# Quantum sensor in a single layer van der Waals material

Rohit Babar1,† Gergely Barcza2,† Anton Pershin2,† Hyoju Park3 Oscar Bulancea
Lindvall1 Gergő Thiering2 Örs Legeza2,4,5 Jamie H. Warner  3 Igor A.
Abrikosov1 Adam Gali2,6,∗ and Viktor Ivády1,2,7,∗

###### Abstract

Point defect qubits in semiconductors have demonstrated their outstanding high
spatial resolution sensing capabilities of broad multidisciplinary interest.
Two-dimensional (2D) semiconductors hosting such sensors have recently opened
up new horizons for sensing in the subnanometer scales in 2D heterostructures.
However, controlled creation of quantum sensor in a single layer 2D materials
with high sensitivity has been elusive so far. Here, we report on a novel 2D
quantum sensor, the VB2 centre in hexagonal boron nitride (hBN), with superior
sensing capabilities. The centre’s inherently low symmetry configuration gives
rise to unique electronic and spin properties that implement a qubit in a 2D
material with unprecedented sensitivity. The qubit is decoupled from its dense
spin environment at low magnetic fields that gives rise to the reduction of
the spin resonance linewidth and elongation of the coherence time. The VB2
centre is also equipped with a classical memory that can be utilized in
storing population information. Using scanning transmission electron
microscopy imaging, we confirm the presence of the point defect structure in
free standing monolayer hBN created by electron beam irradiation. Our results
provide a new material solution towards atomic-scale sensing in low
dimensions.

Department of Physics, Chemistry and Biology, Linköping University, SE-581 83
Linköping, Sweden

Wigner Research Centre for Physics, PO Box 49, H-1525, Budapest, Hungary

Walker Department of Mechanical Engineering and Materials Graduate Program,
Texas Materials Institute, The University of Texas at Austin, Austin, Texas
78712, United States

Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, Germany

Institute for Advanced Study, Technical University of Munich,
Lichtenbergstrasse 2a, 85748 Garching, Germany

Department of Atomic Physics, Budapest University of Technology and Economics,
Budafoki út 8., H-1111, Budapest, Hungary

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Street 38,
D-01187 Dresden, Germany

Contributed equally.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
August 28, 2024

## Introduction

In the new era of quantum sensing, point defect qubits play a crucial role in
revolutionizing measurements in material science, biology, and medicine [1].
In particular, the NV center in diamond [2] has provided the means to detect
magnetic field [3], electric field [4], strain [5], and temperature [6] with
high spatial resolution and high sensitivity. Novel magnetic resonance
protocols [7, 8] with unlimited frequency resolution [9, 10] have opened new
horizons for magnetic resonance in the few-spin limit that provides hitherto
inaccessible information on the structure and functioning of molecules and
proteins [11]. On the other hand, the NV centre is an inherently bulk system,
that implies severe challenges when near surface functioning is demanded [12,
13]. Wide-band gap van der Waals semiconductors with mature exfoliation
possibilities, such as the hexagonal boron nitride (hBN), hosting applicable,
optically addressable point defect qubits on the surface and even in a single
layer may be advantageous in various sensing applications [14].

Indeed, hBN has received considerable attention in the domain of point defect
based quantum information processing and quantum sensing applications [15,
16]. There have been several bright single photon emitters reported with
photon energy ranging from near infrared to ultraviolet [15, 17, 18]. The
photoluminescence signal of these centres often exhibits giant shift upon
applied strain and electric field [19]. Recent optically detected magnetic
resonance (ODMR) and magnetic field dependent photoluminescence studies have
demonstrated optically addressable point defect spins in hBN that provides
additional degrees of freedom for sensing [20, 21, 22, 23]. One of the centres
has already been assigned to the negatively charged boron vacancy
(V${}_{\\_}{\text{B}}$) centre [21, 24, 25, 26]. Very recently, the
V${}_{\\_}{\text{B}}$ centre has been coherently manipulated [27, 28] and
proposed as a high spatial resolution sensor for temperature, deformation, and
magnetic field [29]. The sensitivity of this novel sensors, however, most
often falls behind the sensitivity of the NV centre in diamond [30]. The
fundamental reasons behind the relatively low sensitivity of the
V${}_{\\_}{\text{B}}$ centre are the low photon emission rate due to the
first-order forbidden optical transition [24] and the noisy environment due to
the 100% natural abundance of paramagnetic isotopes in hBN. In order to
significantly enhance sensitivity, novel qubits are required that enable fast
and high contrast read-out and decouple the spin state from the spin bath of
the host material.

Here we experimentally and computationally investigate a point defect complex
created by the fusion of two boron vacancies in the neutral charge state. We
demonstrate by annular dark-field scanning transmission electron microscopy
(ADF-STEM) imaging that the stable configuration of the defect, named as VB2
center, can be created in single layer free-standing hBN samples. We report on
nearly degenerate decoupled singlet-triplet pair states throughout the
spectrum of the defect that enable long-lived storage of population
information in the singlet manifold and give rise to electric dipole allowed
optical transitions for reading out the spin states. Furthermore, we
demonstrate that the transverse zero-field splitting interaction of the ground
state gives rise to efficient coherence protection mechanisms. As a result,
the electron spin resonance linewidth is reduced by a factor of $4.2$ and the
inhomogeneous coherence time is elongated by a factor of $7.5$ at zero
magnetic field that directly enhances the sensitive of spin resonance based
sensing applications. The reported favourable properties of the VB2 defect may
be utilized in novel quantum sensing applications with an expected gain in
sensitivity, exceeding two orders of magnitude relative to the boron vacancy
in hBN.

## Main

###  0.1 Structure and stability

Figure 1:  Structure and stability of the VB2 centre.  a, Unrelaxed structure
of two adjacent boron vacancies. b, The energetically most favourable VB2
configuration where the unstable nitrogen atom of two dangling bonds has
jumped into the upper boron vacancy site forming a
V${}_{\\_}{\text{B}}$V${}_{\\_}{\text{N}}$N${}_{\\_}{\text{B}}$ complex. c,
Formation energies of VB2 (magenta) and V${}_{\\_}B$ (light blue) defects in
h-BN as a function of Fermi level under B-poor (solid line) and B-rich (dashed
line) growth conditions. Colour coded numbers represent the charge states of
the corresponding defect, while vertical dotted gray lines indicate the charge
state transition levels of the VB2 defect. d, A false-coloured ADF-STEM image
of the VB2 configuration in single layer free standing hBN sample. Top and
bottom panels show experiment and simulation of the VB2 configuration. e,
Intensity profile across the bright atoms along the yellow dashed box in d
that reveals an anti-site nitrogen next to the vacancy sites.

