Pierre-François Loos
email: loos@irsamc.ups-tlse.fr

Filippo Lipparini

Devin A Matthews

Aymeric Blondel

Denis Jacquemin
email: denis.jacquemin@univ-nantes.fr

A Mountaineering Strategy to Excited States: Revising Reference Values with EOM-CC4

In the framework of the computational determination of highly-accurate vertical excitation energies in small organic compounds, we explore the possibilities offered by the equation-of-motion formalism relying on the approximate fourth-order coupled-cluster (CC) method, CC4. We demonstrate, using an extended set of more than 200 reference values based on CC including up to quadruples excitations (CCSDTQ), that CC4 is an excellent approximation to CCSDTQ for excited states with a dominant contribution from single excitations with an average deviation as small as 0.003 eV. We next assess the accuracy of several additive basis set correction schemes, in which vertical excitation energies obtained with a compact basis set and a high-order CC method are corrected with lower-order CC calculations performed in larger basis sets. Such strategies are found to be overall very beneficial, though their accuracy depend significantly on the actual scheme. Finally, CC4 is employed to improve several theoretical best estimates of the QUEST database for molecules containing between four and six (non-hydrogen) atoms, for which previous estimates were computed at the CCSDT level.

INTRODUCTION

Defining sets of high-quality reference values that can be employed to reliably assess the pros and cons of lower-cost theoretical methods is a very popular and useful research line in quantum chemistry. [START_REF] Roos | Multiconfigurational Perturbation Theory: Applications in Electronic Spectroscopy[END_REF][START_REF] Piecuch | Recent Advances in Electronic Structure Theory: Method of Moments of Coupled-Cluster Equations and Renormalized Coupled-Cluster Approaches[END_REF][START_REF] Dreuw | Single-Reference ab initio Methods for the Calculation of Excited States of Large Molecules[END_REF][START_REF] Krylov | Spin-Flip Equation-of-Motion Coupled-Cluster Electronic Structure Method for a Description of Excited States, Bond Breaking, Diradicals, and Triradicals[END_REF][START_REF] Sneskov | Excited State Coupled Cluster Methods[END_REF][START_REF] González | Progress and Challenges in the Calculation of Electronic Excited States[END_REF][START_REF] Laurent | DFT Benchmarks: A Review[END_REF][START_REF] Adamo | The calculations of Excited-State Properties with Time-Dependent Density Functional Theory[END_REF][START_REF] Ghosh | Combining Wave Function Methods with Density Functional Theory for Excited States[END_REF][START_REF] Blase | The Bethe-Salpeter Formalism: From Physics to Chemistry[END_REF][START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF] Although experimental values may constitute natural references for some properties (e.g., thermodynamical and kinetic data), [START_REF] Pople | Gaussian-1 Theory: A General Procedure for Prediction of Molecular Energies[END_REF][START_REF] Curtiss | Gaussian-2 Theory for Molecular Energies of First-and Second-row Compounds[END_REF][START_REF] Curtiss | Investigation of the Use of B3LYP Zero-Point Energies and Geometries in the Calculation of Enthalpies of Formation[END_REF][START_REF] Curtiss | Gaussian-3 (G3) Theory for Molecules Containing First and Second-Row Atoms[END_REF][START_REF] Curutchet | How Solvent Controls Electronic Energy Transfer and Light Harvesting: Toward a Quantum-Mechanical Description of Reaction Field and Screening Effects[END_REF] it is often welcome to rely on state-of-the-art electronic structure methods to produce reference values for other properties. Of course, this latter approach is intrinsically limited by the computational cost of these high-accuracy models. However, it has the undeniable advantage to allow well-grounded comparisons within a unique, well-defined set of parameters. [START_REF] Tajti | High Accuracy Extrapolated Ab Initio Thermochemistry[END_REF][START_REF] Bomble | High-Accuracy Extrapolated ab initio Thermochemistry. II. Minor Improvements to the Protocol and a Vital Simplification[END_REF][START_REF] Harding | High-Accuracy Extrapolated ab Initio Thermochemistry. III. Additional Improvements and Overview[END_REF][START_REF] Van Setten | The GW-Method for Quantum Chemistry Applications: Theory and Implementation[END_REF] Indeed, one can perform comparisons with exactly the same geometries, basis set, solvent model, etc. Purely theoretical reference values are especially useful for electronic excited states (ESs), as the most straightforwardly accessible theoretical values, namely, vertical transition energies (VTEs), are not directly measurable experimentally. [START_REF] Loos | The Quest for Highly-Accurate Excitation Energies: A Computational Perspective[END_REF] This explains why the determination of accurate VTEs has been an active and productive avenue of research during the past three decades, with, in particular, valuable works from Roos' [START_REF] Serrano-Andrés | Towards an Accurate Molecular Orbital Theory for Excited States: Ethene, Butadiene, and Hexatriene[END_REF][START_REF] Serrano-Andrés | Theoretical Study of the Electronic Spectra of Cyclopentadiene, Pyrrole, and Furan[END_REF][START_REF] Serrano-Andres | Theoretical Study of the Internal Charge Transfer in Aminobenzonitriles[END_REF] and Thiel's [START_REF] Schreiber | Benchmarks for Electronically Excited States: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Silva-Junior | Basis Set Effects on Coupled Cluster Benchmarks of Electronically Excited States: CC3, CCSDR(3) and CC2[END_REF][START_REF] Silva-Junior | Benchmarks of Electronically Excited States: Basis Set Effects on CASPT2 Results[END_REF] groups.

Since 2018, our groups have made several contributions in this field, [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF][START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Reference Energies and Benchmarks[END_REF][START_REF] Loos | Reference Energies for Double Excitations[END_REF][START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Exotic Molecules and Radicals[END_REF][START_REF] Chrayteh | A Mountaineering Strategy to Excited States: Highly-Accurate Oscillator Strengths and Dipole Moments of Small Molecules[END_REF][START_REF] Loos | Reference Energies for Intramolecular Charge-Transfer Excitations[END_REF] and it eventually led to the creation of the QUEST database [see https://lcpq.github.io/QUESTDB_ website], that contains a large panel of reference VTEs for molecules containing from 1 to 10 non-hydrogen atoms. [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] At the present stage, the QUEST database includes more than 500 theoretical best estimates (TBEs) for diverse ESs (singlet, doublet, and triplet; valence and Rydberg; charge transfer, singly-and doubly-excited states) that have been established with the aug-cc-pVTZ basis set. Typically, the TBEs contained in the QUEST database are produced using VTEs computed with the selected configuration interaction (SCI) algorithm named "Configuration Interaction using a Perturbative Selection made Iteratively" (CIPSI) [START_REF] Huron | Iterative Perturbation Calculations of Ground and Excited State Energies from Multiconfigurational Zeroth-Order Wavefunctions[END_REF][START_REF] Giner | Using Perturbatively Selected Configuration Interaction in Quantum Monte Carlo Calculations[END_REF][START_REF] Giner | Fixed-Node Diffusion Monte Carlo Potential Energy Curve of the Fluorine Molecule F 2 Using Selected Configuration Interaction Trial Wavefunctions[END_REF][START_REF] Garniron | Hybrid Stochastic-Deterministic Calculation of the Second-Order Perturbative Contribution of Multireference Perturbation Theory[END_REF][START_REF] Garniron | Selected Configuration Interaction Dressed by Perturbation[END_REF][START_REF] Garniron | Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs[END_REF] to obtain near full configuration interaction (FCI) quality VTEs for systems containing from 1 to 3 non-hydrogen atoms, coupled cluster (CC) with singles, doubles, triples and, quadruples (CCS-DTQ) [START_REF] Kucharski | Recursive Intermediate Factorization and Complete Computational Linearization of the Coupled-Cluster Single, Double, Triple, and Quadruple Excitation Equations[END_REF] for molecules encompassing 4 non-hydrogen atoms, and CC with singles, doubles, and triples (CCSDT) [START_REF] Noga | The Full CCSDT Model for Molecular Electronic Structure[END_REF][START_REF] Scuseria | A New Implementation of the Full CCSDT Model for Molecular Electronic Structure[END_REF][START_REF] Kucharski | Coupled-Cluster Theory for Excited Electronic States: The Full Equation-Of-Motion Coupled-Cluster Single, Double, and Triple Excitation Method[END_REF][START_REF] Kowalski | The Active-Space Equationof-Motion Coupled-Cluster Methods for Excited Electronic States: Full EOMCCSDt[END_REF][START_REF] Kowalski | Excited-State Potential Energy Curves of CH + : a Comparison of the EOMCCSDt And Full EOMCCSDT Results[END_REF] for larger derivatives. Popular basis set correction schemes have often been applied. For example, the CCSDTQ VTEs computed with a double-𝜁 basis set were corrected thanks to CCSDT values obtained with a triple-𝜁 basis set. Most TBEs included in the QUEST database were estimated to be chemically-accurate (corresponding to 1 kcal.mol -1 or 0.04 eV error), with a typical error bar of ±0.03 eV. The reference values included in QUEST have been used by various groups, for example, to (i) assess the relative accuracies of third-order, [START_REF] Loos | Is ADC(3) as Accurate as CC3 for Valence and Rydberg Transition Energies?[END_REF] multireference, [START_REF] Sarkar | Assessing the Performances of CASPT2 and NEVPT2 for Vertical Excitation Energies[END_REF] and other emerging [START_REF] Giner | Chemically Accurate Excitation Energies With Small Basis Sets[END_REF][START_REF] Otis | A hybrid approach to excited-state-specific variational Monte Carlo and doubly excited states[END_REF][START_REF] Loos | Dynamical correction to the Bethe-Salpeter equation beyond the plasmon-pole approximation[END_REF][START_REF] Kossoski | Excited States from State-Specific Orbital-Optimized Pair Coupled Cluster[END_REF][START_REF] Gould | Double excitations in molecules from ensemble density functionals: Theory and approximations[END_REF][START_REF] Gould | Single Excitation Energies Obtained from the Ensemble "HOMO-LUMO Gap": Exact Results and Approximations[END_REF] methods, (ii) quantify the accuracy of local hybrids for triplet ESs, [START_REF] Grotjahn | Assessment of Hybrid Functionals for Singlet and Triplet Eccitations: Why do Some Local Hybrid Functionals Perform so well for Triplet Excitation Energies[END_REF] (iii) determine the relative performance of several hybrid [START_REF] Liang | Revisiting the performance of time-dependent density functional theory for electronic excitations: Assessment of 43 popular and recently developed functionals from rungs one to four[END_REF] or double hybrid [START_REF] Casanova-Páez | 𝜔B2PLYP and 𝜔B2GPPLYP: The First Two Double-Hybrid Density Functionals with Long-Range Correction Optimized for Excitation Energies[END_REF][START_REF] Mester | A Simple Range-Separated Double-Hybrid Density Functional Theory for Excited States[END_REF][START_REF] Casanova-Páez | Time-Dependent Long-Range-Corrected Double-Hybrid Density Func-tionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet-Singlet and Singlet-Triplet Excitation Energies[END_REF][START_REF] Mester | Spin-Scaled Range-Separated Double-Hybrid Density Functional Theory for Excited States[END_REF] functionals, and (iv) evaluate the potential of orbital-optimized density-functional theory for double excitations. [START_REF] Hait | Excited State Orbital Optimization via Minimizing the Square of the Gradient: General Approach and Application to Singly and Doubly Excited States via Density Functional Theory[END_REF][START_REF] Hait | Orbital Optimized Density Functional Theory for Electronic Excited States[END_REF] Despite burning an unreasonable number of CPU hours during the past four years, we could hardly "do better" than what is described above, as we rapidly hit the computational wall of both high-order CC schemes and/or large CIPSI calculations. For instance, CCSDTQ formally scales as O (𝑁 10 ) (where 𝑁 is the number of basis functions) and determining VTEs in molecules like furan or thiophene is nearly impossible even in a double-𝜁 basis set. In an effort to go one step further we explored very recently, and for the first time, the performance of the approximate fourth-order CC model, CC4, [START_REF] Kállay | Approximate Treatment of Higher Excitations in Coupled-Cluster Theory[END_REF] in the context of ESs. [START_REF] Loos | How Accurate are EOM-CC4 Vertical Excitation Energies?[END_REF] From a theoretical point of view, CC4 can be viewed as an approximation of CCSDTQ that still includes iterative quadruples, but neglects the calculation of the most expensive components (and avoids the storage of the higher-excitation amplitudes), allowing to reduce the formal scaling by one order of magnitude to O (𝑁 9 ). [START_REF] Kállay | Approximate Treatment of Higher Excitations in Coupled-Cluster Theory[END_REF] In our first investigation, [START_REF] Loos | How Accurate are EOM-CC4 Vertical Excitation Energies?[END_REF] we considered very small systems (BH, BF, CO, HCl, H 2 O, H 2 S, N 2 , and NH 3 ) for which wellconverged CIPSI calculations were achievable for 25 ESs. This preliminary study indicates that CC4 indeed provides highly competitive VTEs, as it allows to significantly reduce the CCSDT error as compared to FCI, with final deviations -for ESs with a dominant single excitation characteronly slightly larger than their CCSDTQ counterparts.

