Wen Tan

Liang Zhu

Tomas Mikoviny

Claus J Nielsen

Armin Wisthaler

Barbara D'anna

Simen Antonsen

Yngve Stenstrøm

Naomi Farren

Jacqueline F Hamilton

§ Liang Zhu

Barbara D´anna

Experimental and Theoretical Study on the OH-Induced Degradation of Piperazine under Simulated Atmospheric Conditions

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Piperazine (1,4-diazacyclohexane, PZ) is among the amines considered for use in largescale Carbon capture (CC) to reduce CO 2 emissions from industrial point sources. [START_REF] Wilk | Solvent selection for CO2 capture from gases with high carbon dioxide concentration[END_REF] Tests at the Technology Centre Mongstad (Norway) have established that it is difficult to avoid ppmlevel emissions of amines and their process degradation products to the environment during operation of a large-scale capture plant [START_REF] Morken | Degradation and Emission Results of Amine Plant Operations from MEA Testing at the CO2 Technology Centre Mongstad[END_REF] -the concern being the carcinogenic nitrosamines and nitramines either directly emitted or resulting in the subsequent atmospheric photooxidation of the emitted amines. [START_REF]Health effects of amines and derivatives associated with CO 2 capture, Health effects of amines and derivatives associated with CO 2 capture[END_REF] The major removal processes of PZ in the atmosphere are uptake in aqueous particles and gas phase reaction with OH radicals during day-time and NO 3 radicals during night-time. The OH rate coefficient for reaction with PZ was recently reported to be ~2.4 × 10 -10 cm 3 molecules -1 s -1 at 298 K and to favor C-H abstraction: k N-H /(k N-H + k C-H ) = 0.09 ± 0.06. [START_REF] Onel | Atmospheric Oxidation of Piperazine by OH has a Low Potential To Form Carcinogenic Compounds[END_REF] The PZ nitrosamine (PZ-NO) and nitramine (PZ-NO 2 ) are both carcinogenic; 3 they result in the following sequence of atmospheric gas phase reactions: [START_REF] Nielsen | Atmospheric chemistry and environmental impact of the use of amines in carbon capture and storage (CCS)[END_REF] + OH 

+ H 2 O (1a)  + H 2 O (1b) + O 2  +HO 2 (2) + NO  (3)
+ NO 2  (4)
Although the O 2 reaction with N-centered amino radicals, R 1 R 2 Ṅ, is reported to be around 6 orders of magnitude slower than the corresponding NO and NO 2 reactions, [START_REF] Lindley | Rate Studies of the Reactions of the (CH 3 ) 2 N Radical with O 2 , NO, and NO 2[END_REF] it is still dominating at most atmospheric conditions, and 1-nitrosopiperazine (PZ-NO) and 1nitropiperazine (PZ-NO 2 ) will only be minor products in the natural atmospheric photooxidation of PZ. Both compounds were observed, but not quantified, in previous piperazine photo-oxidation experiments in the ~200 m 3 EUPHORE atmospheric chamber, [START_REF]Atmospheric Degradation of Amines (ADA)[END_REF] and in the more recent experiments employing the ~18 m 3 CSIRO indoor smog chamber. [START_REF] White | Modelling the photooxidation of ULP, E5 and E10 in the CSIRO smog chamber[END_REF] The open literature includes two theoretical studies on the kinetics of the hydrogen abstraction from PZ by OH radicals, in which the branching between the N-H and C-H abstractions 1a and 1b was predicted to be 0.07 [START_REF] Sarma | Hydrogen atom abstraction from Piperazine by hydroxyl radical: a theoretical investigation[END_REF] and 0.01 [START_REF] Ren | Atmospheric Oxidation of Piperazine Initiated by OH: A Theoretical Kinetics Investigation[END_REF] , respectively, at 298 K. The latter theoretical study also includes an investigation of the atmospheric degradation following the initial H-abstraction.

In the present communication we report results from a new series of PZ and PZ-NO 2 photooxidation and PZ-NO photolysis experiments in the EUPHORE chamber as well as a quantum chemistry based evaluation of the primary products formed in the OH initiated photo-oxidation of PZ.

MATERIALS AND METHODS

Experimental Methods and Chemicals

A series of experiments was carried out in chamber B of the EUPHORE facility in Valencia, Spain. The facility and analytical methods have recently been reported in detail; [START_REF] Tan | Theoretical and Experimental Study on the Reaction of tert-Butylamine with OH Radicals in the Atmosphere[END_REF] additional information specific to the present work is found in the Supporting Information.

