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[Satoru Matsuishi](https://orcid.org/0000-0001-8905-0255), [Hidekazu Ikeno](https://orcid.org/0000-0002-3840-4049), [Yukinori Koyama](https://orcid.org/0000-0002-7090-4430), [Takashi Takeda](https://orcid.org/0000-0003-2510-4562)

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[Luminescence Spectra Simulation of Ce                    <sup>3+</sup>                    ‐Activated Phosphors by Accumulating Emission Lines Along First‐Principles Molecular Dynamics Trajectories](https://mdr.nims.go.jp/datasets/3e4a2cf2-e1a5-4e0f-b0c7-a2de1c77fbbd)

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Luminescence Spectra Simulation of Ce3+‐Activated Phosphors by Accumulating Emission Lines Along First‐Principles Molecular Dynamics TrajectoriesAdvanced Theory and Simulations www.advtheorysimul.comRESEARCH ARTICLELuminescence Spectra Simulation of Ce3 + -Activated Phosphors by Accumulating Emission Lines Along First-Principles Molecular Dynamics Trajectories Satoru Matsuishi1 Hidekazu Ikeno2 Yukinori Koyama3 Takashi Takeda4 1 Research Center For Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Ibaraki, Japan 2 Department of Materials Science, Graduate School of Engineering, Osaka Metropolitan University, Sakai, Osaka, Japan 3 Center for Basic Research On Materials, National Institute for Materials Science, Tsukuba, Ibaraki, Japan 4 Research Center For Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Correspondence: Satoru Matsuishi ( matsuishi.satoru@nims.go.jp) Received: 8 December 2025 Revised: 24 April 2026 Accepted: 18 May 2026 Keywords: Ce3 + -activated phosphor | first-principles molecular dynamics | photoluminescence ABSTRACT To simulate the luminescence emission spectra of Ce3 + -activated phosphors, we propose a method based on scalar-relativistic density functional theory with corrections for on-site Coulomb interactions, and first-principles molecular dynamics (FPMD) calculations, accounting for atomic thermal motion beyond harmonic oscillations. The FPMD calculations at finite temperature were performed with the excited-state (ES) electron configuration corresponding to the Ce 4f0 5d1 state. Subsequently, the transition energies and probabilities of 4f0 5d1 →4f1 5d0 transitions in each FPMD snapshot were calculated using the N − 1 electron configuration in which the highest occupied level in the ground-state (GS) configuration is vacated. This is justified because applying Janak’s theorem to a flat-band system allows each Kohn-Sham orbital’s energy to be regarded as the total energy of an ES generated by an excitation from the highest level. The time-averaged spectrum was calculated by accumulating emission lines at the transition energies with amplitudes proportional to the probabilities. The spectra calculated for Ce3 + -activated yttrium aluminum garnet exhibit shapes and positions that resemble those measured at temperatures above 300 K. The peak energies and full widths at half maximum at 300 K for the series of Ce3 + -activated phosphors correlate with the experimental values, demonstrating the potential for predicting emission spectra of novel phosphors. 1R  e  p  s  [  r  p  d  m            To©Ah Introduction esearchers are currently searching for novel phosphors thatxhibit stable emission brightness even at high operating tem-eratures in devices, as well as those that produce narrow-widthingle-line emission, for use in future high-brightness lighting 1–4 ] and 8K televisions defined by the BT.2020 standard [ 5 ],espectively. In the search for rare-earth-activated inorganic com-ounds that could serve as phosphors, data-driven and theory-riven methods, including machine learning (ML), are being usedore and more employed [ 6–8 ]. The ML-based prediction of thehis is an open access article under the terms of the Creative Commons Attribution Licenriginal work is properly cited. 2026 The Author(s). Advanced Theory and Simulations published by Wiley-VCH GmbHdvanced Theory and Simulations , 2026; 9:e02287 ttps://doi.org/10.1002/adts.202502287spectral position and shape of emission bands in candidate com-pounds is expected to speed up the discovery of new materials.However, these properties vary based on the concentration ofrare earth ions, impurities, defects, and measurement conditions,leading to a lack of consistent data for ML. Therefore, a method isrequired to determine the emission spectrum and its temperaturedependence solely from the structural and compositional data ofthe host crystal and the information about the rare-earth dopants.Since the electronic excitation and relaxation of rare-earth ionsare understood as transitions among the multiplets of 4f andse, which permits use, distribution and reproduction in any medium, provided the  1 of 14http://www.advtheorysimul.comhttps://doi.org/10.1002/adts.202502287https://orcid.org/0000-0001-8905-0255https://orcid.org/0000-0002-3840-4049https://orcid.org/0000-0002-7090-4430https://orcid.org/0000-0003-2510-4562mailto:matsuishi.satoru@nims.go.jphttp://creativecommons.org/licenses/by/4.0/https://doi.org/10.1002/adts.202502287http://crossmark.crossref.org/dialog/?doi=10.1002%2Fadts.202502287&domain=pdf&date_stamp=2026-05-275  o  t  (  (  t  i  c  b  e  o  mT  s  o  e  a  t  c  t  A  t  r  (  m  c  c  c  m  r  h  h  eF  t  a  p  c  t  a  s  o  a  q  p  e  b  f  r  lI  s  t  e  i  e  i  o                                                  2 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Cread electrons, their electronic states are calculated using meth-ds that account for many-body effects of electrons, such ashe state-averaged complete active space self-consistent fieldSA-CASSCF) and second-order many-body perturbation theoryCASPT2) methods [ 9–12 ]. However, simulating electronic transi-ion processes that involve dynamic changes in atomic positionss challenging with many-body techniques, which demand highomputational costs. The Ce3 + ion is one such rare-earth ion,ut it contains only one electron in its 4f shell. Therefore, itslectronic state can be obtained even by a low-cost method basedn density-functional theory (DFT), which does not considerany-body effects. he delta self-consistent field ( ΔSCF) approach using the con-trained DFT (cDFT) technique is a DFT-based method previ-usly used to estimate the spectral positions of absorption andmission maxima, i.e., peak energies, due to the 4f1 5d0 →4f0 5d1nd 4f0 5d1 →4f1 5d0 transitions [ 13, 14 ]. This method calculateshe total energy difference between the ground-state electroniconfiguration (GSEC) and the excited-state electronic configura-ion (ESEC) to determine the absorption/emission peak energy.dditionally, by applying the quasi-1D harmonic oscillator modelo the total energy change associated with the atomic configu-ation shift from a ground state (GS) relaxed to an excited stateES) relaxed structure, it becomes possible to estimate the inho-ogeneous broadening of the absorption and emission bandsaused by phonon vibrations [ 15–17 ]. However, this approach onlyonsiders the total energy difference between specific electroniconfigurations and does not provide spectral details involvingultiple optical transitions between various electronic configu-ations. Moreover, since atomic motion is approximated by a 1Darmonic oscillator, the influence of more complex higher-orderarmonic vibrations and anharmonic motion on the emissionnergy and linewidth cannot be captured. irst-principles molecular dynamics (FPMD) using DFT elec-ronic structure calculations can track the time evolution oftomic positions and momentum at