Multiconfigurational nature of electron correlation within nitrogen vacancy centers in diamond

Diamond is a solid-state platform used to develop quantum technologies, but it has been a long-standing problem that the current understanding of quantum states of nitrogen vacancy (NV) centers in diamond is mostly limited to single-electron pictures. Here, we combine the full configuration interaction quantum Monte Carlo method and the density-matrix functional embedding theory to achieve an accurate description of the many-body quantum states of such defects in diamond. More than 30 electrons and 130 molecular orbitals are correlated, revealing the multiconfigurational nature of the wave functions of the many-body quantum states therein. Such a description explains puzzling experimental measurements in intersystem crossing and charge-state transitions at NV centers in diamond. The calculations not only reproduce the available experimental measurements of the energy gaps between quantum states but also provide benchmarks for states that are still subject to considerable uncertainty.

Figure 1

Flowchart of this work. (a) Left: the local structure of the ${\mathrm{NV}}^{-}$ center in diamond. Right: a sketch of the single-particle band structure and related states between the valence band (VB) and the conduction band (CB; degenerate $e_{x,y}$, $a_1$, and $a_1^{\prime }$), where the arrows indicate the spin occupations of the ground state. (b) The partitioning scheme of a cluster ${\mathrm{C}}_{42}{\mathrm{H}}_{42}{\mathrm{N}}^{-}$ containing the ${\mathrm{NV}}^{-}$ center. (c) A sketch of constructing the DFT embedding potential using DMFET, orbital orthogonalization, and orbital projection. (d) The multiconfigurational electronic structure of the ${\mathrm{NV}}^{-}$ center. Left: showing four important low-lying states: the ground triplet state ${}^3\overline{A}_2$, the optical excited triplet state ${}^3\overline{E}$, and the dark intermediate singlet states ${}^1\overline{A}_1$ and ${}^1\overline{E}$. The green dashed arrows highlight the intersystem crossing (ISC) processes ${}^3\overline{E}\leftrightarrow {}^1\overline{A}_1$ and ${}^1\overline{E}\leftrightarrow {}^3\overline{A}_2$. Right: the configurations of the ground state ${}^3\overline{A}_2$. The blue box represents the single-configurational picture, while the red box highlights the strongly correlated multiconfigurational picture obtained by FCIQMC-in-DFT.

Figure 2

(a) Flowchart of FCIQMC-in-DFT. (b) Calculated frontier molecular orbitals $e_{x,y}$ and $a_1$. The translucent orange atoms represent carbon, and blue atoms represent nitrogen. Hydrogen atoms are not plotted. The Kohn-Sham (KS) orbitals are obtained by DFT calculations with the HSE06 exchange correlation functional, 6-31G basis set, and Gaussian smearing. Starting from the initial system ${\mathrm{C}}_{42}{\mathrm{H}}_{42}{\mathrm{N}}^{-}$, the DFT embedding potential $V_{\mathrm{emb}}^A$ of the embedded region ${\mathrm{C}}_3{\mathrm{N}}^{-}$ is obtained through DMFET, orbital orthogonalization, and orbital projection (O. and P., respectively). Finally, FCIQMC is performed using the restricted Hartree-Fock (HF) orbitals with $V_{\mathrm{emb}}^A$ and the 6-31G basis set.

Figure 3

FCIQMC calculations of ${\mathrm{NV}}^{-}$. (a) The FCIQMC energies of ${}^1\overline{E}$ and ${}^3\overline{A}_2$ as a function of the number of walkers. (b) The corresponding excitation energies of ${}^1\overline{E}\leftrightarrow {}^3\overline{A}_2$. The red dashed line shows the final estimate at 0.572 eV. (c) Embedded frontier molecular orbitals $e_{x,y}$ and $a_1$. (d) The amplitude of the 20 most populated configurations from FCIQMC. (e) The configuration graphs of ${}^1\overline{E}$ (left) and ${}^3\overline{A}_2$ (right) states, characterizing the population and connection of configurations sampled from FCIQMC [38]. The size of each circle represents the absolute value of the CI coefficient, and the length of connecting lines correlates inversely with the transition matrix element between two configurations. The four most populated configurations have been highlighted by dashed lines. (f) The amplitude of the four most populated configurations of the spin triplet ${}^3\overline{A}_2$ (orange) and the singlet state ${}^1\overline{E}$ (blue). The corresponding configurations are described by frontier orbitals $x$, $y$, and $a$ [solid lines in the inset, same as (c)], as well as orbitals with lower energies and higher energies (dashed lines in the inset), occupied by four spin electrons. The asterisk represents a normalized conjugate configuration. (g) In multiconfigurational electronic state ${}^1\overline{E}$, the contributions ${\left|C\right|}^2$ of three mixed-in single-configurational states ${}^1{E}^{\prime }$, ${}^{1}A_1$, and ${}^1{A}_1^{\prime }$. $C$ represents the CI expansion coefficient corresponding to the single-configurational state.