In the present contribution, which is only the second work presenting CC4 calculations for ESs, we build on our previous study, [START_REF] Loos | How Accurate are EOM-CC4 Vertical Excitation Energies?[END_REF] and we aim at (i) evaluating the performance of CC4 for a more significant set of molecules and a larger variety of ESs, (ii) assessing the accuracy of additive basis set correction procedures based on CC4, and (iii) providing improved TBEs for many ESs included in the QUEST database.

COMPUTATIONAL DETAILS

All calculations presented here rely on the frozen-core approximation and the high-quality ground-state geometries extracted from the QUEST database. [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] For the sake of reproducibility, cartesian coordinates for all systems displayed in Figure 1 are reproduced in the supporting information (SI). Note that, below, we do not specify the equation-of-motion (EOM) prefix for the CC calculations, although all ES calculations are performed using this formalism.

Most of our CC calculations have been performed with CFOUR, [START_REF] Cfour | Coupled-Cluster Techniques for Computational Chemistry, a Quantum-Chemical Program Pack[END_REF][START_REF] Matthews | Coupled-Cluster Techniques for Computational Chemistry: The CFOUR Program Package[END_REF] that provides an efficient implementation of high-order CC methods up to quadruples. [START_REF] Matthews | Non-Orthogonal Spin-Adaptation of Coupled Cluster Methods: A new Implementation of Methods Including Quadruple Excitations[END_REF][START_REF] Matthews | Developments in Physical & Theoretical Chemistry[END_REF] For all considered systems, we performed, when technically achievable, CC2, [START_REF] Christiansen | The Second-Order Approximate Coupled Cluster Singles and Doubles Model CC2[END_REF][START_REF] Hättig | CC2 Excitation Energy Calculations on Large Molecules Using the Resolution of the Identity Approximation[END_REF] CCSD, [START_REF] Purvis | A Full Coupled-Cluster Singles and Doubles Model: The Inclusion of Disconnected Triples[END_REF][START_REF] Scuseria | The Closed-Shell Coupled Cluster Single and Double Excitation (CCSD) Model for the Description of Electron Correlation. A Comparison with Configuration Interaction (CISD) Results[END_REF][START_REF] Koch | Excitation Energies from the Coupled Cluster Singles and Doubles Linear Response Function (CCSDLR). Applications to Be, CH + , CO, and H 2 O[END_REF][START_REF] Stanton | The Equation of Motion Coupled-Cluster Method -A Systematic Biorthogonal Approach to Molecular Excitation Energies, Transition-Probabilities, and Excited-State Properties[END_REF][START_REF] Stanton | Many-Body Methods for Excited State Potential Energy Surfaces. I: General Theory of Energy Gradients for the Equation-of-Motion Coupled-Cluster Method[END_REF] CC3, [START_REF] Christiansen | Response Functions in the CC3 Iterative Triple Excitation Model[END_REF][START_REF] Koch | Excitation Energies of BH, CH 2 and Ne in Full Configuration Interaction and the Hierarchy CCS, CC2, CCSD and CC3 of Coupled Cluster Models[END_REF][START_REF] Koch | The CC3 Model: An Iterative Coupled Cluster Approach Including Connected Triples[END_REF] CCSDT, [START_REF] Noga | The Full CCSDT Model for Molecular Electronic Structure[END_REF][START_REF] Scuseria | A New Implementation of the Full CCSDT Model for Molecular Electronic Structure[END_REF][START_REF] Kucharski | Coupled-Cluster Theory for Excited Electronic States: The Full Equation-Of-Motion Coupled-Cluster Single, Double, and Triple Excitation Method[END_REF][START_REF] Kowalski | The Active-Space Equationof-Motion Coupled-Cluster Methods for Excited Electronic States: Full EOMCCSDt[END_REF][START_REF] Kowalski | Excited-State Potential Energy Curves of CH + : a Comparison of the EOMCCSDt And Full EOMCCSDT Results[END_REF] CC4, [START_REF] Kállay | Approximate Treatment of Higher Excitations in Coupled-Cluster Theory[END_REF][START_REF] Kállay | Calculation of Excited-State Properties Using General Coupled-Cluster and Configuration-Interaction Models[END_REF] and CCSDTQ [START_REF] Kucharski | Recursive Intermediate Factorization and Complete Computational Linearization of the Coupled-Cluster Single, Double, Triple, and Quadruple Excitation Equations[END_REF][START_REF] Kállay | Calculation of Excited-State Properties Using General Coupled-Cluster and Configuration-Interaction Models[END_REF] calculations using three Gaussian basis sets: 6-31+G(d), aug-cc-pVDZ, and aug-cc-pVTZ. For the sake of conciseness, these basis sets are sometimes dubbed as "Pop", "AVDZ", and "AVTZ" in the following.

In the statistical analysis presented below, we report the usual statistical indicators: the mean signed error (MSE), the mean absolute error (MAE), as well as largest positive and negative deviations [Max(+) and Max(-), respectively]. CCSDTQP/6-31+G(d) calculations have been performed with MRCC [START_REF] Kállay | Quantum Chemical Program[END_REF][START_REF] Kallay | The MRCC Program System: Accurate Quantum Chemistry from Water to Proteins[END_REF] for the molecules encompassing two nonhydrogen atoms (see Figure 1). These values are reported in Table S1 of the SI. Considering all 32 VTEs and taking these CCSDTQP values as references, we obtain a MAE of 0.003 eV for the CCSDTQ data determined with the same basis set. Unsurprisingly, the largest error, 0.023 eV, comes from the pure (𝑛, 𝑛) → (𝜋 ★ , 𝜋 ★ ) doubly-excited state of nitroxyl for which the CC expansion obviously converges slower. Removing this pathological case yields a MAE of 0.002 eV with all errors below 0.010 eV, clearly confirming the quality of the CCSDTQ estimates.

(non-H) atom molecules

METHODOLOGICAL ASPECTS 3.1 How does CC4 compare to CCSDTQ?

Given that CC4 is an approximation of CCSDTQ, and that the accuracy of the latter is recognized as exceptional for ESs with a dominant contribution from singly-excited determinants (see above), it seems natural to assess the performance of CC4 with respect to CCSDTQ. Considering the values Table 1: Statistical analysis obtained using various correction schemes. MSE, MAE and maximal deviations (in eV) are obtained with respect to the reference method given in the leftmost column. The number of reference ESs considered at the CCSDTQ/aug-cc-pVTZ, CC4/aug-cc-pVTZ, and CCSDT/aug-cc-pVTZ levels is 31, 59, and 118, respectively. In the rightmost column, we provide the number of states for which the correction provides smaller (or equal) absolute errors as compared to the non-corrected scheme (rows labeled as "no correction").

listed in the SI (with the three basis sets and all compounds), we have 220 CCSDTQ reference values at hand. Using these reference values, one obtains a MSE of 0.002 eV and a MAE as small as 0.003 eV for CC4. As one could have foreseen, [START_REF] Loos | How Accurate are EOM-CC4 Vertical Excitation Energies?[END_REF] the largest deviations between CC4 and CCSDTQ originate from the genuine double excitations in nitroxyl, nitrosomethane, and glyoxal for which CC-based methods would likely not be chosen in practical applications. Removing these three states, but keeping all other cases including the 2 [START_REF] Roos | Multiconfigurational Perturbation Theory: Applications in Electronic Spectroscopy[END_REF] 𝐴 𝑔 dark state of butadiene, leads to a MSE of 0.001 eV and a MAE of 0.003 eV for CC4. Interestingly, in our earlier work devoted to very small compounds, we obtained the same MAE of 0.003 eV for CC4 (against CCSDTQP). [START_REF] Loos | How Accurate are EOM-CC4 Vertical Excitation Energies?[END_REF] Clearly, one can state that CC4 is a very good approximation of CCS-DTQ for the ESs that are not dominated by doubly-excited determinants.

Is CC4 significantly more accurate than CCSDT and CC3?

The next natural question is to determine if CC4 is worth its cost. Indeed, CC4 formally scales as O (𝑁 [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF][START_REF] Kánnár | Benchmarking Coupled Cluster Methods on Valence Singlet Excited States[END_REF][START_REF] Tajti | Accuracy of Spin-Component-Scaled CC2 Excitation Energies and Potential Energy Surfaces[END_REF] The most valuable observation for our purposes is that CC4 indeed lowers the CCSDT and CC3 deviations by a factor of five, which we consider a very significant improvement when ones aims at producing highly-accurate reference values.

Are CC4 basis set effects transferable?

When defining values for benchmarking purposes, it is obvious from the discussion above that one can safely employ CC4/aug-cc-pVTZ as reference when neither well-converged CIPSI/aug-cc-pVTZ nor CCSDTQ/aug-cc-pVTZ calculations are technically achievable. As detailed below, this is the case, for example, for some of the molecules encompassing four non-hydrogen atoms. However, as customary in the field, [START_REF] Chrayteh | A Mountaineering Strategy to Excited States: Highly-Accurate Oscillator Strengths and Dipole Moments of Small Molecules[END_REF][START_REF] Loos | Reference Energies for Intramolecular Charge-Transfer Excitations[END_REF][START_REF] Casanova-Páez | 𝜔B2PLYP and 𝜔B2GPPLYP: The First Two Double-Hybrid Density Functionals with Long-Range Correction Optimized for Excitation Energies[END_REF][START_REF] Kállay | Calculation of Excited-State Properties Using General Coupled-Cluster and Configuration-Interaction Models[END_REF][START_REF] Balabanov | Basis set Limit Electronic Excitation Energies, Ionization Potentials, and Electron Affinities for the 3𝑑 Transition Metal Atoms: Coupled Cluster and Multireference Methods[END_REF][START_REF] Kamiya | Higher-Order Equation-of-Motion Coupled-Cluster Methods for Ionization Processes[END_REF][START_REF] Peach | Overcoming Low Orbital Overlap and Triplet Instability Problems in TDDFT[END_REF][START_REF] Watson | Excited States of Butadiene to Chemical Accuracy: Reconciling Theory and Experiment[END_REF][START_REF] Feller | A Systematic Approach to Vertically Excited States of Ethylene Using Configuration Interaction and Coupled Cluster Techniques[END_REF][START_REF] Gui | Accuracy Assessment of 𝐺𝑊 Starting Points for Calculating Molecular Excitation Energies Using the Bethe-Salpeter Formalism[END_REF][START_REF] Franke | tert-Butyl Peroxy Radical: Ground and First Excited State Energetics and Fundamental Frequencies[END_REF] one can employ a double-𝜁 VTE computed with a CC model including quadruples in order to correct a triple-𝜁 VTE obtained with a quadruple-free CC method.