finite temperature. Byerforming time-dependent density functional theory (TDDFT)alculations with the nonadiabatic couplings between wavefunc-ions at neighboring MD time steps, the electronic excitationnd relaxation processes of Ce3 + -activated phosphors have beenimulated [ 18, 19 ]. Therefore, FPMD allows the simulation ofptical absorption and emission spectra while accounting fortomic motion. However, previous studies’ FPMD and subse-uent SCF calculations for emission spectral simulation wereerformed solely using GSEC, which is unsuitable for calculatingmission spectra. The atomic motion before emission shoulde described using ESEC, and the electronic states obtainedrom GSEC do not include the electronic levels necessary forealistically representing emission spectra, as will be explainedater. n this paper, to obtain the emission spectra of Ce3 + -activatedystems, we propose using FPMD calculations with ESEC andransition dipole moment (TDM) calculations with the N − 1lectron configuration, which vacates the highest occupied leveln GSEC. Applying Janak’s theorem to the system in which thenergy of each orbital is nearly independent of its occupancy,t can be shown that the energies of each one-electron levelbtained from self-consistent field (SCF) calculations with theof 14tN − 1 electron configuration approximate the total energies ofthe ground state and excited states generated by one-electronexcitations from the ground state. Therefore, the results of theN − 1 electron calculation are suitable for obtaining excitation andemission spectra due to one-electron transitions. By using theN − 1 electron configuration, we obtained the electronic statesof the Ce3 + -activated systems, which include multiple Ce 4flevels below the lowest Ce 5d level, leading to emission spectraconsisting of multiple 5d→4f emission bands, as observed inthe experimental spectra. By accumulating the emission lineslocated at transition energies with amplitudes proportional to thetransition probabilities along the FPMD trajectory, we obtainedan ensemble-averaged emission spectrum that accounts for theeffects of atomic thermal motion at finite temperatures, withoutbeing restricted by the harmonic oscillator approximation. Thespectral positions and shapes of the emission bands calculatedfor a Ce3 + -activated yttrium aluminum garnet (YAG) structuralmodel between 100 and 700 K were compared with those ofexperimental spectra in the literature. Additionally, the emissionpeak energy and full width at half maximum (FWHM) at 300K were calculated for a series of Ce3 + -activated systems andcompared with literature values. 2 Results and Discussion 2.1 Calculation of Ce3 + 4f1 5d0 ↔4f0 5d1 Transition Energies and Probabilities in a Fixed Configuration Coordinate Figure 1a shows the energy level diagram of Ce3 + in the YAGcrystal, based on experimental spectroscopic data [ 20 ]. Due tothe spin-orbit (SO) interaction, the 4f1 5d0 state splits into 2 F5/2 and 2 F7/2 multiplets with an energy difference of 0.290 eV (2340cm− 1 ). The crystal field (CF) interaction further divides thesemultiplets into three and four substates, respectively. Meanwhile,the 4f0 5d1 state splits into 5d1 and higher substates (such as5d2 , 5d3 , etc.), mainly because of the CF interaction. The 2 F5/2 ground state is excited to the 5d1 and higher excited states(ESs) by blue to ultraviolet light, and these higher ESs quicklyundergo a non-radiative transition to the 5d1 state. As a result,the radiative transition mainly occurs through 5d1 →2 F5/2 and5d1 →2 F7/2 transitions. A periodic structural model of Ce3 + -activated YAG crystal withthe composition CeY23 Al40 O96 was constructed by substitutinga Y atom with Ce in the conventional face-centered cubicunit cell, which has a lattice parameter of approximately 12Å. The ground-state relaxed configuration coordinate ( Qg ) wasdetermined through DFT geometry optimization with a GSEC of961 valence electrons ( N = 961), where the Kohn-Sham orbitals(KSOs) are filled from lowest to highest energy. The KSO energylevel diagram for the 𝐴g state, generated using scalar-relativisticPBEsol + U calculations with GSEC and UCe opt = 4.68 eV, appearson the far left of Figure 1b . The level marker colors show whetherthe Ce 4f (blue), Ce 5d (red), or other atomic orbitals mainlycontribute to each level. Between the valence band maximum(VBM), made up of oxygen 2p orbitals, and the lowest Ce 5d leveljust above the conduction band minimum (CBM), there is onlyone Ce 4f level with orbital index i = 1, which is the highestoccupied level. The other unoccupied 4f levels are at least 4.1 eVAdvanced Theory and Simulations, 2026ive Commons LicenseFIGURE 1 (a) Multiplet energy diagram of Ce3 + in YAG. (b) One-electron Kohn-Sham orbital energy level diagrams for the CeY23 Al40 O96 system, respectively calculated for the ground-state electron configuration (GSEC) in the GS-relaxed structure Qg ( Ag ), the excited-state electron configuration (ESEC) in Qg ( Ag * ), ESEC in the ES-relaxed structure Qe ( Ae * ), GSEC in Qe ( Ae ), the N − 1 electron configuration in Qg ( Ag − 1 ), and the N − 1 electron configuration in Qe ( Ae − 1 ). The level marker with a black spot indicates the occupied level, and the color of the marker indicates the contribution of atomic orbitals, with red and blue colors representing larger contributions of Ce 5d and Ce 4f orbitals, respectively. h  4  o  o  a  T  4  a  wU  t  t  s w  e  j A  a  r  ϵ  G  K  p  e                                 A 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creatigher than this occupied level. Therefore, when the occupied Cef level is assigned to the 4f orbital occupied in the lowest substatef 2 F5/2 shown in Figure 1a , the energies of the other substatesf 2 F5/2 and 2 F7/2 , where one of the other 4f levels is occupied,re at least 4.1 eV higher than that of the lowest 2 F5/2 substate.his conflicts with the observation that the absorption due to 4f-f transitions among the 2 F5/2 and 2 F7/2 substates occurs at anbsorption energy below 0.5 eV. The presence of 4f-4f transitionsill be discussed later. nder the independent particle approximation, the TDM fromhe i -th orbital to the j -th orbital is described by Equation ( 1 ) usinghe wave functions φi and φj , and eigenvalues ϵi and ϵj in eachtate. 𝑴𝑖𝑗 = − 𝑖ℏ 𝑚𝑒 (𝜖𝑗 − 𝜖𝑖 ) ⟨𝜑𝑖 |𝒑 |𝜑𝑗 ⟩(1)here p is the momentum operator, and me is the rest mass of anlectron. Then, the oscillator strength fij between the i -th and the -th orbitals is given by Equation ( 2 ). 