# Excited-State Dynamics and Optically Detected Magnetic Resonance of Solid-
State Spin Defects from First Principles

Kejun Li  Department of Physics, University of California, Santa Cruz,
California, 95064, USA  Vsevolod D. Dergachev  Department of Chemistry,
University of Nevada, Reno, Nevada, 89557, USA  Ilya D. Dergachev  Department
of Chemistry, University of Nevada, Reno, Nevada, 89557, USA  Shimin Zhang
Department of Materials Science and Engineering, University of Wisconsin-
Madison, 53706, USA  Sergey A. Varganov  Department of Chemistry, University
of Nevada, Reno, Nevada, 89557, USA  Yuan Ping<EMAIL_ADDRESS>Department of
Materials Science and Engineering, University of Wisconsin-Madison, 53706, USA
Department of Physics, University of California, Santa Cruz, California,
95064, USA

###### Abstract

Optically detected magnetic resonance (ODMR) is an efficient and reliable
method that enables initialization and readout of spin states through spin-
photon interface. In general, high quantum efficiency and large spin-dependent
photoluminescence (PL) contrast are desirable for reliable quantum information
readout. However, reliable prediction of the ODMR contrast from first-
principles requires accurate description of complex spin polarization
mechanisms of spin defects. These mechanisms often include multiple radiative
and nonradiative processes in particular intersystem crossing (ISC) among
multiple excited electronic states. In this work we present our implementation
of the first-principles ODMR contrast, by solving kinetic master equation with
calculated rates from ab initio electronic structure methods then benchmark
the implementation on the case of the negatively-charged nitrogen vacancy (NV)
center in diamond. We show the importance of correct description of multi-
reference electronic states for accurate prediction of excitation energy,
spin-orbit coupling, and the rate of ISC. Moreover, we underscore the
importance of pseudo Jahn-Teller effect for the spin-orbit coupling, and the
dynamical Jahn-Teller effect for electron-phonon coupling, key factors
determining ISC rates and ODMR contrast. We show good agreement between our
first-principles calculations and the experimental ODMR contrast under
magnetic field. We then demonstrate reliable predictions of magnetic field
direction, pump power, and microwave frequency dependency, as important
parameters for ODMR experiments. Our work clarifies the important excited-
state relaxation mechanisms determining ODMR contrast and provides a
predictive computational platform for spin polarization and optical readout of
solid-state quantum defects from first principles.

††preprint: AIP/123-QED

##  I Introduction

Point defects in solids as spin qubits offer multiple revenues to quantum
technologies, in particular quantum sensing and quantum networking [1]. The
promising candidates exhibit long quantum coherence times [2, 3, 4], essential
for performing multi-step quantum operations with high fidelity. They are
relatively scalable and easy to integrate into existing technologies due to
their solid-state nature [5]. Furthermore, they can be optically initialized
and reliably read out in a wide temperature range [6, 7, 8], through the spin-
photon entanglement [9]. While discovering and exploring new spin defects
beyond the extensively-studied negatively-charged nitrogen-vacancy (NV) center
in diamond [10] has been a key interest in recent years, reliable theoretical
predictions of optical readout properties remain challenging, which hinder
rapid progress.

The first outstanding issue is the missing link between experimental
observable and first-principles simulations for the spin polarization
processes. The experimental tool to study spin polarization is through
optically detected magnetic resonance (ODMR) [11, 12, 13, 14], by recording
photoluminescence contrast with and without microwave radiation as a function
of magnetic field. This requires to solve the kinetic equations for excited-
state occupations of different spin sublevels. The excited-state occupation is
determined by kinetic processes in the polarization cycle, including radiative
and nonradiative recombination between different spin states. Previously, only
model simulations with experimental rates and energy levels to describe ODMR
have been reported [11, 12, 13, 14]. A first-principles formalism and
computational tool is yet to be developed in order to interpret experiments or
predict ODMR of new spin defect systems.

To perform first-principles ODMR, predictions of optical excitation energies,
zero-field splitting, radiative and nonradiative recombination rates are
required. Various first-principles electronic structure methods have been
proposed to calculate excitation energies of spin defects. The key
consideration is to accurately capture the electron correlation of defect
states. For instance, mean-field theory such as constrained DFT (CDFT) [15]
could provide reasonable structural and ground state information, but GW and
solving the Bethe-Salpeter Equation (GW-BSE) can provide more accurate
quasiparticle energies and optical properties including electron-hole
interactions [16, 17]. On the other hand, if the defect states have strong
multi-reference nature, such as open-shell singlet excited state of the NV
center, theories like quantum-defect-embedding theory (QDET) for solids [18,
19] or multi-reference wavefunction methods [20] may be more appropriate.

The first-principles theory for excited-state kinetic properties of spin
defects has recent advancement as well. For example, the radiative
recombination rates have been calculated at finite temperature with inputs
from many-body perturbation theory for better description of excitons [18,
21]. The nonradiative recombination transitions, in particular intersystem
crossing (ISC) that requires spin-flip transition and phonon-assisted internal
conversion (IC), can be obtained from phonon perturbation to electronic state
[22, 16, 23, 15]. However, these calculations have not considered the multi-
reference nature of the electronic states and complex Jahn-Teller (JT) effects
coupled in the kinetic processes.

In this work, we have developed the first-principles ODMR theory and
computational tool to directly predict ODMR contrast without prior input
parameters as shown in Fig. 1(a). We note that such tool is fully general to
all solid-state defect systems, although we first use the NV center as a
prototypical example. By combining advanced electronic structure methods and
group theory analysis, we discuss the significance of Coulomb repulsion,
pseudo JT and dynamical JT effects to the SOC and ISC. The consideration of
these effects facilitates the interpretation of the experimentally observed
ISC rates that essentially contribute to spin polarization. We predict the
ODMR contrast as a function of an external magnetic field fully from first-
principles. We demonstrate that these simulations are invaluable for
predicting spin-polarization mechanism and provide control strategy on high
fidelity optical initialization and readout.

##  II Computational Methods

We carried out first-principles calculations of structural and ground-state
properties using the open-source plane-wave code Quantum Espresso [24]. The NV
center defect is introduced into a diamond crystal with $3\times 3\times 3$
supercell size, containing 216 atoms. Only Gamma point is sampled in the
Brilloin zone of the defect supercells. We use the optimized norm-conserving
Vanderbilt (ONCV) pseudopotentials [25] with the wavefunction energy cutoff of
70Ry. The lattice constant is optimized by using the exchange-correlation
functional with the Perdew-Burke-Ernzerhof (PBE) generalized gradient
approximation [26]. For comparison, we also used the range-separated hybrid
function proposed by Heyd, Scuseria, and Ernzerhof (HSE) [27, 28]. The HSE
functional has been shown to better describe the electronic structure of the
NV center in diamond compared to the PBE functional [29, 30, 31]. We calculate
the excitation energies using three different methods, constrained DFT (CDFT),
single-shot GW plus the Bethe-Salpeter Equation
($\text{G}_{0}\text{W}_{0}\text{-}\text{BSE}$), and complete active space
self-consistent field (CASSCF) method in quantum chemistry [32].

The many-body perturbation theory calculations are performed by using the
Yambo-code [33], with PBE eigenvalues and wavefunctions as the starting point
for quasiparticle energies at GW approximation. For the dynamical screening,
we take the Godby-Needs plasmon-pole approximation (PPA) [34, 35]. We then
solve the Bethe-Salpeter Equation (BSE) for two-particle excitation,
specifically for optical excitation energies and transition dipole moments
including explicit electron-hole interactions. A $2\times 2\times 2$ supercell
is used for the $\text{G}_{0}\text{W}_{0}\text{-}\text{BSE}$ calculations, and
the results are comparable to the previous reported values [18].

Due to the multi-reference character of the ${}^{3}E$, ${}^{1}E$, and
${{}^{1}A_{1}}$ excited states of the NV center, a single Slater determinant
description in DFT is incomplete. To overcome this problem, we use the CASSCF
method [32] to construct the state wavefunctions as linear combinations of
multiple Slater determinants. Each determinant describes a particular
occupation of single-electron orbitals. In the CASSCF method, the orbital
space is partitioned into subspaces of the inactive, active, and virtual
orbitals. The occupation of the active orbitals is allowed to change to
generate all possible spin- and symmetry-allowed determinants for a particular
electronic state. A choice of a partition of the orbital space, that is the
choice of a number of active orbitals and active electrons, determines the
CASSCF active space. We use the state-average (SA) version of the CASSCF
method as implemented in ORCA [36, 37] to obtain state wavefunctions for
calculations of SOC matrix elements. Specifically, we perform
SA(10)-CASSCF(4,6) calculations, where SA(10) refers to state-averaging over
ten states, which include five singlet and five triplet states, and
CASSCF(4,6) refers to four electrons and six orbitals included in the active
space. The second order Douglas-Kroll-Hess (DKH2) Hamiltonian [38] and the cc-
pVDZ-DK basis set [39, 40] were used to account for the scalar relativistic
effects. The spin-orbit mean-field operator was used [41].

##  III Results and Discussion

###  III.1 ODMR Theory, Implementation, and Benchmark

Figure 1: Numerical implementation of ODMR contrast and its benchmark with the
NV center in diamond. (a) The continuous-wave (cw) ODMR and its simulation
workflow with first-principles inputs. (b) Schematic diagram depicting the
energy levels and excited-state kinetic processes of the NV center in diamond.
$\ket{0}$, $\ket{-1}$ and $\ket{+1}$ denote the spin sublevels in the triplet
states. The spin sublevels $\ket{+1}$ and $\ket{-1}$ are degenerate under zero
magnetic field. (c) The time-evolved populations of the states reach a steady-
state plateau with sufficiently long time, in the presence (w/MW, left panel)
or absence (w/o MW, right panel) of microwave field. (d) Transition rates
($k$) for different processes vary differently with magnetic field B. (e)
Simulated ODMR contrast as a function of magnetic field (solid line) and
compared with the experiment (dots) [11]. The experimental data is extracted
from a figure by using WebPlotDigitizer tool [42].

The basic mechanism of ODMR is related to the excited-state populations of
spin sublevels. Therefore, we begin with the Hamiltonian that describes the
spin sublevels of triplet states as in Fig. 1(b),

$\displaystyle H_{\text{s}}$
$\displaystyle=D(S_{z}^{2}-\frac{\mathbf{S}^{2}}{3})+E(S_{x}^{2}-S_{y}^{2})+g_{e}\mu_{B}\mathbf{B}\cdot\mathbf{S}$
(1)

where $S_{i}$ is the spin-1 operator along $i$ axis, $D$ in the first term is
the axial zero-field splitting (ZFS) parameter, $E$ in the second term is the
rhombic ZFS parameter, $g_{e}$ is the electron g-factor or gyromagnetic ratio
whose value is ${\sim}2$ in the NV center, $\mu_{B}$ is Bohr magneton, and
$\mathbf{B}$ is the external magnetic field. The ZFS lifts the spin
degeneracy. The third term describes the Zeeman effect occuring under magnetic
field ($\mathbf{B}$) and further leading to the mixing of the original spin
sublevels. Let $\alpha_{ij}$ be the mixing coefficient of original spin
sublevels $\ket{i}$ and $\ket{j}$, then we can write the transition rates
under magnetic field $k_{ij}(\mathbf{B})$ to be a function of mixing
coefficients and the rates at zero magnetic field ($k^{0}_{pq}$),

$\displaystyle k_{ij}(\mathbf{B})$
$\displaystyle=\sum_{p,q}|\alpha_{ip}(\mathbf{B})|^{2}|\alpha_{jq}(\mathbf{B})|^{2}k^{0}_{pq}.$
(2)

The spin mixing will be shown below to be the source of the magnetic field-
dependent ODMR.