To test such strategies, we report in Table 1 the statistical analysis obtained with various correction schemes. We consider as reference the actual results obtained with the target method (leftmost column of Table 1). These reference values are available in Tables S3, S6, and S9 of the SI. In the rightmost column, we provide the number of states for which the correction yields smaller (or equal) absolute errors as compared to the non-corrected scheme (rows labeled as "no correction").

Globally, the data listed in Table 1 are quite appealing and one notices three general trends. First, adding corrections for higher-order excitations systematically decrease the MAE and improves the (vast) majority of the estimates. For example, while CC2/aug-cc-pVTZ returns a MAE of 0.215 eV as compared to CCSDTQ/aug-cc-pVTZ, correcting them with the differences between the VTEs computed with these two models but with the much smaller 6-31+G(d) basis set allows reducing the error to 0.051 eV and improves 29 of the 31 VTEs. Second, the more refined the starting point, the more accurate the final estimate. For example, using the same approach as above but starting from CC3/aug-cc-pVTZ rather than CC2/aug-cc-pVTZ would cut down the MAE from 0.051 eV to 0.004 eV. Third, performing the correction with aug-cc-pVDZ typically yields smaller statistical deviations than with 6-31+G(d), though the difference between the two approaches is small.

Given that CC4 is such a stunning approximation of CCS-DTQ for ESs with a single excitation character, it is particularly relevant to investigate CC4/aug-cc-pVTZ estimates based on CCSDT and CC3, i.e.,

Δ ẼCC4 AVTZ Δ𝐸 CCSDT AVTZ + Δ𝐸 CC4 AVDZ -Δ𝐸 CCSDT AVDZ , (1a)
Δ ẼCC4 AVTZ Δ𝐸 CCSDT AVTZ + Δ𝐸 CC4 Pop -Δ𝐸 CCSDT Pop , (1b)
Δ ẼCC4 AVTZ Δ𝐸 CC3 AVTZ + Δ𝐸 CC4 AVDZ -Δ𝐸 CC3 AVDZ , (1c)
Δ ẼCC4 AVTZ Δ𝐸 CC3 AVTZ + Δ𝐸 CC4 Pop -Δ𝐸 CC3 Pop . (1d)
Looking at Table 1, one notices that all four approaches deliver very comparable error patterns, with MAE not larger than 0.004 eV, and global improvement over the uncorrected CCSDT/aug-cc-pVTZ and CC3/aug-cc-pVTZ data, the errors being cut down by a factor of three. Of course, as can be deduced from the last column, not all values are more accurate when adding corrections obtained with smaller basis sets, but improvements are observed for the vast majority of the cases. Additionally, although this is not the focus of the present work, we stress that the following expressions

Δ ẼCCSDT AVTZ Δ𝐸 CC3 AVTZ + Δ𝐸 CCSDT AVDZ -Δ𝐸 CC3 AVDZ , (2a)
Δ ẼCCSDT AVTZ Δ𝐸 CC3 AVTZ + Δ𝐸 CCSDT Pop -Δ𝐸 CC3 Pop , (2b)
are extremely effective with MAE of 0.003 and 0.005 eV and small maximal deviations as compared to the true CCSDT/aug-cc-pVTZ transition energies. The fact that these two approximations are very accurate, is certainly valuable, as both CC3/aug-cc-pVTZ and CCSDT/6-31+G(d) calculations are technically achievable on systems with up to approximately 10-12 non-hydrogen atoms, allowing to significantly expand the number of TBEs based on CCSDT/aug-cc-pVTZ that are included in the QUEST database for larger systems. The take-home message of this Section is that, while these correction schemes (which are widespread in the CC community) are generally valuable to produce accurate VTEs, their overall accuracy depends ultimately on the quality of the starting point.

REFINED THEORETICAL BEST ESTIMATES

Given the above observations, it is worth reexamining our previous TBEs with the additional accuracy provided by CC4.

Linear systems

The QUEST database includes three linear systems encompassing four non-hydrogen atoms: cyanoacetylene, cyanogen, and diacetylene. [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF][START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] For these three systems CC4/aug-cc-pVTZ calculations could be performed, whereas CCSDTQ/aug-cc-pVTZ calculations remain beyond reach (see Table S9 in the SI for raw data). We have, therefore, computed new TBEs on the basis of these CC4 data, applying the very trustworthy basis set correction (see top lines of Table 1)

Δ𝐸 TBE AVTZ = Δ𝐸 CC4 AVTZ + Δ𝐸 CCSDTQ AVDZ -Δ𝐸 CC4 AVDZ . (3)
The results for seven ESs of these three linear molecules are listed in Table 2. It can be seen that the variations are negligible, with three TBEs unchanged, and four decreasing by 0.01 eV.

Four non-hydrogen compounds

For most non-linear compounds encompassing four nonhydrogen nuclei, the TBEs of the QUEST database were obtained with CCSDT/aug-cc-pVTZ corrected using CCSDTQ/6-31+G(d) to take into account the quadruples contribution. [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF][START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] Given the results gathered in Table 1, it is not clear that applying CC4/aug-cc-pVDZ corrections would improve significantly these original estimates. However, for a few compounds, the QUEST TBEs do not include corrections from the quadruples as they were computed at the CCSDT/aug-cc-pVTZ level "only". Hence, we have considered these derivatives and employed Eq. (1a) to determine new TBEs. The results are listed in Table 3 and compared to the previous estimates. Although the differences listed in the rightmost column of Table 3 have different signs, the impact of CC4 is typically a small decrease of the previous estimates, hinting that CCSDT tends to overestimate the VTEs. This is consistent with the 0.003 eV MSE reported in Table 1 for CCSDT as compared to CC4. For the 13 states listed in Table 3, the average correction is -0.011 eV and the average absolute change is 0.022 eV only. More importantly, for the vast majority of cases, the changes are not larger than ±0.03 eV which was the estimated error bar in Ref. 11. Nevertheless, a molecule worth discussing is acrolein. For its lowest hallmark 𝑛 → 𝜋 ★ transition, the CCSD, CC3, and CCSDT VTEs obtained with aug-cc-pVTZ are 3.913, 3.743, and 3.725 eV. The new TBE, 3.72 eV, is consistent with this trend, but is significantly smaller than the original CIPSI-based estimate of 3.78 eV. It is likely that the CIPSI extrapolation error bar was underestimated previously, and we believe that our current TBE is more accurate. Indeed, CC4/6-31+G(d) and CCSDTQ/6-31+G(d) values are very consistent (see Table S7). The second ES, a strongly dipoleallowed 1 𝐴 (𝜋 → 𝜋 ★ ) transition, was slightly blueshifted in our original work, but the error, -0.02 eV, remains low, a statement also holding for the higher-lying Rydberg ES of the same symmetry. The original estimate for the second 1 𝐴 (𝑛 → 𝜋 ★ ) transition, 6.72 eV, was labeled as unsafe in Ref. 11 due to its significant double excitation character. Indeed, its percentage of single excitations (%𝑇 1 ) is only 79.4% according to CC3/aug-cc-pVTZ. Nevertheless, the CC4 correction obtained with aug-cc-pVDZ is not very large and one can likely claim that the new TBE of 6.69 eV stands as the most trustworthy estimate published to date for this particular ES. The higher-lying 1 𝐴 (𝜋 → 𝜋 ★ ) transition of acrolein has a nature similar to the famous 𝐴 𝑔 state of butadiene with a %𝑇 1 of 75%. At the CIPSI/6-31+G(d) level, a value of 8.00 ± 0.03 eV was obtained previously, [START_REF] Loos | Reference Energies for Double Excitations[END_REF] and our CC4 value obtained with the same basis set is consistent with this result: 8.035 eV (see Table S7 in the SI). Based on CC4/aug-cc-pVDZ, i.e, using Eq. (1a) the new TBE is 7.93 eV, upshifted by 0.06 eV as compared to the original one. Given that for the similar transition in butadiene, the changes between CC4 and CCSDTQ are trifling (Table S7), we consider this new TBE as more trustworthy than the original one, though giving a reliable error bar is not straightforward. Finally, one can also note that, for carbonylfluoride, the current TBE is 0.02 eV smaller than the original CIPSI-based one, for which the estimated extrapolation error bar was 0.02 eV. [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF] For the largest systems of Figure 1, CC4 has the advantage to provide a systematic path towards high-accuracy for systems that are beyond reach at the CCSDTQ level. Let us first illustrate this with a set of 30 ESs extracted from systems with five (non-hydrogen) atom (Table 4). To the best of our knowledge, for all five molecules, our results provide the first VTEs including quadruples corrections. As one can see, the differences with the previous TBEs are small, and are typically slightly negative, confirming the outcome of the previous section showing that CCSDT/aug-cc-pVTZ yields slightly too large VTEs. The average absolute correction for the data of Table 4 is nonetheless very small, 0.02 eV, which confirms that quality of the original TBEs from the QUEST database.

Let us now take a look at the most problematic cases. Unsurprisingly, the 1 𝐴 1 (𝜋 → 𝜋 ★ ) transition of cyclopentadiene is significantly redshifted by the CC4 correction (-0.07 eV). This transition was previously classified as unsafe, [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF] due to its %𝑇 1 value of 78.9% and its similar nature to the famous 1 𝐴 𝑔 ES of butadiene. For the latter compound, the difference between the CIPSI/6-31+G(d) and CCSDT/6-31+G(d) estimates is -0.08 eV, [START_REF] Loos | Reference Energies for Double Excitations[END_REF] which suggests that the CC4 correction of -0.07 eV for the 1 𝐴 1 ES of cyclopentadiene is very reasonable. The lowest 𝐵 2 ES of thiophene is subject to a -0.04 eV shift despite having also a very large single excitation character (%𝑇 1 = 91.5%). All other ESs are even less affected by the CC4 corrections with variations of ±0.03 eV at most.

We now consider three highly-symmetric six-membered rings, namely benzene, pyrazine, and tetrazine, for which CC4/6-31+G(d) calculations are still doable. The results obtained for a large number of ESs are listed in Table 5. For these three compounds, this Table is, as far as we are aware of, the first to propose CC-based VTEs including corrections from iterative quadruples.

For benzene, we consider three valence and four Rydberg ESs. These ESs have a very strong single excitation character with %𝑇 1 > 92%, except for the lowest 1 𝐵 2𝑢 transition (%𝑇 1 = 86%). In all cases, one finds, as most of the five-membered cycles, that CC4/6-31+G(d) provides small negative or null corrections, with a maximal amplitude of -0.02 eV, even for the lowest-energy transition.

Eleven ESs of pyrazine are listed in Table 5, including four Rydberg states. The corrections to previous CCSDT-based TBEs are either positive, null or negative, but most are within the expected error bar of the original estimates. Nevertheless, for the lowest 1 1 𝐵 1𝑔 ES, corresponding to a 𝑛 → 𝜋 ★ excitation, a relatively large correction of -0.04 eV is noticed. This transition has a rather low %𝑇 1 value of 84%, which however does not translate into a large CC3/CCSDT difference (ca. 0.01 eV for the three basis sets, see the SI). Likewise the highest-lying 1 1 𝐵 1𝑢 ES also undergoes a -0.04 eV correction, but this state was rated unsafe previously, [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF][START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] due to an unusually large CC3/CCSDT difference. Four other ESs have their VTEs corrected by ±0.03 eV by CC4, a shift corresponding to the expected error bar of the original TBEs.