𝑓𝑖𝑗 =2𝑚𝑒 (𝜖𝑗 − 𝜖𝑖 )3ℏ2 |||𝑀𝑖𝑗 |||2 (2)s shown by the red line in Figure 2a , the qualitative opticalbsorption spectrum of the 𝐴 g state in the visible to ultravioletegion was created by superimposing Gaussian peaks centered atj − ϵ1 with area intensities of f1, j ( j > 1). The FWHM w1, j of allaussian peaks was fixed at 0.234 eV, a value estimated at 300 using the ΔSCF approach described in the next section. Theosition and the height of the blue bar indicate the transitionnergy and oscillator strength of each transition, respectively. Fordvanced Theory and Simulations, 2026icomparison, the experimental fluorescence excitation spectrumat 300 K, as reported in the literature [ 21 ], is also shown as thedashed black line, rather than an absorption spectrum. In theexperimental spectrum, two major peaks are observed at 2.70 eV,corresponding to the 2 F5/2 →5d1 transition, and at 3.62 eV, corre-sponding to the 2 F5/2 →5d2 transition. However, the lowest Ce 5dstate is hybridized with delocalized Y 4d states that make up theCBM, and such peaks do not appear in the calculated spectrum.Similarly, the split fluorescence emission spectrum resulting fromthe coexistence of 5d1 →2 F5/2 and 5d1 →2 F7/2 transitions cannot beexplained by the levels shown in this diagram, because 4f levelsother than the lowest 4f level are located above the lowest 5d state.In the ΔSCF approach, the absorption peak energy 𝜖ΔSCF abs forthe 4f1 5d0 →4f0 5d1 transition is estimated as the total energydifference between the Ag state with GSEC and the Ag * statewith ESEC in Qg , rather than the difference in one-electron levelenergy [ 13, 14 ]. In the Ag * state obtained by the cDFT calculationwith ESEC, there are 14 unoccupied levels below the highestoccupied level, as shown in the second diagram from the left inFigure 1b . The highest occupied level comprises Ce 5d orbitals,while the lower unoccupied levels mainly consist of Ce 4f orbitals.Similarly, the emission peak energy 𝜖ΔSCF em for the 4f0 5d1 →4f1 5d0 transition is estimated as the difference in total energy betweenthe Ae * state with ESEC and the Ae state with GSEC in the ES-relaxed configuration coordinate Qe . This structure is determinedby the cDFT structural optimization with the ESEC. From theobtained energy differences, 𝜖ΔSCF abs , 𝜖ΔSCF em and the Stokes shift 𝛥𝑆calculated as 𝜖ΔSCF abs − 𝜖ΔSCF em were evaluated to be 2.75, 2.19, and0.56 eV, respectively. These values are comparable to those ofthe experimental excitation and emission spectra, 2.70, 2.36, and0.31 eV, respectively. In Figure 2a–c , the calculated absorptionspectrum consists of a single Gaussian peak centered at 𝜖ΔSCF abs (0)3 of 14ve Commons LicenseFIGURE 2 Calculated absorption spectra (red curves) for YAG:Ce using TDM from the lowest Ce 4f level to higher levels in (a) Ag , (b) Ag * , and (c) Ag − 1 states: Blue bars indicate the emission energies and the oscillator strengths; Black dashed lines shows the experimental luminescence excitation spectrum at 300 K for 2.31 eV emission taken from the literature, and the vertical magenta line indicates the ΔSCF absorption energy 𝜖ΔSCF abs at 0 K; Calculated luminescence emission spectra for YAG: Ce (red curve) using TDM from the lowest Ce 5d level to lower Ce 4f levels in (d) Ae * and (e) Ae − 1 states. Black dashed lines show the experimental emission spectrum at 5 K for 2.75 eV excitation, and the magenta line indicates the ΔSCF emission energy 𝜖ΔSCF em (0) at 0 K. w  T p  s s  o  5A  F  o  w  1  c  f  o  e  aO  o  =  w  T  e  n  f  G  f  a  p  t  1  e               4 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creith FWHM of 𝑤Δ𝑆𝐶 𝐹 𝑎 𝑏 𝑠 (300) is plotted as magenta dotted lines.he absorption peak at 2.75 eV overlapped with the 2 F5/2 →5d1eak at 2.70 eV of the experimental spectrum. Since the ΔSCFcheme considers only GSEC corresponding to the lowest 2 F5/2ubstate and ESEC corresponding to the 5d1 state, we cannotbtain information about other transitions such as 2 F5/2 →5d2 andd1 →2 F7/2 . s shown in the second and third diagrams from the left inigure 1b , the unoccupied 4f levels are positioned below theccupied Ce 5d level in the Ag * and Ae * states, as calculatedith the ESEC configuration. By using ϵj – ϵ1 and f1, j for j > obtained from the TDM calculation for the Ag * state, wean derive the absorption spectrum shown in Figure 2b , whicheatures an absorption band peaking at 3.54 eV, consisting of threeverlapping transition lines. However, this absorption energyxceeds the measured value by more than 0.8 eV, and the ΔSCFbsorption energy 𝜖Δ𝑆𝐶𝐹 𝑎 𝑏 𝑠 (0). n the other hand, by using the transition energies ϵ15 – ϵi andscillator strength fi ,15 from the lowest occupied Ce 5d level ( j 15) to the unoccupied 4f levels ( i = 1–14) in the Ae * state,e also obtain the emission spectrum as shown in Figure 2d .he FWHM 𝑤𝑖 ,15 of all transitions was set to 0.270 eV, a valuestimated at 5 K using the ΔSCF approach described in theext section. For comparison, the measured spectrum at 5 Krom literature [ 21 ] and the calculated spectrum with a singleaussian peak at 𝜖ΔSCF em (0) = 2.19 eV are also plotted in thisigure. Similar to the measured spectrum, which contains a peakt 2.36 eV corresponding to the 5d1 → 2 F5/2 transition and aeak at 2.17 eV corresponding to the 5d1 → 2 F7/2 transition,he spectrum of the Ae * state consists of two peaks at 1.92 and.55 eV. However, the peak positions deviate significantly from thexperimental positions of 2.36 and 2.17 eV by − 0.44 and − 0.62 eV, of 14atirespectively. This contrasts with the agreement of the 𝜖ΔSCF em (0)with the experimental 5d1 → 2 F5/2 energy to within 0.17 eV. Here, we demonstrate that SCF calculations using the N − 1 elec-tron configuration, which vacates the highest occupied orbital inGSEC, as shown in the fifth diagram from the left in Figure 1b ,produce electronic states that are more suitable for obtaining theabsorption and emission spectra. According to Janak’s theorem[ 22 ], which describes the relationship between one-electronorbital energy and total energy in DFT, the energy ϵi of the i -thband is equal to the total energy Etot partially differentiated withrespect to the occupation number ni of the i -th orbital. 𝜖𝑖 =𝜕𝐸tot 𝜕𝑛𝑖 (3)Therefore, in the A− 1 state with an N − 1 electron configurationcorresponding to the Ce 4f0 5d0 state and a total energy ofEtot [4f0 5f0 ], the one-electron energy of the lowest Ce 4f band ϵ4f is expressed by 𝜖4f = lim 𝛿→0 𝐸tot [4f 𝛿5d0 ]− 𝐸tot [4f 0 5d0 ]𝛿(4)Assuming the flat band where the total energy is proportional tothe 4f band occupancy, Equation ( 4 ) is approximated by 𝜖4f ∼ lim 𝛿→0 𝐸tot [4f 1 5d0 ]𝛿 + 𝐸tot [4f 0 5d0 ]( 1 − 𝛿) − 𝐸tot [4f 0 5d0 ]𝛿(5)∼ 𝐸tot [4f 1 5d0 ]− 𝐸tot [4f 0 5d0 ]Advanced Theory and Simulations, 2026ve Commons LicenseS  g T  m T  (  a  t  f  l  =  s  ϵ i i  s  t  w  c  mT  N  f  a  1  l  b  2 t  o  c  u  a  s                                                    A 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Cimilarly, the one-electron energy of the lowest Ce 5d band ϵ5d isiven by 𝜖5d = lim 𝛿→0 𝐸tot [4f 0 5d𝛿]− 𝐸tot [4f 0 5d0 ]𝛿∼ 𝐸tot [4f 0 5d1 ]− 𝐸tot [4f 0 5d0 ](6)hen, the difference between the one-electron energies is esti-ated by 𝜖5d − 𝜖4f ∼ 𝐸tot [4f 0 5d1 ]− 𝐸tot [4f 1 5d0 ](7)his equals the transition energy from the A state with GSEC4f1 5d0 ) to the A* state with ESEC (4f0 5d1 ). The same conceptpplies to any single-electron excitation and relaxation betweenhe i -th and j -th bands in the A− 1 state, where the orbital indexor the highest occupied band with GSEC is 1. Each one-electronevel corresponds to either the multielectron ground state (GS) ( i 1) or the multielectron excited state (ES) ( i > 1) resulting from aingle-electron excitation from the ground state. For comparison,4f , ϵ5d , and ϵ5d − ϵ4f were approximated by 𝜖4f ∼ 𝐸tot [4f 1 5d0 ]− 𝐸tot [4f 0 5d0 ], 𝜖5d ∼ 𝐸tot [4f 1 5d1 ]− 𝐸tot [4f 1 5d0 ](8)𝜖5d − 𝜖4f ∼ 𝐸tot [4f 1 5d1 ]+ 𝐸tot [4f 0 5d0 ]− 2𝐸tot [4f 1 5d0 ]n the A state with GSEC (4f1 5d0 ) and 𝜖4f ∼ 𝐸tot [4f 1 5d1 ]− 𝐸tot [4f 0 5d1 ], 𝜖5d ∼ 𝐸tot [4f 0 5d1 ]− 𝐸tot [4f 0 5d0 ](9)𝜖5 𝑑 − 𝜖4 𝑓 ∼ 2𝐸tot [4f 0 5d1 ]− 𝐸tot [4f 0 5d0 ]− 𝐸tot [4f 1 5d1 ]n the A* state with ESEC (4f0 5d1 ). The ϵ5d − ϵ4f in A and A*tates have no direct relationship with the transition energy fromhe A state to the A* state. Therefore, the eigenvalues andavefunctions in A− 1 are the most suitable for approximatelyalculating the transition energies and probabilities betweenany-electron states. he energy level diagram of the Ag − 1 state, obtained through the − 1 electron calculation for Qg , is shown in the fifth columnrom the left of Figure 1b . Corresponding to the Ce 4f0 5d0 state,ll Ce 4f and 5d levels become