We then numerically implement the ODMR contrast based on the kinetic master
equation [11, 43], which integrates all transition rates and allows to
simulate the dynamics of states populations, photoluminescence (PL) intensity,
as well as continuous wave (cw) or time-resolved ODMR. The model starts with
the conventional definition of ODMR contrast,

$\displaystyle\begin{split}C(\mathbf{B})=1-\frac{\bar{I}(t,\mathbf{B},k_{\mathrm{MW}})}{\bar{I}(t,\mathbf{B},k_{\mathrm{MW}}=0)}\end{split}$
(3)
$\displaystyle\begin{split}\bar{I}(t,\mathbf{B})=\eta\sum_{i\in\mathrm{ES}}\sum_{j\in\mathrm{GS}}k_{ij}(\mathbf{B})\bar{n}_{i}(t,\mathbf{B})\end{split}$
(4)

where $\bar{I}(t,\mathbf{B},k_{\mathrm{MW}})$ is the magnetic-field dependent
PL intensity at the steady state in the presence of microwave resonance (MW),
which is different from the case in the absence of microwave resonance
($k_{\mathrm{MW}}=0$). Here, $k_{\mathrm{MW}}$ is the Rabi frequency of a
microwave field that is applied for rotating the populations of spin
sublevels, and is a parameter in the model as it scales with the amplitude of
the microwave field [44]. $\eta$ is the collection coefficient parameter for
PL intensity which depends on experimental setup but does not affect ODMR. The
optical saturation parameter $\beta$ seen in Fig. 1(b) and SM Sec. I plays a
role in the optical excitation. Both $k_{\mathrm{MW}}$ and $\beta$ are
determined according to the experimental range [11]. The PL intensity evolves
with time depending on $n_{i}(t,\mathbf{B})$, the population of spin level
$\ket{i}$ at time $t$, and can be solved numerically using the Euler method.

To validate the ODMR contrast implementation for triplet systems, we firstly
simulate the ODMR of the prototypical system NV center, as shown in Fig. 1(b),
using the experimental values of ZFS and rates [11, 45, 43]. The purpose of
this benchmark is to confirm the numerical implementation of ODMR contrast
from Eq. (1) to Eq. (4), independent of the accuracy of electronic structure
and kinetic rates, which will be discussed in detail in the next sections.
Values for the ODMR simulation parameters are tabulated in SM Table S1, and
the microwave resonance is applied to drive the rotation between
$\ket{\prescript{3}{}{A}_{2}^{0}}$ and $\ket{\prescript{3}{}{A}_{2}^{-1}}$.
Since the simulated cw-ODMR is observed at steady state, arbitrary initial
populations of the spin sublevels can be used. The system reaches the steady
state after ${\sim}10^{4}$ ns, as shown in Fig. 1(c). From the steady-state
populations, we can find that $\ket{\prescript{3}{}{A}_{2}^{-1}}$ has larger
population due to the Rabi oscillation in the presence of the microwave field
compared to the absence of the microwave field. Because the optical excitation
is mostly spin-conserving [46], the population of
$\ket{\prescript{3}{}{E}^{-1}}$ is subsequently larger. Because the nonaxial
ISC $\prescript{3}{}{E}\rightarrow\prescript{1}{}{A}_{1}$ is symmetrically
allowed and fast, as can be seen in Fig. 1(d), it competes with the radtiative
recombination $\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$, leading
to overall smaller excited-state population. Therefore, the PL intensity in
the presence of the microwave field becomes smaller, and the ODMR contrast is
positive as shown in Fig. 1(e). The decrease of ODMR contrast with the
magnetic field is a consequence of the smaller difference between axial ISC
rates and nonaxial ISC rates, compared Fig. 1(d) to Fig. 1(e). The fundamental
reason is the mixing of spin sublevels, which will be discussed in details in
Sec. III.6. To understand the magnetic field effect, first, when the magnetic
field is misaligned with the NV axis, there is mixing between spin sublevels
$m_{s}=0$ and $m_{s}=\pm 1$. Second, the spin mixing will further mix ISC
rates between different transitions as Eq. (2).For the NV center, the spin
polarization is mainly a result of
$k_{\perp}(\prescript{3}{}{E}\rightarrow\prescript{1}{}{A}_{1})\gg
k_{z}(\prescript{3}{}{E}\rightarrow\prescript{1}{}{A}_{1})$. When
$k_{\perp}(\prescript{3}{}{E}\rightarrow\prescript{1}{}{A}_{1})$ is similar to
$k_{z}(\prescript{3}{}{E}\rightarrow\prescript{1}{}{A}_{1})$ due to spin
mixing, spin polarization is weaker, ODMR contrast is smaller. The excellent
agreement of our simulation with experiment in Fig. 1(e) validates the
numerical implementation of ODMR.

Besides triplet spin defects, singlet defects that have a triplet metastable
state can also show ODMR signal, like ST1 defect in diamond [47, 48]. In SM
Sec. I, we additionally include the detailed benchmark of the ODMR model for
singlet spin defect. The benchmark for both triplet and singlet spin defects
demonstrates the generality of the ODMR model.

Next, we will discuss in detail the first-principles calculations of
electronic structure and excited-state kinetic rates at solid-state spin
defects, specifically the NV center here.

###  III.2 Electronic Structure, Excitation Energies and Radiative
Recombination of NV Center

Table 1: The excitation energies of the NV center from various theories along with experiments. The energy of $\prescript{3}{}{A}_{2}$ is aligned to 0 eV, and the energies of the other states are entered with respect to $\prescript{3}{}{A}_{2}$. $\Sigma$ is the excitation energy of $\prescript{1}{}{E}$ with respect to $\prescript{3}{}{A}_{2}$, and $\Delta$ is the energy difference between $\prescript{3}{}{E}$ and $\prescript{1}{}{A}_{1}$, as illustrated in Fig. 1(b).  Method |  $\prescript{1}{}{E}$ or $\Sigma$ (eV) |  $\prescript{1}{}{A}_{1}$ (eV) |  $\prescript{3}{}{E}$ (eV) |  $\Delta$ (eV)
---|---|---|---|---
Expt. (ZPL) | 0.325-0.411111The range of $\prescript{1}{}{E}$ or $\Sigma$ was determined according to the triplet ZPL $\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$, the singlet ZPL $\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$, and $\Delta$. | 1.515-1.601 [49, 50, 45] | 1.945 [51, 52, 53] | 0.344-0.430222The range of $\Delta$ was determined by comparing the ISC model to the experiment. [52, 53]
Expt. (abs) | – | 1.76333This energy is obtained by adding the approximate energy (0.40 eV) of $\prescript{1}{}{E}$ to the absorption energy $\prescript{1}{}{E}\rightarrow\prescript{1}{}{A}_{1}$ 1.36 eV (912 nm). [45] | 2.180 [51] | –
CDFT (ZPL, HSE, this work) | 0.37 | – | 1.95 | –
GW-BSE@PBE (this work, absorption) | - | - | 2.25 | -
CASSCF (this work) | 0.59 | 1.68 | 2.05 | 0.36

The NV center comprises a substitutional nitrogen atom and a vacancy with one
extra electron in the faced-center cubic diamond, with its local structure
described by the $C_{3v}$ symmetry point group. The three-fold symmetry axis
is the nitrogen-vacancy axis along the [111] direction. The ground and low-
lying excited electronic states of the NV center can be determined from two
one-electron orbitals, $a_{1}$ and $e$ ($e_{x}$, $e_{y}$), which transform as
the $A_{1}$ and $E$ irreducible representations of the $C_{3v}$ point group,
respectively. These one-electron orbitals are formed by the carbon and
nitrogen dangling bonds near the vacancy defect. The $a_{1}^{2}e^{2}$
configuration gives rise to the ${{}^{3}A_{2}}$, ${{}^{1}E}$ and
${{}^{1}A_{1}}$ states, and $a_{1}e^{3}$ to the ${{}^{3}E}$ and
${{}^{1}E^{\prime}}$ states. The states above are listed in the ascending
energy order determined experimentally for all the states except
${{}^{1}E^{\prime}}$ whose relative energy is determined from the group theory
[50, 45, 52, 15]. The total spin-orbit irreducible representations of these
states are given in SM Table S3. The derivation of the representations can be
found in literature [54, 55, 56].

Excitation energies play an important role in energy conservation in radiative
(electron-photon interaction) and nonradiative (electron-phonon interaction)
recombination rates. As we discussed earlier, the excited states of defects
can have multi-reference character which is difficult to describe by
conventional DFT. Therefore we show the calculated excitation energies from
several different electronic structure methods, CDFT,
$\text{G}_{0}\text{W}_{0}\text{-}\text{BSE}$@PBE and CASSCF, as listed in
Table 1. With all three methods, we obtain excitation energies of triplet
excited-state $\prescript{3}{}{E}$ in good agreement with experiments.
However, only in the CASSCF calculation we can obtain the excitation energy of
the $\prescript{1}{}{A}_{1}$ state, which has a multi-reference character. We
also include the higher-lying ${{}^{1}E^{\prime}}$ state in the state-
averaging to account for the Coulomb repulsion that couples the ${{}^{1}E}$
and ${{}^{1}E^{\prime}}$ states. More discussion about the excitation energies
can be found in SM Sec. II A.

With the excitation energies, we employ the Fermi’s golden rule to calculate
radiative recombination rate. The details of calculation methods can be found
in our past work [21, 22, 16] and results can be found in SM Sec. III B. For
the convenience of comparing our calculated results with previous experiment
and calculations, here we adopt lifetime ($\tau$), which is the inverse of
rate ($\tau=1/k$). Using the optical dipole moments and excitation energy from
$\text{G}_{0}\text{W}_{0}\text{-}\text{BSE}$@PBE calculations which includes
excitonic effect accurately, we obtain $\tau_{\text{R}}=$16.8 ns for the
$\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$ transition and 3061 ns
for the $\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ transition. Our
calculated radiative lifetimes are consistent with experiment 12 ns for
$\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$ [52], and previous
calculations, ${\sim}10$ ns for
$\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$ and 1878 ns for
$\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ [18, 57].

###  III.3 Spin-Orbit Coupling and Pseudo Jahn-Teller Effect

Spin-orbit coupling Hamiltonian can be separated into the axial/nonaxial SOC
components ($\lambda_{z/\perp}$) with the angular momentum ladder operators
($L_{\pm}$ and $S_{\pm}$) and $z$ components ($L_{z}$ and $S_{z}$),

$\displaystyle H_{\text{soc}}$
$\displaystyle=\frac{\lambda_{\perp}}{2}(L_{+}S_{-}+L_{-}S_{+})+\lambda_{z}L_{z}S_{z}.$
(5)