A pair of adjacent in-plane boron vacancies results in a metastable
configuration where the nitrogen atom between the two vacancies possess two
dangling bonds and binds only to a single boron atom, see Fig. 1a. The loosely
bonded nitrogen atom between the boron vacancies migrates into one of the
vacancy sites and forms a new point defect complex consisting of a boron
vacancy, a nitrogen vacancy, and a nitrogen anti-site
(V${}_{\\_}{\text{B}}$V${}_{\\_}{\text{N}}$N${}_{\\_}{\text{B}}$), see Fig 1b.
The defect complex is referred hereinafter as the VB2 configuration. Due to
the appearance of strong nitrogen-nitrogen bonds in the VB2 configuration, the
formation of the complex is highly favourable. Indeed, the binding energy of
the vacancy pair is found to be as large as $5.14$ eV [31].

The formation energy curves of the VB2 and the V${}_{\\_}\text{B}$ defects as
a function of the Fermi energy are depicted in Fig. 1c. The figure
demonstrates that the VB2 configuration exhibits four stable charge states and
three charge-state transition levels, $(+1/0)$, $(0/\\!\\!-\\!\\!1)$, and
$(-1/\\!\\!-\\!\\!2)$ at $1.78$ eV, $3.61$ eV, and $4.90$ eV, respectively, in
the band gap of hBN. The neutral charge state is stable in the middle of the
band gap and photo-stable up to $3.61$ eV photon energy, which is the
threshold for exciting an electron from the conduction band to a defect
ionisation state, see Fig. 1c.

The VB2 configuration is directly observed using ADF-STEM imaging of a single
layer free standing hBN sample, see Fig. 1d. The image of the VB2
configuration resembles the boron vacancy-nitrogen vacancy complex, however,
the intensity profile across the bright atoms shows an excellent match to the
VB2 complex with an N atom contrast from one of the B atom sites, by comparing
to the multi-slice simulated STEM image in Fig. 1e. We note that the
appearance of the VB2 configuration is a direct consequence of the STEM
measurement itself. In our measurements, a nearly defect free hBN layer is
suspended on a metallic grid. As the electron beam scans through the sample
defects, more likely boron vacancies are created [32]. Prior work showed that
electron beam irradiation of monolayer hBN causes ejection of B and N atoms
around point defects. A range of different small point defects result, and in
some cases the VB2 defect is produced. We note also that high positioning
accuracy may be achieved in defect fabrication in STEM by using focused
electron beam pulses at specific points instead of scanning the surface [32].
The VB2 configuration may also be realized in bulk hBN by either electron or
neutron irradiation and subsequent annealing at $1000$ K. For further details
on fabrication, see Supplementary Note 2 .

###  0.2 Excited state properties

The in-plane and perpendicular to plane dangling bonds of the vacancy sites of
the VB2 configuration give rise to numerous localized defect states, most of
which can be found in the band gap of hBN. A comprehensive analysis of the
single particle electronic structure can be found Supplementary Note 3. As a
consequence of the complicated electronic structure, several excited states
can be found below the ionisation threshold that results in diverse optical
excitation and decay pathways.

Figure 2:  Optical properties, initialisation, and non-destructive readout of
VB2.  a, Energy level structure of the VB2 defect exhibiting quasi-degenerate
singlet-triplet pairs throughout the spectrum. b, PL emission spectra of the
VB2 defect at room temperature. The emission maximum is red-shifted by $60$ nm
compared to the PL emission peak of the boron vacancy center. c, Schematic
diagram of the possible radiative and non-radiative relaxation pathways. Solid
arrows indicate optical transition with parallel to $c$-axis polarization,
dashed arrows indicate spin-conserving non-radiative relaxation, dashed
double-line arrows indicate direct spin-conserving relaxation to the ground
state, and dotted arrows indicate spin non-conserving non-radiative
transitions through inter-system crossings.

The energy level spectrum including the eight lowest lying many-particle
energy levels obtained at the corresponding relaxed geometries is depicted in
Fig. 2a. The spectrum exhibits a peculiar feature not seen in point defect
qubits before: it is built up from nearly degenerate pairs of singlet and
triplet states, referred as excited manifolds hereinafter. The reason behind
the observed quasi-degenerate singlet-triplet pairs can be understood by
examining the ground state triplet. Due to the defect’s low symmetry, the
observed high-spin ground state is the result of an accidental degeneracy in
the single particle electronic structure, see Supplementary Note 3. The
unpaired electrons occupy two defect states that belong to two different
irreducible representations and are localized on distinct first neighbour
atoms of the defect, see the spin density in Fig. 3a. As a result, the
exchange energy is marginal in the ground state and gives rise only to a 3.3
meV splitting between the ground state triplet and the corresponding lowest
energy singlet state. The wave-functions of the low energy optical excited
states exhibit similar characteristics that explain the spectrum depicted in
Fig. 2a.

Since the triplet and singlet states in each manifold have nearly the same
spatial wave-functions and atomic configurations, spin-orbit coupling and
phonon mediated inter-system crossing (ISC) between these states are forbidden
in leading order (El-Sayed’s rule). Consequently, the lowest lying S0 singlet
excited state close to the triplet ground state T0 is long-lived. Indeed, we
find the lifetime of the S0 state to be as long as $39.5$ ms at low
temperature, which remains in the same order of magnitude at room temperature.
This indicates that the S0 singlet state can serve as a classical memory to
store population information.

By analysing the potential decay pathways from the excited states, we find
that the states in the lowest lying ES1 manifold can directly decay non-
radiatively to the ground state and its lifetime is expected to be in the
pico-second range, see Supplementary Note 4. The fast decay suppresses any
optical emission from the T1 triplet and S1 singlet states. On the other hand,
the ES2 manifold is optically active and gives rise to zero-phonon
luminescence (ZPL) lines from the T2 triplet and the S2 singlet states at
$1.578$ eV and $1.586$ eV, respectively. The high temperature PL spectrum of
the triplet emission is depicted in Fig. 2b, which shows a $60$ nm red-shifted
PL phonon side-band compared to the spectrum of the boron vacancy center. The
ES3 manifold decays dominantly to the ES2 manifold via non-radiative
processes.