PZ (Sigma-Aldrich, ReagentPlus®, 99%), acetonitrile (Sigma-Aldrich, LiChrosolv®, ≥99.8%), pyrrole (Sigma-Aldrich, 98 %), 1,3,5-trimethylbenzene (Sigma-Aldrich, 98 %), isoprene (Sigma-Aldrich, analytical standard) and limonene (R-(+), Sigma-Aldrich, analytical standard) were used without further purification. 2-propyl nitrite (isopropyl nitrite, IPN) was synthesized from isopropanol, hydrochloric acid and sodium nitrite, and purified by repeated washing with ice water. The 1:1 nitric acid salt of PZ was prepared by adding a small excess of diluted nitric acid (HNO 3 ) to diluted PZ followed by rotary evaporation to dryness at 80 °C. Synthesis of 1-nitrosopiperazine and 1-nitropiperazine is described in the Supporting Information.

Computational Methods

Stationary points on the potential energy surfaces for the atmospheric degradation of piperazine were characterized in M06-2X [START_REF] Zhao | The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals[END_REF] calculations employing the aug-cc-pVTZ [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF][START_REF] Kendall | Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions[END_REF] basis set. Pre and post reaction complexes were located by following the intrinsic reaction coordinate [START_REF] Fukui | The path of chemical reactions -the IRC approach[END_REF][START_REF] Hratchian | Accurate reaction paths using a Hessian based predictor-corrector integrator[END_REF][START_REF] Hratchian | Theory and Applications of Computational Chemistry: The First 40 Years[END_REF][START_REF] Hratchian | Using Hessian Updating To Increase the Efficiency of a Hessian Based Predictor-Corrector Reaction Path Following Method[END_REF] from the saddle points. Reaction enthalpies and proton affinities were calculated using the G4 model chemistry. [START_REF] Curtiss | Gaussian-4 theory[END_REF] Additional dipole moments and isotropic polarizabilities, serving as input to prediction of ion-molecule reaction rate coefficients, [START_REF] Su | Parametrization of kinetic energy dependences of ion-polar molecule collision rate constants by trajectory calculations[END_REF] were obtained in B3LYP/aug-cc-pVTZ calculations, see Table S1 in the Supporting Information. All calculations were performed with Gaussian 09. [START_REF] Frisch | Revision B.01[END_REF] Master equation calculations were carried out using the program MESMER 3.0 [START_REF] Glowacki | MESMER: An Open-Source Master Equation Solver for Multi-Energy Well Reactions[END_REF] (Master Equation Solver for Multi-Energy-well Reactions) to simulate the reactions at atmospheric conditions. The required input parameters for molecules, intermediate species and products were obtained from the ab initio calculations.

RESULTS

Computational results

The kinetics of the initial step in the PZ + OH reaction is complicated by PZ existing in two conformations close in energy; a detailed theoretical account of the kinetics is outside the scope of the present work and will be presented elsewhere. 23 [START_REF] Pitts | Photooxidation of Aliphatic Amines under Simulated Atmospheric Conditions -Formation of Nitrosamines, Nitramines, Amides, and Photochemical Oxidant[END_REF][START_REF]Summary Report: Photo-oxidation of Methylamine, Dimethylamine and Trimetahylamine. Climit project no. 201604, Summary Report: Photo-oxidation of Methylamine, Dimethylamine and Trimetahylamine[END_REF] and diethylamine [START_REF]Atmospheric Degradation of Amines (ADA)[END_REF][START_REF] Pitts | Photooxidation of Aliphatic Amines under Simulated Atmospheric Conditions -Formation of Nitrosamines, Nitramines, Amides, and Photochemical Oxidant[END_REF] photo-oxidation experiments. A recent theoretical study on the OH initiated degradation of piperazine employing the G4 chemistry model did not include the atmospheric fate of the piperazine aminyl radical. [START_REF] Ren | Atmospheric Oxidation of Piperazine Initiated by OH: A Theoretical Kinetics Investigation[END_REF] The present results also differ with respect to the atmospheric reactions following C-H abstraction from PZ; we find the cyclic alkoxy radical, that ultimately follows C-H abstraction, to be metastable resulting in spontaneous ring opening, and that the major fraction of the resulting CHONHCH 2 CH 2 NHĊH 2 radical will end up as a diamide. The branching between ring-opening and formation of 1,2,3,6-tetrahydropyrazine (PZ-Im) depends on the NO mixing ratio; master equation calculations employing 5 ppb of NO results in a 15 : 85 branching, whereas a 35 : 65 branching is obtained with 50 ppb of NO.

Scheme 1. Quantum chemistry prediction of the major primary products in the atmospheric photo-oxidation of piperazine. # The branching between ring-opening and formation of 1,2,3,6-tetrahydropyrazine (PZ-Im) depends on the NO mixing ratio.