In tetrazine, we consider 12 ESs, including two with a pure double excitation character, for which NEVPT2 was employed in the QUEST database to obtain TBEs, whereas, for the other transitions, CCSDT/aug-cc-pVTZ or basis set corrected CCSDT/aug-cc-pVDZ values were originally selected as TBEs. [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF] All the latter were rated as safe as the energy differences between CCSDT and CC3 were smaller than 0.03 eV despite %𝑇 1 values often smaller than 90%. [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF][START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] For several ESs, small negative CC4 corrections, -0.01 or -0.02 eV, are obtained. For the lowest 𝜋 → 𝜋 ★ transition, the correction is only slightly larger, i.e., -0.04 eV. However, for kcal.mol -1 ). We also showed that neither CC3 nor CCSDT could deliver the same level of accuracy for the very same set of ESs.

Second, we investigated the performance of additive basis set correction schemes in the EOM-CC context using the same set of ESs as in the first part. It appeared that these popular correction strategies indeed statistically improve the quality of the VTEs. More interestingly, such basis set corrections were found particularly powerful when starting from CC3/aug-cc-pVTZ or CCSDT/aug-cc-pVTZ, as they allowed estimating the true CC4/aug-cc-pVTZ or CCSDTQ/aug-cc-pVTZ VTEs with average errors of approximately 0.005 eV. Of course, it is essential to confirm these trends for larger compounds, but the cost of CC4/aug-cc-pVTZ and/or CCS-DTQ/aug-cc-pVTZ unfortunately prevents us of doing it as of today.

Finally, CC4 was employed to improve previous CCSDTbased TBEs of the QUEST database. [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] For benzene, cyclopentadiene, difluorodiazirine, furan, hexatriene, pyrazine, pyrrole, tetrazine, and thiophene, it was possible to perform CC4/6-31+G(d) and/or CC4/aug-cc-pVDZ calculations whereas CCSDTQ remains beyond reach. The corrections obtained were, in most cases, smaller than the expected average error of the original TBEs (ca. ±0.03 eV). Nevertheless, most corrections were null or negative, hinting at a slight overestimation trend in our original TBEs. In addition, some of the estimates originally viewed as unsafe can be now considered as trustworthy (2 1 𝐴" in acrolein, 2 [START_REF] Roos | Multiconfigurational Perturbation Theory: Applications in Electronic Spectroscopy[END_REF] 𝐴 1 in cyclopentadiene, 3 1 𝐴 1 in thiophene, etc) whereas a few optimistic assessments have been revised, resulting in more accurate (in particular for the valence 𝐵 1𝑔 and 𝐵 2𝑔 ESs of tetrazine).

We are currently pursuing our efforts to improve the quality, size, and diversity of the QUEST database, so as to provide the most trustworthy reference values possible to the community. We plan to publish, within a year or two, an expanded and improved version of the QUEST database.

Figure 1 :

 1 Figure 1: Representation of the systems investigated in the present study.

Table 2 :

 2 TBE/aug-cc-pVTZ (in eV) established for the linear systems encompassing four non-hydrogen atoms.𝑎

			TBEs
	Molecule	Transition	Nature New Prev. Diff.
	Cyanoacetylene 1 Σ -(𝜋 → 𝜋 ★ )	Val 5.79 5.80 -0.01
		1 Δ (𝜋 → 𝜋 ★ )	Val 6.07 6.07 0.00
	Cyanogen	1 Σ -𝑢 (𝜋 → 𝜋 ★ )	Val 6.38 6.39 -0.01
		1 Δ 𝑢 (𝜋 → 𝜋 ★ )	Val 6.65 6.66 -0.01
		1 Σ -

𝑢 (𝜋 → 𝜋 ★ )[F] 𝑏 Val 5.04 5.05 -0.01 Diacetylene 1 Σ - 𝑢 (𝜋 → 𝜋 ★ ) Val 5.33 5.33 0.00 1 Δ 𝑢 (𝜋 → 𝜋 ★ ) Val 5.61 5.61 0.00 𝑎 "Val" stands for valence ESs. TBEs obtained with Eq. (3) using data from the SI. The previous TBEs taken from Ref. 11 are given for comparison. 𝑏 Fluorescence from the optimized excited-state geometry.

Table 3 :

 3 TBE/aug-cc-pVTZ (in eV) established for non-linear systems containing four non-hydrogen atoms obtained using Eq. (1a). 𝑎 𝐴 (𝑛 → 𝜋 ★ ) Val 3.72 3.78 𝑏 -0.06 1 𝐴 (𝜋 → 𝜋 ★ ) Val 6.67 6.69 -0.02 1 𝐴 (𝑛 → 𝜋 ★ ) Val 6.69 6.72 𝑐 -0.03 1 𝐴 (𝑛 → 3𝑠) Ryd 7.11 7.08 +0.03 1 𝐴 (𝜋 → 𝜋 ★ ) Val 𝑑 7.93 7.87 𝑏 +0.06 Carbonylfluoride 1 𝐴 2 (𝑛 → 𝜋 ★ ) Val 7.29 7.31 𝑒 -0.02 Cyanoformaldehyde 1 𝐴 (𝑛 → 𝜋 ★ ) Val 3.82 3.81 +0.01 1 𝐴 (𝜋 → 𝜋 ★ ) Val 6.43 6.46 -0.03 𝐴 (𝑛 → 𝜋 ★ ) Val 3.81 3.80 +0.01 1 𝐴 (𝜋 → 𝜋 ★ ) Val 5.51 5.54 -0.03 Thioacrolein 1 𝐴 (𝑛 → 𝜋 ★ ) Val 2.10 2.11 -0.01 Thiopropynal 1 𝐴 (𝑛 → 𝜋 ★ ) Val 2.02 2.03 -0.01 𝑎 "Val" and "Ryd" stand for valence and Rydberg ESs. Raw data are given in the SI. The previous TBE/aug-cc-pVTZ values taken from Refs. 29, 11, 30, and 33 are given for comparison. 𝑏 TBEs obtained from a basis set corrected CIPSI/6-31+G(d) estimate. The CCSDT/aug-cc-pVTZ value for this particular state is 3.73 eV. 𝑐 Considered as unsafe in Ref. 11. 𝑑 ES with a significant double excitation character.

	TBEs

𝑒

TBE obtained from a basis set corrected CIPSI/6-31+G(d) estimate.

ACKNOWLEDGEMENTS

The authors would like to warmly thank Martial Boggio-Pasqua, Anthony Scemama, and Claudia Filippi for numerous fruitful discussions. PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 863481) for financial support. DJ is indebted to the CCIPL computational center installed in Nantes for (the always very) generous allocation of computational time.

Larger systems

three 𝑛 → 𝜋 ★ transitions (1 1 𝐵 1𝑔 , 1 1 𝐵 2𝑔 , and 2 1 𝐵 2𝑔 ), much larger changes (-0.06, -0.05, and -0.07 eV) are induced by the inclusion of quadruples, indicating that our original assessment of the TBE quality was optimistic. Interestingly, for these three transitions, one has %𝑇 1 < 85%, in contrast with the other singly-excited ESs for which %𝑇 1 > 85%. This observation suggests that the 85% barrier might be the limit for which CCSDT can be considered trustworthy, irrespective of the difference between CC3 and CCSDT. For the two doubly-excited states of tetrazine listed in Tables 5 andS11, that are both characterized by %𝑇 1 < 1%, [START_REF] Loos | A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules[END_REF] the differences between CC4/6-31+G(d) and CCSDT/6-31+G(d) are very significant, that is, -0.79 eV (𝐴 𝑔 ) and -0.99 eV (𝐵 3𝑔 ). These values are nonetheless typical for these challenging ESs with a dominant contribution from the doubly-excited determinants. Applying Eq. (1b) delivers VTEs of 5.17 and 6.44 eV. These two results logically remain higher in en-ergy than the corresponding NEVPT2/aug-cc-pVTZ values of 4.61 and 6.15 eV, that we consider as the most accurate TBEs available to date, though with an error bar of approximately ±0.1 eV. Finally, let us briefly discuss the challenging case of hexatriene (Table 6). The original TBEs for this small polyene have been obtained via Eq. (2a): 5.37, 5.62, 5.79, and 5.94 eV for the lowest 1 𝐵 𝑢 , 1 𝐴 𝑔 , 1 𝐴 𝑢 , and 1 𝐵 𝑔 ESs, respectively. [START_REF] Véril | QUESTDB: a Database of Highly-Accurate Excitation Energies for the Electronic Structure Community[END_REF] All were considered safe, except the second one that has a very significant contribution from the double excitations with %𝑇 1 = 65% (i.e., 10% less than in butadiene). Using Eq. (1d) to include CC4 corrections leads to improved TBEs of 5.34, 5.46, 5.79, and 5.93 eV. Only the 1 𝐴 𝑔 VTE undergoes a significant downshift (-0.16 eV) due to the quadruples. Logically, this 5.46 eV value can be considered as an upper bound and we roughly approximate the exact value to be of the order of 5.43 eV. To support this crude estimate, we looked at the equivalent transition in the smaller polyene, butadiene. Indeed, in butadiene, the difference between the CC4/6-31+G(d) and FCI/6-31+G(d) values is -0.01 eV; the correction should be larger for hexatriene given its smaller %𝑇 1 value. In addition, as can be seen in Ref. 29, the NEVPT2/6-31+G(d) value is 0.20 eV larger than the FCI/6-31+G(d) result for butadiene, and applying such rigid shift to NEVPT2/aug-cc-pVTZ VTE of hexatriene, [START_REF] Loos | Reference Energies for Double Excitations[END_REF] would yield an estimate of 5.44 eV. Although it would not be suited to rate this new TBE/aug-cc-pVTZ of 5.43 eV as safe, it is likely one of the most accurate estimate published to date for this dark transition.

CONCLUSIONS AND OUTLOOK

Three main problems were tackled in the present study. Our first aim was to confirm that CC4 provides very accurate VTEs. To this end, we considered 43 molecules containing between two and four (non-hydrogen) atoms and computed their ESs with three Gaussian basis sets containing diffuse functions, i.e., 6-31+G(d), aug-cc-pVDZ, and aug-cc-pVTZ. We started by defining more than 200 CCSDTQ reference values. Excluding the few pathological pure doubly-excited states of (𝑛, 𝑛) → (𝜋 ★ , 𝜋 ★ ) nature but conserving all other transitions that have a non-negligible double excitation character (2 1 𝐴 𝑔 in butadiene, 2 [START_REF] Roos | Multiconfigurational Perturbation Theory: Applications in Electronic Spectroscopy[END_REF] 𝐴" in acrolein, etc), we showed that CC4 is an excellent approximation of CCSDTQ with a mean absolute error of 0.003 eV. This is a totally negligible deviation for the vast majority of chemical studies (< 0.1

SUPPORTING INFORMATION AVAILABLE

Raw VTEs. Geometries.

Graphical TOC Entry


Quang Nguyen
email: quang-chuc.nguyen@irt-saintexpery.com

Chuc

Patrick Tounsi
email: patrick.tounsi@laas.fr

Jean-Pierre Fradin
email: jean-pierre.fradin@icam.fr

Jean-Michel Reynes
email: jean-michel.reynes@irt-saintexupery.com

Development of SiC MOSFET electrical model and experimental validation: improvement and reduction of parameter number

Keywords: SPICE model, Silicon Carbide (SiC), MOSFET I

In this work, a new approach for electrical modeling of Silicon Carbide (SiC) MOSFET is presented. The developed model is inspired from the Curtice model which is using a mathematic function reflecting MOSFET output characteristics. The first simulation results showed good agreement with measurements. Improvement is needed in order to increase model accuracy and to take into account the influence of the junction temperature on device characteristics.

INTRODUCTION

In power electronics field, engineers must simulate electrical circuit before building an electrical system. During this phase, a good electrical model of transistors is mandatory because it gives the necessary indications on the behaviors of the systems under different conditions. That is why having a good transistor model is critical. Often, a complex model provides a better accuracy, but it has a longer simulation time. Conversely, a simple model has much shorter simulation time, but often with a mediocre accuracy. Electrical engineers always have to compromise these two aspects.