Fig. 2c depicts the most relevant radiative and non-radiative decay pathways
of the VB2 defect. Besides the optical decay the system can decay non-
radiatively from the ES2 manifold through the ES1 manifold to the ground state
manifold. This process reduces the quantum efficiency of the centre for which
we obtain $0.15$ and $0.12$ for the triplet and the singlet transitions,
respectively. Furthermore, since the ISC rates between the singlet and triplet
channels are at least three orders of magnitude smaller than the spin-
conserving decay rates, see Fig. 2c, the singlet and triplet emissions are
decoupled from each other to a large degree. This effect and the differences
in the optical lifetimes and the emission energies of the T2 triplet and the
S2 singlet states can be used to realize single-shot read-out of the singlet-
triplet occupation, see Supplementary Note 4 for more details.

Since the electronic structure of the VB2 configuration does not exhibit dark
long-lived metastable states the photon-count observed in experiment is
limited by the radiative lifetime, two photon excitation processes, and
collection efficiency of the emitted photons in saturation. Due to the high
photon collection efficiency achievable in hBN [33, 34] and the $\approx$3 MHz
radiative rate of the VB2 centre, the saturated photon-count of a single VB2
center is expected to be in the order of $100$ kcps.

The ISC transitions between the singlet and triplet states are found to be
strictly selective on the spin sub-levels of the triplet states, i.e. only the
$\left|m_{\\_}{\text{S}}=0\right\rangle$ state is connected with the singlet
channel. Consequently, continuous optical excitation of the VB2 center leads
to spin-polarization in the triplet ground state. Furthermore, since the
triplet and singlet radiative relaxation rates are slightly different in the
ES2 manifold, see Fig. 2c, the luminescence of the VB2 defect depends on the
spin state. The saturated spin read-out contrast is found to be 4.6%, see
Supplementary Note 4. Since the contrast is set by the relative luminescence
intensity of the singlet and triplet manifolds, for which we do not expect
considerable temperature dependence, and not by the relative rates of various
non-radiative decay processes, the observed 4.6% contrast may be preserved
even at room temperature. Finally, we note that fast ISC transitions and fast
optical spin initialisation may be achieved by pumping though the ES3
manifold, where ISC rates are found in the MHz range. Such excitation, on the
other hand may not be advantageous for read-out due to the enhanced non-
radiative decay rates that may substantially reduces the quantum efficiency of
the defect.

###  0.3 VB2 centre as a qubit

Figure 3:  Qubit properties of the VB2 centre.  a, Spin density of the triplet
ground state of VB2 centre. b, Rabi oscillation of the electron spin at zero
magnetic field. c Magnetic resonance map of the VB2 centre as a function of
the applied magnetic field (horizontal axis) and the microwave frequency
(vertical axis). The system is driven by an in-plane polarized microwave
field. The broad frequency independent signal at $160$ G is due to the
hyperfine mixing of the spin states observed at the level anti-crossing of the
electron spin levels. d, Magnetic field dependence of the free induction decay
time ($T_{\\_}2^{\star}$). The coherence time enhances by a factor of $7.5$
close to $B=0$ due to the transverse zero-field splitting that partially
suppresses the hyperfine coupling. e, Magnetic resonance signal at $0$ G and
$500$ G magnetic field values. The width of the zero-field electron spin
resonance signal is 23.8% of the width of the signal at $500$ G. This effect
gives rise to a $4.2$ times enhancement of the sensitivity at zero magnetic
field.

As we have shown the spin states of the ground state triplet of the VB2 centre
can be initialised and read-out optically; therefore the spin states of the
defect complex implement an optically addressable qubit in hBN. In the
following, we study the spin and qubit properties of the centre. Fig. 3a
depicts the spin density of the triplet ground state that clearly shows low
symmetric characteristics, in contrast to the existing point defect qubits
that exhibit (quasi-)three-fold rotation symmetry. The spin Hamiltonian of the
triplet ground state is discussed in Supplementary Note 5 in details.

The low symmetry configuration has a significant effect on the zero-field
splitting (ZFS) interaction whose preferential quantization axis is in-plane
and the corresponding conventional ZFS parameters are $D=1.394$ GHz and
$E=78.2$ MHz. Note, however, that the preferential quantization axis of the
spin-orbit interaction is perpendicular to the plane. Hereinafter, we use this
latter quantization axis, in which the electron spin is polarized in the
$\left|m_{\\_}{\text{S}}=0\right\rangle=\left|0\right\rangle$ spin state. In
this basis a new set of non-conventional ZFS parameters can be defined as
$\widetilde{D}=-814$ MHz and $\widetilde{E}=-658$ MHz that provides the same
level structure as the conventional ZFS parameters in the corresponding
quantization basis.

The large effective transverse ZFS $\widetilde{E}$ mixes the
$\left|m_{\\_}{\text{S}}=\pm 1\right\rangle=\left|\pm 1\right\rangle$ spin
states and gives rise to new quantum states
$\left|\pm\right\rangle=\left|+1\right\rangle\pm\left|-1\right\rangle$. By
applying a microwave field that is polarized in perpendicular to plane
direction, spin transitions can be resonantly driven between the
$\left|0\right\rangle$ and $\left|\pm\right\rangle$ state at zero magnetic
field. Rabi oscillation can be observed in the optical signal due to the
microwave drive, see Fig. 3b. In Fig. 3c, we depict the magnetic field
dependence of the resonance signal. As the magnetic field enhances, the zero-
field mixing of the $m_{\\_}{\text{S}}=\pm 1$ states reduces, the states
become the eigenstates of the Zeeman Hamiltonian term, and eventually depend
linearly on the magnetic field. On the other hand, as can be seen in Fig. 3c,
the spin states are independent of the magnetic field in first order at
$B\approx 0$. The zero first-order Zeeman (ZEFOZ) transitions arising at this
field are protected from magnetic fluctuations that results in an elongated
coherence time of the qubit states. We study this effect through the magnetic
field dependence of the free induction decay time ($T_{\\_}2^{\star}$) in Fig.
3d. As can be seen, the coherence time enhances by a factor of $7.5$ at $B=0$
and reaches as high as $128$ ns. Note that the coherence time may be further
elongated by driving a transition between $\left|+\right\rangle$ and
$\left|-\right\rangle$ states with a parallel to plane polarized microwave
field. These coherence protection schemes are indispensable in hBN as it
possesses nuclear spins in its lattice with 100% natural abundance. We note
furthermore that the electron spin resonance linewidth is reduced by a factor
of $4.2$ at zero magnetic field, see Fig. 3e.

Note, furthermore, that the magnetic resonance map in Fig, 3c exhibits a broad
driving frequency independent signal at $B\approx 160$ G. This signal is due
to hyperfine coupling of the environment whose effect is enhanced at the anti-
crossing of the ground state spin levels observed at $B\approx 160$ G, see
Supplementary Note 5. The interplay of the optical pumping and the strong
hyperfine coupling at this magnetic field may give rise to efficient
polarization of surrounding spin bath. Low magnetic field hyperpolarization of
a hBN sample can be utilized in sensitivity enhanced NMR and MRI measurements
[35].