The atmospheric fate of PZ-NO 2 was also investigated, and the major photo-oxidation routes for PZ-NO 2 , outlined in Scheme 2, parallel those of PZ (Scheme 1) with one exception the alkyl-radical formed upon ring-opening ejects NO 2 forming an imine. Details of the quantum chemistry study of the OH radical initiated atmospheric PZ-NO 2 photo-oxidation are collected in the Supporting Information.

Scheme 2. Quantum chemistry prediction of the major primary products in the atmospheric photo-oxidation of 1-nitropiperazine (PZ-NO 2 ).

Experimental results

We first report results from new kinetic studies of the PZ + OH reaction. We then present results from PZ-NO photolysis and from PZ-NO 2 photo-oxidation experiments facilitating interpretation of the PZ photo-oxidation experiments. Finally, we present results from studies of the aerosol formed in the PZ photo-oxidation experiments.

Piperazine + OH reaction kinetics

Two relative rate experiments were carried out in the EUPHORE chamber B in which isoprene, limonene, 1,3,5-trimethylbenzene and pyrrole were used as reference compounds.

Acetonitrile was added as inert tracer to monitor the apparent dilution by purified air that is constantly added to compensate for leakage and continuous sampling by the air monitors (k OH+CH 3 CN = 2.2 × 10 -14 cm 3 molecule -1 s -1 at 298 K). [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] Figure 1 displays the time evolution of compound-specific PTR-ToF-MS ion signals measured during the second experiment (the first experiment is documented in Figure S6, Supporting Information). The dilution rate due to air replenishment was 8.6 × 10 -6 s -1 in the two experiments; PZ wall loss rates (derived from the reagent decay prior to adding the OH precursor IPN) ranged from 1 to 4 × 10 -5 s -1 .

Initial mixing ratios were 100 ppb for the reference compounds and 200 ppb for PZ.

Average OH densities in the EUPHORE chamber during the periods selected for analyses s -1 at 307 ± 2 K and 1014 ± 2 hPa. Significant amounts of PZ are, however, transferred from the gas to the particle phase during the periods selected for analysis. Figures S9 andS10 (Supporting Information) show the time evolutions of aerosol mass and the PZ content, and approximately 6.3 and 1.2 % of PZ were actually lost to the aerosol during the two kinetic experiments. Correction for PZ loss to particles during the kinetic experiments was therefore (T/298) -(1.76±0.08) and k = (2.25  0.28) × 10 -10 cm 3 molecule -1 s -1 at 307 K from flash photolysis/resonance fluorescence experiments.

1-Nitropiperazine photo-oxidation studies

The atmospheric fate of PZ-NO 2 was investigated in two photo-oxidation experimentsone under high NO 2 levels, the other under high NO levels. Figure S17

1-Nitrosopiperazine photolysis studies

Nitrosamines have a characteristic n  * transition in the UV-A region and photolyse rapidly in natural sunlight; the quantum yield to photo-dissociation of (CH 3 ) 2 NNO following S 0  S 1 (n*) excitation at 363.5 nm was reported to be 1.03  0.10, [START_REF] Geiger | Photodissociation of dimethylnitrosamine[END_REF][START_REF] Geiger | Photolysis of Dimethylnitrosamine in the Gas-Phase[END_REF] and theory shows that the excited S 1 state is repulsive leading to swift dissociation following excitation. [START_REF] Pelaez | A complete active space self-consistent field study of the photochemistry of nitrosamine[END_REF] Previous experiments at EUPHORE show that nitroso-dimethylamine, -diethylamine, -methylethylamine, -methylpropylamine, -methylbutyamine, -morpholine, -piperazine,piperidine and -pyrazine all photolyze equally fast with j Nitrosamine /j NO 2 = 0.34 ± 0.04. [START_REF]Atmospheric Degradation of Amines (ADA)[END_REF]30 Three photolysis experiments were carried out in the EUPHORE chamber B under conditions that did not allow quantification of the solar radiation. Cyclohexane was added to the chamber (~2 ppm) to enable an estimate of the amount of OH radicals formed following PZ-NO photolysis; the derived OH radical mixing ratio varied between 1 and 4  10 5 cm -3 (for details, see the Supporting Information).