In most applications, Silicon (Si) power devices play a dominant role because technologies are mature. However, foreseeing their limits [START_REF] Bhatnagar | Comparison of 6H-SiC, 3C-SiC and Si for power devices[END_REF], a lot of efforts were put in order to find alternative devices. Wide band gap materials such as Silicon Carbide (SiC) and Gallium Nitride (GaN) are two most promising materials to replace Silicon in power devices. Presently, SiC power devices are targeted for replacing Si power devices in high voltage applications. SiC MOSFETs and SiC Schottky diodes are already successfully commercialized and replaced their Si counterparts in some applications with higher overall performances [START_REF] Glaser | Direct comparison of silicon and silicon carbide power transistors in high-frequency hardswitched applications[END_REF], [START_REF] She | Review of silicon carbide power devices and their applications[END_REF]. Electrical models of these devices are necessary for engineers to contemplate which solutions to choose while considering all aspects: performance, price, harsh conditions … A great amount of works are put into accurately modeling these devices, particularly SiC MOSFET.

In the literature, many models of SiC MOSFET have been proposed [START_REF] Mantooth | Modeling of wide bandgap power semiconductor devices-Part I[END_REF]. McNutt et al. [START_REF] Mcnutt | Silicon carbide power MOSFET model and parameter extraction sequence[END_REF] have efficiently adapted a physical model of Si IGBTs for SiC MOSFETs. While the advantage of this model is presenting the MOSFET with physical meaning, it is still limited because of its complexity and its accuracy.

On the industrial side, SiC MOSFET manufacturers also provide electrical models for their own power devices. Some of them (STM, ROHM and CREE Wolfspeed devices) are analyzed and validated by measurements [START_REF] Stefanskyi | Review of commercial SiC MOSFET models: Validity and accuracy[END_REF]. These die manufacturers have used the level 1 Si Model (STM), the EKV model (ROHM and CREE Wolfspeed ) and the Curtice model (CREE Wolfspeed) for their devices, even though the level 1 Si model and EKV model do not properly adapt to SiC devices, The Curtice model gives a better accuracy and adaptation for these devices. Since its creation which was initially used for GaAs FET [START_REF] Curtice | A nonlinear GaAs FET model for use in the design of output circuits for power amplifiers[END_REF], consequential works have been completed to adapt this model to MOSFETs. An improved modeling based on Curtice model for SiC MOSFET is proposed [START_REF] Stefanskyi | Universal Behavioural Model for SiC Power MOSFETs Under Forward Bias[END_REF].

Taking a step back, the idea behind Curtice model is that the hyperbolic tangent function has the same form as the output characteristics of MOSFET. Using this function, we can easily represent output characteristics of SiC MOSFET with a good accuracy by adjusting fewer parameters in the MOSFET equations.

It is natural to think about other functions having the same form and to extend the model by using these functions. The idea here is to see if we can minimize the number of parameters and so decrease the complexity of the model while maintaining the same level of accuracy.

In this particular work, we build our model using the following function instead of the hyperbolic tangent function:

𝑓 𝑑 (𝑥) = 𝑥 |𝑥| + 𝑏 (1)
This function is relatively simple comparing to the hyperbolic tangent function.

II. SIC MOSFET MODELING

A. State of the art of Curtice model for SiC MOSFET

In this section, we will present our model of SiC MOSFET. The original Curtice model of MOSFET channel is the following [START_REF] Maas | Fixing the Curtice model[END_REF]:

𝐼 𝐷𝑆 = 𝑓 𝑔 (𝑉 𝐺𝑆 ) × 𝑓 𝑑 (𝑉 𝐷𝑆 ) (2)
Where:

𝑓 𝑔 (𝑉 𝐺𝑆 ) = 𝐴 0 + 𝐴 1 𝑉 𝐺𝑆 + 𝐴 2 𝑉 𝐺𝑆 2 + 𝐴 3 𝑉 𝐺𝑆 3 (3) 𝑓 𝑑 (𝑉 𝐷𝑆 ) = tanh(𝛾𝑉 𝐷𝑆 ) (1 + 𝜆𝑉 𝐷𝑆 ) (4)
The function 𝑓 𝑔 (𝑉 𝐺𝑆 ) represents transfer characteristics while the function 𝑓 𝑑 (𝑉 𝐷𝑆 ) represents output characteristics of MOSFET. Some modifications have been realized in order to adapt the model to SiC MOSFET [START_REF] Stefanskyi | Review of commercial SiC MOSFET models: Validity and accuracy[END_REF].

The function of 𝑓 𝑔 has been changed from a cubic polynomial to an algebraic fraction function [START_REF] Stefanskyi | Review of commercial SiC MOSFET models: Validity and accuracy[END_REF]. The idea behind this change is that the new function 𝑓 𝑔 suits better the transfer characteristics of SiC MOSFET.

The function 𝑓 𝑑 , on the other hand, has been the same hyperbolic tangent function as the Curtice model's one.

B. New model for SiC MOSFET

Firstly, we separate the MOSFET equation by blocks and normalize the two functions that represent transfer characteristics and output characteristics. The MOSFET equation will have the following formula:

𝐼 𝐷𝑆 = 𝐶 × 𝑓 𝑔 (𝑉 𝐺𝑆 ) × 𝑓 𝑑 (𝑉 𝐷𝑆 ) × (1 + 𝜆𝑉 𝐷𝑆 ) (5)
Where:

𝐶 is a constant which relates to the saturation current of SiC MOSFET. 1 + 𝜆𝑉 𝐷𝑆 is the modulation factor with λ is channel-length modulation parameter. This factor allows the IDS compensate the small augmentation of drain current when VDS becomes very large (in saturation mode). The value of 𝑓 𝑑 (𝑉 𝐷𝑆 ) tends to 1 when VDS becomes very large. The smaller 𝑏 is, the faster 𝑓 𝑑 (𝑉 𝐷𝑆 ) approaches its asymptote.

𝑓 𝑔 (𝑉 𝐺𝑆 ) = (

Analysis of the model:

In order to have a complete MOSFET equation, we need to find totally 6 mentioned coefficients: 𝐶, 𝑎, 𝑛, 𝑚, 𝑏, 𝜆.

The conventional procedure is to acquire the output characteristics of devices by experimental measurement. Afterwards, using curve fitting techniques, we will be able to find a set of coefficients that minimize the difference between the simulation curve and the measurement one (method of Least-Squares). The found coefficients will be then employed in the model to plot the simulation curve. A comparison between the simulation curve and measurement will be made in order evaluate the accuracy of the model.

III. SIMULATION AND MEASUREMENT

In this third section, in order to validate the accuracy of the built model, we will make use of this model on different SiC MOSFETs from different manufacturers. Two selected SiC MOSFETs are: SCT50N120 (from STM) and SCT2280KE (from ROHM).

Forward output characteristics of these devices are measured by using the curve tracer HP4142b. Here, the drainsource voltage VDS was measured under different values of the gate-source voltage VGS (from 12V to 20V) when the drain current IDS varies from:

• 0 to 10A for forward conduction.

• 0 to -10A for reverse conduction.

In figure 3 As we can see in figure 3, the model gives a pretty good accuracy of forward characteristics at high gate-source voltage VGS for both devices. While it still maintains a good precision at low gate source voltage VGS for STM device, its precision decreased for ROHM device under these conditions (VGS = 12-13V).

Figure 4 shows the simulation and measurement comparison for reverse output characteristics (ISD as function of VSD). The model shows a promising accuracy for ROHM device. However, it only give a good precision for STM device under high gate-source voltage VGS. In the case of low gatesource voltage (VGS = 12V), the model still needs to be improved.

IV. CONCLUSION

This paper presents a behavioral electrical model of SiC MOSFETs. Following the funding idea of Curtice model, a different function is used instead of the hyperbolic tangent function in order to represent the output characteristics of MOSFETs. As illustrated in figure 2, the model showed a promising modeling accuracy vs. measurement. However, the model still needs to be improved to better address static output characteristics at medium gate-source voltage.

The model only uses 6 parameters for the functionality of MOSFET channels. For comparison, another model [START_REF] Stefanskyi | Universal Behavioural Model for SiC Power MOSFETs Under Forward Bias[END_REF], which was also built on the Curtice model, has 12 parameters. However, this model already takes into consideration the variations of characteristics with the junction temperature.

Taking into account the influence of the junction temperature in our model is a first development to follow.

Secondly, this work should be extended to a fully completed model of SiC MOSFET with dynamic characteristics, body diode characteristics and other aspects.

  with coefficients 𝑎, 𝑛, 𝑚. 𝑓 𝑑 (𝑉 𝐷𝑆 ) = 𝑉 𝐷𝑆 |𝑉 𝐷𝑆 |+𝑏 (7), is a function of VDS, with coefficient 𝑏.

  In the function 𝑓 𝑔 (𝑉 𝐺𝑆 ), the added coefficient 𝑎 allows having a better dynamic while fitting the model with measurement. By adjusting the values of 𝑎, 𝑛 and 𝑚, we can have a different threshold voltage and different curves of the function 𝑓 𝑔 . The curve of 𝑓 𝑔 (𝑉 𝐺𝑆 ) with different sets of parameters (𝑎, 𝑛, 𝑚) is illustrated in figure 1. The function 𝑓 𝑑 (𝑉 𝐷𝑆 ) has the same curve as the hyperbolic tangent function in the Curtice model. The figure 2 shows several forms of the function 𝑓 𝑑 (𝑉 𝐷𝑆 ) with different values of b.

Fig. 1 .

 1 Fig. 1. Curves of the function 𝑓 𝑔 (𝑉 𝐺𝑆 ), with different set values of coefficients (𝑎, 𝑛, 𝑚).

Fig. 2 .

 2 Fig. 2. Curves of the function 𝑓 𝑑 (𝑉 𝐷𝑆 ) with different values of 𝑏.

Fig. 3 .

 3 Fig. 3. SiC MOSFET forward output characteristics: simulation vs measurement (a) SCT50N120, (b) SCT2280KE.

Fig. 4 .

 4 Fig. 4. SiC MOSFET reverse output characteristics: simulation vs measurement (a) SCT50N120, (b) SCT2280KE.


Quang Chuc Nguyen
email: quang-chuc.nguyen@irt-saintexupery.com

Patrick Tounsi

Jean-Pierre Fradin

Jean-Michel Reynes

Investigation on the use of the MOSFET SiC body diode for junction temperature measurement

As Silicon-based semiconductors approach their limits in different areas, wide bandgap devices, such as Silicon Carbide components, offer an excellent alternative in many applications. Recently, SiC MOSFETs are replacing Si -based IGBT in various fields as automotive, solar energy… The junction temperature is important to evaluate the performance and the reliability of these components. For Silicon MOSFET, the body diode is usually used as a thermal sensitive electrical parameter (TSEP) for junction temperature measurement. For SiC MOSFET, however, in gate-source short configuration (VGS=0V), some current still flows through the channel, which, with interface trapping, prevents from accurately estimating SiC MOSFET junction temperature. Practically, a sufficiently negative gate voltage VGS (-5V or below, depending on devices) must to applied to eliminate the current part through the MOS channel so that the body diode forward voltage is immune from interface trapping.

Background

Silicon Carbide MOSFETs are becoming more frequently used in different power electronic applications such as solar inverters, power supply, railway traction inverter, electric vehicle and induction heating because of their numerous advantages: high blocking voltage, high working temperature and high switching speed [START_REF] She | Review of silicon carbide power devices and their applications[END_REF]. Measuring the SiC MOSFET junction temperature is a mean to evaluate the performance and reliability of SiC-based power modules.