###  0.4 VB2 qubit as a sensor

Finally, we study the sensitivity of the VB2 centre. To this end, we first
determine the pressure and electric field dependence of the ZFS parameters,
for which we obtain $d\widetilde{D}/dp=9.98$ MHz/kBar ($-8.61$ MHz/kBar) and
$d\widetilde{E}/dp=8.50$ MHz/kBar ($-9.36$ MHz/kBar) for pressure applied
parallel (perpendicular) to the V${}_{\\_}{\text{B}}$-V${}_{\\_}{\text{N}}$
axis and $d\widetilde{D}/d\mathcal{E}=0.04$ Hz cm/V ($-3.45$ Hz cm/V) and
$d\widetilde{E}/d\mathcal{E}=3.81$ Hz cm/V ($7.52$ Hz cm/V) for static
electric field $\mathcal{E}$ set parallel (perpendicular) to the
V${}_{\\_}{\text{B}}$-V${}_{\\_}{\text{N}}$ axis. The stress coupling
parameters of the VB2 centre and the V${}_{\\_}{\text{B}}$ centre [29] are
comparable, however, these defects couple two orders of magnitude stronger to
the stress field than the NV center in diamond [36]. The electric field
coupling parameter of the VB2 centre along special directions is comparable to
the NV centre’s electric field coupling strength, however, in general it is
less than an order of magnitude smaller.

The shot noise limited sensitivity of a single VB2 sensor in continuous wave
magnetic resonance measurement scheme can be obtain according to the formula
[37, 38]

$\eta_{\\_}{\text{MR}}=\left(\frac{d\nu_{\\_}{0+}}{d\mathcal{P}}\right)^{-1}\frac{\Delta\nu}{C\sqrt{\mathcal{R}}}\text{,}$
(1)

where $\nu_{\\_}{0+}$ is the resonance frequency, $\mathcal{P}$ is the
perturbation to be measured, $\Delta\nu$ is the full width at half maximum of
the magnetic resonance curve, which takes $49.5$ MHz at zero magnetic field
for electric field, pressure, and temperature sensing and $205$ MHz at high
magnetic field values for DC magnetic field sensing, $C\approx 0.05$ is the
spin read-out contrast, and $\mathcal{R}$ is photon-count that can reach $100$
kcps in saturation. Using this formula and the values specified above, for
pressure, DC electric field, and DC magnetic field sensitivity we obtain $30$
MPa/Hz1/2, $907$ kV/cm/Hz1/2, and $463$ $\mu$T/Hz1/2, respectively, for a
single VB2 defect in hBN in continuous wave sensing mode.

## Discussion

The sensitivity of the VB2 centre in hBN in continuous wave measurement scheme
is lower than that of the NV centre in diamond, dominantly due to the enhanced
magnetic resonance linewidth, typical in hBN, and the slightly lower photon-
count of the VB2 centre. On the other hand, it may still become the leading
contender for high spatial resolution sensing in near surface and low
dimension applications. The dipole allowed optical transition, the relatively
short radiative lifetime, the saturated high-spin read-out contrast, and the
narrowed-down magnetic resonance signal at zero magnetic field are all in
favour of the VB2 centre. Indeed, the reported $3$ MHz radiative recombination
rate is $33$ times faster [26], the 4.6% high temperature read-out contrast is
$30$ times larger [29, 39], and the magnetic resonance linewidth is about $4$
times narrower [21] than that of the V${}_{\\_}{\text{B}}$ centre in hBN.
Combining all these factors in Eq. (1), the saturated high temperature single
defect sensitivity of the VB2 centre may exceed two orders of magnitude
enhancement compared to the leading contender in hBN.

Pulsed sensing protocols and decoherence protection techniques can be utilized
to further enhance the sensitivity of the VB2 centre. The low-symmetry
structure offers new functionalities that have not been reported in two
dimensional wide-band gap semiconductors to date. The long-lived bright
singlet manifold can be utilized as a memory and may lead to the storage of
spin state population information beyond the spin relaxation time of the
ground state triplet. This may be utilized in single-shot read-out of the spin
states when the centre is coupled with an optical cavity. Furthermore,
dressing of the $\left|\pm\right\rangle$ states with resonant microwave drive
may be used in filtering out low frequency magnetic field fluctuations [40]
and gives rise to a highly protected subspace in hBN.

Fabrication of point defect qubits in 2D materials offers new possibilities in
nanometer scale sensing. For example, few-layer hBN samples with pre-
fabricated sensors may be moved on top of nano-scale structures or installed
in van der Waals heterostructure in order to obtain high spatial resolution
information not accessible with other means. The VB2 centre, already
fabricated in single layer hBN, with superior qubit properties provides a
suitable sensor for such applications.

## Summary

In summary, we investigated the peculiar properties of the VB2 centre directly
observed in our STEM measurements in a free-standing single-sheet hexagonal
boron nitride. We showed that the triplet ground state of the VB2 defect
implements an optically addressable spin qubit with novel characteristics. The
possibility of efficient coherence protection mechanism and dipole allowed
optical transition makes VB2 configuration the most suitable candidate today
for emerging point defect qubit applications such as sub-nanoscale sensing in
van der Waals nano and heterostructure.

## Methodology

In this study, we use scanning transmission electron microscopy, three state-
of-the-art electronic structure computational methods, see also Supplementary
Note 1, and exact spin dynamics simulations to form and determine the
properties of VB2 defect.

Sample and scanning transmission electron microscopy measurements The hBN film
was spin coated with PMMA, etched in ammonium borane to remove the Cu, and
then transferred to a TEM grid. The PMMA was removed by acetone overnight and
then the samples were annealed in vacuum overnight. Annular dark-field
scanning transmission electron microscopy (ADF-STEM) was performed using a
JEOL ARM200CF STEM equipped with a CEOS probe corrector operated at an
accelerating voltage of 80 kV. ADF-STEM with a low collection angle was
acquired to increase the signal from light elements and discriminate B and N
atoms as darker and brighter contrast. Dwell times of $5-20$ $\mu$s and a
pixel size of $0.006$ nm/px were used for imaging with a convergence semi-
angle of $22.5$ mrad and a beam current of $35$ pA. Camera length of $20$ cm
was used to provide large collection of low angle scattered electrons,
increasing the contrast of hBN. ImageJ was used for line profile measurements
and image analysis. Multislice image simulations were using QSTEM software
with the accelerating voltage of $80$ kV, the convergence semiangle of $21$
mrad, the detector angle of $40-110$ mrad, the spherical aberration of $0.001$
nm, the chromatic aberration of $1$ mm, and the defocus of $-0.001$ nm. The
ADF-STEM images were colourized with Gem in lookup tables using ImageJ for the
better visibility.