Table 2: Summary of SOC matrix elements predicted by group theory, calculated at the theory levels of CASSCF and TDDFT, and from experiment. The unit is GHz. CASSCF ($C_{3v}$) SOCs are consistent with prediction. TDDFT ($C_{3v}$) SOCs are inconsistent with group theory. “CR” stands for Coulomb repulsion that mixes $\prescript{1}{}{E}$ and ${{}^{1}E^{\prime}}$. Pseudo JT effect mixes $\prescript{1}{}{A}_{1}$ and $\prescript{1}{}{E}$ through electron-phonon coupling, leading to nonzero SOCs. “–” denotes value that does not exist or that is not found. | $\lambda_{z}(\prescript{3}{}{E},\prescript{3}{}{E})$ | $\lambda_{\perp}(\prescript{1}{}{A}_{1},\prescript{3}{}{E})$ | $\lambda_{z}(\prescript{1}{}{A}_{1},\prescript{3}{}{E})$ | $\lambda_{\perp}(\prescript{3}{}{A}_{2},{{}^{1}\bar{E}})$ | $\lambda_{z}(\prescript{3}{}{A}_{2},{{}^{1}\bar{E}})$
---|---|---|---|---|---
Group Theory $(C_{3v})$111The matrix elements here are expressed in terms of the reduced one-particle matrix elements, $\lambda_{z}=-i\hbar\matrixelement{e|}{L^{A_{2}}}{|e}$ and $\lambda_{\perp}=-(i/\sqrt{2})\hbar\matrixelement{e|}{L^{E}}{|a_{1}}$. | $\pm\lambda_{z}$ | $i\lambda_{\perp}$ | 0 | nonzero by CR | 0
CASSCF ($C_{3v}$) | 7.56 | 2.04 | 0 | 4.29222The matrix element is nonzero because $\prescript{1}{}{E}$ coupled with singlet excited state ${{}^{1}E^{\prime}}$ through Coulomb repulsion, and $\lambda_{z}(\prescript{3}{}{A}_{2},{{}^{1}E^{\prime}})$ is symmetry-allowed. This coupling exists in CASSCF solutions, as can be seen in SM Table S6.  | 0
TDDFT ($C_{3v}$) | – | 6.12 | 0.001 | 8.18 | 50.37
(w/ pseudo JT) | $\lambda_{z}({{}^{3}\widetilde{E}},{{}^{3}\widetilde{E}})$ | $\lambda_{\perp}({{}^{1}\widetilde{A}_{1}},{{}^{3}\widetilde{E}})$ | $\lambda_{z}({{}^{1}\widetilde{A}_{1}},{{}^{3}\widetilde{E}})$ | $\lambda_{\perp}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$ | $\lambda_{z}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$
Group Theory | $\pm\lambda_{z}$ | nonzero | nonzero by pseudo JT | nonzero by CR | nonzero by pseudo JT
Effective SOC CASSCF | 7.56 | 8.17 | 1.48 | 5.59 | 7.11
HSE [23, 15] | 15.8333$\lambda_{z}({{}^{3}E},{{}^{3}E})$ was calculated by $\matrixelement{e_{+}}{H_{\text{soc}}}{e_{+}}$ with $\ket{e_{+}}=\frac{1}{\sqrt{2}}(\ket{e_{x}}+i\ket{e_{y}})$ at HSE [23]. The matrix element reduced to 4.8 by a reduction factor. | 56.3444$\lambda_{\perp}(\prescript{1}{}{A}_{1},\prescript{3}{}{E})$ was calculated by $\matrixelement{e_{+}}{H_{\text{soc}}}{a_{1}}$ with $\ket{e_{+}}=\frac{1}{\sqrt{2}}(\ket{e_{x}}+i\ket{e_{y}})$ at HSE [23], assuming single-particle picture and that KS wavefunctions constructed ${{}^{3}E}$ and ${{}^{1}A_{1}}$ did not differ [53]. | – | 18.96555$\lambda_{\perp}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$ was a parameter estimated by taking $\lambda_{\perp}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})/\lambda_{z}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})=1.2$ for matching it with expt [15]. | 15.8666$\lambda_{z}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$ was calculated according to $\matrixelement{{}^{3}E}{H_{soc}}{{}^{3}E}=\matrixelement{e_{+}}{H_{\text{soc}}}{e_{+}}$ [23, 15].
Expt. [52, 58, 53] | 5.33 | 6.4777The matrix element is esitmated by using the ratio $\lambda_{\perp}(\prescript{1}{}{A}_{1},\prescript{3}{}{E})/\lambda_{z}(\prescript{1}{}{A}_{1},\prescript{3}{}{E})=1.2$ [53] | – (nonzero) | – (nonzero) | – (nonzero)

Group theory provides important information of whether a SOC matrix element is
allowed or forbidden, from the symmetry point of view. And the multi-particle
representation of the SOC matrix elements can be reduced to the single-
particle representation by using the Wigner-Eckark theorem [56]. Some SOC
matrix elements were previously evaluated at the single-particle level at HSE,
as shown in Table 2. However, because of their multi-reference character, the
excited states cannot be properly described with a single Slater determinant
[20], as can be seen in SM Table S6. Therefore, we perform CASSCF for the
evaluation of SOC to take into account the multi-reference nature of the
states, in comparison with TDDFT [59] and previous DFT results [23, 15].

As shown in Table 2, the SOC matrix elements from CASSCF are in accordance
with the group theory prediction. Specifically, within the important SOC
matrix elements listed, only
$\lambda_{z}(\prescript{3}{}{E},\prescript{3}{}{E})$ and
$\lambda_{\perp}(\prescript{3}{}{E},\prescript{1}{}{A}_{1})$ are symmetry-
allowed. Accordingly we obtain finite values for these SOC matrix elements at
CASSCF. We note that the nonzero
$\lambda_{\perp}(\prescript{3}{}{A}_{2},\prescript{1}{}{E})$ at CASSCF does
not contradict to the zero value from the group theory prediction. This is
because $\prescript{1}{}{E}$ couples with the higher singlet excited state
${{}^{1}E^{\prime}}$ under Coulomb repulsion, as also discussed in Refs [45,
56, 55], resulting in
${{}^{1}\bar{E}}=C\prescript{1}{}{E}+(1-C)\prescript{1}{}{E}^{\prime}$ ($C$
the mixing coefficient). This leads to symmetry-allowed
$\lambda_{\perp}(\prescript{3}{}{A}_{2},{{}^{1}\bar{E}})$. In comparison, the
SOC matrix elements from TDDFT do not agree with the group theory prediction.
The issue could still be that TDDFT as a single-determinant method does not
describe the multi-reference state correctly. An apparent error is that the
symmetry of the wavefunction is not preserved, as can be seen in SM Fig. S6.

However, experiments show allowed axial ISC for
$\prescript{3}{}{E}\rightarrow\prescript{1}{}{A}_{1}$ and
$\prescript{1}{}{E}\rightarrow\prescript{3}{}{A}_{2}$ [52, 11, 60, 46]. This
is contradictory to the current group theory prediction that these axial ISC
transitions are forbidden by zero
$\lambda_{z}(\prescript{3}{}{E},\prescript{1}{}{A}_{1})$ and
$\lambda_{z}(\prescript{3}{}{A}_{2},\prescript{1}{}{E})$. It implies that an
additional mechanism may have modified the symmetry of wavefunctions. Previous
studies attribute it to the pseudo JT effect, which particularly couples
nondegenerate electronic states through electron-phonon coupling [61, 62, 15,
63, 31],

$\displaystyle H$ $\displaystyle=\underbrace{\begin{pmatrix}\Lambda&0&0\\\
0&0&0\\\
0&0&0\end{pmatrix}}_{H_{\text{e}}}+\underbrace{\frac{K}{2}(Q_{x}^{2}+Q_{y}^{2})}_{H_{\text{osc}}}+\underbrace{\begin{pmatrix}0&FQ_{x}&FQ_{y}\\\
FQ_{x}&0&0\\\ FQ_{y}&0&0\end{pmatrix}}_{H_{\text{PJT}}}$ (6) $\displaystyle F$
$\displaystyle=\matrixelement{{}^{1}A_{1}}{\frac{\partial V}{\partial
Q_{x}}}{{{}^{1}E_{x}}}=\matrixelement{{}^{1}A_{1}}{\frac{\partial V}{\partial
Q_{y}}}{{{}^{1}E_{y}}}$ (7)

Here, $\Lambda$ is the energy gap between $\ket{{}^{1}A_{1}}$ and
$\ket{{}^{1}E}$ in the electronic Hamiltonian $H_{\text{e}}$ . $K$ is the
elastic vibronic constant in the harmonic oscillator Hamiltonian
$H_{\text{osc}}$. $F$ is the linear vibronic coupling constant in the pseudo
JT Hamiltonian $H_{\text{PJT}}$. $\ket{{}^{1}A_{1}}$, $\ket{{}^{1}E_{x}}$ and
$\ket{{}^{1}E_{y}}$ form the basis of the Hamiltonian $H$. $Q_{x}$ and $Q_{y}$
are the nuclear coordinate transform as $x$ and $y$, respectively. Under the
pseudo JT distortion, $\prescript{1}{}{A}_{1}$ is coupled with
$\prescript{1}{}{E}$ with the assistance of the Jahn-Teller active $e$ phonon,
denoted as $(A_{1}+E)\otimes e$. The coupling of electronic states results in
the vibronic states ${{}^{1}\widetilde{E}}$ and ${{}^{1}\widetilde{A}_{1}}$ as
the linear combination of $\prescript{1}{}{E}$ and $\prescript{1}{}{A}_{1}$,

$\displaystyle\begin{split}\ket{{}^{1}\widetilde{E}_{\pm}}&=\sum_{i=1}^{\infty}\Bigg{[}c_{i}\ket{{}^{1}\bar{E}_{\pm}}\otimes\ket{\chi_{i}^{A_{1}}}+d_{i}\ket{{}^{1}A_{1}}\otimes\ket{\chi_{i}^{E_{\pm}}}\\\
&\quad+f_{i}\ket{{}^{1}\bar{E}_{\pm}}\otimes\ket{\chi_{i}^{E_{\pm}}}+g_{i}\ket{{}^{1}\bar{E}_{\pm}}\otimes\ket{\chi_{i}^{A_{2}}}\Bigg{]}\end{split}~{}$
(8)
$\displaystyle\begin{split}\ket{{}^{1}\widetilde{A}_{1}}&=\sum_{i=1}^{\infty}\Bigg{[}c^{\prime}_{i}\ket{{}^{1}A_{1}}\otimes\ket{\chi_{i}^{A_{1}}}\\\
&\quad+\frac{d^{\prime}_{i}}{\sqrt{2}}\bigg{(}\ket{{}^{1}\bar{E}_{+}}\otimes\ket{\chi_{i}^{E_{-}}}+\ket{{}^{1}\bar{E}_{-}}\otimes\ket{\chi_{i}^{E_{+}}}\bigg{)}\Bigg{]}\end{split}~{}$
(9)

where $\ket{\chi_{i}^{\Gamma}}$ is the $i$-th phonon wavefunction transforming
as irreducible representation $\Gamma$, and $c_{i}$, $d_{i}$, $f_{i}$,
$g_{i}$, $c^{\prime}_{i}$ and $d^{\prime}_{i}$ are the amplitude of vibronic
states. The states coupling give rise to a significant change in the SOC
matrix elements as listed below:

$\displaystyle\begin{split}\lambda_{z}({{}^{1}\widetilde{A}_{1}},{{}^{3}\widetilde{E}})&=d^{\prime}_{eff}\matrixelement{{{}^{1}\bar{E}}}{H_{\text{soc}}}{\prescript{3}{}{E}}_{z}\end{split}~{}$
(10)
$\displaystyle\begin{split}\lambda_{\perp}({{}^{1}\widetilde{A}_{1}},{{}^{3}\widetilde{E}})&=c^{\prime}_{eff}\matrixelement{\prescript{1}{}{A}_{1}}{H_{\text{soc}}}{\prescript{3}{}{E}}_{\perp}+d^{\prime}_{eff}\matrixelement{{{}^{1}\bar{E}}}{H_{\text{soc}}}{\prescript{3}{}{E}}_{\perp}\end{split}~{}$
(11)
$\displaystyle\begin{split}\lambda_{z}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})&=d_{eff}\matrixelement{\prescript{3}{}{A}_{2}}{H_{\text{soc}}}{\prescript{1}{}{A}_{1}}_{z}\end{split}~{}$
(12)
$\displaystyle\begin{split}\lambda_{\perp}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})&=(c_{eff}+f_{eff})\matrixelement{\prescript{3}{}{A}_{2}}{H_{\text{soc}}}{{{}^{1}\bar{E}}}_{\perp}\end{split}~{}$
(13)

where $c^{\prime}_{eff}$, $d^{\prime}_{eff}$, $c_{eff}$, $d_{eff}$, $f_{eff}$
are the normalized effective state mixing coefficients, e.g.
$|c^{\prime}_{eff}|^{2}+|d^{\prime}_{eff}|^{2}=1$.

We complete the derivation for all effective SOC matrix elements of the NV
center, in complement to Ref. [15] which only provides
$\lambda_{\perp/z}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$. The
derivation details can be found in SM Sec. IV A. The pseudo JT effect results
in the finite values of
$\lambda_{z}({{}^{1}\widetilde{A}_{1}},{{}^{3}\widetilde{E}})$ and
$\lambda_{z}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$, and a
modification to
$\lambda_{\perp}({{}^{1}\widetilde{A}_{1}},{{}^{3}\widetilde{E}})$ and
$\lambda_{\perp}({{}^{3}\widetilde{A}_{2}},{{}^{1}\widetilde{E}})$, as can be
seen in Table 2. The resulting effective SOC shows much better agreement with
experiments, in contrast to other calculations by DFT or TDDFT. As will be
shown in Sec. III.4, the nonzero SOC matrix elements lead to allowed ISC,
consistent with experimental observations.

Finally, another approach based on the Taylor expansion of SOC matrix element
in terms of coupling with phonon vibration is commonly used for the study of
ISC in molecules [64, 65]. We examined this approach by computing its first-
order derivative of SOC with respect to nuclear coordinate change. The result
indicates that this approach does not apply to the NV center under the 1D
effective phonon approximation. More details can be seen in the SM Sec. IV B.