empty with orbital indices from to 16, including the lowest 4f level. Seven doubly degenerateevels, consisting of Ce 4f orbitals, are located 2.25 to 2.62 eVelow the lowest level composed of Ce 5d orbitals. Similar to the F5/2 and 2 F7/2 states in Figure 1a , these levels are divided intohree lower and four higher levels, with a centroid separationf 0.37 eV. Using ϵj − ϵ1 and f1,j ( j > 3) obtained from TDMalculation for Ag − 1 , an absorption spectrum in the visible toltraviolet range was calculated with prominent peaks at 2.59nd 3.69 eV due to 4f-5d transitions, as shown in Figure 2c . Thispectrum resembles the experimental excitation spectrum withdvanced Theory and Simulations, 2026peaks at 2.70 and 3.62 eV. Furthermore, as shown in Figure S1 ,an infrared absorption spectrum with weak peaks due to 4f-4ftransitions below 0.43 eV was also calculated, consistent with theexperimental spectrum reported in the literature. By performing the N − 1 electron calculation for Qe , we obtainedthe energy level diagram for the Ae − 1 state shown on the far rightof Figure 1b . Table 1 lists the energy 𝜖𝑖 of the Ce 4f levels and thelowest 5d levels with indices i from 1 to 16, relative to the energyof the lowest unoccupied orbitals ( 𝑖 = 1, 2) in the diagram. Eachodd-numbered orbital with index i = 1, 3, . . . , 13 degenerates withthe even-numbered orbital with index i + 1. For comparison, theexperimental multiplet energies of Ce3 + in YAG, 𝜖𝑛Γ5 ( n = 1, 2,. . . , 8) based on the results of the CF theory fitting of the infraredabsorption data and the luminescence measurement are listed inTable 1 [ 20 ]. Additionally, the theoretical literature values of 𝜖𝑛Γ5 are also listed. These were calculated using the SA-CASSCF andCASPT2 methods with a (CeO8 )13 − embedded cluster model in theGS-relaxed structure, while considering many-body effects [ 12 ].Each degenerated one-electron level corresponds to a sublevelof the multiplet, 1 Γ5 -3 Γ5 of 2 F5/2 , 4 Γ5 -7 Γ5 of 2 F7/2 , and 8 Γ5 of 5d1 .The separation between the centroids of levels i = 1–6 and levels𝑖 = 7–14, corresponding to the SO splitting of 2 F5/2 and 2 F7/2 isoverestimated at 2858 cm− 1 . This value is 1.22 times greater thanthe experimental value of 2334 cm− 1 and 1.12 times greater thanthe SA-CASSCF/CASPT2 calculation value of 2543 cm− 1 . Thepossible causes of the overestimation are the neglect of the multi-electron effect and the error in estimating the SO interactionin the PAW method, which depends on the selection of theexchange-correlation functional [ 23 ]. Table 1 also shows the oscillator strengths fi ,15 ( i = 1–14) betweenCe 4f levels and the lowest Ce 5d level obtained from TDMcalculations, as well as the sum of those from degenerate levelsfi ,15 + fi + 1,15 ( i = 1, 3, . . . , 13). For comparison, the table alsolists the oscillator strengths 𝑓𝑛Γ5 , 8Γ5 between 2 F5/2 and 2 F5/2 substates and 5d1 state calculated using the SA-CASSCF andCASPT2 methods in a past paper [ 12 ]. Each fi ,15 + fi + 1,15 hasthe same magnitude as the 𝑓𝑛Γ5 , 8Γ5 with n = i , indicatingthat the KSOs in the Ae − 1 state inherit the character of thecorresponding GS and ES multi-electron orbitals. Furthermore,the ratio of ∑14 𝑖= 7 𝑓𝑖, 15 ∕∑6 𝑖= 1 𝑓𝑖, 15 corresponding to the intensityratio of 5d1 →2 F5/2 and 5d1 →2 F7/2 emissions is 0.90, which is alsocomparable to ∑7 𝑛= 4 𝑓𝑛Γ5 , 8Γ5 ∕∑3 𝑛= 1 𝑓𝑛Γ5 , 8Γ5 = 0.81 of SA-CASSCFcalculation. As shown in Figure 2e , the emission spectrum features two peaksat 2.28 and 1.90 eV, which are obtained using ϵ15 − ϵi and fi ,15 ( i= 1–14) in the Ae − 1 state in Table 1 . The deviations of the peakenergies from the experimental values are − 0.06 and − 0.27 eV,respectively, indicating better agreement with the experimentaldata compared to results for the Ae * state. Furthermore, thedeviation of the higher peak energy from 𝜖ΔSCF em (0) also decreasesfrom 0.27 to − 0.09 eV, which further improves the consistencywith the result obtained by the ΔSCF method. From the results ofTDM calculations of Ag − 1 and Ae − 1 states, the peak energies of thelowest 4f1 5d0 →4f0 5d1 absorption and the highest 4f0 5d1 →4f1 5d0 emission are estimated to be 2.59 and 2.28 eV, respectively. Thus,the Stokes shift is evaluated to be 0.31 eV, which almost agreeswith the experimental value of 0.34 eV. 5 of 14reative Commons LicenseTABLE 1 Relative energies of Ce 4f ( i = 1–14) and Ce 5d ( i = 15,16) KSO levels ( ϵi , cm− 1 ), oscillator strengths of 4f→5d transitions ( fi ,15 ) and sum oscillator strength of degenerated transitions ( fi ,15 + fi + 1,15 ) from N − 1 electron calculation for ES-relaxed structure of YAG:Ce structure model. Experimental multi-electron orbital energies ( 𝜖exp 𝑛Γ5 , cm− 1 ) of 2 F5/2 , 2 F7/2 , and 2 D3/2 multiplet terms measured by CF theory fitting for IR absorption data and theoretical orbital energies ( 𝜖cal 𝑛Γ5 , cm− 1 ) with oscillator strengths ( 𝑓cal 𝑛Γ5 , 8Γ5 ) obtained by SA-CASSCF calculation are also listed. i ϵi fi ,15 fi ,15 + fi + 1,15 n 𝝐𝐞𝐱𝐩 𝒏𝚪𝟓 𝝐𝐜𝐚𝐥 𝒏𝚪𝟓 𝒇𝐜𝐚𝐥 𝒏𝚪𝟓 , 𝟖𝚪𝟓 Term 1 0 2.3 × 10− 2 4.2 × 10− 2 1 0 0 3.2 × 10− 2 2 F5/2 2 1.9 × 10− 2 3 486 2.1 × 10− 3 3.9 × 10− 3 2 273 430 3.3 × 10− 3 4 1.8 × 10− 3 5 586 1.8 × 10− 3 5.2 × 10− 3 3 786 770 1.1 × 10− 2 6 3.5 × 10− 3 7 3018 5.2 × 10− 3 2.9 × 10− 2 4 2097 2260 2.5 × 10− 2 2 F7/2 8 2.4 × 10− 2 9 3518 2.4 × 10− 3 8.6 × 10− 3 5 2343 2500 9.9 × 10− 3 10 6.3 × 10− 3 11 3615 3.8 × 10− 3 4.9 × 10− 3 6 2485 2780 2.0 × 10− 3 12 1.2 × 10− 3 13 3648 2.9 × 10− 3 3.8 × 10− 3 7 3822 4230 7.8 × 10− 4 14 8.9 × 10− 4 15 18493 — — 8 19 275 18 700 — 2 D3/2 (5d1 ) 16 —U  c  t  i T  w  I  4  t  t T  e  b  tA  t  e  4  T  e                       6 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creatsing the TDM for Ae − 1 as defined in Equation ( 1 ), Einstein’s Aoefficient Aji , which represents the probability of transition fromhe j -th level to the i -th level due to spontaneous photon emission,s given by Equation ( 10 ). 𝐴𝑗𝑖 =𝑒3 (𝜖𝑗 − 𝜖𝑖 )3 3 𝜋𝜀0 ℏ3 𝑐3 |||𝑀𝑖𝑗 |||2 (10)he number of photons emitted per unit time is given by Aji ρj ,hich is the product of Aji and the occupancy ρj of the j -th level.f the transitions from the lowest Ce 5d level ( j = 15) to the Cef levels ( i = 1–6 and 7–14) are the dominant radiative processes,hen the reciprocal of the radiative transition time τr is given byhe sum of A15, i for i = 1–14: 𝜏r − 1 =14 ∑𝑖= 1 𝐴15 ,𝑖 (11)he τr is estimated to be 53 ns, which closely matches thexperimental value of approximately 60 ns at low temperatureselow 300 K, where radiative transitions dominate non-radiativeransitions [ 21, 24 ]. s described above, using one-electron wavefunctions obtainedhrough DFT calculations with the N − 1 electron configurationnables the calculation of transition energies and TDMs for thef1 5d0 →4f0 5d1 excitation and the 4f0 5d1 →4f1 5d0 emission lines.his approach aligns with the results from photoluminescencexcitation and emission spectra measurements, high-precisionof 14calculations considering multi-electron configurations, and theΔSCF method. Here, it is noted that TDM calculations using the N − 1 electronconfiguration are only valid for systems such as Ce3 + -activatedphosphors, where the energies of each KSO are hardly dependenton the occupied states and can be treated as a one-electronsystem. In other words, this method is not suitable for describingthe excited states of systems that contain strongly correlatedmultiple 4f electrons, such as Eu2 + and other rare-earth activatedphosphors. 