DFT calculations We apply density functional theory (DFT) calculations to
study the energetic, ground state electronic and spin properties, as well as
selected excited state properties of the VB2 defect. Throughout this work, we
use a plane wave basis set of $450$ eV and PAW [41] core potentials as
implemented in VASP [42] and HSE06 hybrid exchange-correlation functional [43]
with $0.32$ exact exchange fraction [44]. For the charge transition level
calculations we use a 768 atom supercell model of bulk hBN and the charge
correction scheme by Freysoldt and colleagues [45]. For all the other
calculations, we use a 162 atom single sheet model of hBN embedding a single
defect complex. In perpendicular direction, we use $30$ Å supercell size.
Furthermore, in single layer calculations we use DFT-D3 correction method of
Grimme and colleagues [46]. We use a ZFS tensor calculation routine in the PAW
formalism as implemented in VASP and spin contamination error correction as
proposed in Ref. [[47]]. We note that the spin contamination error is
comparable with the largest element of the ZFS tensor, thus accurate ZFS
parameters can only be achieved after decontamination. The DFT values are
confirmed by NEVPT2 results for the non-strained configuration.

NEVPT2 calculations

The N-Electron Valence State Perturbation Theory (NEVPT2) [48] calculations
were carried out using cc-pVDZ basis set [49] by the ORCA program package
[50]. To this end, we employed a flake model of hBN, consisting of $38$ B-,
$40$ N-, and $22$ H-atoms. The underlining complete active space self-
consistent field (CASSCF) calculations were performed based on the
unrestricted Hartree-Fock orbitals, while we employed the CAS of $8$ electrons
and $8$ orbitals and considered $6$ triplet and $6$ singlet states for the
state-averaging. The spin-orbit coupling was introduced in the framework of
quasi-degenerate perturbation theory, as implemented in ORCA.

DMRG calculations DMRG [51] is a reference-free variational post-Hartree–Fock
method which relies on an iterative local optimization scheme based on Schmidt
decomposition [52]. The method is natively suitable for describing multi-
reference problems of several dozens of strongly correlated orbitals [53, 54].
As DMRG is capable of treating significantly larger completely active spaces
compared to CASSCF-based methods, it may be used as a reference method for
strongly correlated problems.

DMRG computations are performed applying the Budapest DMRG package [55]
restricting the spin of the target states to $0$ and $1$. In the DMRG
truncation [54], the density matrix is formed of the equally weighted linear
combination of the targeted spin-states and the quantum information loss is
preferred to be kept below threshold value $\chi=10^{-5}$. Considering that
only six defect orbitals display multi-reference character whereas the rest of
the orbitals are found to be contributing to the dynamical correlation
effects, a variant of the standard dynamically extended active space approach
(DEAS) [56] is implemented and applied to initialize environment DMRG block
states. Namely, the orbitals in the DMRG chain are arranged according to their
localization on the defect atoms in descending order and the DMRG system block
is optimized in the configuration space of the environment orbitals which is
composed of determinants allowing up to all single orbital excitations.
Contrary to the standard DEAS implementation, where environment block states
tend to capture the essence of the multi-reference features of the
corresponding orbitals, the modified initialisation scheme focuses on the
description of dynamical correlations of the complete environment from the
initial DMRG micro iterations. The application of the novel DEAS protocol is
favourable for the particular orbital space setups of current interest, i.e.,
orbitals display well distinguished static or dynamical correlation character.
In practice, the relative vertical energies are found to approach convergence
within the first macro iterations of DMRG owing to the effective environmental
initialisation.

The input required for the DMRG calculations, i.e., Hamiltonian matrix
elements, were expanded in the active space of Kohn-Sham DFT orbitals using
program suite ORCA [50, 57]. The spin-restricted Kohn-Sham orbitals expanded
in cc-pVDZ basis set [49] were obtained assuming PBE correlation-exchange
functional and fixing half-occupation on the highest-lying valence defect
orbital pair. Matrix elements were computed by MRCC program [58, 59] applying
the complete active space protocol [60, 61]. In our analysis various active
space setups were tested providing consistent excitation spectra, while the
actual results are presented for 100 canonical orbitals filled with 188
electrons. The NEVPT and the DMRG calculations were performed on the same
flake geometry.

Spin dynamics simulations Exact spin dynamics simulations were carried out by
using a spin Hamiltonian parametrized by our DFT spin coupling parameter
calculations [62]. Magnetic resonance (MR) spectra were obtained by
propagating the states of a four spin model, including the triplet electron
spin and three closest 14N triplet nuclear spins, under oscillating magnetic
field of various frequency and polarization direction. The MR figures show the
change of the initial population of the $m_{\\_}{\text{S}}=\pm 1$ states. The
$T_{\\_}2^{\star}$ coherence time was obtained by simulating the coherent
oscillation of 178 electron-spin–nuclear-spin two-spin systems. The overall
coherence function is obtained as a product of the individual coherence
functions [63]. The decaying coherence function of the electron spin was
fitted by a Gaussian function to obtain the $T_{\\_}2^{\star}$. No nuclear
spin nuclear spin flip-flop processes are included here.

## Data availability

The data that support the findings of this study are available from the
authors upon reasonable request.

## Acknowledgments

We acknowledge support from the Knut and Alice Wallenberg Foundation through
WBSQD2 project (Grant No. 2018.0071). Support from the Swedish Government
Strategic Research Area SeRC and the Swedish Government Strategic Research
Area in Materials Science on Functional Materials at Linköping University
(Faculty Grant SFO-Mat-LiU No. 2009 00971) is gratefully acknowledged. V.I.
acknowledges the support from the MTA Premium Postdoctoral Research Program.
A.G. acknowledges the Hungarian NKFIH grants No. KKP129866 of the National
Excellence Program of Quantum-coherent materials project and the Quantum
Information National Laboratory sponsored by Ministry of Innovation and
Technology of Hungary as well as the support of EU Commission for the project
ASTERIQS (Grant No. 820394). G.B. acknowledges the support from NKFIH
FK-20-135496 project and from the Bolyai Research Scholarship of the Hungarian
Academy of Sciences. Ö.L. acknowledges financial support from the Hungarian
National Research, Development and Innovation Office (NKFIH) through Grants
No. K120569 and No. K134983 and the Hans Fischer Senior Fellowship programme
funded by the Technical University of Munich – Institute for Advanced Study.
The development of DMRG libraries has been supported by the Center for
Scalable and Predictive methods for Excitation and Correlated phenomena
(SPEC), which is funded as part of the Computational Chemical Sciences Program
by the U.S. Department of Energy (DOE), Office of Science, Office of Basic
Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences
at Pacific Northwest National Laboratory. G.T. was supported by the János
Bolyai Research Scholarship of the Hungarian Academy of Sciences and the
ÚNKP-20-5 New National Excellence Program for Ministry Innovation and
Technology from the source of the National Research, Development and
Innovation Fund. The calculations were performed on resources provided by the
Swedish National Infrastructure for Computing (SNIC) at the National
Supercomputer Centre (NSC) and by Wigner Research Centre for Physics. We
acknowledge KIFÜ for awarding us access to computational resource based in
Hungary.