Table 3: Electron-phonon coupling and rates of ISC ${{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}}$ and ${{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}}$ at 300 K. The degree of geometry degeneracy $g$ is specified as 3. $\Delta Q$ is the nuclear coordinate change between the initial and final states. $\hbar\omega_{i}$ and $\hbar\omega_{f}$ represent the phonon energy of the initial and final states, respectively. $S_{f}$ denotes the 1D Huang-Rhys (HR) factor of the final state, which is a measure of e-ph coupling strength. $\widetilde{X}_{if}$ is the phonon term. In these calculations, the JT-distorted geometries of $C_{s}$ symmetry are used for the initial states ${{}^{3}\widetilde{E}}$ and ${{}^{1}\widetilde{E}}$, and the geometry of $C_{3v}$ symmetry of ${{}^{3}\widetilde{A}_{2}}$ is used for the final states ${{}^{1}\widetilde{A}_{1}}$ and ${{}^{3}\widetilde{A}_{2}}$, considering the fact that the geometry and phonon modes of ${{}^{1}\widetilde{A}_{1}}$ is similar to that of ${{}^{3}\widetilde{A}_{2}}$ [63, 45].  | Transition | $\Delta Q$ | $\hbar\omega_{i}$ | $\hbar\omega_{f}$ | $S_{f}$ | $\widetilde{X}_{if}$ | $k_{\perp}$ | $k_{z}$
---|---|---|---|---|---|---|---|---
|  | (amu${}^{1/2}$\mathrm{\SIUnitSymbolAngstrom}$$) | (meV) | (meV) |  | ($\mathrm{eV}^{-1}$) | (MHz) | (MHz)
expt. | ${{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}}$ | 0.64111$\Delta Q$ is estimated by the Huang-Rhys factor and phonon energy, $S=\omega\Delta Q^{2}/2\hbar$, under one-dimensional effective phonon approximtion. | – | 71 [45] | 3.49 [45] | – | 24.3 [52] |  $<0.62$ [52]
Calc. (our work) | 0.65 | 73.7 | 67.7 | 3.38 | 1.34 | 43.8 | 1.43
Calc. [23, 15] | – | 77.6 | $66-91.8$ | 2.61222$S=2.61$ when approximating $e_{x}e_{x}$ singlet determinant for ${{}^{1}\widetilde{A}_{1}}$, and $S=3.11$ when using $\prescript{3}{}{A}_{2}$ geometry for ${{}^{1}\widetilde{A}_{1}}$. | – | 243 | –
expt. | ${{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}}$ | 0.34111$\Delta Q$ is estimated by the Huang-Rhys factor and phonon energy, $S=\omega\Delta Q^{2}/2\hbar$, under one-dimensional effective phonon approximtion. | – | 64 [45] | 0.9333This HR factor is for ${{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}}$. Considering the similarity of geometry and potential surfaces between ${{}^{3}\widetilde{A}_{2}}$ and ${{}^{1}\widetilde{A}_{1}}$, we put it here for comparison with our calculations.  [45] | – | 2.61 [46] | 3.0 [46]
Calc. (our work) | 0.23 | 75.9 | 71.0 | 0.45 | $2.71\times 10^{-3}$ | $4.55\times 10^{-2}$ | $7.35\times 10^{-2}$
Calc.555By using the experimental Huang-Rhys factor to correct the phonon term at HSE, $\widetilde{X}_{if}$ increases by a factor of 23.6. (our work, corr. w/ expt. HR factor) | 0.33111$\Delta Q$ is estimated by the Huang-Rhys factor and phonon energy, $S=\omega\Delta Q^{2}/2\hbar$, under one-dimensional effective phonon approximtion. | 75.9 | 71.0 | 0.9333This HR factor is for ${{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}}$. Considering the similarity of geometry and potential surfaces between ${{}^{3}\widetilde{A}_{2}}$ and ${{}^{1}\widetilde{A}_{1}}$, we put it here for comparison with our calculations.  [45] | $6.40\times 10^{-2}$ | 1.07 | 1.74
Calc. [15] | – | – | 66 | 0.07 | – | 0.90444The corresponding SOC can be found in Table 2. | 4.95444The corresponding SOC can be found in Table 2.

###  III.4 Electron-Phonon Coupling and ISC

The ISC rate between electronic states with different spin multiplicities can
be calculated as

$\displaystyle k_{\text{ISC}}$
$\displaystyle=\frac{2\pi}{\hbar}g\sum_{n,m}p_{in}|\matrixelement{fm}{H_{\text{soc}}}{in}|^{2}\delta(E_{fm}-E_{in}).~{}$
(14)

Here, we make the 1D effective phonon approximation [66, 22, 16] for the ISC
rate. Compared to the full-phonon method used in Ref. [23, 15], an advantage
of this 1D effective phonon method is that it enables the use of different
values for the phonon energy of initial state ($\hbar\omega_{i}$) and the one
of final state ($\hbar\omega_{f}$). As already discussed in Sec. III.3 SM Sec.
IV A, the SOC matrix element $\matrixelement{fm}{H_{\text{soc}}}{in}$ that
couples the initial vibronic state $\ket{in}$ and final vibronic state
$\ket{fm}$ can be separated into the effective SOC and effective phonon
overlap. Therefore, the ISC rate equation is expressed as

$\displaystyle k_{\text{ISC}}$
$\displaystyle=\frac{2\pi}{\hbar}g\lambda^{2}\widetilde{X}_{if}~{}$ (15)
$\displaystyle\lambda$
$\displaystyle=\matrixelement{\psi_{f}}{H_{\text{soc}}}{\psi_{i}}$ (16)
$\displaystyle\widetilde{X}_{if}$
$\displaystyle=\sum_{n,m}p_{in}(T)|\innerproduct{\phi_{m}}{\phi_{n}}|^{2}\delta(m\hbar\omega_{f}-n\hbar\omega_{i}+\Delta
E_{if})$ (17)

where $g$ is the degree of degeneracy on equivalent structural configuration,
$\lambda$ is the effective SOC matrix element, and $\widetilde{X}_{if}$ is the
temperature-dependent phonon term representing the phonon contribution.
$\ket{\psi}$ is the linear combination of possible electronic states after
considering pseudo JT effect, and $\ket{\phi}$ denotes the phonon wavefunction
under the harmonic oscillator approximation. $\lambda$ has been discussed in
detail in Sec. III.3. Therefore, our primary focus in this section will be on
the phonon term $\widetilde{X}_{if}$. We will show that dynamical JT effect is
important to include for electron-phonon coupling in the nonradiative
intersystem-crossing transitons. Otherwise the transition rates can be order
of magnitude smaller.

In Fig. 2(a), we show the lower branch adiabatic potential energy surface
(APES) of $\prescript{1}{}{E}$ by fitting the Hamiltonian in Eq. (18) [62] to
the calculated potential energy curves,

$\displaystyle\begin{split}H&=\underbrace{\frac{K}{2}(Q_{x}^{2}+Q_{y}^{2})}_{H_{\text{osc}}}\\\
&\quad+\underbrace{F(Q_{x}\sigma_{z}-Q_{y}\sigma_{x})+G[(Q_{x}^{2}-Q_{y}^{2})\sigma_{z}+2Q_{x}Q_{y}\sigma_{x}]}_{H_{\text{DJT}}}.~{}\end{split}$
(18)

Here $K$, $F$ and $G$ are elastic, linear and quadratic vibronic constants,
respectively. $\sigma_{z}$ and $\sigma_{x}$ are Pauli’s matices. $Q_{x}$ and
$Q_{y}$ denote nuclear coordinate transform as $x$ and $y$, respecitvely. The
APES appears as a tricorn Mexican hat under the $E\otimes e$ JT distortion,
which splits the doubly degenerate electronic states via the coupling with $e$
phonon. The symmetry of the geometry is $C_{3v}$ at the conical intersection,
and $C_{s}$ at the three equivalent geometries of energy minima. The three-
fold degeneracy of the geometry indicates that $g$ should be 3 in the ISC
calculation. We find that the three energy minima are separated by a minor
energy barrier ($\delta=2F^{2}|G|/(K^{2}-4G^{2})$) at ${\sim}30$ meV. Because
of the small energy barrier, the system of the vibronic ground state may be
delocalized and tend to undergo a hindered internal rotation among the energy
minima like $\prescript{3}{}{E}$ [67], which is the so-called “dynamical JT
effect”.

The dynamical JT effect distorts the geometries from $C_{3v}$ to lower
symmetry. Then the geometries of $C_{s}$ symmetry become the starting points
for the system to relax from the initial states ${{}^{1}\widetilde{E}}$
(${{}^{3}\widetilde{E}}$) to the final state ${{}^{3}\widetilde{A}_{2}}$
(${{}^{1}\widetilde{A}_{1}}$) through ISC. The nonradiative process is
depicted in Fig. 2(b), the configuration coordinate diagram of
${{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}}$ as an example. In
the nonradiative process, the JT-active $e$ phonon breaks the symmetry of
${{}^{1}\widetilde{E}}$ and alters the potential energy curve of
${{}^{1}\widetilde{E}}$. This results in a small energy barrier between the
initial and final electronic states, which is easy to overcome. Finally, the
small energy barrier consequences in the fast nonradiative relaxation [68,
22]. In contrast, if not considering the dynamical JT effect, only the totally
symmetric $a_{1}$ type phonon would participate in the ISC, and the transition
rate would be orders of magnitude smaller due to the large energy barrier
between the initial and final electronic states. The dramatic difference
between $C_{3v}$-symmetry geometry and $C_{s}$-symmetry geometry as starting
point indicates that dynamical JT effect plays an essential role in the
nonradiative processes.

By listing the ISC calculation details in Table 3, we first find
$k_{\perp}({{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}})$ shows
good agreement with experimental values. And
$k_{z}({{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}})$ becomes
allowed in the first order due to the finite SOC by the pseudo JT effect,
consistent with the nonzero ISC in experiment [52, 11, 60, 46]. On the other
hand, the
$k_{\perp/z}({{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}})$ ISC
rates are 1-2 orders of magnitude smaller than experiment, but the ratio
$k_{z}({{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}})/k_{\perp}({{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}})=1.62$
is similar to the experimental value $1.15\pm 0.05$ or $1.6\pm 0.4$ [46] (the
ratio between $k_{z}$ and $k_{\perp}$ is more critical for spin polarization).
The underestimation of the rates may be originated from the underestimated
electron-phonon coupling related to $\widetilde{X}_{if}$, the phonon
contribution to ISC. Using the relation $\widetilde{X}_{if}\propto
e^{-S}S^{k}/k!$ that follows the Poisson distribution of emitting phonons [69,
70, 22], we can proceed to determine the degree to which the phonon
contribution is underestimated by using the experiment Huang-Rhys factor. Here
$k=\Delta E_{if}/\hbar\omega$ denotes the number of absorbed or emitted
phonons needed for satisfying energy conservation between initial and final
states. With replacing the experimental $S$ for the calculated $S$, we find
that $k_{\perp/z}({{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}})$
increases by a factor of 23.6, giving the rates in good agreement with
experiment, as shown in Table 3. This factor explains why the ISC rates are
1-2 orders of magnitude smaller, and suggests the need for a better
theoretical description of the $\prescript{1}{}{E}$ geometry.

Figure 2: (a) Adiabatic potential energy surface (APES) of
$\prescript{1}{}{E}$ of the NV center with the dynamical JT effect. $\Delta Q$
denotes the nuclear coordinate change, the mass weighted total nuclear
displacement. $x$ and $y$ represent the displacement directions due to the
degenerate $e_{x}$ and $e_{y}$ phonons, respectively. (b) The configuration
coordinate diagram for the
${{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}}$ ISC. The ISC
starts from ${{}^{1}\widetilde{E}}$, whose geometry at local energy minima can
have $C_{3v}$ and $C_{s}$ symmetries, and ends at ${{}^{3}\widetilde{A}_{2}}$
of $C_{3v}$ symmetry. The solid line represents the potential energy curve
created by electron coupling with $e$ phonon, and the dashed line represent
that due to electron coupling with $a_{1}$ phonon. The shaded area between the
solid and dashed lines include electron coupling with the mix of $a_{1}$ and
$e$ phonons. The inset is the visualization of effective $a_{1}$ and $e$
phonons, whose arrows represent the vibration amplitude larger than the
threshold $0.05~{}\mathrm{amu}^{1/2}$\mathrm{\SIUnitSymbolAngstrom}$$ and
$0.1~{}\mathrm{amu}^{1/2}$\mathrm{\SIUnitSymbolAngstrom}$$, respectively.