Piperazine photo-oxidation studies

Previous PZ photo-oxidation experiments in the EUPHORE chamber were severely affected by wall adsorption/desorption and particle formation, [START_REF]Atmospheric Degradation of Amines (ADA)[END_REF] and a new series of piperazine photo-oxidation experiments was carried out under different, warmer conditions reducing wall adsorption (see Table S8 in the Supporting Information). Piperazine, dissolved in water, was injected into the chamber in a stream of N 2 . The detected ion signals can be separated according to their time evolution: (1) signals that appear upon injection of piperazine along with that of protonated piperazine, (2) signals that grow and decrease again S9. Other major ions in the spectrum include m/z 83.059, m/z 74.023 and m/z 46.029. We assigned the peak at m/z 83.059 to C 4 H 7 N 2 + whose profile as well as that of m/z 81.045 highly correlates with the one of the imine (m/z 85.078). This is consistent with the observation in aforementioned Nitrosopiperazine photolysis section. The peak at m/z 74.023 was assigned to N-formylformamide (CHONHCHO), a product from the reaction initiated by hydrogenabstraction from one of the -CH 2groups in piperazine (Scheme 2). The yield of m/z 74.023 was estimated to be 5.1%. The signal at m/z 46.029 is identified as formamide (NH 2 CHO).

Piperazine is expected to condense with formaldehyde, forming C 5 H 11 N 2 + at m/z 99.092.

, whereas S18 and Table S8.

The present piperazine experiments are hampered by wall adsorption/desorption and particle formation, and the data are insufficient to allow and unambiguously determine the neutral molecules corresponding to the detected ion signals. However, none of the molecular formulae can be related to alkyl nitramines.

Piperazine + OH particle analysis

The aerosol formed in the PZ+OH photo-oxidation was simultaneously detected by three instruments: (1) a PTR-ToF 8000 instrument coupled with a CHemical Analysis of aeRosol ON-line (CHARON) inlet; [START_REF] Eichler | A novel inlet system for online chemical analysis of semi-volatile submicron particulate matter[END_REF] (2) a compact time-of-flight Aerosol Mass Spectrometer (C-ToF-AMS); and (3) a Scanning Mobility Particle Sizer (SMPS). A detailed description of experimental and instrumental settings is available elsewhere [START_REF] Tan | Theoretical and Experimental Study on the Reaction of tert-Butylamine with OH Radicals in the Atmosphere[END_REF] . A relatively good agreement was achieved amongst the results obtained by these three instruments (Figure S19). The aerosol formation proceeded fast and heavily (>10 5 #/cm 3 during all the experiments), as depicted in Figure S20 The C-ToF-AMS measurements indicated that aminium salt is the main consititute in particle phase. In addition to the PZ-nitrate salt, there exists a large fraction of organic species in the aerosol phase. This is in consistence with C-ToF-AMS observations.

To better understand the aerosol composition, we scrutinized the molecular information by The PZ portioning in aerosol varied from 0.2% (slow IPN, no NOx) to 6% (quick IPN, 90 ppbv NO 2 ) in PZ+OH photooxidation experiment (see Table S9). Under the assumption that all organic carbon speices in principle should originate from piperazine, this would further increase the portioning of PZ into aerosols to 15.5%. This is justified due to the expected heterogeneous reactions involving primary amines, imines, aldehydes, ketones and carboxylic acids happening on aerosol surfaces.

CONCLUSIONS

H-law

ATMOSPHERIC IMPLICATIONS

We should mention that there is no experimental value for the corresponding NO 3 rate coefficient, but the empirical correlation between OH and NO 3 rate coefficients for reaction with amines suggests a PZ + NO 3 rate coefficient around 4.6 × 10 -11 cm 3 molecules -1 s -1 at 298 K. [START_REF] Nielsen | Atmospheric chemistry and environmental impact of the use of amines in carbon capture and storage (CCS)[END_REF] Other



( 9 :

 9 10 -9:30 and 14:10 -14:35 UTC) were around 3  10 6 cm -3 ; average pressure and temperature in the two experiments were 1014 ± 2 mbar and 307 ± 2 K. The temporal profile of PZ recorded by PTR-ToF-MS matches well the one obtained by a home-built high temperature PTR-MS, indicating an adequate rinstrument esponse time (Figure S7 in the Supporting Information).

Figure 1 .

 1 Figure 1. (A): Time evolution of the acetonitrile, pyrrole, piperazine and limonene ion signals at m/z 42.034, 68.050, 87.092 and 137.133, respectively, during the second kinetic experiment on 2016.07.28. (B):Relative rate plot showing the decays of isoprene, limonene, pyrrole and piperazine at 1014 hPa and 307 K in the presence OH radicals. For the sake of clarity the data have been displaced along the abscissa. The data have been corrected for dilution due to chamber air replenishment, for wall loss and for loss to the aerosol, see Supporting Information.