## Author contributions

The DFT calculations were performed by R.B., V.I., O.B.L., and G.T. with
inputs from A.G. and I.A.A. The DFT-CAS-DMRG and NEVPT2 calculations were
performed by G.B. and A.P., respectively, with inputs from V.I., A.G., and
Ö.L. The STEM measurements were carried out by H.P. and J. H. W. The spin
dynamics simulations were carried out by V.I. The results were analysed with
contributions from all authors. V.I. wrote the manuscript with inputs from the
co-authors.

## Competing interests

The authors declare no competing interests.

## References

  * [1] Degen, C., Reinhard, F. & Cappellaro, P.  Quantum sensing.  _Reviews of Modern Physics_ 89, 035002 (2017).  URL https://link.aps.org/doi/10.1103/RevModPhys.89.035002.  Publisher: American Physical Society.
  * [2] Doherty, M. W. _et al._ The nitrogen-vacancy colour centre in diamond.  _Physics Reports_ 528, 1 – 45 (2013).  URL http://www.sciencedirect.com/science/article/pii/S0370157313000562.
  * [3] Taylor, J. M. _et al._ High-sensitivity diamond magnetometer with nanoscale resolution.  _Nature Physics_ 4, 810–816 (2008).  URL https://www.nature.com/articles/nphys1075.  Number: 10 Publisher: Nature Publishing Group.
  * [4] Michl, J. _et al._ Robust and Accurate Electric Field Sensing with Solid State Spin Ensembles.  _Nano Letters_ 19, 4904–4910 (2019).  URL https://doi.org/10.1021/acs.nanolett.9b00900.  Publisher: American Chemical Society.
  * [5] Ovartchaiyapong, P., Lee, K. W., Myers, B. A. & Jayich, A. C. B.  Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator.  _Nature Communications_ 5, 4429 (2014).  URL https://www.nature.com/articles/ncomms5429.  Number: 1 Publisher: Nature Publishing Group.
  * [6] Kucsko, G. _et al._ Nanometre-scale thermometry in a living cell.  _Nature_ 500, 54–58 (2013).  URL https://doi.org/10.1038/nature12373.
  * [7] DeVience, S. J. _et al._ Nanoscale NMR spectroscopy and imaging of multiple nuclear species.  _Nature Nanotechnology_ 10, 129–134 (2015).  URL https://www.nature.com/articles/nnano.2014.313.  Number: 2 Publisher: Nature Publishing Group.
  * [8] Bucher, D. B. _et al._ Quantum diamond spectrometer for nanoscale NMR and ESR spectroscopy.  _Nature Protocols_ 14, 2707–2747 (2019).  URL https://www.nature.com/articles/s41596-019-0201-3.  Number: 9 Publisher: Nature Publishing Group.
  * [9] Schmitt, S. _et al._ Submillihertz magnetic spectroscopy performed with a nanoscale quantum sensor.  _Science_ 356, 832–837 (2017).  URL https://science.sciencemag.org/content/356/6340/832.
  * [10] Boss, J. M., Cujia, K. S., Zopes, J. & Degen, C. L.  Quantum sensing with arbitrary frequency resolution.  _Science_ 356, 837–840 (2017).  URL https://science.sciencemag.org/content/356/6340/837.  Publisher: American Association for the Advancement of Science Section: Reports.
  * [11] Lovchinsky, I. _et al._ Nuclear magnetic resonance detection and spectroscopy of single proteins using quantum logic.  _Science_ 351, 836–841 (2016).  URL https://science.sciencemag.org/content/351/6275/836.  Publisher: American Association for the Advancement of Science Section: Report.
  * [12] Bluvstein, D., Zhang, Z., McLellan, C. A., Williams, N. R. & Jayich, A. C. B.  Extending the quantum coherence of a near-surface qubit by coherently driving the paramagnetic surface environment.  _Phys. Rev. Lett._ 123, 146804 (2019).  URL https://link.aps.org/doi/10.1103/PhysRevLett.123.146804.
  * [13] Dwyer, B. L. _et al._ Probing spin dynamics on diamond surfaces using a single quantum sensor.  _arXiv:2103.12757 [cond-mat, physics:quant-ph]_ (2021).  URL http://arxiv.org/abs/2103.12757.  ArXiv: 2103.12757.
  * [14] Tetienne, J.-P.  Quantum sensors go flat.  _Nature Physics_ 1–2 (2021).  URL https://www.nature.com/articles/s41567-021-01338-5.  Bandiera_abtest: a Cg_type: Nature Research Journals Primary_atype: News & Views Publisher: Nature Publishing Group Subject_term: Qubits;Two-dimensional materials Subject_term_id: qubits;two-dimensional-materials.
  * [15] Tran, T. T., Bray, K., Ford, M. J., Toth, M. & Aharonovich, I.  Quantum emission from hexagonal boron nitride monolayers.  _Nature Nanotechnology_ 11, 37–41 (2016).  URL https://www.nature.com/articles/nnano.2015.242.  Number: 1 Publisher: Nature Publishing Group.
  * [16] Fröch, J. E. _et al._ Coupling Hexagonal Boron Nitride Quantum Emitters to Photonic Crystal Cavities.  _ACS Nano_ 14, 7085–7091 (2020).  URL https://doi.org/10.1021/acsnano.0c01818.  Publisher: American Chemical Society.
  * [17] Hayee, F. _et al._ Revealing multiple classes of stable quantum emitters in hexagonal boron nitride with correlated optical and electron microscopy.  _Nature Materials_ 19, 534–539 (2020).  URL https://www.nature.com/articles/s41563-020-0616-9.  Number: 5 Publisher: Nature Publishing Group.
  * [18] Sajid, A., Ford, M. J. & Reimers, J. R.  Single-photon emitters in hexagonal boron nitride: a review of progress.  _Reports on Progress in Physics_ 83, 044501 (2020).  URL https://doi.org/10.1088/1361-6633/ab6310.  Publisher: IOP Publishing.
  * [19] Li, S. _et al._ Giant shift upon strain on the fluorescence spectrum of V N N B color centers in h -BN.  _npj Quantum Information_ 6, 1–7 (2020).  URL https://www.nature.com/articles/s41534-020-00312-y.  Number: 1 Publisher: Nature Publishing Group.