###  III.5 Internal Conversion

The IC represents spin-conserving phonon-assisted nonradiative transition.
Under the static coupling approximation and one-dimensional effective phonon
approximation, the equation of nonradiative transition rates is written as in
Refs. [66, 22].

$\displaystyle\begin{split}k_{\text{IC}}&=\frac{2\pi}{\hbar}g|W_{if}|^{2}X_{if}(T)\\\
\end{split}$ (19)
$\displaystyle\begin{split}W_{if}&=(\varepsilon_{f}-\varepsilon_{i})\innerproduct{\psi_{i}(\mathbf{r,R})}{\frac{\partial\psi_{f}(\mathbf{r,R})}{\partial
Q}}\bigg{|}_{\mathbf{R=R_{a}}}\\\ \end{split}$ (20)
$\displaystyle\begin{split}X_{if}&=\sum_{n,m}p_{in}(T)\Big{|}\matrixelement{\phi_{fm}(\mathbf{R})}{Q-Q_{a}}{\phi_{in}(\mathbf{R})}\Big{|}^{2}\\\
&\quad\times\delta(m\hbar\omega_{f}-n\hbar\omega_{i}+\Delta
E_{if}).\end{split}$ (21)

This equation is similar to that of ISC except that the electronic term
$W_{if}$ replaces SOC $\lambda$ with single particle wavefunctions $\psi_{i}$
to approximate many-electron wavefunctions, and that the phonon term becomes
$X_{if}$ with additional expectation values of nuclear coordinate change
($Q-Q_{a}$).

From our calculations reported in Table 4,
$\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$ shows long IC lifetime
(${\gg}1$ s) because a great number of emitted phonons is needed to fulfill
the energy conservation between $\prescript{3}{}{E}$ and
$\prescript{3}{}{A}_{2}$. Therefore, this process is a radiative-dominant
process. On the other hand, the
$\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ transition is likely to
be nonradiative-dominant according to past experiments [50], but does not
appear in the calculation where we obtain long IC lifetime. This could be a
consequence of inaccurate description of the electron-phonon coupling of
multi-reference states $\prescript{1}{}{A}_{1}$ and $\prescript{1}{}{E}$,
currently calculated at DFT.

Although the calculated IC rate of
$\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ is underestimated, we
can correct the IC rate. We estimate $W_{if}$ and $X_{if}$ by considering the
pseudo JT-induced state mixing coefficients of ${{}^{1}\widetilde{E}}$ and
${{}^{1}\widetilde{A}_{1}}$, and the experimental Huang-Rhys factor [15, 63,
69, 70, 22]. The detailed procedure can be found in SM Sec. V C. With the
corrected $W_{if}$ and $X_{if}$, the
$k_{\text{IC}}({{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}})$ can
be enhanced by orders of magnitude and closer to experiment, as can be seen in
Table 4.

Without correcting the IC rate, we find that the PL rate
($k_{\text{PL}}=k_{\text{R}}+k_{\text{IC}}$) of
$\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ is underestimated, but
still allows us to obtain qualitatively correct ODMR contrast. When we
consider the corrected
$k_{\text{IC}}({{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}})$,
this transition becomes dominant by nonradiative recombination, and ODMR
contrast is in better agreement with experiments. More details can be seen in
Sec. III.6.

Table 4: Calculation details of internal conversion of the NV center at 300 K. The degree of geometry degeneracy $g$ is specified as 3. The ZPL is the energy change in the complete nonradiative process. $\Delta Q$ denotes the nuclear coordinate change between the initial and final states. $\hbar\omega_{f}$ is the phonon energy of the final state. $S_{f}=\omega\Delta Q^{2}/2\hbar$ denotes the Huang-Rhys factor under 1D effective phonon approximation. $W_{if}$ and $X_{if}$ are the electronic and phonon terms, respectively.  Transition | ZPL | $\Delta Q$ | $\hbar\omega_{f}$ | $S_{f}$ | $W_{if}$ | $X_{if}$ | $\tau_{\mathrm{IC}}$ | $k_{\mathrm{IC}}$
---|---|---|---|---|---|---|---|---
| (eV) | (amu${}^{1/2}$\mathrm{\SIUnitSymbolAngstrom}$$) | (meV) |  | (eV/(amu${}^{1/2}$\mathrm{\SIUnitSymbolAngstrom}$$)) | (amu$\cdot$\mathrm{\SIUnitSymbolAngstrom}$^{2}$/eV) |  | (MHz)
$\prescript{3}{}{E}\rightarrow\prescript{3}{}{A}_{2}$ | 1.95 | 0.65 | 67.7 | 3.38 | 16.6 | $3.17\times 10^{-19}$ |  ${\sim}1$ s | ${\ll}1$
$\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ (w/o corr.) | 1.13 | 0.22 | 76.4 | 0.46 | $5.48*10^{-4}$ | $8.84*10^{-12}$ |  ${\gg}1$ s | ${\ll}1$
${{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}}$ (w/ corr.) | 1.13 | 0.33 | 76.4 | 0.9 |  ${\sim}1$111The estimation is performed by considering pseudo JT effect according to Eq. (20), Eq. (8), Eq. (9) and the state mixing coefficients from Ref. [15, 63]. |  $4.19*10^{-10}$ 222By using the experimental Huang-Rhys factor to correct the phonon term, $X_{if}$ increases by a factor of 47.5.  | 83.2 ns | 12.0
$\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E}$ (Expt.) | 1.19 | 0.34333$\Delta Q$ is estimated by the Huang-Rhys factor and phonon energy, $S=\omega\Delta Q^{2}/2\hbar$, under one-dimensional effective phonon approximtion. | 64 [45] | 0.9 [45] | – | – | 0.9 ns444They are the PL lifetime and rate of $\prescript{1}{}{A}_{1}$, respectively. Since the transition is claimed to be nonradiative dominant [50], they are approximately IC lifetime and rate. [49] |  $1.11\times 10^{3}$ 444They are the PL lifetime and rate of $\prescript{1}{}{A}_{1}$, respectively. Since the transition is claimed to be nonradiative dominant [50], they are approximately IC lifetime and rate. [49]

###  III.6 Angle-Dependent and Magnetic-Field Dependent ODMR

The rates of radiative recombination, internal conversion and intersystem
crossing obtained from the calculations above set the prerequisite for the
simulation of ODMR contrast (spin-dependent PL contrast). Additionally, the
ZFS of triplet ground state and excited state are entered to account for their
spin sublevels. Our calculated ZFS of $\prescript{3}{}{A}_{2}$ is $D=3.03$
GHz, similar to the experimental value $D=2.87$ GHz [16, 11, 71]. We currently
use the experimental value of $D=1.42$ GHz for excited triplet state
$\prescript{3}{}{E}$ [71] given methods for accurate prediction of excited
state ZFS remain to be developed. The optical saturation parameter $\beta$ and
Rabi frequency $\Omega$ are entered as parameters into the model, and their
values are selected within the experimental range [43]. cw-ODMR, which is
evaluated at the steady state when the populations no longer change as a
function of time, is independent on the initial spin state. We then apply an
oscillatory microwave field to drive the population between $\ket{-1}$ and
$\ket{0}$ spin sublevels in $\prescript{3}{}{A}_{2}$.

Figure 3: ODMR contrast from the simulation using the calculated rates
compared with Expt. A [11] and Expt. B [72]. The angle refers to the angle of
magnetic field with respect to the NV axis along [111]. Expt. A shows an angle
of $74\pm 1$$\mathrm{\SIUnitSymbolDegree}$. The angle of magnetic field in
Expt. B is unknown. The dashed blue curve is the ODMR simulation using all the
rates from calculations. The solid blue curve is the simulation with
calculated rates and corrected
$k_{\perp/z}(\prescript{1}{}{E}\rightarrow\prescript{3}{}{A}_{2})$ and
$k_{\text{IC}}(\prescript{1}{}{A}_{1}\rightarrow\prescript{1}{}{E})$ as
discussed in Sec. III.4 and Sec. III.5.

Using the input from our first-principles calculations, we plot the simulated
ODMR contrast against magnetic field in comparison with the experiments [11,
72] in Fig. 3. The simulated ODMR contrast decreases with increasing the
magnetic field, consistent with the trend shown by the experiments (orange
dots and squares). Such decrease of ODMR contrast is originated from the
mixing of the spin sublevels under magnetic field, as can be seen in SM Fig.
S11. The mixing of the spin sublevels leads to smaller contrast of the axial
and nonaxial ISC rates, especially
$k_{\perp/z}({{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}})$.
Consequently, the spin polarization is less pronounced, manifesting as reduced
ODMR contrast.

Our simulated ODMR contrast is overestimated compared to experiments. This is
because of the underestimated ISC rates
$k_{\perp/z}({{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}})$ and
IC rate
$k_{\text{IC}}({{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}})$,
with detailed explanation in earlier sections. Under continuous optical
excitation, the system is easier to populate ${{}^{1}\widetilde{E}}$ and
${{}^{1}\widetilde{A}_{1}}$ through the channel ${{}^{3}\widetilde{E}}^{\pm
1}\rightarrow{{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}}$ when
there is an applied microwave field driving populations from spin sublevel
$\ket{0}$ to $\ket{-1}$. When
$k_{\perp/z}({{}^{1}\widetilde{E}}\rightarrow{{}^{3}\widetilde{A}_{2}})$ and
$k_{\text{IC}}({{}^{1}\widetilde{A}_{1}}\rightarrow{{}^{1}\widetilde{E}})$ are
underestimated, the population accumulates at the singlet states
${{}^{1}\widetilde{A}_{1}}$ and ${{}^{1}\widetilde{E}}$. This results in the
enhanced PL intensity contrast between the two situations, namely in the
presence and absence of a microwave field. With the corrected rates discussed
in Sec. III.5 and Sec. III.4, we obtain ODMR contrast in better agreement with
the experiments.

Finally, we study the ODMR contrast dependence on the magnetic field
direction. How spin sublevels mix depends on the magnetic field direction with
respect to the NV axis, which is quantified by polar angle $\theta$. In Figs.
4(a) and 4(b), we show both the angle dependency and magnetic-field dependency
of normalized PL intensity and ODMR contrast. The simulated normalized PL
intensity at $\theta=1$\mathrm{\SIUnitSymbolDegree}$$ exhibits similar sharp
reduction character as the experiment [73]. This reduction shares the same
origin as the ODMR, as elaborated below. When the magnetic field is perfectly
aligned with the NV axis, the ODMR contrast is maximal since there is no spin
mixing. When magnetic field is slightly misaligned to the NV axis, we can see
two positions of sharp reductions of the ODMR contrast at $B={\sim}50$ mT and
${\sim}100$ mT, which are related to the ZFS in $\prescript{3}{}{E}$ and the
one in $\prescript{3}{}{A}_{2}$, respectively. This is resulted from the
excited state level-anticrossing (ESLAC) and the ground-state level-
anticrossing (GSLAC). As can be seen in SM Fig. S11 that shows the extend to
which the spin sublevels mix, there is strong spin mixing of $\ket{0}$ to
$\ket{-1}$ at ESLAC and GSLAC. When the magnetic field is more misaligned with
the NV axis, i.e.
$10$\mathrm{\SIUnitSymbolDegree}$<\theta<90$\mathrm{\SIUnitSymbolDegree}$$,
the ODMR contrast becomes highly sensitive to the magnetic field, nearly
vanishing after the ESLAC. In addition, we find that the GSLAC and ESLAC of
the NV center gradually disappear with increasing magnetic field after $B>25$
mT. The vanishing GSLAC and ESLAC can also be reflected by the ODMR frequency
plots in SM Fig. S11. In general, this result suggests our theory can reliably
predict the ODMR dependence on magnetic field direction, which is useful for
setting up experiments.

Figure 4: (a) Normalized PL intensity simulated using calculated rates against
the applied magnetic field $\mathbf{B}$ whose direction is
$1$\mathrm{\SIUnitSymbolDegree}$$ relative to the NV axis. The experiment for
comparison is from Ref. [73]. (b) ODMR contrast against magnetic field of an
angle with respect to the NV axis [111], which is the spin quantization axis
at $B=0$. The strong magnetic field dependency is originated from spin mixing.
In both the normalized PL intensity and ODMR contrast, the sharp reduction at
$B={\sim}50$ and ${\sim}100$ mT corresponds to level-anticrossing (LAC) in the
excited state $\prescript{3}{}{E}$ and ground state $\prescript{3}{}{A}_{2}$,
respectively. $B={\sim}50$ and ${\sim}100$ are related to the ZFS in
$\prescript{3}{}{E}$ and $\prescript{3}{}{A}_{2}$, respectively. At
$\theta=0$\mathrm{\SIUnitSymbolDegree}$$, the ODMR contrast is optimized with
no spin mixing.