The power module package often prevents from directly measuring the junction temperature with optical or physical contact methods such as IR camera, thermocouple… Consequently, the junction temperature is usually indirectly measured by electrical methods via a thermal sensitive electrical parameter (TSEP): voltage under low current (diode, IGBT, MOSFET), threshold voltage, peak gate current… [START_REF] Avenas | Temperature measurement of power semiconductor devices by thermo-sensitive electrical parameters-A review[END_REF], [START_REF] Brandelero | Online junction temperature measurements for power cycling power modules with high switching frequencies[END_REF]. The thermal sensitive electrical parameters are well understood for Silicon components but some phenomena related to TSEP are not yet clarified for SiC MOSFET.

In order to have a correct measurement, the change of the TSEP response before and after heating phase should only come from the change of the junction temperature. If any other electrical aspects can influence the TSEP response, then the measurement is not accurate [START_REF] Funaki | Difficulties in characterizing transient thermal resistance of SiC MOSFETs[END_REF], [START_REF] Kato | Effect of forward voltage change depending on gate voltage in body diode of SiC-MOSFET at thermal transient testing for analysing SiC power module package[END_REF]. Therefore, the TSEP response compliance was first checked before being used for the junction temperature measureme. 1 depicts the test bench electrical circuit dedicated to measuring the SiC MOSFET junction temperature using its body diode as a TSEP. The power source IP (high current of tens of amps to hundreds of amps) is used to heat the MOSFET. The measurement current IM (low current of 10mA) is applied to measure the forward voltage of the MOSFET body diode, which gives an estimation of the junction temperature.

The power module used in this study is designed and produced by Institute of Technology Saint-Exupery. It consists of 12 SiC MOSFETs, which are split in 6 switching cells (see fig. 2a). In our tests, the module is mounted on a fancooled heat sink with a thermal paste within the contact. The control system contains a motherboard and 12 gate drivers, which are fixed on the upper side of the power module (see fig. 2b). The electrical system of the test bench consists of two power transistors (T1, T2) that drive the power current into the SiC MOSFET during the heating phase. A system of National Instruments chassis and control module allows us to send the control signal to the gate drivers. The forward voltage of SiC MOSFET body diode is measured by a voltage probe and a NI oscilloscope (see fig 2 .c).

The measurement procedure consists of three phases:

(i) The switch T1 is turn-on, T2 is turn-off, the SiC MOSFET is controlled by a gate voltage VGS = VGSM (VGS during measurement), only the current I M goes through the MOSFET, the junction temperature is also the ambient temperature Ta. The body diode forward voltage under low current is VF = VFM_Ta;

(ii) The switch T1 is turn-off, T2 is turn-on, the gate voltage VGS = VGSP (VGS during power phase), both currents IP and IM go through the MOSFET and heat it to a temperature TJ. The body diode forward voltage under high current is VF = VFP;

(iii) The switch T1 is turn-on, T2 is turn-off again, the gate voltage VGS = VGSM, only the current IM goes through the MOSFET. At the switching moment, the MOSFET is still at heating temperature TJ, the body diode forward voltage under low current: VF = VFM_Tj.

The difference between VFM_TJ and VFM_Ta corresponds to the change of the junction temperature before and after heating phase:

𝑉 𝐹𝑀_𝑇 𝐽 -𝑉 𝐹𝑀_𝑇 𝑎 = 𝑘 × (𝑇 𝐽 -𝑇 𝑎 )
where 𝑘 is the temperature coefficient of the body diode forward voltage (usually between -2mV/°C and -3mV/°C). This coefficient is calibrated beforehand by using a curve tracer HP4142B and the thermostream Temptronic TP04200 (see fig. 3).

When power is dissipated in the SiC MOSFET (IP >> 0), the junction temperature increases (from Ta to TJ) so that the body diode forward voltage in phase (i) and phase (iii) (VFM_Ta and VFM_Tj) are different Obviously, if there is no dissipated power in the SiC MOSFET (IP = 0), the junction temperature is the same as the ambient temperature. Thus: VFM_Tj = VFM_Ta.

Results

Fig. 3 illustrates the body diode voltage VF under VGSM = 0V, VGSP = 20V (fig. 4a), VGSM=-5V, VGSP=20V (fig. 4b) and IP=0A, IM=10mA (no heating). Notice that under this condition, the MOSFET is not heated at all because of the absent of the heating power. As discussed above, in this case, we expected that under the same measurement current (10mA), the body diode forward voltage does not change between measurement phases (phase (i) and phase (iii)). In SiC MOSFET's reverse configuration, even when the gate voltage VGS = 0V, because of the body effect [START_REF] Pala | Physics of bipolar, unipolar and intermediate conduction modes in Silicon Carbide MOSFET body diodes[END_REF], [START_REF] Huerner | Analytical model for the influence of the gate-voltage on the forward conduction properties of the body-diode in SiC-MOSFETs[END_REF], the current is composed of two parts: Ibody (through the body diode) and Ich (through the channel) (see fig. 5a). When the gate voltage V GS changes value (from 0V to 20V then back to 0V), the charges are trapped in the interface between the SiC layer (p-region) and the gate oxide SiO2 layer (see fig. 5b). This interface trapping changes the conductivity of the channel and so affects this channel current part Ich during temperature measurement procedure, which ultimately ma kes the body diode forward voltage VF unstable and distorts the measurement.

IF(VF) as a function of the gate bias voltage VGS

As the gate voltage becomes progressively more negative (from 0V to -5V), the body diode voltage becomes higher until VGS = -5V (fig. 6), which means that the channel current part I ch is slowly eliminated. From then, V F remains constant, even if VGS becomes more negative (-5V to -8V), which means Ich has completely disappeared and only Ibody exists when VGS=-5V or higher. In this case, when the gate is controlled at a sufficient negative level (-5V or more for this component), the body diode is fully turned on and can be used for junction temperature measurement.

The junction temperature of power devices is an important parameter to evaluate the performance and reliability of power systems. Because of the package, the junction temperature of power module is often measured indirectly via thermal sensitive electrical parameters. These parameters are well known and studied for Silicon-based devices. However, the mechanism of using TSEP for Silicon Carbide components such as SiC MOSFET still need to be studied. This work helps clarify the use of SiC MOSFET body diode as thermal sensitive electrical parameter. The current path in reverse configuration of SiC MOSFET includes two parts: a part through the body diode and a part through the channel.

When the gate voltage changes, the charges are trapped in the interface between the gate oxide and the silicon. This phenomenon changes the conductivity of the channel and influences the current part through the channel. In order to address this issue, the gate voltage needs to be controlled at a sufficient negative level (-5V or more) to eliminate the current part through the channel. Only then, the body diode can be used as thermal sensitive parameter for SiC MOSFET junction temperature measurement.

Fig. 1 .

 1 Fig. 1. (a) Electrical circuit for junction temperature measurement (a), (b) diagram of different phases of the thermal characterization.

Fig.

  Fig.1depicts the test bench electrical circuit dedicated to measuring the SiC MOSFET junction temperature using its body diode as a TSEP. The power source IP (high current of tens of amps to hundreds of amps) is used to heat the MOSFET. The measurement current IM (low current of 10mA) is applied to measure the forward voltage of the MOSFET body diode, which gives an estimation of the junction temperature.

Fig. 2 .

 2 Fig. 2. Presentation of the test bench for thermal characterization: (a) SiC power module with control pins and power connections, (b) SiC power module with control system mounted on the heat sinks, (c) PC with Labview and NI components (chassis, control module, oscilloscope) for control and signal acquisisition.

Fig. 3 .

 3 Fig. 3. Electrical test bench for power module characterization.

Fig. 4 .

 4 Fig. 4. Forward voltage of the body diode under a current of 10mA and control voltages: V GSP = 20V, V GSM = 0V (a); V GSP = 20V, V GSM = -5V.

Fig. 5 .

 5 Fig. 5. (a) Cross-section of current paths through the body diode and (b) a zoom of the interface between gate oxide (SiO 2 ) and p-region (SiC).

Fig. 6 .

 6 Fig. 6. IF(VF) as a function of the gate bias voltage VGS

Acknowledgements

The authors would like to thank to the Institute of Technology Antoine de Saint-Exupéry for providing the SiC power module and their help in setting up the control system combined with the module.


Yohann Dabi

Samia Ouasti

Hélène Didelot

Henri Wohrer

Dounia Skalli

Grégoire Miailhe

Jennifer Uzan

Clément Ferrier

Sofiane Bendifallah

Bassam Haddad

Emile Daraï

Cyril Touboul
email: cyril.touboul@gmail.com

Prof Cyril Touboul

Total hysterectomy by low-impact laparoscopy to decrease opioids consumption: A prospective cohort study

Keywords: Low impact laparoscopy, hysterectomy, postoperative pain, opioids

Our objective was to evaluate postoperative pain and opioid consumption in patients undergoing hysterectomy by low-impact laparoscopy and compare these parameters with conventional laparoscopy. We conducted a prospective study in two French gynecological surgery departments from May 2017 to January 2018. The primary endpoint was the intensity of postoperative pain evaluated by a validated numeric rating scale (NRS), and opioid consumption in the postoperative recovery unit on Day 0 and Day 1. Thirty-two patients underwent low-impact laparoscopy and 77 had conventional laparoscopy. Most of the patients (90.6%) who underwent low-impact laparoscopy were managed as outpatients. There was a significantly higher consumption of strong opioids in the conventional compared to the lowimpact group on both Day 0 and Day 1: 26.0% and 36.4% vs 3.1% and 12.5%, respectively (p=0.02 and p<0.01). Over two-thirds of the patients in the low-impact group did not require opioids postoperatively. Two factors were predictive of lower postoperative opioid consumption: low-impact laparoscopy (OR 1.38, 95%CI 1.13-1.69, p=0.002) and a mean intraoperative peritoneum <below 10 mmHg (OR 1.25, 95%CI 1.03-1.51). Total hysterectomy by low-impact laparoscopy is feasible in an outpatient setting and is associated with a marked decrease in opioid consumption compared to conventional laparoscopy.

Introduction

Around 33,000 hysterectomies are performed each year in the UK which means that one in five women will undergo this major gynecological surgery at some point [START_REF] Madhvani | Route of Hysterectomy: A Retrospective, Cohort Study in English NHS Hospitals from 2011 to 2017[END_REF] . In the past, hysterectomies were performed by the vaginal route or by open surgery (laparotomy) but the development of laparoscopy has supplanted these approaches over the years in developed countries [START_REF] Moawad | Movement to Outpatient Hysterectomy for Benign Indications in the United States, 2008-2014[END_REF] . The emergence of robotic-assisted surgery, which reduces the surgical complexity, has further promoted the laparoscopic approach allowing more gynecologists to perform the procedure [START_REF] Luchristt | Trends in Operative Time and Outcomes in Minimally Invasive Hysterectomy from 2008 to 2018[END_REF] . These combined improvements have led to reduced hospital stays and even to outpatient management in certain cases. Overall, there is a trend for increased outpatient management for surgical procedures as they are associated with greater patient satisfaction and lower costs compared to inpatient management [START_REF] Ahmad | A Meta-Analysis of Ambulatory versus Inpatient Laparoscopic Cholecystectomy[END_REF][START_REF] Crawford | Clinical and Cost Implications of Inpatient Versus Outpatient Orthopedic Surgeries: A Systematic Review of the Published Literature[END_REF][START_REF] Lee | Outpatient versus Inpatient Superficial Parotidectomy: Clinical and Pathological Characteristics[END_REF] .