  * [20] Exarhos, A. L., Hopper, D. A., Patel, R. N., Doherty, M. W. & Bassett, L. C.  Magnetic-field-dependent quantum emission in hexagonal boron nitride at room temperature.  _Nature Communications_ 10, 222 (2019).  URL https://www.nature.com/articles/s41467-018-08185-8.  Number: 1 Publisher: Nature Publishing Group.
  * [21] Gottscholl, A. _et al._ Initialization and read-out of intrinsic spin defects in a van der Waals crystal at room temperature.  _Nature Materials_ 19, 540–545 (2020).  URL https://www.nature.com/articles/s41563-020-0619-6.  Number: 5 Publisher: Nature Publishing Group.
  * [22] Chejanovsky, N. _et al._ Single spin resonance in a van der Waals embedded paramagnetic defect.  _arXiv:1906.05903 [cond-mat, physics:physics]_ (2019).  URL http://arxiv.org/abs/1906.05903.  ArXiv: 1906.05903.
  * [23] Stern, H. L. _et al._ Room-temperature optically detected magnetic resonance of single defects in hexagonal boron nitride.  _arXiv:2103.16494 [cond-mat, physics:quant-ph]_ (2021).  URL http://arxiv.org/abs/2103.16494.  ArXiv: 2103.16494.
  * [24] Ivády, V. _et al._ Ab initio theory of the negatively charged boron vacancy qubit in hexagonal boron nitride.  _npj Computational Materials_ 6, 1–6 (2020).  URL https://www.nature.com/articles/s41524-020-0305-x.  Number: 1 Publisher: Nature Publishing Group.
  * [25] Sajid, A., Thygesen, K. S., Reimers, J. R. & Ford, M. J.  Edge effects on optically detected magnetic resonance of vacancy defects in hexagonal boron nitride.  _Communications Physics_ 3, 1–8 (2020).  URL https://www.nature.com/articles/s42005-020-00416-z.  Number: 1 Publisher: Nature Publishing Group.
  * [26] Reimers, J. R. _et al._ Photoluminescence, photophysics, and photochemistry of the ${{\mathrm{V}}_{\\_}{\mathrm{B}}}^{-}$ defect in hexagonal boron nitride.  _Phys. Rev. B_ 102, 144105 (2020).  URL https://link.aps.org/doi/10.1103/PhysRevB.102.144105.
  * [27] Gottscholl, A. _et al._ Room temperature coherent control of spin defects in hexagonal boron nitride.  _Science Advances_ 7, eabf3630 (2021).  URL https://advances.sciencemag.org/content/7/14/eabf3630.  Publisher: American Association for the Advancement of Science Section: Research Article.
  * [28] Liu, W. _et al._ Rabi oscillation of V$_\text{B}^-$ spin in hexagonal boron nitride.  _arXiv:2101.11220 [quant-ph]_ (2021).  URL http://arxiv.org/abs/2101.11220.  ArXiv: 2101.11220.
  * [29] Gottscholl, A. _et al._ Spin defects in hBN as promising temperature, pressure and magnetic field quantum sensors.  _Nature Communications_ 12, 4480 (2021).  URL https://www.nature.com/articles/s41467-021-24725-1.  Bandiera_abtest: a Cc_license_type: cc_by Cg_type: Nature Research Journals Number: 1 Primary_atype: Research Publisher: Nature Publishing Group Subject_term: Electronic properties and materials;Qubits Subject_term_id: electronic-properties-and-materials;qubits.
  * [30] Gottscholl, A. _et al._ Sub-nanoscale Temperature, Magnetic Field and Pressure sensing with Spin Centers in 2D hexagonal Boron Nitride.  _arXiv:2102.10890 [cond-mat, physics:quant-ph]_ (2021).  URL http://arxiv.org/abs/2102.10890.  ArXiv: 2102.10890.
  * [31] Strand, J., Larcher, L. & Shluger, A. L.  Properties of intrinsic point defects and dimers in hexagonal boron nitride.  _Journal of Physics: Condensed Matter_ 32, 055706 (2019).  URL https://doi.org/10.1088/1361-648x/ab4e5d.
  * [32] Park, H. _et al._ Atomically Precise Control of Carbon Insertion into hBN Monolayer Point Vacancies using a Focused Electron Beam Guide.  _Small_ 17, 2100693 (2021).  URL https://onlinelibrary.wiley.com/doi/abs/10.1002/smll.202100693.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/smll.202100693.
  * [33] Vogl, T., Lu, Y. & Lam, P. K.  Room temperature single photon source using fiber-integrated hexagonal boron nitride.  _Journal of Physics D: Applied Physics_ 50, 295101 (2017).  URL https://doi.org/10.1088/1361-6463/aa7839.  Publisher: IOP Publishing.
  * [34] Li, X., Scully, R. A., Shayan, K., Luo, Y. & Strauf, S.  Near-Unity Light Collection Efficiency from Quantum Emitters in Boron Nitride by Coupling to Metallo-Dielectric Antennas.  _ACS Nano_ 13, 6992–6997 (2019).  URL https://doi.org/10.1021/acsnano.9b01996.  Publisher: American Chemical Society.
  * [35] Bucher, D. B., Glenn, D. R., Park, H., Lukin, M. D. & Walsworth, R. L.  Hyperpolarization-Enhanced NMR Spectroscopy with Femtomole Sensitivity Using Quantum Defects in Diamond.  _Physical Review X_ 10, 021053 (2020).  URL https://link.aps.org/doi/10.1103/PhysRevX.10.021053.  Publisher: American Physical Society.
  * [36] Udvarhelyi, P., Shkolnikov, V. O., Gali, A., Burkard, G. & Pályi, A.  Spin-strain interaction in nitrogen-vacancy centers in diamond.  _Phys. Rev. B_ 98, 075201 (2018).  URL https://link.aps.org/doi/10.1103/PhysRevB.98.075201.
  * [37] Dréau, A. _et al._ Avoiding power broadening in optically detected magnetic resonance of single NV defects for enhanced dc magnetic field sensitivity.  _Physical Review B_ 84, 195204 (2011).  URL https://link.aps.org/doi/10.1103/PhysRevB.84.195204.
  * [38] Pham, L. M.  Magnetic Field Sensing with Nitrogen-Vacancy Color Centers in Diamond (2013).  URL https://dash.harvard.edu/handle/1/11051173.  Accepted: 2013-09-18T14:29:19Z.