###  III.7 Optimization for ODMR Contrast

The optimization of ODMR contrast is a critical step experimentally, as we
show next can be realized through tuning optical saturation parameter $\beta$
(related to excitation efficiency $\beta k_{R}$) and tuning Rabi frequency
$k_{\text{MW}}$ [43]. Such parameters are cumbersome to search for
experimentally but our theory can be much more efficient.

In Fig. 5, we show the optimization of ODMR through the two methods,
separately. By Fig. 5(a), we find that ODMR contrast is optimized when the
optical excitation power is ${\sim 0.1}$ of the saturation. The part in
$\beta>0.1$ is consistent with the experiment [43], and we additionally show
the part in $\beta<0.1$, which might be difficult to measure due to low PL
intensity under weak optical excitation. Fig. 5(b) indicates that the ODMR
contrast can reach its maximum at $k_{\text{MW}}\approx 5$ MHz. When the
populations of the spin sublevels are driven by the Rabi oscillation, the
nonaxial ISC ${{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}}$
becomes the preferential path for the relaxation from ${{}^{3}\widetilde{E}}$.
Thus, the upper bound of the ODMR contrast in this case is limited by the
contrast between
$k_{\perp}({{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}})$ and
$k_{z}({{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}})$.

Figure 5: ODMR contrast can be comptimized by tuning (a) optical saturation
parameter $\beta$ and (b) Rabi frequency $k_{\text{MW}}$. The ODMR is
simulated at $B=0$ and $\theta=0$. The corrected rates are used for the ODMR
simulation to show the optimization of ODMR contrast.

##  IV Conclusion

In conclusion, we have developed general first-principles computational
platform for spin-dependent PL contrast and cw-ODMR, for triplet and singlet
spin defects in solids. We solved the kinetic master equation with excited-
state relaxation rates calculated from first-principles. We validated our
implementation by comparing with the experimental data on NV center and ST1
defect in diamond.

The ODMR simulation from first principles requires electronic structure of
spin defects including spin sublevels due to ZFS, and the rates of all
possible transitions including radiative recombination, ISC (spin-flip
nonradiative recombination) and IC (spin-conserving nonradiative
recombination). Our calculations overall show good agreement with experiments
on the protopytical solid-state spin defect, NV center in diamond. The results
showcase the rich physics underlying the entire excitation and relaxation
cycle. In particular, we identify the coupling of $\prescript{1}{}{E}$ and
$\prescript{1}{}{E^{\prime}}$ due to the Coulomb repulsion from CASSCF
calculation and group theory, and show the corresponding effect on SOC matrix
elements. We complete the derivation and calculation for the effective SOC
matrix elements of the NV center considering pseudo JT effect.As a result, we
unequivocally clarify the experimental observation of the axial ISC
${{}^{3}\widetilde{E}}\rightarrow{{}^{1}\widetilde{A}_{1}}$, which was thought
to be forbidden in the first order by symmetry. Importantly, we show dynamical
JT effect in the degenerate states $\prescript{3}{}{E}$ and
$\prescript{1}{}{E}$ significantly enhances the nonradiative recombination by
reducing the potential energy barrier.

Finally, we simulate cw-ODMR of the NV center from first-principles
calculations. We provide the optimization strategies for ODMR contrast with
respect to magnetic field, optical saturation parameter and Rabi frequency.
These are found to be informative especially for the range of conditions
difficult to reach experimentally. This shows the critical role of our
computational technique can play in guiding experiments, which also enables
deeper understanding to the spin polarization of spin defects.

The study emphasizes the importance of accounting for the multi-reference
character of the electronic states, SOC and electron-phonon coupling for spin
defects. It unveils the challenge that there is a need of developing advanced
first-principles theory for accurate prediction of SOC and electron-phonon
coupling in solid-state spin defects. On the other hand, the atomic structure
of many spin defects has not been determined yet, such as ST1 defect in
diamond and quantum defects in two-dimensional wide-band gap semiconductors.
Our developed computational platform can be essential for identifying existing
spin defects, and potentially applied to the design of new solid-state spin
defect important for nanophotonics and quantum information science.

###### Acknowledgements.

Ping, Li, and Zhang acknowledge the support by the National Science Foundation
under grant no. DMR-2143233. Varganov, V. Dergachev, and I. Dergachev
acknowledge the support by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences Established Program to Stimulate Competitive
Research under Award Number DE-SC0022178. This research used resources of the
Scientific Data and Computing Center, a component of the Computational Science
Initiative, at Brookhaven National Laboratory under Contract No. DE-SC0012704,
the Lux Supercomputer at UC Santa Cruz, funded by NSF MRI Grant No. AST
1828315, the National Energy Research Scientific Computing Center (NERSC), a
U.S. Department of Energy Office of Science User Facility operated under
Contract No. DE-AC02-05CH11231, and the Extreme Science and Engineering
Discovery Environment (XSEDE), which is supported by National Science
Foundation Grant No. ACI-1548562 [74].

## References

  * Mamin  _et al._ [2013] H. Mamin, M. Kim, M. Sherwood, C. Rettner, K. Ohno, D. Awschalom, and D. Rugar, Sci. 339, 557 (2013).
  * Ryan  _et al._ [2010] C. A. Ryan, J. S. Hodges, and D. G. Cory, Phys. Rev. Lett. 105, 200402 (2010).
  * Dolde  _et al._ [2014] F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, _et al._ , Nat. Commun. 5, 3371 (2014).
  * Gottscholl  _et al._ [2021] A. Gottscholl, M. Diez, V. Soltamov, C. Kasper, A. Sperlich, M. Kianinia, C. Bradac, I. Aharonovich, and V. Dyakonov, Sci. Adv. 7, eabf3630 (2021).
  * Bernien  _et al._ [2013] H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, _et al._ , Nat. 497, 86 (2013).
  * Gruber  _et al._ [1997] A. Gruber, A. Drabenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. v. Borczyskowski, Sci. 276, 2012 (1997).
  * Toyli  _et al._ [2012] D. Toyli, D. Christle, A. Alkauskas, B. Buckley, C. Van de Walle, and D. Awschalom, Phys. Rev. X 2, 031001 (2012).
  * Scheidegger  _et al._ [2022] P. J. Scheidegger, S. Diesch, M. L. Palm, and C. L. Degen, Appl. Phys. Lett. 120 (2022).
  * Togan  _et al._ [2010] E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, _et al._ , Nat. 466, 730 (2010).
  * Doherty  _et al._ [2013] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, Phys. Rep. 528, 1 (2013).
  * Tetienne  _et al._ [2012] J. Tetienne, L. Rondin, P. Spinicelli, M. Chipaux, T. Debuisschert, J. Roch, and V. Jacques, New J. Phys. 14, 103033 (2012).
  * Ernst  _et al._ [2023a] S. Ernst, P. J. Scheidegger, S. Diesch, and C. L. Degen, Phys. Rev. B 108, 085203 (2023a).
  * Ernst  _et al._ [2023b] S. Ernst, P. J. Scheidegger, S. Diesch, L. Lorenzelli, and C. L. Degen, Phys. Rev. Lett. 131, 086903 (2023b).
  * Baber  _et al._ [2021] S. Baber, R. N. E. Malein, P. Khatri, P. S. Keatley, S. Guo, F. Withers, A. J. Ramsay, and I. J. Luxmoore, Nano Lett. 22, 461 (2021).
  * Thiering and Gali [2018] G. Thiering and A. Gali, Phys. Rev. B 98, 085207 (2018).
  * Smart  _et al._ [2021] T. J. Smart, K. Li, J. Xu, and Y. Ping, npj Comput. Mater. 7, 1 (2021).
  * Wu  _et al._ [2017] F. Wu, A. Galatas, R. Sundararaman, D. Rocca, and Y. Ping, Phys. Rev. Mater. 1, 071001(R) (2017).
  * Ma  _et al._ [2010] Y. Ma, M. Rohlfing, and A. Gali, Phys. Rev. B 81, 041204 (2010).
  * Sheng  _et al._ [2022] N. Sheng, C. Vorwerk, M. Govoni, and G. Galli, J. Chem. Theory Comput. 18, 3512 (2022).
  * Bhandari  _et al._ [2021] C. Bhandari, A. L. Wysocki, S. E. Economou, P. Dev, and K. Park, Phys. Rev. B 103, 014115 (2021).
  * Wu  _et al._ [2019a] F. Wu, D. Rocca, and Y. Ping, J. Mater. Chem. C 7, 12891 (2019a).
  * Wu  _et al._ [2019b] F. Wu, T. J. Smart, J. Xu, and Y. Ping, Phys. Rev. B 100, 081407(R) (2019b).
  * Thiering and Gali [2017] G. Thiering and A. Gali, Phys. Rev. B 96, 081115 (2017).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.: Condens. Matter 21, 395502 (2009).
  * Hamann [2013] D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).
  * Heyd  _et al._ [2006] J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of Chemical Physics 124, 219906 (2006).
  * Deák  _et al._ [2010] P. Deák, B. Aradi, T. Frauenheim, E. Janzén, and A. Gali, Phys. Rev. B 81, 153203 (2010).
  * Gali  _et al._ [2009] A. Gali, E. Janzén, P. Deák, G. Kresse, and E. Kaxiras, Phys. Rev. Lett. 103, 186404 (2009).
  * Razinkovas  _et al._ [2021] L. Razinkovas, M. W. Doherty, N. B. Manson, C. G. Van de Walle, and A. Alkauskas, Phys. Rev. B 104, 045303 (2021).
  * Roos [2005] B. O. Roos, in  _Multiconfigurational quantum chemistry_ (Elsevier, 2005) pp. 725–764.
  * Marini  _et al._ [2009] A. Marini, C. Hogan, M. Grüning, and D. Varsano, Comput. Phys. Commun. 180, 1392 (2009).
  * Godby and Needs [1989] R. W. Godby and R. Needs, Phys. Rev. Lett. 62, 1169 (1989).
  * Oschlies  _et al._ [1995] A. Oschlies, R. Godby, and R. Needs, Phys. Rev. B 51, 1527 (1995).
  * Neese [2012] F. Neese, Wiley Interdiscip. Rev. Comput. Mol. Sci. 2, 73 (2012).
  * Neese  _et al._ [2020] F. Neese, F. Wennmohs, U. Becker, and C. Riplinger, J. Chem. Phys. 152, 224108 (2020).
  * Reiher [2006] M. Reiher, Theor. Chem. Acc. 116, 241 (2006).
  * Dunning [1989] T. H. J. Dunning, J. Chem. Phys. 90, 1007 (1989).
  * Pritchard  _et al._ [2019] B. P. Pritchard, D. Altarawy, B. Didier, T. D. Gibson, and T. L. Windus, J. Chem. Inf. Model. 59, 4814 (2019).
  * Neese [2005] F. Neese, J. Chem. Phys. 122, 034107 (2005).
  * Rohatgi [2014] A. Rohatgi, URL http://arohatgi. info/WebPlotDigitizer/app , 1 (2014).
  * Dréau  _et al._ [2011] A. Dréau, M. Lesik, L. Rondin, P. Spinicelli, O. Arcizet, J.-F. Roch, and V. Jacques, Phys. Rev. B 84, 195204 (2011).
  * Knight and Milonni [1980] P. L. Knight and P. W. Milonni, Phys. Rep. 66, 21 (1980).
  * Kehayias  _et al._ [2013] P. Kehayias, M. Doherty, D. English, R. Fischer, A. Jarmola, K. Jensen, N. Leefer, P. Hemmer, N. Manson, and D. Budker, Phys. Rev. B 88, 165202 (2013).
  * Robledo  _et al._ [2011] L. Robledo, H. Bernien, T. Van Der Sar, and R. Hanson, New J. Phys. 13, 025013 (2011).
  * Lee  _et al._ [2013] S.-Y. Lee, M. Widmann, T. Rendler, M. W. Doherty, T. M. Babinec, S. Yang, M. Eyer, P. Siyushev, B. J. Hausmann, M. Loncar, _et al._ , Nat. Nanotechnol. 8, 487 (2013).
  * Balasubramanian  _et al._ [2019] P. Balasubramanian, M. H. Metsch, P. Reddy, L. J. Rogers, N. B. Manson, M. W. Doherty, and F. Jelezko, Nanophotonics 8, 1993 (2019).
  * Acosta  _et al._ [2010] V. Acosta, A. Jarmola, E. Bauch, and D. Budker, Phys. Rev. B 82, 201202 (2010).
  * Rogers  _et al._ [2008] L. Rogers, S. Armstrong, M. Sellars, and N. Manson, New J. Phys. 10, 103024 (2008).
  * Davies and Hamer [1976] G. Davies and M. Hamer, Proc. Math. Phys. Eng. Sci. 348, 285 (1976).
  * Goldman  _et al._ [2015a] M. L. Goldman, A. Sipahigil, M. Doherty, N. Y. Yao, S. Bennett, M. Markham, D. Twitchen, N. Manson, A. Kubanek, and M. D. Lukin, Phys. Rev. Lett. 114, 145502 (2015a).
  * Goldman  _et al._ [2015b] M. L. Goldman, M. Doherty, A. Sipahigil, N. Y. Yao, S. Bennett, N. Manson, A. Kubanek, and M. D. Lukin, Phys. Rev. B 91, 165201 (2015b).
  * Tinkham [2003] M. Tinkham,  _Group theory and quantum mechanics_ (Courier Corporation, 2003).
  * Maze  _et al._ [2011] J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, New J. Phys. 13, 025025 (2011).
  * Doherty  _et al._ [2011] M. W. Doherty, N. B. Manson, P. Delaney, and L. C. Hollenberg, New J. Phys. 13, 025019 (2011).
  * Bockstedte  _et al._ [2018] M. Bockstedte, F. Schütz, T. Garratt, V. Ivády, and A. Gali, npj Quantum Mater. 3, 31 (2018).
  * Batalov  _et al._ [2009] A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. Rogers, R. McMurtrie, N. Manson, F. Jelezko, and J. Wrachtrup, Phys. Rev. Lett. 102, 195506 (2009).
  * de Souza  _et al._ [2019] B. de Souza, G. Farias, F. Neese, and R. Izsak, J. Chem. Theory Comput. 15, 1896 (2019).
  * Gupta  _et al._ [2016] A. Gupta, L. Hacquebard, and L. Childress, J. Opt. Soc. Am. B 33, B28 (2016).
  * Bersuker [2017] I. Bersuker, in  _J. Phys. Conf. Ser._ , Vol. 833 (IOP Publishing, 2017) p. 012001.
  * Bersuker [2006] I. Bersuker, in  _The Jahn-Teller Effect_ (Cambridge University Press, 2006).
  * Jin  _et al._ [2022] Y. Jin, M. Govoni, and G. Galli, npj Comput. Mater. 8, 238 (2022).
  * Lawetz  _et al._ [1972] V. Lawetz, G. Orlandi, and W. Siebrand, J. Chem. Phys. 56, 4058 (1972).
  * Tatchen  _et al._ [2007] J. Tatchen, N. Gilka, and C. M. Marian, Phys. Chem. Chem. Phys. 9, 5209 (2007).
  * Alkauskas  _et al._ [2014] A. Alkauskas, Q. Yan, and C. G. Van de Walle, Phys. Rev. B 90, 075202 (2014).
  * Abtew  _et al._ [2011] T. A. Abtew, Y. Sun, B.-C. Shih, P. Dev, S. Zhang, and P. Zhang, Phys. Rev. Lett. 107, 146403 (2011).
  * Brawand  _et al._ [2015] N. P. Brawand, M. Vörös, and G. Galli, Nanoscale 7, 3737 (2015).
  * Davies [1981] G. Davies, Rep. Prog. Phys. 44, 787 (1981).
  * Alkauskas  _et al._ [2016] A. Alkauskas, M. D. McCluskey, and C. G. Van de Walle, J. Appl. Phys. 119 (2016).
  * Neumann  _et al._ [2009] P. Neumann, R. Kolesov, V. Jacques, J. Beck, J. Tisler, A. Batalov, L. Rogers, N. Manson, G. Balasubramanian, F. Jelezko, _et al._ , New J. Phys. 11, 013017 (2009).
  * Janitz  _et al._ [2022] E. Janitz, K. Herb, L. A. Völker, W. S. Huxter, C. L. Degen, and J. M. Abendroth, J. Mater. Chem. C 10, 13533 (2022).
  * Epstein  _et al._ [2005] R. Epstein, F. Mendoza, Y. Kato, and D. Awschalom, Nat. Phys. 1, 94 (2005).
  * Towns  _et al._ [2014] J. Towns, T. Cockerill, M. Dahan, I. Foster, K. Gaither, A. Grimshaw, V. Hazlewood, S. Lathrop, D. Lifka, G. D. Peterson, R. Roskies, J. R. Scott, and N. Wilkins-Diehr, Comput. Sci. Eng. 16, 62 (2014).