  Two primary products are expected following nitrosamine photolysis: the imine (PZ-Im, 1,2,3,6-tetrahydropyrazine) and the nitramine PZ-NO 2 , Scheme 1. Attempts to synthesise PZ-Im were furtile, and fragmentation of protonated PZ-NO and PZ-NO 2 complicates the unambiguous identification of PZ-Im by PTR-TOF-MS. Inspection of the ion signals observed in the PZ-NO photolysis experiments during the time period before opening the chamber canopy (see Figure S11) reveals the [PZ-NO]H + fragmentation; the fragmentation of [PZ-NO 2 ]H + was determined in separate experiments with the same instrument parameters, see later. In summary, the ion signals at m/z 44.052, 85.080 and 86.085 all have an origin in both PZ-NO and PZ-NO

Figure 2 .

 2 Figure 2. Top: Fragmentation-corrected normalized ion signals during the 1-nitrosopiperazine photolysis experiment on 2016.09.20. Ion signals increasing by less than 2% of the m/z 116.080 decrease have been omitted for the sake of clarity. Bottom: Derived volume mixing ratios (ppbV) of 1-nitroso-, 1-nitropiperazine and 1,2,3,6-tetrahydropyrazine during the experiment.

  Volume mixing ratio /ppbV 2016.09.20 /hh:mm Open chamber canopy during the photo-oxidation experiment (intermediate products), and (3) signals that grow steadily after opening the chamber canopy. Figure 3 examplifies the observed time evolution of PZ and the major products during a photo-oxidation experiment; the ion signals observed in the new as well as in the previous experiments are collected in Table

Figure 3 .

 3 Figure 3. Time evolution of ion signals during the piperazine photo-oxidation experiment on 2016.07.25. Ion signals increasing by less than 2% of the m/z 87.090 decrease have been omitted for the sake of clarity.

Figure 4 .

 4 Figure 4. Time evolution of ion signals during the piperazine photo-oxidation experiment on 2016.07.25. Derived volume mixing ratios (ppbV) of piperazine and xxxxx during the experiment. Product yields, calculated after the 30 mins of reaction, are shown in parentheses.

Figure 4 .

 4 Figure 4. A typical spectrum of particle composition obtained at the highest aerosol loading of piperazine photooxidation under high NOx initial condition, the mass concentration of each m/z is calculated by CHARON-PTR-ToF-MS

  implemented in the data analysis (Supporting Information) resulting in an average k OH+Piperazine = (2.8  0.6)  10 -10 cm 3 molecule -1 s -1 at 307 ± 2 K and 1014 ± 2 hPa. The present result agrees well with those of Onel et al.,[START_REF] Onel | Atmospheric Oxidation of Piperazine by OH has a Low Potential To Form Carcinogenic Compounds[END_REF] who reported k(T) = (2.37 ± 0.03) × 10 -10

		8k		Open Canopy	IPN added		Close Canopy	×t	0,9
	Normalized counts /s -1	1k 2k 3k 4k 5k 6k 7k	(A)		to Chamber		Pyrrole Acetonitrile Limonene Piperazine	ln{[Reference] 0 /[Reference] t } -k Wall+Dilution	0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8	1,3,5-Trimethylbenzene Pyrrole Isoprene Reference compounds: Limonene (B)
		0							0,0
		13:00	13:30	14:00	14:30	15:00			0,0	0,2	0,4	0,6	0,8	1,0	1,2
				2016.07.28 /hh:mm				ln{[Piperazine] 0 /[Piperazine] t } -k	Wall+Dilution ×t -aerosol(t)

  in the Supporting Information shows time profiles of the relevant PTR-ToF-MS ion signals observed during a photo-oxidation experiment demonstrating that protonated PZ-NO 2 fragments severely at the instrumental settings employed: 15% m/z 132.076 (protonated molecule), 38% m/z 86.085 (NO 2 ejection), 30% m/z 85.080 (HNO 2 ejection), 4% m/z 57.059 (ring fragment), and 13% m/z 44.052 (ring fragment). The observed ion signals related to PZ-NO 2 photo-oxidation are summarized in Table S7 in the Supporting Information. 84.069 (NO 2 ejection), 83.061 (HNO 2 ejection), 55.042 (CH 2 CH 2 NHCH + ), and 42.034 (CH 2 CH 2 N + ). Of these, the m/z 83.061, 55.042 and 42.034 are observed.

PZ-NO 2 is a very "sticky" compound that transfers relatively fast to particles and wall surfaces (Figure

S17

) making quantitative conclusions arduous. In line with the extensive fragmentation of protonated PZ-NO 2 , most of the ion signals observed during the photooxidation corresponds to fragments. It is, however, clear that 1,2-dinitropiperazine (m/z 177.055 ([PZ-(NO 2 ) 2 ,]H + ) and fragment ions) is a major product, and that also 1-nitro-4nitrosopiperazine (m/z 161.064 ([NO 2 -PZ-NO]H + ) and fragment ions) is formed in the experiments. Surprisingly, the expected key product, 1-nitro-1,2,3,6-tetrahydropyrazine (NO 2 -PZ-Im), is not detected as the protonated ion at m/z 130.061. Assuming a similar fragmentation of protonated PZ-Im as observed for protonated PZ-NO 2 , fragment ions are expected at m/z

Table 1 .