2.2 Ce3 + 4f0 5d1 →4f1 5d0 Emission Spectrum Simulation Considering the Effect of Atomic Thermal Motion In FPMD calculations, interatomic forces at time t are calculatedusing quantum mechanics of the electron system on fixed atomiccoordinates. Using the atomic coordinates, velocities, and thecalculated interatomic forces at time t , a classical mechanicscalculation of the atoms over the time interval Δt was performedto find the coordinates and velocities at time t + Δt . FPMDdoes not consider the quantum mechanical motion of atoms,such as zero-point motion of harmonic oscillators, and thereforecannot precisely simulate atomic motion at low temperatures.Meanwhile, FPMD monitors the time evolution of configurationcoordinates, capturing not only harmonic vibrational modes butalso all atomic displacement modes accessible through thermalmotion. This allows for the consideration of anharmonic motionsthat become important at high temperatures. Advanced Theory and Simulations, 2026ive Commons LicenseFIGURE 3 Time evolution of KSO energies calculated for N − 1 electron system on the trajectory of FPMD performed for YAG:Ce structure model with ESEC corresponding to 4f0 5d1 state of Ce3 + ion at a temperature of (a) 100 K, (b) 200 K, (c) 300 K, (d) 400 K, (e) 500 K, (f) 600 K, and (g) 700 K. The line’s color indicates the contribution of Ce 4f and 5d orbitals. F  E  s  b  e  T  a  1  a  t                                  A 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creator the YAG:Ce model, MD calculations were performed usingSEC across a temperature range of 100–700 K. Figure 3hows the time evolution of the KSO energies at the Γ pointased on scalar relativistic SCF calculations using the N − 1lectron configuration at each snapshot of the MD trajectory.he energies of the Ce 4f levels ( i = 1, 2, . . . , 14), whichre 1 to 2 eV above the VBM, and the Ce 5d levels ( i = 15,6), located 1 to 2 eV below the CBM, fluctuate periodicallynd vary over time. Their oscillation amplitudes increase withemperature. dvanced Theory and Simulations, 2026As previously noted, transitions from orbital j = 15 to orbitals i= 1–6 (15→1-6) and 7–14 (15→7-14) correspond to 5d1 →2 F5/2 and5d1 →2 F7/2 transitions in the multiplet energy diagram. Figure S2shows how the mean radiative lifetime ‹ τr › depends on temper-ature. This lifetime is obtained as the reciprocal of the meanradiative transition rate, which is the average of τr calculatedusing Equation ( 11 ) at each snapshot of the MD trajectory. ⟨𝜏r ⟩− 1 = 1 𝑇 ∫𝑇 0 14 ∑𝑖= 1 𝐴15 ,𝑖 ( 𝑡) 𝑑𝑡 (12)The calculated ‹ τr ›− 1 increases linearly with temperature, rangingfrom 56 ns at 100 K to 93 ns at 700 K. Values below 400 K closelymatch the experimentally observed luminescence lifetime [ 24 ].Below 400 K, the experimental non-radiative relaxation rate isless than 106 s− 1 , which is negligible compared to the experimentalradiative relaxation rate, which is higher than 107 s− 1 [ 24 ]. Thehomogeneous linewidth estimated from the lifetime ranges from4.4 × 10− 8 to 7.5 × 10− 8 eV, which is negligible compared to theinhomogeneous linewidth caused by atomic motion. A typical fluorescence spectrophotometer detects the radiationenergy, which is the product of the number of photons and thephoton energy, at each wavelength. Consequently, the theoreticaltime-averaged emission spectrum I ( ϵ), which can be compared tothe measured one, is expressed by Equation ( 13 ) as the sum ofemission lines located at ϵ15 − ϵi with the amplitude of A15, i ρ15 ( ϵ15 − ϵi ). 𝐼 ( 𝜖) = 1 𝑇 ∫𝑇 0 14 ∑𝑖= 1 𝐴15 ,𝑖 ( 𝑡) 𝜌15 ( 𝑡) { 𝜖15 ( 𝑡) − 𝜖𝑖 ( 𝑡) } × 𝛿 { 𝜖 − 𝜖15 ( 𝑡) + 𝜖𝑖 ( 𝑡) } 𝑑𝑡 (13)If the occupancy of the lowest Ce 5d orbital ρ15 ( t ) remains constantdue to continuous excitations, Equation ( 13 ) is rewritten to 𝐼 ( 𝜖) = 1 𝑇 ∫𝑇 0 14 ∑𝑖= 1 𝐴15 ,𝑖 ( 𝑡) { 𝜖15 ( 𝑡) − 𝜖𝑖 ( 𝑡) } 𝛿 { 𝜖 − 𝜖15 ( 𝑡) + 𝜖𝑖 ( 𝑡) } 𝑑𝑡 (14)As shown in Table 1 , the energy differences between the 4fsublevels with orbital indices 𝑖 from 1 to 6 or from 7 to 14, whichcorrespond to 2 F5/2 or 2 F7/2 states, are less than 0.1 eV. This valueis smaller than the FWHM of each 15→i emission line causedby atomic motions estimated by the ΔSCF method ( > 0.3 eV).Therefore, the six emission lines due to 5d1 → 2 F5/2 transitions andthe eight lines due to 5d1 → 2 F7/2 transitions are each combinedinto a single emission band. To evaluate the FWHM induced bythermal atomic motion, the spectra of emission bands for the5d1 → 2 F5/2 and 5d1 → 2 F7/2 components were counted separatelyusing Equations ( 15 ) and ( 16 ). 𝐼5∕2 ( 𝜖) = 1 𝑇 ∫𝑇 0 6 ∑𝑖= 1 𝐴15 ,𝑖 ( 𝑡) { 𝜖15 ( 𝑡) − 𝜖𝑖 ( 𝑡) } × 𝛿 { 𝜖 − 𝜖15 ( 𝑡) + 𝜖𝑖 ( 𝑡) } 𝑑𝑡 (15)7 of 14ive Commons License T  a F  1  (  b  d  𝜀  𝑤  d  o  f  t  F  t  o  e  𝑤  e  t  iI  e  t  n  e  𝜖o  d  t  s  ΔC  A  a  c  e w  c  a  Q  a  Ω  /  a  FIGURE 4 Ce 4f0 5d1 →4f1 5d0 emission spectra calculated by ELA method for YAG:Ce structure model at 100–700 K. Light blue and orange parts of stacked bar graphs respectively indicate intensities of 5d1 →2 F5/2 ( I5/2 ( ϵ)) and the 5d1 →2 F7/2 ( I7/2 ( ϵ)) components. Blue and red lines indicate peak fitting curves for I5/2 ( ϵ) and I7/2 ( ϵ) components, respectively, and the black dashed line indicates the sum of fitting curves.     8 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creat𝐼7∕2 ( 𝜖) = 1 𝑇 ∫𝑇 0 14 ∑𝑖= 7 𝐴15 ,𝑖 ( 𝑡) { 𝜖15 ( 𝑡) − 𝜖𝑖 ( 𝑡) } × 𝛿 { 𝜖 − 𝜖15 ( 𝑡) + 𝜖𝑖 ( 𝑡) } 𝑑𝑡 (16)herefore, the overall spectrum averaged over time is found bydding these spectra. 𝐼 ( 𝜖) = 𝐼5∕2 ( 𝜖) + 𝐼7∕2 ( 𝜖) (17)igure 4 displays the stacked bar chart for I5/2 ( ϵ) and I7/2 ( ϵ) from00 to 700 K, obtained using the emission lines accumulationELA) method. It is represented as histograms of ϵ15 − ϵi weightedy A15, i ( ϵ15 − ϵi ) with a bin size of 0.01 eV. As shown by the 100 Kata, I5/2 ( ϵ) and I7/2 ( ϵ) consist of single Gaussian peaks centered at5∕2cal and 𝜖cal 7∕2 , respectively, with almost equal FWHMs 𝑤cal 5∕2 andcal 7∕2 . Below 200 K, the FWHM is smaller than the peak separationue to SO interaction, Δcal SO = 𝜖cal 5∕2 − 𝜖cal 7∕2 ∼ 0.38 eV, so the presencef two peaks is recognized even in the whole spectrum I ( ϵ). Byitting two Gaussian functions to I5/2 ( ϵ) and I7/2 ( ϵ), respectively,he transition energies at emission maxima 𝜖cal 5∕2 and 𝜖cal 7∕2 , theWHM 𝑤cal em = ( 𝑤cal 5∕2 + 𝑤cal 7∕2 )∕2 , and the area intensity ratio of thewo spectral components 𝑠cal 7∕2 ∕𝑠cal 5∕2 were obtained as a functionf temperature as shown in Figure 5a–c . For comparison, thexperimental peak energies 𝜖exp 5∕2 and 𝜖exp 7∕2 , FWHM 𝑤exp em = ( 𝑤exp 5∕2 +exp 7∕2 )∕2 , and ratio 𝑠exp 5∕2 ∕𝑠exp 7∕2 were also plotted in Figure 5a–c . Thexperimental values were evaluated by fitting the peak functionso the measured spectra reported in the literature [ 21 ], as shownn Figure S3 . n Figure 5a , the peak energy 𝜖cal 5∕2 shifts from 𝜖cal 5∕2 (0) = 2.29(2)V to lower energies with increasing temperature, exhibiting aemperature coefficient 𝑑 𝜖cal 5∕2 ∕𝑑 𝑇 of − 2.6(4) × 10− 4 eV/K (Theumber in parentheses indicates the standard deviation). Thexperimental peak energy 𝜖exp 5∕2 also shifts to lower energy fromexp 5∕2 (0) = 2.361(1) eV with a temperature coefficient 𝑑 𝜖exp 5∕2 ∕𝑑 𝑇f − 1.8(6) × 10− 5 eV/K. In previous studies, the temperatureependence of emission peak energy was estimated by applyinghe quasi-1D harmonic oscillator model to potential energyurfaces in the configuration coordinate diagram obtained bySCF approach, as shown in Figure S4 [ 15–17 ]. The Franck–ondon energy shifts from Ag to Ae states ( 𝐸ΔSCF FC , g ) and from Ae * tog * states ( 𝐸ΔSCF FC , e ) are evaluated by the ΔSCF approach to be 0.347nd 0.210 eV, respectively. On the other hand, the total normaloordinate change ΔQ from Qg to Qe defined by Equation ( 18 ) wasvaluated to be 1.423 amu1/2 ⋅Å. Δ𝑄 =√ ∑𝛼𝜉𝑚𝛼(𝑟e; 𝛼𝜉 − 𝑟g; 𝛼𝜉)2 (18)here α ( = 1, 2, . . . ) and ( ξ = 𝑥 , 𝑦 , 𝑧 ) label atoms in the unitell and coordinate in Cartesian axes, respectively, mα representstomic mass, and re; αξ and rg; αξ are atomic coordinates in Qe andg , respectively. Using the values of 𝐸ΔSCF FC , g , 𝐸ΔSCF FC , e , and ΔQ , thessociated effective vibrational energies, Ωg = (2 𝐸ΔSCF FC , g )1/2 / ΔQ ande = (2 𝐸ΔSCF FC , e )1/2 / ΔQ , and the Huang-Rhys parameters, Sg = 𝐸ΔSCFFC , g  ℏΩg and Se = 𝐸ΔSCF FC , e / ℏΩe for the A and the A* states are givens 37.8 meV and 29.4 meV, and 9.17 and 7.13, respectively. Then,of 14the temperature dependence of the emission peak energy 𝜖ΔSCF em isobtained by using Equation ( 19 ) [ 15–17 ]. 