Nevertheless, surgeons need to be sure that their patients will not undergo any pain once home. Pain typically leads to over-prescription and over-consumption of opioids. In a cohort of 40,000 patients who had undergone surgery, Clarke et al. reported that 49% were discharged with an opioid prescription and that 3% were still using opioids 90 days after the operation [START_REF] Clarke | Rates and Risk Factors for Prolonged Opioid Use after Major Surgery: Population Based Cohort Study[END_REF] . The opioid crisis that has come to light in the United States over the past few years has largely been driven by postoperative prescriptions for pain management. In a study by Brummet et al. of a large cohort of privately insured patients in the United States, the rates of new persistent postoperative opioid use ranged from 5.9% to 6.5% [START_REF] Brummett | New Persistent Opioid Use After Minor and Major Surgical Procedures in US Adults[END_REF] , even after minor surgery. The rates of new persistent opioid use was close to 6% after hysterectomy in their study. Postoperative opioid use is thus a current public health crisis and surgeons can play their part by improving surgical procedures.

Laparoscopic surgery has evolved significantly since the first laparoscopic hysterectomy by Reich in 1989 [START_REF] Reich | Laparoscopic Hysterectomy[END_REF] . Mini-invasive surgery now encompasses techniques such as robotic-assisted surgery, single-port laparoscopy and, more recently, the concept of "low-impact laparoscopy". Low-impact laparoscopy combines low peritoneal pressure with miniaturization of laparoscopic ports from 5-15 mm to 3-5 mm in diameter. While this technique has been shown to reduce postoperative pain and increase esthetic satisfaction for several surgical procedures [START_REF] De'angelis | The Protocol of Low-Impact Laparoscopic Cholecystectomy: The Combination of Mini-Laparoscopy and Low-Pressure Pneumoperitoneum[END_REF][START_REF] Hua | Low-Pressure versus Standard-Pressure Pneumoperitoneum for Laparoscopic Cholecystectomy: A Systematic Review and Meta-Analysis[END_REF] , there are very few data concerning gynecologic surgery. In a cohort of 60 patients, Sroussi et al. reported the feasibility and effectiveness of low-impact laparoscopy for reducing shoulder pain after benign gynecological surgeries [START_REF] Sroussi | Low Pressure Gynecological Laparoscopy (7mmHg) with AirSeal® System versus a Standard Insufflation (15mmHg): A Pilot Study in 60 Patients[END_REF] . To date, little attention has been paid to postoperative analgesic and opioid consumption in patients undergoing a hysterectomy by low-impact laparoscopy.

The main objective of our study was thus to evaluate postoperative pain and opioid consumption in patients undergoing a hysterectomy by low-impact laparoscopy. Our second objective was to compare these parameters with those of patients undergoing a conventional laparoscopic hysterectomy.

Methods

Population

We conducted an analysis of patients prospectively included from May 2017 to January 2018 in two French surgical gynecology departments (Centre Hospitalier Intercommunal de Creteil and Tenon Hospital, AP-HP). All consecutive adult patients undergoing laparoscopic hysterectomy were included in the study. Patients with additional procedures were excluded from analysis. The research protocol was approved by the Institutional Review Board (IRB) of the French College of Obstetrics and Gynecology (CEROG 2021 -GYN -0101).

Surgical Details

All the low-impact laparoscopic procedures were performed by two senior surgeons.

Conventional laparoscopy was performed by other surgeons in the departments. Patients in the low-impact group were all prescribed analgesics including paracetamol, a nonsteroidal anti-inflammatory drug (NSAID), and a weak opioid (tramadol) at their preoperative appointment. All the low-impact laparoscopy hysterectomies were planned as outpatient procedures.

Anesthesia Protocol

A formalized anesthesia protocol was applied for patients who underwent low-impact laparoscopy. Anesthesia was achieved by a remifentanil/propofol rapid induction technique, and curarization by rocuronium bromide (quick-acting). All the patients received an infusion of ketamine (0.15 mg/kg) at the beginning of the surgery and were administered an antiemetic (dexamethasone) during the procedure. Peroperative analgesia also included paracetamol and nefopam. Vascular filling was restricted to 2 to 3 mL/kg/h. A bladder catheter was inserted systematically for both procedures.

For the low-impact procedures, pneumoperitoneum was created at the umbilicus using a Verress needle up to 15 mmHg. An AirSeal® 5 mm valve-free trocar -which provides stable pneumoperitoneum even under constant suction-was then introduced [START_REF] Luketina | Comparison of a Standard CO₂ Pressure Pneumoperitoneum Insufflator versus AirSeal: Study Protocol of a Randomized Controlled Trial[END_REF][START_REF] Horstmann | Prospective Comparison between the AirSeal® System Valve-Less Trocar and a Standard Versaport TM Plus V2 Trocar in Robotic-Assisted Radical Prostatectomy[END_REF] . We then inserted a 3 or 4 mm 30° optical lens. Under visual control, three other ports were introduced: in the right and left iliac fossae, and in the right hypochondrium. The optical lens was then moved to the right hypochondrium port for the rest of the surgery and intraperitoneal pressure was lowered to 6 -8 mmHg.

For the conventional laparoscopic procedure, a 10 mm optical lens was introduced through the umbilicus. The first insufflation was at 15 mmHg. It was then mediated either by a conventional insufflator or by an Airseal® device. Three other 5 mm ports were introduced under direct visual control in the right and left iliac fossae and medially. The pressure was then lowered to 10 -12 mmHg.

The operative technique was similar in the two groups. Hysterectomy was performed using Ultracision® Harmonic forceps and coagulation using a bipolar grasp. At the end of surgery, the ports were infiltrated by ropivacaine (75mg) at the surgeon's discretion.

Postoperative management was similar in the two groups especially in terms of pain evaluation and administration of analgesics. The nurses in the postoperative recovery unit (PACU) and outpatient/gynecologic unit were completely blinded to the surgical approach used.

The bladder catheter was removed at the end of surgery for the outpatients and on Day 1 for inpatients.

Outcome Measures

The primary endpoint was opioid consumption for postoperative pain. Pain was evaluated in the PACU on Day 0 and Day 1 by a validated numeric rating scale (NRS) ranging from 0 to 10 (where 0 is no pain and 10 the worst imaginable pain) [START_REF] Hjermstad | (EPCRC). Studies Comparing Numerical Rating Scales, Verbal Rating Scales, and Visual Analogue Scales for Assessment of Pain Intensity in Adults: A Systematic Literature Review[END_REF][START_REF] Ferreira-Valente | Validity of Four Pain Intensity Rating Scales[END_REF][START_REF] Breivik | A Comparison of Pain Rating Scales by Sampling from Clinical Trial Data[END_REF][START_REF] Thong | The Validity of Pain Intensity Measures: What Do the NRS, VAS, VRS, and FPS-R Measure?[END_REF] . Opioid consumption was recorded at each time point. Weak opioids included tramadol 50 mg 4 times a day and strong opioids consisted of morphine titration by 1 mg doses. The outpatients were all called on Day 1 as standard protocol to ensure the absence of uncontrolled pain.

Statistical Analysis:

Databases were managed using Excel (Microsoft Corporation, Redmond, WA, USA) and statistical analyses were performed using R studio software (1.1.463 version, available online).

After descriptive analysis the patients in the low-impact laparoscopy group were compared to those in the conventional laparoscopy group. Data are presented as n (%) unless otherwise specified. Statistical analysis was based on the Student's t test for continuous variables and the χ2 test or Fisher's exact test for categorical variables. All the factors that could influence postoperative pain and analgesic consumption were then investigated in univariate analysis. Values of p <0.05 were considered to denote significant differences.

Results

Clinical Characteristics and Surgical Parameters

During the study period, of the 109 patients who had a hysterectomy by laparoscopy, 32 underwent a low-impact laparoscopic procedure and 77 a conventional laparoscopy. The data are shown in Table 1. The main clinical characteristics were comparable between the two groups with no significant difference for age, obesity, American Society of Anesthetists (ASA) physical status score, comorbidities, or history of surgery (including cesarean sections) (p >0.05). There was no significant difference in the duration of surgery, but the duration of anesthesia was greater in patients the conventional procedure group (193 versus 174 minutes, p = 0.03).

None of the patients of the total cohort experienced significant peroperative bleeding.

Postoperative Pain and Opioid Consumption

The main postoperative outcomes are displayed in Table 3. Most of the patients in the lowimpact group were managed as outpatients (90.6%, 29/32). In the whole cohort, two patients (one in each group) experienced operative complications that required a second procedure:

one patient in the low-impact group experienced a late vesicovaginal fistula which was successfully repaired 3 months later; and the other (in the conventional group) a hemorrhage on the umbilical port. The maximum pain evaluated in the PACU was significantly higher in the patients that had conventional laparoscopy (p = 0.04) but similar on Days 0 and 1 postoperatively (p = 0.92 and p = 0.32, respectively). NSAIDs were used for two patients (2.6%) in the conventional group and for one (3.1%) in the low-impact group.

No patient (0.0%) consumed weak opioids in the PACU or on Day 0 in the low-impact group.

There was a significantly higher consumption of strong opioids in the conventional group vs the low-impact group on Day 0 and Day 1: 26.0% (20/77) and 36.4% (28/77) vs 3.1% (1/32) and 12.5% (4/32), respectively (p = 0.02 and p <0.01). Of note, more than 2/3 of the patients who underwent a low-impact procedure did not use any opioids in the postoperative period.

Postoperative opioid consumption and pain evaluation are plotted in the figure 1.

Factors Associated with Postoperative Opioid Consumption

Two factors were predictive of lower post-operative opioid consumption: a surgical approach by low-impact laparoscopy (OR 1.38 95%CI 1.13 -1.69), p = 0.002) and a mean intraoperative peritoneum pressure below 10 mmHg (OR 1.25 95%CI 1.03 -1.51). None of the other parameters, including uterus size and duration of pneumoperitoneum, had an influence of postoperative opioid consumption, (Table 4).

Discussion

Main Findings

We report here that patients who underwent a hysterectomy by low-impact laparoscopy experienced less postoperative pain and used significantly fewer opioids than those who had conventional laparoscopy. More than two thirds of the patients who underwent a low-impact procedure did not use any opioids postoperatively. Low-impact laparoscopy was successfully managed in an outpatient setting for around 90% of the patients. Two factors were predictive of lower postoperative opioid consumption: low-impact laparoscopy, and a mean intraoperative peritoneum pressure below 10 mmHg.

Interpretation

The implementation of low-impact procedures in our centers was motivated by the double objective of achieving outpatient management and reducing (and even completely suppressing) postoperative opioid consumption.

It is now accepted that hysterectomy by laparoscopy is associated with lower morbidity and a shorter hospital stay than hysterectomy by laparotomy or by the vaginal route, for both cancerous and benign indications [START_REF] Bijen | Costs and Effects of Abdominal versus Laparoscopic Hysterectomy: Systematic Review of Controlled Trials[END_REF][START_REF] Walker | Laparoscopy Compared with Laparotomy for Comprehensive Surgical Staging of Uterine Cancer: Gynecologic Oncology Group Study LAP2[END_REF] . For many years, the challenge has been to improve surgical skills to decrease the number of cases requiring a laparotomy. During the last decade, the exponential use of robotic-assisted surgery has democratized minimally invasive surgery to less experienced surgeons [START_REF] Madhvani | Route of Hysterectomy: A Retrospective, Cohort Study in English NHS Hospitals from 2011 to 2017[END_REF][START_REF] Desai | An Update on Inpatient Hysterectomy Routes in the United States[END_REF] . Several techniques have been developed with a view to reducing surgical morbidity, including a reduction in the number of ports as described by Tyan et al. [START_REF] Tyan | Safety and Feasibility of the Three-Port Robot-Assisted Hysterectomy across Uterine Weights[END_REF] . Miniaturization of laparoscopic instruments -an approach known as minilaparoscopy-represents little difficulty for trained surgeons as the lower grasping ability is rapidly overcome. In our setting, we chose to use a 3 mm port in the upper right region which enables the use of Ultracision® Harmonic forceps energy in the 5 mm umbilical port or even larger bipolar forceps in case of uncontrolled hemorrhage or large uterine pedicles. Minilaparoscopy has been used by general surgeons for various procedures, such as cholecystectomy, with good outcomes. Casarin J. et al in a recent review concluded that minilaparoscopy could be a valid alternative to conventional laparoscopy in gynecologic surgery including major surgical and oncological procedures [23] . In our cohort, the two groups (lowimpact and conventional laparoscopy) did not differ for the main clinical characteristics, with the same proportion of older / obese patients at higher risk of complex procedures and postoperative complications. We observed one case of vesico-vaginal fistula in the lowimpact group caused by a vaginal vault stich. This rare complication could have been avoided by a more distal vesico-vaginal cleavage and was certainly independent of the approach.