 1 Taking k PZ-NO+OH  k PZ+OH as a conservative estimate of the PZ-NO reactivity towards OH places an upper level of 2% PZ-NO removal by OH radicals during the photolysis. Table 1 summarises the observed PTR-TOF-MS ion signals relevant to the PZ-NO photolysis experiments; a more complete list of ion signals observed in the present experiments is found in Table S6 in the Supporting Information, which also includes data obtained in our previous studies. Selected ion signals observed during 1-nitrosopiperazine photolysis experiments.

	m/z	Ion sum formula Interpretation
	44.052 83.061 85.080 86.085 116.079 132.074 [PZ-NO]H + fragmentation: m/z 116.079 (68% protonated molecule); m/z 86.085 (11% NO C 2 H 6 N + Fragment of [PZ-NO]H + , [PZ-NO 2 ]H + and [PZ-Im]H + C 4 H 7 N 2 + H 2 elimination from [PZ-Im]H + C 4 H 9 N 2 + Fragment of [PZ-Im]H + , [PZ-NO]H + , [PZ-NO 2 ]H + C 4 H 10 N 2 + Fragment of [PZ-NO]H + , [PZ-NO 2 ]H + [PZ-NO]H + C 4 H 10 N 3 O + C 4 H 10 N 3 O 2 + [PZ-NO 2 ]H +
	ejection); m/z 85.080 (17% HNO ejection); m/z 44.052 (4% ring scission). [PZ-NO 2 ]H +
	fragmentation: m/z 132.076 (17% protonated molecule); m/z 86.085 (40% NO 2 ejection); m/z
	85.080 (32% HNO 2 ejection); m/z 44.052 (11% ring scission).

[START_REF]Atmospheric Degradation of Amines (ADA)[END_REF]

Figure

S15

in the Supporting Information illustrates the raw ion signal time profiles; the ion signals growing in upon photolysis fall in 2 categories: (1) the m/z 44.052, 83.061, 85.080, 86.085 and 132.074 having time profiles typical of primary products, and (2) the less intense m/z 46.030, 74.024, 81.045 and 97.045 with time profiles more resembling those of secondary products.

  2 , Table 1. That is, the expected ion signal of protonated PZ-Im at m/z 85.080 (C 4 H 9 N 2 + ) is also a major fragment of both protonated PZ-NO and PZ-NO 2 . Figure 2 also includes the derived volume mixing ratios of PZ-NO, PZ-NO 2 and PZ-Im. The mass balance in the photolysis experiments is only around 60% and the discrepancy is primarily due to "missing" PZ-Im, which in part can be explained by partitioning to surfaces and subsequent hydrolysis and/or condensation reactions. The corresponding CHARON PTR-TOF-MS results (see later) suggests a minor presence of PZ-imines in co-existing aerosols.

	The PZ-NO and PZ-NO 2 fragmentation corrected ion signals are presented in Figure 2,
	from which it can be seen that there remains significant ion intensities at m/z 44.052 and
	85.080, whereas m/z 86.085 is fully accounted for by PZ-NO and PZ-NO 2 . The m/z 44.052
	(C 2 H 6 N + ) is obviously also a ring scission fragment of [PZ-Im]H + and m/z 83.061 (C 4 H 7 N 2	+ )
	is explained by H 2 -loss from [PZ-Im]H + .

Table 2

 2

	summarises the major PTR-TOF-MS ion signals observed. The [M-H 2 ] + fragment
	(4.1%) of piperazine upon protonation was taken into consideration

Table 2 .