Figure 4

Vertical excitation energies (VEEs) relative to the ground state ${}^3\overline{A}_2$ (0 eV) of excited states ${}^1\overline{E}$, ${}^1\overline{A}_1$, and ${}^3\overline{E}$ of ${\mathrm{NV}}^{-}$. Here, clust., cluster model; ext. Hub., extended Hubbard model; Hub., Hubbard model; period., periodic model. The energy of ${}^3\overline{E}$ obtained in this paper (FCIQMC-in-DFT, highlighted in orange) is marked with an asterisk due to the influence of image states. Refs. a and b, Refs. [32] and [40] (experimental values: the VEE for ${}^3\overline{E}\leftrightarrow {}^3\overline{A}_2$ is 2.20 eV, and the zero phonon line (ZPL) is 1.945 eV [32]; the ZPL for ${}^1\overline{A}_1\leftrightarrow {}^1\overline{E}$ is 1.19 eV [40]); Ref. c, Ref. [18] (DFT calculations of ${\mathrm{C}}_{284}{\mathrm{H}}_{144}{\mathrm{N}}^{-}$ and stochastic CI of ${\mathrm{C}}_{42}{\mathrm{H}}_{42}{\mathrm{N}}^{-}$); Ref. d, Ref. [17] (extended Hubbard cluster model); Ref. e, Ref. [16] ($\mathrm{GW}+\mathrm{BSE}$); Ref. f, Ref. [19] (CI-cRPA); Ref. g, Refs. [20, 39] (variational quantum eigensolvers); Ref. h, Ref. [9] (extended Hubbard model fitted to GW).

Figure 5

Multiconfigurations of ${\mathrm{NV}}^0$ and ${\mathrm{NV}}^{+}$. (a) Amplitude of the 20 most populated configurations for the ground state ${}^2\overline{E}$ of ${\mathrm{NV}}^0$. The inset shows the corresponding configuration graph. (b) The three dominating configurations for ${}^2\overline{E}$ of ${\mathrm{NV}}^0$ with orbital occupation indicated. (c) and (d) Same as (a) and (b) for the ground state ${}^1\overline{A}_1$ and excited state ${}^3\overline{E}$ of ${\mathrm{NV}}^{+}$.

Figure 6

Calculations on different charge states of NV centers. (a) For the ground states ${}^3\overline{A}_2$, ${}^2\overline{E}$, and ${}^1\overline{A}_1$ of the three charged NV centers, the relative contribution of the sum of the type II and type III configurations with respect to the type I configuration (the first three most populated configurations only involving the $x$, $y$, and $a$ orbitals). The coefficient 0.5 in the definition of the relative contribution (RC) results from the assumption that the depletion contribution of type II is half that of type III. (b) Total natural occupancy (spin up plus spin down) for the ground states of the three charged NV centers. Red points highlight the natural atomic orbital (NAO) occupancy corresponding to the $a$ orbital. (c) The blue histogram (left axis) shows the difference between the experimental and DFT-calculated values for nuclear spin quadrupole splittings ($\mathrm{\Delta }|C_q|$ in units of MHz) via data reported in Ref. [10]. The red dashed line (right axis) shows the natural occupancy depletion of $a$ ($\mathrm{\Delta }|O_a|$) calculated from FCIQMC. (d) The NAO occupancy of the spin-majority channel (spin up, left) and the spin-minority channel (spin down, right) of three charged ground states [same as (b)]. Dep., the spin channel of main charge depletion.