𝜖em ( 𝑇) = 𝜖em ( 0) +⎛ ⎜ ⎜ ⎜ ⎝ Ω𝑔 2 −Ω𝑒 2 Ω𝑒 2 +8Ω𝑔 4 Ωe 2 (Ω𝑔 2 +Ω𝑒 2 ) Δ𝑆 ( 0) 𝜖em ( 0) ⎞ ⎟ ⎟ ⎟ ⎠ 𝑘B 𝑇 (19)The magenta line in Figure 5a shows the temperature dependenceof 𝜖ΔSCF em with 𝜖ΔSCF em (0) = 2.19 eV and a temperature coefficientAdvanced Theory and Simulations, 2026ive Commons LicenseFIGURE 5 Temperature dependences of peak profile parameters of the emission spectra calculated by the ELA method for YAG:Ce structure model: Peak energies 𝜖cal 5∕2 and 𝜖cal 7∕2 (a), FWHM 𝑤cal em = (𝑤cal 5∕2 + 𝑤cal 7∕2 ) ∕2 (b), and areal intensity ratio 𝑠cal 7∕2 ∕𝑠cal 5∕2 (c) of 5d1 →2 F5/2 and 5d1 →2 F7/2 peaks, and peak energy 𝐸cal em (d) and FHWM 𝑊cal em of the whole emission band. For comparison, the parameters, 𝜖exp 5∕2 and 𝜖exp 7∕2 , 𝑤exp em , 𝑠exp 7∕2 ∕𝑠exp 5∕2 , 𝐸exp em , and 𝑊exp em obtained by the spectral fitting to the experimental spectra in the literature were plot in the panels. Emission energy 𝜖ΔSCF em and FWHM 𝑤ΔSCF em estimated by using the ΔSCF method are plotted in (a,b), respectively. o  t  o  o  a  t  a a  t a  e  𝐸  0 a  aT  i  K  t  0                                              A 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creaf + 2.38 × 10− 4 eV/K. Compared to the experimental results,he ΔSCF approach not only overestimates the absolute valuef the temperature coefficient but also inverts the sign. On thether hand, the ELA method yields the correct sign, while itlso overestimates the absolute value. The discrepancy betweenhe experimental and calculated temperature coefficients isttributed to errors in estimating the Franck-Condon shift 𝐸FC,gnd 𝐸FC,e . By fitting the curve of Equation ( 19 ) to 𝜖exp em , notinghat 𝐸FC,g + 𝐸FC,e = ( Ωg 2 + Ωe 2 )/ ΔQ2 = ΔS (0), 𝐸FC,g and 𝐸FC,ere found to be 0.122(5) and 0.218(5) eV, respectively, where thexperimental Δ𝑆 (0) is 0.34 eV. Also, the curve fitting to 𝜖cal em yieldsFC,g = 0.031(80) eV and 𝐸FC,e = 0.279(80) eV, where Δ𝑆 (0) =.31 eV. In the electronic state obtained by DFT calculation, 𝐸FC,gnd 𝐸FC,e are estimated to deviate from the actual values bybout 0.1 eV. he splitting of the high- and low-energy bands due to the SOnteraction in Ce 4f1 5d0 state, Δcal SO = 𝜖cal 5∕2 − 𝜖cal 7∕2 , is 0.38 eV at 100 and 0.37 eV at 700 K. These values are 58% overestimated fromhe experimental values Δexp SO = 𝜖exp 5∕2 − 𝜖exp 7∕2 = 0.24 eV at 5 K and.23 eV at 600 K. dvanced Theory and Simulations, 2026tSimilar to peak energy, the temperature dependence of the emis-sion peak FWHM has also been estimated using the harmonicoscillator approximation [ 15–17 ]. 𝑤em ( 𝑇) =√8 ln 2 ℏΩg √ 𝑆𝑔 Ωg Ωe √ coth ℏΩe 2𝑘𝐵 𝑇 (20)Furthermore, at classical limit, 2 kB T > ℏΩe , wem ( T ) is approxi-mated by 𝑤em ( 𝑇) ∼√8 ln 2 Ωg Ωe √ 2𝑘𝐵 𝑇ℏΩg 𝑆𝑔 (21)In Figure 5b , the magenta solid line shows the calculatedtemperature dependence of 𝑤ΔSCF em , and the magenta dotted lineshows its classical limit curve. As the agreement of the two curvesindicates, the classical limit holds at temperatures above 300 K(2 kB T = 51.8 meV > ℏΩe = 37.8 meV). The 𝑤cal em of the simulatedspectra overlaps with the classical limit curve, indicating that theFPMD spectral accumulation method estimates transition energyfluctuation widths comparable to those obtained by the ΔSCFmethod. However, in the temperature range from 300 to 700 K,both 𝑤cal em and 𝑤ΔSCF em are ∼ 1.6 times larger than 𝑤exp em . By fitting thecurve of Equation ( 20 ) to 𝑤exp em , 𝐸FC,g and 𝐸FC,e are found to be0.174 and 0.166 eV, respectively. Furthermore, fitting the curve ofEquation ( 21 ) to 𝑤cal em gives 𝐸FC,g = 0.219 eV and 𝐸FC,e = 0.091 eV.Thus, like the previous considerations regarding the temperaturecoefficient of the emission peak energy, the overestimation of𝑤cal em is caused by the overestimation of 𝐸FC,g by ∼ 0.1 eV andunderestimation of 𝐸FC,e by ∼ 0.1 eV compared to the actualvalues. As shown in Figure 5c , the calculated intensity ratio 𝑠cal 7∕2 / 𝑠cal 5∕2 decreases slightly with temperature from 0.54 at 100 K, while thetemperature coefficient is negligibly small at − 4.2(7) × 10− 5 K− 1 .On the other hand, the ratio 𝑠exp 7∕2 ∕𝑠exp 5∕2 obtained by spectral fittingof observed spectra increases with temperature from 0.46 at 5 Kwith the relatively large coefficient + 4.2(7) × 10− 4 K− 1 . However,as the temperature increases, the two emission bands overlap,making separation by fitting difficult. The intensity ratios tend tobe estimated closer to 1 than the true value. Therefore, the actualvalue of 𝑠exp 7∕2 ∕𝑠exp 5∕2 is thought to be constant at approximately 0.5,such as 𝑠exp 7∕2 ∕𝑠exp 5∕2 . Figure 5d,e shows the temperature dependence of the calculatedpeak energy 𝐸cal em and the FWHM 𝑊cal em of the whole emissionband consisting of the superposition of the 5d1 →2F7/2 and the5d1 →2 F5/2 bands. At low temperatures where the two emissionbands are separated, the 𝐸cal em coincides with the peak energy𝜖cal 5∕2 of the 5d1 →2 F5/2 component. As 𝑤cal em increases with increas-ing temperature, the two bands merge, and the 𝐸cal em linearlyapproaches the midpoint between 𝜖cal 5∕2 and 𝜖cal 7∕2 . This behavior islike that of the experimental whole emission band peak energy𝐸exp em , while the 𝐸cal em is slightly underestimated at 96% and 93%of the 𝐸exp em at 100 and 600 K, respectively. Since zero-pointvibrations are not considered, as the temperature is lowered,the FWHM 𝑤cal em of each band approaches zero, and the 𝑊cal em asymptotes to the Δcal SO = 0.38 eV. On the other hand, the FWHM𝑊exp em of experimental whole spectrum linearly increased withtemperature from 0.37 eV at 5 K, since zero-point motion of atoms9 of 14ive Commons LicenseFIGURE 6 Emission spectra at 300 K calculated by the ELA method for a series of Ce3 + -activated phosphors; (a) Y3 Al5 O12 :Ce, (b) YAlO3 :Ce, (c) YAl3 B4 O12 :Ce, (d) YBO3 :Ce, (e) CaYAl3 O7 :Ce, (f) LiYF4 :Ce, (g) CaF2 :Ce,F, (h) CaAl2 O4 :Ce,Na, (i) CaSrAl2 O7 :Ce,Na, (j) Sr2 Al2 O7 :Ce,Na, (k) LiBaPO4 :Ce,Na, and (l) Sr3 B2 O6 :Ce,Na. Light blue and orange parts of stacked bar graphs respectively indicate intensities of 5d1 →2 F5/2 ( I5/2 ( ϵ)) and the 5d1 →2 F7/2 ( I7/2 ( ϵ)) components. Blue and red lines indicate peak fitting curves for I5/2 ( ϵ) and I7/2 ( ϵ) components, respectively, and the black dashed line indicates the sum of fitting curves. g  Δ  a  v  m2CA  a  i  L [  i               1 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creives the FWHM 𝑤exp em ∼ 0.216 eV comparable to the SO splittingexp so = 0.29 eV even at 0 K. Above 300 K, 𝑊cal em increases linearlyt a rate of 2.6(8) × 10− 4 eV/K, which is close to that of 𝑊exp em . Thealue of 𝑊cal em at 300 K is 0.660 eV, 1.54 times larger than theeasured value (1.2(1) × 10− 4 eV/K). .3 Spectral Calculations for Various Known e3 + -Activated Phosphors s shown in Figure 6 , the Ce3 + 4f0 5d1 →4f1 5d0 emission spectrat 300 K were also simulated for the 11 phosphor host systems,ncluding YAlO3 [ 25 ], YAl3 B4 O12 [ 26 ], YBO3 [ 26 ], CaYAl3 O7 [ 27 ],iYF4 [ 28 ], CaF2 [ 29 ], CaAl4 O7 [ 30 ], CaSrAl2 SiO7 [ 31 ], Sr2 Al2 SiO7 32 ], LiBaPO4 [ 33 ], and Sr3 B2 O6 [ 34 ], in addition to YAG listedn Table S1 . This table contains the 𝑈opt Ce used for structure0 of 14optimization, total energy, and TDM calculations, along withthe parameters obtained by ΔSCF analysis, including 𝜖ΔSCF abs (0) ,𝜖ΔSCF em (0) , EFC,g , EFC,e , and ΔQ . Additionally, the results of harmonicoscillator model analysis, ℏΩg , ℏΩe , Sg , Se , the emission peakenergy 𝜖Δ𝑆𝐶𝐹 𝑒𝑚 (300 ) , and FWHM 𝑤Δ𝑆𝐶 𝐹 𝑒𝑚 (300 ) at 300 K are listedin Table S2 . By the spectral fitting of skew normal distributionfunctions to the simulated spectra I5/2 ( ϵ) and I7/2 ( ϵ) shown inFigure 6 , we also obtained 𝜖cal 5∕2 , 𝜖cal 7∕2 , 𝑤cal 5∕2 , 𝑤cal 7∕2 and 𝑠cal 7∕2 / 𝑠cal 5∕2 at300 K listed in Table S3 , and 𝐸cal em and 𝑊cal em listed in Table S4 .For comparison, the experimental peak energies 𝐸exp em and FWHM𝑊exp em read from spectra at 300 K in the literature, as shown inFigure S5 , are also listed in Table S4 . As shown in the 𝜖ΔSCF em (300 ) versus 𝜖cal 5∕2 (300 ) plot in Figure 7a ,all data points are distributed around the line of 𝜖cal 5∕2 = 𝜖ΔSCF em .Advanced Theory and Simulations, 2026ative Commons LicenseFIGURE 7 Comparison between the results of ΔSCF analysis and ELA method at 300 K for a series of Ce3 + -activated phosphors: (a) Peak energy of 5d1 →2 F5/2 band ( 𝜖cal 5∕2 ) versus ΔSCF emission energy ( 𝜖ΔSCF em ) (b) FWHM of 5d1 →2 F5/2 emission band ( 𝑤cal 5∕2 ) versus ΔSCF FWHM ( 𝑤ΔSCF em ). Comparison between calculated and experimental whole emission spectra at 300 K for a series of Ce3 + -activated phosphors: (c) maximum emission energy 𝐸cal em versus 𝐸exp em , (d) FWHM of emission band 𝑊cal em versus 𝑊exp em . T  i  w  d  a  t  c  w  𝑤  e  t  t  r  F  t  p  p  sF  s  l                   A 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Crehe result of the statistical analysis, summarized in Table S5 ,ndicates the 𝜖cal 5∕2 (300 ) agrees well with 𝜖ΔSCF em (300 ) in accuracyithin 0.16(11) eV or 5(3)%. This indicates that the total energyifference between the GSEC and ESEC obtained by the ΔSCFpproach is nearly identical to the energy difference betweenhe lowest 5d level and the lowest 4f level in the N − 1 electronalculations. Similarly, as shown in Figure 7b , 𝑤ΔSCF em (300 ) agreesith 𝑤cal 5∕2 (300 ) in accuracy within 0.07(10) eV or 13(14)%, whileΔSCF em (300 ) in CaAl4 O7 :Ce,Na and CaYAl3 O7 :Ce systems arexceptionally 1.6 and 1.9 times larger than 𝑤cal 5∕2 (300 ) . It implieshat the harmonic oscillator approximation for the shape ofhe ES and GS energy surface is not valid in these systems,esulting in the overestimation of the line broadening. ThePMD spectra accumulation method generally provides resultshat are consistent with the ΔSCF method in calculating theeak energy and FWHM of the emission bands and may alsorovide more accurate predictions for the FWHM in anharmonicystems. igure 7c shows the peak energies of the calculated total emissionpectra 𝐸cal em versus the experimental values 𝐸exp em read from theiterature spectra shown in Figure S5 . The data points are dvanced Theory and Simulations, 2026distributed around the line of 𝐸cal em = 𝐸exp em , and the statisticalanalysis and linear fitting results are summarized in Table S6 ,indicating 𝐸cal em coinciding with 𝐸exp em to within 0.30(22) eV or 9(6)%.Figure 7d shows the calculated FWHM of the total emission spec-tra 𝑊𝑐 𝑎 𝑙 𝑒𝑚 versus the experimental values 𝑊exp em . Each data point islocated at a position significantly different from the line of 𝑊cal em =𝑊exp em , and according to the statistical data shown in Table S6 , 𝑊cal em is overestimated by approximately 1.6 times compared to 𝑊exp em .As shown in Figure S6 , 𝑊cal em has a linear relationship with 𝑤cal em ,𝑊cal em − Δ𝑐 𝑎 𝑙 𝑆𝑂 = 0 . 75(2) 𝑤cal em , using the almost system-independentSO splitting Δcal SO of 0.38(2) eV. Assuming this relationship isvalid for experimental data, i.e., 𝑊exp em − Δexp SO = 0 . 75(2) 𝑤exp em holdsunder Δ𝑒𝑥𝑝 𝑆𝑂 = 0.29 eV, we can see how much 𝑤cal em is overestimatedcompared to 𝑤exp em by comparing 𝑊exp em − Δ𝑒𝑥𝑝 𝑆𝑂 and 𝑊cal em − Δ𝑐 𝑎 𝑙 𝑆𝑂 .The statistical data for 𝑊cal em − Δ𝑐 𝑎 𝑙 𝑆𝑂 shown in Table S6 indicate𝑤cal em = 2 . 2(6) 𝑤exp em . Since 𝑤cal em and 𝑤ΔSCF em are approximately equalin most systems, 𝑤Δ𝑆𝐶 𝐹 𝑒𝑚 is also overestimate by about 2 comparedto 𝑤exp em . The extent of overestimation is the same as that suggestedwhen discussing the temperature dependence of the FWHM forYAG:Ce in the previous section. In all the systems, the DFTcalculations are thought to overestimate 𝐸FC . g and underestimate 11 of 14ative Commons License𝐸  C  E  a3B  t  t  i  t  m  c  w  e  j  s  aT  t  f  c  e  a  p  o  T  t  r  o  aB  m  d  c44  T  u  T  d  w  e  w  a  5  d  F3  Å  a (                                                1 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable CFC . e relative to the measurements by about 0.1 eV. For a series ofe3 + -activated systems, DFT calculations appear to overestimateFC.g and underestimate EFC.g by about 0.1 eV compared to thectual values.  Conclusions y conducting SCF calculations with the N − 1 electron configura-ion on the crystal structure models of the Ce3 + -activated system,he KSO eigenvalues and wavefunctions were obtained, provid-ng transition energies and probabilities for the 4f0 5d1 →4f1 5d0ransition that align with experimental emission energies andany-body calculation results in the literature. Applying thisalculation method to each snapshot of the FPMD performedith ESEC allowed us to analyze the time evolution of transitionnergies and probabilities. As a result of ELA along the MD tra-ectory, we derived emission spectra similar to the experimentalpectra, showing the SO splitting and peak broadening caused bytomic thermal motion. he emission peak energy and emission FWHM caused by atomichermal motion calculated by the ELA method align with thoserom the ΔSCF method above 300 K. However, the temperatureoefficient of peak energy is roughly ten times higher than thexperimental value. The peak energies of all emission bandst 300 K obtained via the ELA method match the measuredeak energies within a 9(7)% error margin, while the FWHMsf these bands at 300 K are 158(19)% of the measured values.his overestimation results from the calculated SO splitting andhermal FWHM being 130% and 223(65)% of the actual values,espectively. The latter is likely because the Franck-Condon shiftsf the ground and excited states from DFT calculations are over-nd underestimated by about 0.1 eV, respectively. y considering the systematic error between the predicted andeasured values, this calculation method can be used to pre-ict the luminescence properties of new candidate phosphorompounds.  Computational Methods .1 SCF and Structure Optimization Calculationshis study’s DFT and FPMD calculations were carried outsing the Vienna ab initio simulation package (VASP) [ 35–38 ].he electronic structure of the periodic boundary system wasetermined using the projector-augmented-wave (PAW) methodith the revised Perdew–Burke–Ernzerhof (PBEsol) correlation-xchange functional [ 39 ]. The high-energy cutoff of the planeave basis, the energy threshold for SCF convergence, and thetomic force threshold for structure optimization were set to00 eV, 1 × 10− 7 eV, and 5 × 10− 3 eV, respectively. The k-meshensity dk used for Brillouin zone (BZ) integration was 5.0 Å.or example, a k-mesh of 2 πdk / a × 2 πdk / a × 2 πdk / a ∼ 3 × 3 × was used for a cubic system with the lattice parameter a = 12. The contributions of the Ce 4f and Ce 5d orbitals to each KSOre evaluated by ( Σm CCe,2, m − Σm CCe,3, m ) / Σml αCαlm , where CCe,2, 𝑚 m = − 2, − 1, . . . , 2) and CCe,3, 𝑚 ( m = − 3, − 2, . . . , 3) are the Ce d-2 of 14reaand Ce f-projected wavefunction characters, and 𝐶𝛼𝑙 𝑚 is the spdf-and site-projected character for atom index α, and the angularmoment and magnetic quantum numbers, l ( = 0, 1, . . . , 3) andm ( = − l , − l + 1, . . . , l ). When the KSO is composed mainly of Ce4f orbitals, the value will be close to − 1, and when it is composedprimarily of Ce 5d orbitals, it will be close to + 1. 