simulation cell as a function of the DMC time step, with DMC error bars. The dashed lines represent linear fits to the state energies.

Figure 5

DFT results of the absorption energy, zero-phonon line, Stokes and anti-Stokes shifts obtained with HSE06 hybrid functional for cubic 63-(red), 215-(cyan), and 512-(green) atom simulation cells. For comparison, also HSE06 results with $512\text{−}\mathrm{atom}$ simulation cell by Gali et al. [26] and experimental results are presented.

Figure 6

Results of the QMC energies of the states of the ${\mathrm{NV}}^{-}$ center against the lowest obtained ground-state energy in the 63- (a) and 215-atom (b) simulation cells with the Slater-Jastrow (SJ) wave function, and in the $63\text{−}\mathrm{atom}$ cell with a backflow correction (c). The SJ wave function results are computed with electronic orbitals prepared using PBE and HSE06 correlation functionals in S.U. and S.R. DFT calculations. The results with backflow are computed with HSE06 S.R orbitals for the $S_z=0$ and with HSE06 S.U. orbitals for the $S_z=1$ spin manifold. Results for the states $3A_2$ (blue), $1Ex$ and $y$ components (violet and pink), $1A_1$ (green), and $3Ex$ and $y$ components (orange and red) are shown. For the triplets represented with HSE06 S.R. orbitals, also the triplet states with $S_z=0$ spin projections are shown (lighter orange). In the $216\text{−}\mathrm{atom}$ simulation cell, we do not present results for the triplets with PBE S.R. orbitals and for the singlets with HSE06 S.U. orbitals. For some of the results, such as for the energies of the $x$ and $y$ components of the $3E$ state with HSE06 S.U. orbitals, the energies are overlapping although it is not visible from the figure.

Figure 7

Energies compared against the $S_z=1$ projection of the $3A_2$ ground state with HSE06 S.R. orbitals. Results are presented for Slater-Jastrow wave functions as a function of the inverse number of electrons in the simulation cell (right panels), for SJ wave functions with energies extrapolated to infinite system size (vertical line at 0), and for the extrapolated results with a backflow correction [left panel, results at $0-SJB\left(63\right)]$. In (a), the energies of the $S_z=0$ and $Sz=1$ projections of the $3A_2$ state, represented with S.R. orbitals, are compared. In (b), the vertical excitation energies of the $1E, 1A_1$, and $3E$ states, drawn with pink, green, and red, respectively, are presented. The singlets were represented with S.R. orbitals, and the $3E$ with S.U. orbitals and the $S_z=1$ projection of the triplet.

Figure 8

Vertical excitation energies of the $1E$ (pink), $1A_1$ (green), and $3E$ states (red) against the ground state (blue). Results are shown from experiments [48, 49, 52, 53] and QMC (this paper), compared against results from a number of theoretical methods: hybrid time-dependent DFT (hybrid TDDFT) [50], complete active space multiconfiguration SCF (CASSCF) [31], spin-flip Bethe-Salpeter equation approach (Spin-flip BSE) [79], GW approximation and BSE ($\mathrm{GW}+\mathrm{BSE}$) [64], and constrained random-phase approximation (cRPA) [32].