A recent report from the West Sussex NHS Trust highlighted that the UK is now heading toward a US, opioid-style crisis with five people dying every day from opioid overdose [24] . There is now abundant literature on this matter reporting that a significant proportion of opioid naïve patients prolong postoperative consumption [START_REF] Hinther | Chronic Postoperative Opioid Use: A Systematic Review[END_REF][START_REF] Neuman | Inappropriate Opioid Prescription after Surgery[END_REF][START_REF] Feinberg | Best Practice in Surgery Group. Opioid Use After Discharge in Postoperative Patients: A Systematic Review[END_REF] . The responsibility of postoperative pain management is shared between surgeons and anesthesiologists. While it is true that not all surgeries are equally painful and therefore do not require the same amount of analgesics, little is known about the quantity each procedure could require. One major issue is the lack of patient reporting of opioid consumption and the lack of specific prescription guidelines either for the recovery room or on hospital discharge.

A recent systematic review by Brenton et al. concluded that there was an absence of evidence for opioid-free hysterectomy [START_REF] Blanton | Non-Opioid Pain Management in Benign Minimally Invasive Hysterectomy: A Systematic Review[END_REF] . In our cohort, five patients (15.6%) used opioids on the day of surgery (Day 0) or the day after (Day 1), which was significantly lower than in the conventional laparoscopy group (51 patients, 66%; p <0.001). The use of opioids did not differ by surgical indication (oncological or benign). The use of non-opioid medication was also significantly lower in the low-impact group. Several authors have studied the performance of hysterectomy by mini-laparoscopy but with a focus on feasibility and complications as well as cosmetic results [START_REF] Luchristt | Trends in Operative Time and Outcomes in Minimally Invasive Hysterectomy from 2008 to 2018[END_REF][START_REF] Ng | Minilaparoscopic Hysterectomy Made Easy: First Report on Alternative Instrumentation and New Integrated Energy Platform[END_REF][START_REF] Uccella | Minilaparoscopic versus Standard Laparoscopic Hysterectomy for Uteri ≥ 16 Weeks of Gestation: Surgical Outcomes, Postoperative Quality of Life, and Cosmesis[END_REF] . While the miniaturization of instruments from 5 to 3 mm does indeed provide a cosmetic advantage, we feel that the main benefit of mini-laparoscopy lies in the clinically relevant reduction in postoperative pain. Our lowimpact approach associates mini-laparoscopy with low pressure (<10 mmHg). Intraperitoneal pressure is commonly set between 12 and 15 mmHg [START_REF] Hypólito | Creation of Pneumoperitoneum: Noninvasive Monitoring of Clinical Effects of Elevated Intraperitoneal Pressure for the Insertion of the First Trocar[END_REF] , and sometimes higher. While guidelines recommend that "the lowest pressure allowing adequate exposure of the operative field" is to be used [START_REF] Neudecker | The European Association for Endoscopic Surgery Clinical Practice Guideline on the Pneumoperitoneum for Laparoscopic Surgery[END_REF] , our work clearly demonstrates that lower pressures are feasible and indeed beneficial, with comparable surgical duration, pneumoperitoneum duration and no increased rates of per-or postoperative complications. Low pressure, which is usually defined as a pressure ranging from 6 to 10 mmHg [START_REF] Esmat | Combined Low Pressure Pneumoperitoneum and Intraperitoneal Infusion of Normal Saline for Reducing Shoulder Tip Pain Following Laparoscopic Cholecystectomy[END_REF][START_REF] Ibraheim | Lactate and Acid Base Changes during Laparoscopic Cholecystectomy[END_REF][START_REF] Eryılmaz | The Effects of Different Insufflation Pressures on Liver Functions Assessed with LiMON on Patients Undergoing Laparoscopic Cholecystectomy[END_REF][START_REF] Schietroma | Changes in the Blood Coagulation, Fibrinolysis, and Cytokine Profile during Laparoscopic and Open Cholecystectomy[END_REF] , potentializes the effect of instrument miniaturization.

D'Angelis et al. reported that low-impact laparoscopy in the setting of cholecystectomy provided improved postoperative outcomes and less pain [START_REF] De'angelis | The Protocol of Low-Impact Laparoscopic Cholecystectomy: The Combination of Mini-Laparoscopy and Low-Pressure Pneumoperitoneum[END_REF] . Some authors argue that mini-laparoscopy is only a valid technique for highly selected patients [23] . In our cohort, 32 patients benefited from the low-impact procedure and they were comparable with those elected for conventional surgery. This implies that the limits of low-impact or miniprocedures might be the same as those of laparoscopy in general: comorbidities and surgical history, very large uterus, etc…

Limitations

Our study has some limitations. Two experienced surgeons performed all the low-impact procedures which might have resulted in an overestimation of the benefit of the technique since surgical duration and gestures performed are based on a surgeon's skills. However, in their experience, the main barriers for undertaking a low-impact procedure were concerns about lacking adequate visibility with low pressure and 3 mm optical lens, and concerns about mastering the mini-instruments. It deserves to be mentioned that the transition in our centers from conventional to micro-instruments was responsible for some material difficulties, and the operative room staff were unfamiliar with the new instruments for the first few cases.

These data are not registered in this work. Another limitation is the short follow-up for postoperative pain. We only evaluated pain on Day 0 and Day 1 postoperatively and some patients might experience prolonged or delayed pain. However, acute pain requiring opioids is usually the most intense on the first two days following surgery. Another limitation of our work was the inability to evaluate the impact of per-operative bleeding on postoperative pain since none of the patients experienced significant bleeding. This could be explained by two factors: 1) the experience of the surgeons that performed the surgeries 2) the relatively limited sample size Another limitation was the lack of data regarding anesthetic management especially pulmonary recruitment maneuver that could reduce significantly the need for postoperative opioids.

This data was not part of our data collection initially and should be included in the future RCT that will be conducted. Finally, bladder catheterization ablation on postoperative day 1 versus immediately at the end of surgery in the low-impact group may be a confounding factor as it is a source of discomfort for patients.

The most appropriate method to formally assess the benefit of low impact procedures would be to conduct a randomized controlled trial. In the figure 2, we present the theoretical number of subjects that would be necessary to include in order to demonstrate the benefit of the lowimpact procedure on the reduction of ever consuming opioids in the postoperative period. In the setting of 60% of patients consuming opioids in the conventional group, showing a 50% reduction of opioid consumption would require the inclusion of 112 patients.

Conclusion

Our study shows that, for women undergoing total hysterectomy, low-impact laparoscopy and an intraperitoneal pressure of <10 mmHg are the main determinants of reduced postoperative pain and analgesic consumption. Total hysterectomy by low-impact laparoscopy is feasible and can be performed in an outpatient setting. Low-impact procedures could constitute the best option for reducing (and possibly eliminating) opioid consumption in the postoperative period. A randomized controlled trial should now be performed to definitely assess the potential benefit of low impact procedures.

Tables/Figures Caption list

  Appropriate analgesics were administered according to the NRS. Postoperative analgesic prescriptions included paracetamol, nefopam, NSAIDs, and tramadol to be used only if the other drugs were not sufficient for pain relief. According to the NRS: Paracetamol for light pain (NRS < 4), Ibuprofen or weak opioids (Tramadol) for moderate pain (4 ≤ NRS < 7) and Morphine for intense pain (NRS ≥ 7)

Figure 1 :Figure 2 :

 12 Figure 1: Postoperative opioid consumption and pain evaluation by surgical approach

Table 1 : Main patient characteristics

 1

	Conventional	Low-impact	P -value
	Laparoscopy	Laparoscopy
	N = 77 (%)	N = 32 (%)

*Missing data for 2 patients ** Some patients had more than one comorbidity Surgical variables are displayed in

Table 2 .

 2 The indication for hysterectomy was more likely to be of an oncologic nature for the patients in the low-impact group (43.8% (14/32) versus 23.4% (14/77), p = 0.01). The size of the uterus was comparable in the two groups with 63

	(21.3) mm and 65 (22.5) mm and in the low impact and conventional groups, respectively
	(p=0.61).
	Patients in the low-impact group had more complementary surgery such as ureterolysis
	(15.6% (5/32) versus 1.3% (1/77)), or omentectomy / appendicectomy (p <0.001).

Table 2 : Surgical factors

 2

	Variables	Conventional	Low impact	P -value
		N = 77 (%)	N = 32 (%)
	Indication for surgery			0.01
	Endometrial neoplasm
	Endometrial cancer	12 (15.6)	10 (31.3)
	Atypical endometrial hyperplasia	2 (2.6)	4 (12.5)
	Other conditions
	Cervical dysplasia	6 (7.8)	1 (3.1)
	Symptomatic Fibroma	37 (48.1)	10 (31.3)
	Adenomyosis	13 (16.9)	4 (12.5)
	Other	7 (9.1)	3 (9.4)
	Mean intraoperative pressure			< 0.001
	< 10	18 (23.4)	28 (87.4)
	10 -12	59 (76.6)	2 (6.3)
	> 12	0	2 (6.3)
	Parietal adherences	20 (26.0)	6 (18.8)	0.57
	Pelvic adherences	17 (22.1)	8 (25)	0.93
	Deep pelvic endometriosis	1 (1.3)	3 (9.4)	0.08
	Associated peroperative surgery*
	Uterosacral ligament resection	1 (1.3)	0

Table 3 : Postoperative outcomes and pain

 3

		Conventional	Low-impact laparoscopy	P -value
		laparoscopy	N = 32 (%)
		N = 77 (%)
	Median length of stay in PACU	136.5 (115 ; 161.2)	133 (114 ; 169)	0.74
	in minutes (Q1 ; Q3)
	Length of hospitalization (days)			< 0.001
	Outpatient	1 (1.3)	29 (90.6)

Table 4 : Analysis of the factors associated with ever consuming opioids in the postoperative period (PACU, Day 0 or Day 1).

 4

	Variables	Univariable analysis
		OR (95% CI)	p -value
	Age	1.0 (0.99 -1.01)	0.76
	BMI	1.01 (0.99 -1.03)	0.20
	Comorbidities	0.94 (0.77 -1.16)	0.58
	Indication for hysterectomy	1.04 (0.98 -1.10)	0.20
	Surgical approach	1.38 (1.13 -1.69)	0.002
	Mean intraoperative
	pneumoperitoneum pressure
	6 -8	ref
	≥9	1.25 (1.03 -1.51)	0.026
	Mean intraoperative pressure (continuous)	1.043 (1.001 -1.087)	0.046
	Uterus size	1.00 (0.99 -1.00)	0.58
	Adherences	1.01 (0.83 -1.23)	0.92
	Duration of pneumoperitoneum	1.00 (0.99 -1.00)	0.44
	Ropivacaine port infiltration	1.11 (0.90 -1.37)	0.33

Table 1 :

 1 Main patient characteristics

Table 2 :

 2 Surgical factors

Table 3 :

 3 Postoperative outcomes and pain

Table 4 :

 4 Analysis of the factors associated with ever consuming opioids in the postoperative period (PACU, Day 0 or Day 1).

Acknowledgments:

We thank Felicity Nelson for her advices in language editing.

Disclosure of Interest: CT reported receiving personal fees as an occasional consultant for

Data availability statement: All relevant data are available within manuscript or its supplementary file.