4.2 Construction of Structural Models of Ce3 + -Activated Phosphors As calculation targets, we chose twelve known phosphor systemscontaining only one crystallographic site of Y, La, or alkalineearth metal elements (Ca, Sr, and Ba) that Ce can substitute. Afteroptimizing the primitive cell structure of the host compound,it was expanded into a supercell containing about 100 atoms,and one of the Y, La, or alkaline earth sites was substituted byCe. When an alkaline earth site with a formal charge of + 2 wassubstituted, the nearest alkaline earth site was substituted byNa with a charge of + 1, preserving the + 3 charge of Ce. Thegeometry of the Ce-substituted model was optimized using non-spin-polarized calculations with a pseudopotential (PP) for Ce,treating the 4f states as core levels. 4.3 Determination of On-Site Coulomb Parameter on Ce Sites The subsequent DFT calculations were performed using the PPfor Ce, treating the 4f states as valence levels and an effective on-site Coulomb interaction ( + U ) at the Ce site ( UCe ). To determinean appropriate value of 𝑈Ce ( UCe opt ) such that the position ofthe occupied Ce 4f level in the band gap satisfies the resultsof spectroscopic experiments, we used the procedure referringto the result of SCF calculation using Heyd–Scuseria–Ernzerhof(HSE06) hybrid functional calculation. Although HSE06 calcu-lations are more expensive than PBEsol + U calculations, theyare known to yield Ce 4f positions consistent with experimentalabsorption and emission data, without the need for the + Uapproach [ 40–42 ]. By performing the Γ-point-only spin-polarizedSCF calculations with UCe ranging from 0 to 9 eV, we obtainedthe KSO energies as a function of UCe as shown in Figure S7 .The energies of the lowest Ce 4f orbital ( ϵCe4f ) vary with 𝑈Ce between the valence band maximum ( ϵVBM ) and the conductionband minimum ( ϵCBM ). Here, the ratio 𝑟 of the energy differencebetween the lowest 4f level and the VBM ( ϵCe4f − ϵVBM ) to theband gap ( ϵCBM − ϵVBM ) is defined as a function of UCe : 𝑟( 𝑈Ce ) =[ ϵCe4f ( 𝑈Ce ) − ϵVBM ] / [ ϵCBM − ϵVBM ]. The SCF calculation usingthe HSE06 without + U was also performed for the same crystalstructure model and obtained the reference ratio rHSE . By fitting alinear function to the r versus UCe plot, the UCe opt was determinedsuch that r ( UCe opt ) = rHSE . Figure S8a shows the KSO energyat the Γ point for the Ae − 1 state in the YAG:Ce system as afunction of UCe . The 5d1 →2 F5/2 and 5d1 →2 F7/2 transition energies,calculated from the energy differences between these levels,decrease linearly with UCe (Figure S8b ). When UCe changes by1 eV from the optimal value UCe opt , the emission energy changesby approximately 0.16 eV. Therefore, to predict the emissionenergy with an accuracy of ± 0.1 eV ( ± 10 nm at a wavelength of500 nm), UCe must be optimized to within ± 0.6 eV. Advanced Theory and Simulations, 2026tive Commons License4T  w  t  [4A  1  t  a  t  7  t  vAS  t  u  T  S  aAT  (  w  IFT  (CTDT  sR P  P  T  22  Y  ES3  H  S  0                                           A 25130390, 2026, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adts.202502287 by National Institute For, Wiley Online Library on [01/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable C.4 TDM Calculation he VASPBandUnfolding code was used to convert the pseudo- avefunctions 𝜑PS 𝑖 generated by VASP into all-electron wavefunc-ions φ and calculate the transition dipole moments from them 43 ]. .5 FPMD Calculation ll FPMD runs in this study were performed with a timestep of fs and the ESEC. The ES relaxed structures Qe were subjectedo an equilibration run of 257 fs in the NVT (fixed number oftoms, volume, and temperature) ensemble using a Nose-Hooverhermostat with a convergence temperature ranging from 100 to00 K. The production runs, which lasted 1025 fs to collect therajectory, were performed in the NVE (fixed number of atoms,olume, and total energy) ensemble. uthor Contributions atoru Matsuishi , Yukinori Koyama , and Takashi Takeda devisedhe basic idea of this study. Satoru Matsuishi conducted calculationsnder the guidance of Yukinori Koyama and Hidekazu Ikeno , andakashi Takeda contributed to the selection of calculation targets.atoru Matsuishi prepared the original draft of the manuscript. All theuthors approved the final version of the manuscript. cknowledgements his work was supported by the Japan Science and Technology AgencyJST), CREST Grant Number JPMJCR19J2. The calculations in this studyere performed on the Numerical Materials Simulator at the Nationalnstitute for Materials Science (NIMS). unding his work was supported by the Japan Science and Technology AgencyJST), CREST Grant Number JPMJCR19J2. onflicts of Interest he authors declare no conflicts of interest. ata Availability Statement he data that support the findings of this study are available in theupplementary material of this article. eferences 1 . K. Morimoto, H. Kasugai, T. 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Supporting Information Additional supporting information can be found online in the SupportingInformation section. Supporting File : adts70428-sup-0001-SuppMat.pdf. Advanced Theory and Simulations, 2026 Creative Commons Licensehttps://doi.org/10.1021/cm8030768https://doi.org/10.1103/PhysRevB.18.7165https://doi.org/10.1103/PhysRevB.93.224425https://doi.org/10.1016/0022-2313(91)90116-Dhttps://doi.org/10.1143/JPSJ.64.4442https://doi.org/10.1149/1.2407508https://doi.org/10.1016/j.ijleo.2018.10.213https://doi.org/10.1109/NSSMIC.2008.4774629https://doi.org/10.7566/JPSJ.91.124713https://doi.org/10.1063/1.1456955https://doi.org/10.1039/C5TC01629Khttps://doi.org/10.1007/s10854-021-06593-zhttps://doi.org/10.1007/s00340-012-4969-xhttps://doi.org/10.1016/j.matlet.2012.07.094https://doi.org/10.1103/PhysRevB.47.558https://doi.org/10.1016/0927-0256(96)00008-0https://doi.org/10.1103/PhysRevB.54.11169https://doi.org/10.1103/PhysRevB.59.1758https://doi.org/10.1103/PhysRevLett.100.136406https://doi.org/10.1063/1.1564060https://doi.org/10.1063/5.0024499https://doi.org/10.1063/5.0102219https://github.com/QijingZheng/VaspBandUnfolding Luminescence Spectra Simulation of Ce3+-Activated Phosphors by Accumulating Emission Lines Along First-Principles Molecular Dynamics Trajectories 1 | Introduction 2 | Results and Discussion 2.1 | Calculation of Ce3+ 4f15d04f05d1 Transition Energies and Probabilities in a Fixed Configuration Coordinate 2.2 | Ce3+ 4f05d14f15d0 Emission Spectrum Simulation Considering the Effect of Atomic Thermal Motion 2.3 | Spectral Calculations for Various Known Ce3+-Activated Phosphors 3 | Conclusions 4 | Computational Methods 4.1 | SCF and Structure Optimization Calculations 4.2 | Construction of Structural Models of Ce3+-Activated Phosphors 4.3 | Determination of On-Site Coulomb Parameter on Ce Sites 4.4 | TDM Calculation 4.5 | FPMD Calculation Author Contributions Acknowledgements Funding Conflicts of Interest Data Availability Statement References Supporting Information