 2 Major PTR-TOF-MS ion signals observed during OH initiated piperazine photooxidation experiments.

	m/z	Ion sum formula	Interpretation
	43.0551 46.0301 59.0490 69.0447 74.0243 81.045 83.0604 85.0756 87.0900 99.0561 99.0923 115.0862 116.0787 119.0855 132.0750	+ C 3 H 7 CH 4 NO + C 3 H 7 O + + C 3 H 5 N 2 C 2 H 4 NO 2 + C 4 H 5 N 2 C 4 H 7 N 2 + C 4 H 9 N 2 + + C 4 H 11 N 2 C 4 H 7 N 2 O + + C 5 H 11 N 2 + C 5 H 11 N 2 O + C 4 H 10 N 3 O + + C 4 H 11 N 2 O 2 C 4 H 10 N 3 O 2 +	Fragment of organic species [NH 2 CHO]H + [CHONHCH 2 OH]H + -NO 2 Imidazole ? [CHONHCHO]H + [Pyrazine]H + PZ, H 2 elimination from [PZ-Im]H + From piperazine a and PZ-Im, Fragment of [PZ-Im]H + , [PZ-NO]H + , [PZ-NO 2 ]H + PZ ? PZ + CH 2 O condensation product PZ-Im + CH 2 O condensation product PZ-NO ? PZ-NO 2
	The predicted major products originated from one of the -NH-groups, PZ-NO 2 (m/z
	132.077), PZ-NO (m/z 116.084) and the imine, PZ-Im, (m/z 85.078) showed lower presence
	in spectrum. The former two fragments extensively upon protonation, and the co-existing
	photolysis caused further signal drop. Upon calibration, the PZ-NO and PZ-NO 2 yield was
	estimated to be 1.1% and 8.4% in this experiment. It should be note that the numbers present
	lower limits due to the underestimated wall losses.
	The ion pair m/z 85.078 and 83.059 partially rationalized the low signal of protonated imine
	signal at m/z 85.078. Taking all factors into consideration, the yield of imine and its
	derivatives are approximately 17.2%.
	One uncertainty factor in data analysis is the peak at m/z 43.055 (C 3 H 7	+ ), potentially a
	fragment of higher order of organic species. We decided to leave it out of yield estimation due

  as the main component in aerosols, which accounts for 52.6% in total aerosol mass loading. The other major peaks are m/z 72.042 (C 3 H 6 NO + ), m/z 99.078 (C 5 H 11 N 2 + , PZ-formaldehyde condensation product), m/z 101.071 (C 4 H 9 N 2 O + ), m/z 115.082 (C 5 H 11 N 2 O + ), m/z 117.067 (C 4 H 9 N 2 O 2 + , CHONHCH 2 CH 2 NHCHO), and m/z 142.092 (tentatively assigned to PZ•(H 2 O) 2 H 3 O + ). In addition, nitramine were detected in particles, m/z 132.076 and m/z 177.059, assigned to PZ-NO 2 and di-PZ-NO 2 , respectively. It is worth to note that the PZ-NO 2 and di-PZ-NO 2 account for 1.7% and 0.88% in total aerosol mass, after fragmentation corrections.

	CAHRON-PTR-ToF measurement of the piperazine photo-oxidation experiments. The
	experiments carried out on 26 July 2016, with the highest aerosol yield, was taken as an
	example. Figure 4 illustrates the quantified mass spectrum of particle-phase products. The
	most abundant peaks are at m/z 87.092 (piperazine) and m/z 45.993 (NO 2	+ ), suggesting
	piperazine nitrate salt

  Associated content, S Supporting InformationThe Supported Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs/jpca.xxxxxxxx.

	 Author information
	Corresponding Author
	ORCID , iD
	Armin Wisthaler: 0000-0001-5050-3018
	Claus Nielsen: 0000-0002-2962-2634
	Notes

*E-mail: armin.wisthaler@kjemi.uio.no. Phone: +47-22859139. *E-mail: claus.nielsen@kjemi.uio.no. Phone: +47 22855680.

The authors declare no competing financial interest  Acknowledgments (Word Style "TD_Acknowledgments"). This work is part of the Atmospheric Chemistry of Amines project (ACA) supported by the CLIMIT program under contract 244055 and has received additional support from the Research Council of Norway through its Centres of Excellence scheme, project number 262695.


Arthur Hagopian

Marie-Liesse Doublet

Jean-Sébastien Filhol

Tobias Binninger
email: tobias.binninger.science@gmx.de

Advancement of the Homogeneous Background Method for the Computational Simulation of Electrochemical Interfaces

Computational studies of electrochemical interfaces based on density-functional theory (DFT) play an increasingly important role in present research on electrochemical processes for energy conversion and storage. The homogeneous background method (HBM) offers a straightforward approach to charge the electrochemical system within DFT simulations, but it typically requires the specification of the "active" fraction of excess electrons based on a certain choice of the electrode-electrolyte boundary location, which can be difficult in presence of electrode-surface adsorbates or explicit solvent molecules. In this work, we present a methodological advancement of the HBM, both facilitating and extending its applicability. The advanced version neither requires energy corrections nor the specification of the "active" fraction of excess electrons, providing a versatile and readily available method for the simulation of charged interfaces also when adsorbates or explicit solvent molecules are present. Our computational DFT 1

by a general joint density-functional theory (JDFT) [START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF][START_REF] Petrosyan | Joint Density-Functional Theory: Ab Initio Study of Cr 2 O 3 Surface Chemistry in Solution[END_REF] that provides a combined DFT description of the electron density together with the ionic and dielectric densities of the electrolyte. It has been noted, however, that the level of complexity of the ionic countercharge model had rather little influence on the numerically obtained value of the interface capacitance, which has been attributed to the dominating role of the underlying dielectric solvent model and the corresponding definition of the solvent boundary [START_REF] Melander | Grand-canonical approach to density functional theory of electrocatalytic systems: Thermodynamics of solid-liquid interfaces at constant ion and electrode potentials[END_REF] .