  * [39] Liu, W. _et al._ Temperature-Dependent Energy-Level Shifts of Spin Defects in Hexagonal Boron Nitride.  _ACS Photonics_ 8, 1889–1895 (2021).  URL https://doi.org/10.1021/acsphotonics.1c00320.  Publisher: American Chemical Society.
  * [40] Miao, K. C. _et al._ Universal coherence protection in a solid-state spin qubit.  _Science_ 369, 1493–1497 (2020).  URL https://science.sciencemag.org/content/369/6510/1493.
  * [41] Blöchl, P. E.  Projector augmented-wave method.  _Phys. Rev. B_ 50, 17953–17979 (1994).  URL http://link.aps.org/doi/10.1103/PhysRevB.50.17953.
  * [42] Kresse, G. & Hafner, J.  ¡i¿ab initio¡/i¿ molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium.  _Phys. Rev. B_ 49, 14251–14269 (1994).  URL http://link.aps.org/doi/10.1103/PhysRevB.49.14251.
  * [43] Heyd, J., Scuseria, G. E. & Ernzerhof, M.  Hybrid functionals based on a screened coulomb potential.  _J. Chem. Phys._ 118, 8207 (2003).  URL http://dx.doi.org/10.1063/1.1564060.
  * [44] Weston, L., Wickramaratne, D., Mackoit, M., Alkauskas, A. & Van de Walle, C. G.  Native point defects and impurities in hexagonal boron nitride.  _Phys. Rev. B_ 97, 214104 (2018).  URL https://link.aps.org/doi/10.1103/PhysRevB.97.214104.
  * [45] Freysoldt, C., Neugebauer, J. & Van de Walle, C. G.  Fully Ab Initio finite-size corrections for charged-defect supercell calculations.  _Phys. Rev. Lett._ 102, 016402 (2009).  URL http://link.aps.org/doi/10.1103/PhysRevLett.102.016402.
  * [46] Grimme, S., Antony, J., Ehrlich, S. & Krieg, H.  A consistent and accurate ab initio parametrization of density functional dispersion correction (dft-d) for the 94 elements h-pu.  _The Journal of Chemical Physics_ 132, 154104 (2010).  URL https://doi.org/10.1063/1.3382344.  https://doi.org/10.1063/1.3382344.
  * [47] Biktagirov, T., Schmidt, W. G. & Gerstmann, U.  Spin decontamination for magnetic dipolar coupling calculations: Application to high-spin molecules and solid-state spin qubits.  _Physical Review Research_ 2, 022024 (2020).  URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.022024.  Publisher: American Physical Society.
  * [48] Angeli, C., Cimiraglia, R., Evangelisti, S., Leininger, T. & Malrieu, J.-P.  Introduction of n-electron valence states for multireference perturbation theory.  _The Journal of Chemical Physics_ 114, 10252–10264 (2001).  URL https://aip.scitation.org/doi/10.1063/1.1361246.  Publisher: American Institute of Physics.
  * [49] Dunning, T. H.  Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.  _The Journal of Chemical Physics_ 90, 1007–1023 (1989).  URL https://aip.scitation.org/doi/10.1063/1.456153.  Publisher: American Institute of Physics.
  * [50] Neese, F.  Software update: the orca program system, version 4.0.  _Wiley Interdisciplinary Reviews: Computational Molecular Science_ 8, e1327 (2018).
  * [51] White, S. R. & Martin, R. L.  Ab initio quantum chemistry using the density matrix renormalization group.  _The Journal of Chemical Physics_ 110, 4127–4130 (1999).  URL https://doi.org/10.1063/1.478295.  https://doi.org/10.1063/1.478295.
  * [52] Schollwöck, U.  The density-matrix renormalization group.  _Rev. Mod. Phys._ 77, 259–315 (2005).  URL https://link.aps.org/doi/10.1103/RevModPhys.77.259.
  * [53] Olivares-Amaya, R. _et al._ The ab-initio density matrix renormalization group in practice.  _J. Chem. Phys._ 142 (2015).  URL https://aip.scitation.org/doi/10.1063/1.4905329.
  * [54] Szalay, Sz. _et al._ Tensor product methods and entanglement optimization for ab initio quantum chemistry.  _Int. J. Quantum Chem._ 115, 1342–1391 (2015).  URL http://dx.doi.org/10.1002/qua.24898.
  * [55] Legeza, Ö., Veis, L. & Mosoni, T.  QC-DMRG-Budapest, a program for quantum chemical DMRG calculations (2020).
  * [56] Legeza, O. & Sólyom, J.  Optimizing the density-matrix renormalization group method using quantum information entropy.  _Phys. Rev. B_ 68, 195116 (2003).  URL https://link.aps.org/doi/10.1103/PhysRevB.68.195116.
  * [57] Neese, F.  The orca program system.  _Wiley Interdiscip. Rev.: Comput. Mol. Sci._ 2, 73–78 (2012).  URL https://onlinelibrary.wiley.com/doi/abs/10.1002/wcms.81.
  * [58] Kallay, M. _et al._ Mrcc, a quantum chemical program suite (2020).  URL www.mrcc.hu.
  * [59] Kallay, M. _et al._ The mrcc program system: Accurate quantum chemistry from water to proteins.  _J. Chem. Phys._ 152, 074107 (2020).  URL https://doi.org/10.1063/1.5142048.
  * [60] Jensen, F.  _Introduction to Computational Chemistry_ (John Wiley and Sons, Inc., Hoboken, NJ, USA, 2006).
  * [61] Barcza, G. _et al._ DMRG on Top of Plane-Wave Kohn-Sham Orbitals: A Case Study of Defected Boron Nitride.  _J Chem Theory Comput_ 17, 1143–1154 (2021).
  * [62] Ivády, V., Abrikosov, I. A. & Gali, A.  First principles calculation of spin-related quantities for point defect qubit research.  _npj Computational Materials_ 4, 76 (2018).  URL https://doi.org/10.1038/s41524-018-0132-5.
  * [63] Seo, H. _et al._ Quantum decoherence dynamics of divacancy spins in silicon carbide.  _Nature Communications_ 7, 1–9 (2016).  URL https://www.nature.com/articles/ncomms12935.  Number: 1 Publisher: Nature Publishing Group.