|  fr |  86.59 |  93.13 |  86.92 |  59.77 |  57.17 |  55.41 |  99.62 |  99.6 |  99.54 |  99.79 |  99.54 |  99.83
|  es |  87.17 |  83.57 |  93.55 |  59.26 |  56.16 |  54.9 |  99.6 |  99.58 |  99.62 |  99.75 |  99.62 |  99.81
|  hi |  65.05 |  62.53 |  64.29 |  90.61 |  71.67 |  64.21 |  99.45 |  99.58 |  99.33 |  98.66 |  97.61 |  98.66
|  gu |  59.09 |  58.0 |  58.76 |  68.4 |  92.04 |  65.88 |  99.45 |  99.6 |  99.54 |  99.06 |  97.63 |  98.97
|  bn |  59.51 |  58.93 |  59.93 |  67.98 |  70.41 |  90.36 |  99.52 |  99.48 |  99.35 |  98.81 |  97.44 |  98.3
RL |  en |  63.79 |  52.14 |  52.39 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.47 |  69.82 |  52.72 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.81 |  52.64 |  68.15 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  70.41 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.22 |  66.64 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.14 |  52.05 |  68.48 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0

Table 72:  The table represents the $G_{S}$ and $S_{S}$ using FT over fine-
tuned XLM-RoBERTa on the fever ‘es’ dataset.

|  |  $G_{S}$ ($x^{\prime}$) $\rightarrow$ |  $S_{S}$ ($\hat{x}$) $\rightarrow$
---|---|---|---
Set |  $x$ $\downarrow$ |  en |  fr |  es |  hi |  gu |  bn |  en |  fr |  es |  hi |  gu |  bn
IL |  en |  94.89 |  83.66 |  84.66 |  53.48 |  54.4 |  54.74 |  87.05 |  87.97 |  87.66 |  99.37 |  98.24 |  97.95
|  fr |  82.31 |  92.37 |  83.24 |  52.89 |  53.65 |  54.15 |  90.13 |  88.24 |  88.77 |  99.5 |  98.53 |  98.18
|  es |  83.49 |  84.07 |  93.21 |  53.31 |  54.15 |  54.23 |  90.03 |  87.76 |  87.15 |  99.5 |  98.41 |  98.24
|  hi |  60.52 |  63.03 |  62.61 |  91.2 |  57.0 |  57.59 |  95.08 |  93.53 |  93.55 |  98.83 |  97.34 |  96.96
|  gu |  58.93 |  61.44 |  60.6 |  53.65 |  95.81 |  58.93 |  95.75 |  93.69 |  93.57 |  99.35 |  95.66 |  96.52
|  bn |  58.93 |  61.69 |  60.44 |  53.4 |  56.92 |  94.8 |  95.94 |  94.03 |  93.84 |  99.25 |  97.07 |  95.66
ML |  en |  88.85 |  67.14 |  64.46 |  52.14 |  52.05 |  52.05 |  87.78 |  91.49 |  93.42 |  100.0 |  100.0 |  100.0
|  fr |  78.62 |  87.43 |  66.81 |  52.14 |  52.05 |  52.05 |  88.98 |  88.66 |  93.23 |  99.96 |  100.0 |  100.0
|  es |  86.59 |  79.72 |  86.25 |  52.05 |  52.05 |  52.05 |  85.41 |  88.5 |  90.34 |  99.98 |  100.0 |  100.0
|  hi |  84.16 |  82.98 |  74.1 |  91.2 |  55.91 |  65.21 |  77.6 |  77.56 |  84.3 |  87.55 |  98.89 |  95.12
|  gu |  86.17 |  88.77 |  78.29 |  83.57 |  85.67 |  78.29 |  72.74 |  72.09 |  79.69 |  84.58 |  95.75 |  90.09
|  bn |  81.31 |  80.39 |  71.25 |  69.07 |  55.32 |  89.27 |  77.6 |  78.25 |  85.58 |  91.89 |  98.95 |  93.55
LL |  en |  93.8 |  81.06 |  81.73 |  56.24 |  53.23 |  53.56 |  99.33 |  99.02 |  99.31 |  99.77 |  99.79 |  99.79
|  fr |  81.22 |  93.13 |  81.98 |  55.16 |  53.23 |  53.98 |  99.37 |  98.99 |  99.18 |  99.79 |  99.81 |  99.79
|  es |  80.13 |  81.47 |  92.46 |  54.57 |  52.89 |  53.9 |  99.48 |  99.2 |  99.33 |  99.79 |  99.79 |  99.77
|  hi |  57.25 |  56.58 |  57.33 |  90.78 |  56.58 |  58.09 |  99.62 |  99.35 |  99.54 |  99.6 |  99.71 |  99.45
|  gu |  54.65 |  55.83 |  55.32 |  59.43 |  87.09 |  59.01 |  99.62 |  99.35 |  99.35 |  99.45 |  99.56 |  99.33
|  bn |  54.9 |  54.99 |  54.74 |  57.17 |  55.66 |  91.28 |  99.75 |  99.52 |  99.69 |  99.73 |  99.79 |  99.48
RL |  en |  87.43 |  63.29 |  61.19 |  52.05 |  52.05 |  52.05 |  90.0 |  92.81 |  94.11 |  100.0 |  100.0 |  100.0
|  fr |  73.43 |  85.75 |  65.46 |  52.05 |  52.05 |  52.05 |  91.28 |  90.17 |  93.71 |  100.0 |  100.0 |  100.0
|  es |  80.55 |  74.43 |  86.34 |  52.05 |  52.05 |  52.05 |  88.87 |  90.11 |  90.8 |  100.0 |  100.0 |  100.0
|  hi |  83.07 |  80.39 |  75.02 |  88.01 |  54.57 |  58.0 |  80.51 |  80.55 |  84.09 |  93.17 |  99.2 |  96.88
|  gu |  81.64 |  82.15 |  75.11 |  65.3 |  83.15 |  62.95 |  78.77 |  78.73 |  83.47 |  94.7 |  97.46 |  95.01
|  bn |  78.12 |  74.35 |  68.82 |  58.68 |  54.15 |  85.33 |  82.23 |  83.36 |  87.3 |  96.94 |  99.33 |  96.23

Table 73:  The table represents the $G_{S}$ and $S_{S}$ using FT over fine-
tuned XLM-RoBERTa on the fever ‘hi’ dataset.

|  |  $G_{S}$ ($x^{\prime}$) $\rightarrow$ |  $S_{S}$ ($\hat{x}$) $\rightarrow$
---|---|---|---
Set |  $x$ $\downarrow$ |  en |  fr |  es |  hi |  gu |  bn |  en |  fr |  es |  hi |  gu |  bn
IL |  en |  88.01 |  74.85 |  76.61 |  55.41 |  52.14 |  52.89 |  84.62 |  89.12 |  86.67 |  97.4 |  99.96 |  98.81
|  fr |  81.31 |  90.7 |  81.47 |  54.57 |  52.05 |  52.56 |  85.52 |  87.87 |  86.23 |  97.99 |  99.98 |  99.12
|  es |  81.39 |  77.95 |  90.36 |  54.74 |  52.05 |  52.56 |  84.97 |  88.08 |  85.98 |  97.67 |  99.98 |  99.02
|  hi |  59.43 |  56.83 |  59.43 |  94.72 |  52.89 |  60.02 |  90.67 |  94.72 |  92.44 |  92.92 |  99.83 |  96.75
|  gu |  55.83 |  55.24 |  57.75 |  61.19 |  90.78 |  61.02 |  92.33 |  95.7 |  93.44 |  95.16 |  99.12 |  95.16
|  bn |  56.83 |  56.08 |  57.59 |  60.69 |  52.56 |  93.88 |  92.06 |  95.28 |  93.29 |  95.1 |  99.83 |  96.12
ML |  en |  53.06 |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.14 |  60.69 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.05 |  52.05 |  60.02 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  59.18 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.05 |  58.84 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  59.26 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
LL |  en |  70.33 |  52.14 |  52.47 |  52.05 |  52.05 |  52.05 |  99.96 |  100.0 |  99.89 |  100.0 |  100.0 |  100.0
|  fr |  52.22 |  76.36 |  53.06 |  52.05 |  52.05 |  52.05 |  100.0 |  96.96 |  99.98 |  100.0 |  100.0 |  100.0
|  es |  52.05 |  52.05 |  72.51 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  99.81 |  100.0 |  100.0 |  100.0
|  hi |  53.14 |  52.22 |  52.64 |  83.91 |  52.72 |  53.56 |  99.67 |  99.89 |  99.6 |  95.1 |  99.85 |  99.54
|  gu |  52.05 |  52.05 |  52.05 |  56.33 |  91.95 |  55.32 |  100.0 |  100.0 |  100.0 |  96.54 |  77.14 |  97.46
|  bn |  52.14 |  52.05 |  52.14 |  52.72 |  52.39 |  84.07 |  99.98 |  100.0 |  100.0 |  99.77 |  100.0 |  94.7
RL |  en |  52.72 |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.14 |  60.86 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.05 |  52.05 |  60.02 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  59.01 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.05 |  58.76 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  58.93 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0

Table 74:  The table represents the $G_{S}$ and $S_{S}$ using FT over fine-
tuned XLM-RoBERTa on the fever ‘gu’ dataset.

|  |  $G_{S}$ ($x^{\prime}$) $\rightarrow$ |  $S_{S}$ ($\hat{x}$) $\rightarrow$
---|---|---|---
Set |  $x$ $\downarrow$ |  en |  fr |  es |  hi |  gu |  bn |  en |  fr |  es |  hi |  gu |  bn
IL |  en |  52.39 |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.14 |  60.02 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.05 |  52.05 |  59.09 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  57.42 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.05 |  57.17 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  56.92 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
ML |  en |  67.39 |  52.47 |  53.4 |  52.05 |  52.05 |  52.05 |  99.98 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.89 |  71.5 |  53.48 |  52.05 |  52.05 |  52.05 |  100.0 |  99.98 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  53.31 |  52.98 |  70.41 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  99.98 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  80.64 |  52.72 |  52.3 |  100.0 |  100.0 |  100.0 |  98.51 |  99.92 |  99.94
|  gu |  52.05 |  52.05 |  52.05 |  53.81 |  79.3 |  52.64 |  100.0 |  100.0 |  100.0 |  99.56 |  99.22 |  99.98
|  bn |  52.05 |  52.14 |  52.05 |  57.84 |  53.73 |  76.87 |  100.0 |  100.0 |  100.0 |  98.09 |  99.67 |  98.58
LL |  en |  97.57 |  98.32 |  95.31 |  87.09 |  72.84 |  82.48 |  60.79 |  58.09 |  63.16 |  69.7 |  82.61 |  73.34
|  fr |  66.47 |  94.89 |  67.56 |  63.96 |  60.27 |  62.87 |  87.26 |  72.19 |  87.22 |  90.42 |  94.15 |  91.47
|  es |  87.43 |  92.71 |  96.56 |  80.13 |  73.09 |  77.28 |  68.94 |  64.61 |  66.41 |  75.61 |  83.32 |  77.98
|  hi |  92.2 |  97.32 |  91.11 |  99.16 |  94.22 |  94.64 |  66.3 |  57.38 |  66.22 |  57.65 |  63.39 |  61.82
|  gu |  85.67 |  96.73 |  89.86 |  97.07 |  99.67 |  95.47 |  71.42 |  58.63 |  67.98 |  58.97 |  55.13 |  59.45
|  bn |  87.09 |  95.47 |  86.76 |  94.22 |  93.21 |  97.9 |  70.37 |  59.54 |  70.64 |  60.5 |  62.13 |  57.96
RL |  en |  61.86 |  52.14 |  52.14 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.3 |  67.06 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.14 |  52.05 |  67.22 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  68.65 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.05 |  64.71 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  66.3 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0

Table 75:  The table represents the $G_{S}$ and $S_{S}$ using FT over fine-
tuned XLM-RoBERTa on the fever ‘bn’ dataset.

|  |  $G_{S}$ ($x^{\prime}$) $\rightarrow$ |  $S_{S}$ ($\hat{x}$) $\rightarrow$
---|---|---|---
Set |  $x$ $\downarrow$ |  en |  fr |  es |  hi |  gu |  bn |  en |  fr |  es |  hi |  gu |  bn
IL |  en |  95.39 |  80.64 |  80.22 |  50.96 |  49.45 |  48.79 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  79.21 |  95.64 |  83.15 |  50.96 |  49.54 |  48.95 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  80.89 |  80.81 |  95.81 |  50.96 |  49.45 |  48.79 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  51.13 |  51.13 |  51.38 |  94.72 |  53.65 |  51.89 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  50.13 |  50.96 |  50.54 |  52.81 |  97.48 |  52.22 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  49.54 |  50.13 |  50.38 |  54.65 |  54.15 |  96.14 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
ML |  en |  92.04 |  78.12 |  79.72 |  55.32 |  52.72 |  53.73 |  98.55 |  98.16 |  97.23 |  99.69 |  99.79 |  99.62
|  fr |  77.87 |  89.69 |  80.05 |  54.9 |  52.89 |  53.73 |  98.32 |  96.52 |  95.73 |  99.45 |  99.67 |  99.62
|  es |  78.21 |  80.39 |  89.94 |  55.32 |  52.72 |  53.73 |  98.81 |  97.74 |  96.56 |  99.48 |  99.71 |  99.71
|  hi |  56.83 |  56.08 |  56.75 |  88.27 |  59.09 |  60.6 |  99.6 |  99.54 |  98.78 |  96.67 |  99.35 |  97.92
|  gu |  52.64 |  52.72 |  53.06 |  56.75 |  87.26 |  57.42 |  99.89 |  99.92 |  99.71 |  99.33 |  98.22 |  97.57
|  bn |  53.06 |  53.4 |  52.98 |  57.75 |  57.25 |  87.59 |  99.83 |  99.83 |  99.64 |  99.22 |  99.04 |  96.04
LL |  en |  91.95 |  87.34 |  86.08 |  77.95 |  71.17 |  75.11 |  99.73 |  99.71 |  99.85 |  99.98 |  100.0 |  99.98
|  fr |  84.16 |  92.12 |  85.16 |  78.62 |  71.75 |  75.27 |  99.85 |  99.58 |  99.81 |  99.98 |  100.0 |  100.0
|  es |  86.0 |  88.43 |  92.46 |  77.37 |  71.08 |  74.27 |  99.83 |  99.67 |  99.75 |  99.92 |  100.0 |  100.0
|  hi |  75.02 |  76.7 |  76.03 |  91.37 |  79.55 |  80.22 |  99.87 |  99.81 |  99.94 |  99.83 |  100.0 |  100.0
|  gu |  70.41 |  73.34 |  71.17 |  81.89 |  92.12 |  82.9 |  99.89 |  99.81 |  99.89 |  99.87 |  99.89 |  99.96
|  bn |  70.41 |  72.42 |  69.66 |  79.38 |  79.46 |  91.45 |  99.94 |  99.92 |  100.0 |  99.94 |  99.98 |  99.89
RL |  en |  89.86 |  71.58 |  71.92 |  54.4 |  52.64 |  52.72 |  99.22 |  99.73 |  99.89 |  100.0 |  100.0 |  100.0
|  fr |  76.95 |  89.02 |  75.61 |  54.57 |  52.64 |  52.47 |  99.39 |  99.48 |  99.89 |  100.0 |  100.0 |  99.96
|  es |  76.7 |  75.86 |  89.1 |  54.57 |  52.64 |  52.56 |  99.39 |  99.62 |  99.81 |  100.0 |  100.0 |  99.96
|  hi |  54.74 |  54.06 |  53.9 |  88.6 |  55.57 |  56.24 |  99.31 |  99.69 |  99.77 |  99.69 |  99.98 |  99.94
|  gu |  53.06 |  52.72 |  52.89 |  55.41 |  89.52 |  55.99 |  99.06 |  99.5 |  99.85 |  99.67 |  99.89 |  99.56
|  bn |  53.48 |  53.23 |  53.23 |  54.99 |  54.57 |  89.02 |  99.48 |  99.67 |  99.98 |  99.85 |  99.98 |  99.81

Table 76:  The table represents the $G_{S}$ and $S_{S}$ using FT over fine-
tuned XLM-RoBERTa on the fever ‘mixed’ dataset.

|  |  $G_{S}$ ($x^{\prime}$) $\rightarrow$ |  $S_{S}$ ($\hat{x}$) $\rightarrow$
---|---|---|---
Set |  $x$ $\downarrow$ |  en |  fr |  es |  hi |  gu |  bn |  en |  fr |  es |  hi |  gu |  bn
IL |  en |  53.65 |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.22 |  60.86 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.05 |  52.05 |  60.44 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  60.94 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.05 |  60.77 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  61.11 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
ML |  en |  58.09 |  52.14 |  52.05 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  fr |  52.39 |  63.62 |  52.3 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  es |  52.05 |  52.05 |  62.53 |  52.05 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  hi |  52.05 |  52.05 |  52.05 |  66.39 |  52.05 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  gu |  52.05 |  52.05 |  52.05 |  52.3 |  66.81 |  52.05 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
|  bn |  52.05 |  52.05 |  52.05 |  52.05 |  52.05 |  66.47 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0 |  100.0
LL |  en |  94.64 |  83.74 |  85.08 |  55.99 |  52.89 |  53.06 |  98.95 |  98.99 |  99.16 |  99.87 |  99.87 |  99.87
|  fr |  83.82 |  94.13 |  85.33 |  56.41 |  53.06 |  52.56 |  98.89 |  98.74 |  99.04 |  99.81 |  99.87 |  99.81
|  es |  84.24 |  86.17 |  93.46 |  58.0 |  52.81 |  53.14 |  98.68 |  98.78 |  98.93 |  99.81 |  99.85 |  99.81
|  hi |  59.01 |  59.85 |  60.35 |  91.62 |  55.74 |  59.01 |  99.22 |  99.43 |  99.33 |  99.37 |  99.62 |  99.54
|  gu |  53.4 |  53.56 |  53.81 |  53.81 |  90.86 |  53.56 |  99.67 |  99.69 |  99.67 |  99.87 |  99.45 |  99.83
|  bn |  54.48 |  54.9 |  54.57 |  59.18 |  55.57 |  91.11 |  99.58 |  99.45 |  99.64 |  99.69 |  99.77 |  99.1
RL |  en |  87.93 |  67.98 |  68.9 |  52.89 |  52.05 |  52.14 |  99.25 |  99.75 |  99.73 |  99.92 |  99.87 |  99.79
|  fr |  72.84 |  88.94 |  72.59 |  52.72 |  52.3 |  52.14 |  99.45 |  99.62 |  99.5 |  99.98 |  99.94 |  99.73
|  es |  71.5 |  71.58 |  87.93 |  52.98 |  52.22 |  52.14 |  99.41 |  99.56 |  99.58 |  99.83 |  99.96 |  99.73
|  hi |  53.4 |  53.06 |  53.06 |  87.76 |  52.56 |  53.73 |  99.77 |  99.83 |  99.85 |  99.45 |  99.73 |  98.93
|  gu |  52.39 |  52.39 |  52.39 |  52.47 |  87.34 |  52.98 |  99.77 |  99.87 |  99.89 |  99.56 |  99.71 |  99.39
|  bn |  52.39 |  52.22 |  52.56 |  52.81 |  52.14 |  87.43 |  99.75 |  99.96 |  99.92 |  99.81 |  100.0 |  98.6

Table 77:  The table represents the $G_{S}$ and $S_{S}$ using FT over fine-
tuned XLM-RoBERTa on the fever ‘inverse’ dataset.