To model a charged interface within DFT-based simulations, the electron number of the simulation cell, which contains the electrode slab embedded in solvent, is changed with respect to the electron number of the neutral electrode slab. In order to preserve overall charge neutrality, a counter-charge distribution must be included that compensates the electronic excess charge and mimics the ionic charge of the electrolyte in the electrochemical double-layer. The homogeneous background method (HBM) [START_REF] Filhol | Elucidation of the Electrochemical Activation of Water over Pd by First Principles[END_REF][START_REF] Mamatkulov | An abinitio study of electrochemical vs. electromechanical properties: the case of CO adsorbed on a Pt(111) surface[END_REF][START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] utilizes a compensating homogeneous background charge, ρ bg = const, across the entire simulation cell, which generally is a "built-in" functionality of periodic DFT-codes. The uniform counter-charge distribution must be regarded as rather unphysical from the perspective of an electrochemical interface, in particular because it extends across the bulk of the electrode slab. This aspect complicates the application of the HBM for two reasons: First, an a posteriori energy correction is typically required, and second, the fraction of "active" excess electrons at the electrode surface must be estimated based on a certain choice of the electrode-electrolyte boundary location [START_REF] Filhol | Elucidation of the Electrochemical Activation of Water over Pd by First Principles[END_REF][START_REF] Mamatkulov | An abinitio study of electrochemical vs. electromechanical properties: the case of CO adsorbed on a Pt(111) surface[END_REF][START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] . The latter aspect, in particular, makes the use of the HBM difficult in cases where the definition of the electrode-electrolyte boundary becomes ambiguous, e.g. due to the presence of surface adsorbate species or explicit solvent molecules that become, to a certain extent, "metallized" [START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] and therefore cannot be uniquely ascribed to either the electrode or the electrolyte side. In the present work, we resolve this shortcoming by developing an advanced version of the HBM that can be straightforwardly applied without the typically employed correction schemes and without requiring the knowledge of the "active" fraction of excess electrons, thus significantly extending the applicability of the HBM.

As a computational validation, we compare the advanced HBM with the linearized Poisson-Boltzmann model (PBM) [START_REF] Mathew | Implicit self-consistent electrolyte model in plane-wave density-functional theory[END_REF] , both in combination with a PCM for the implicit solvent as implemented in the VASPsol package [START_REF] Mathew | Implicit solvation model for density-functional study of nanocrystal surfaces and reaction pathways[END_REF] . The PBM employs a physically inspired counter-charge distribution based on the Gouy-Chapman theory for the electrochemical double-layer [START_REF] Schmickler | Interfacial Electrochemistry[END_REF] , where the ion densities n i (r) = n i,0 exp (-z i eφ(r)/k B T ) in the electrolyte are determined by the local electrostatic potential φ(r) via Boltzmann statistics. In the linearized version, an ionic charge density ρ ion (r) = -0 r κ 2 (φ(r) -φ elyte ) is obtained, where r is the relative permittivity of the electrolyte, κ = 1/λ D is the inverse Debye length, and φ elyte is the reference electrostatic potential in the bulk electrolyte. Combined with the dielectric charge density of the linear PCM solvent, this leads to the linearized Poisson-Boltzmann equation ∇[ r ∇φ]r κ 2 (φ -φ elyte ) = -(ρ ext + ρ e )/ 0 , the solution of which produces an ionic counter-charge distribution ρ ion (r) with an essentially exponential decay, as exp(-z/λ D ), from the electrolyte boundary towards the bulk of the electrolyte. However, it should be noted that, strictly speaking, the mean-field thermodynamics underlying the PBM do not apply to the description of the atomistic interactions at an electrochemical interface [START_REF] Gauthier | Challenges in Modeling Electrochemical Reaction Energetics with Polarizable Continuum Models[END_REF] . In spite of their very different counter-charge distributions, we demonstrate that both the HBM and PBM yield essentially equal results for the free energy and capacitance in DFT-based simulations of charged electrochemical interfaces. We rationalize this finding by an electrostatic model, highlighting the central role of the gap region [START_REF] Hörmann | Grand canonical simulations of electrochemical interfaces in implicit solvation models[END_REF][START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF][START_REF] Lespes | Using Implicit Solvent in Ab Initio Electrochemical Modeling: Investigating Li+/Li Electrochemistry at a Li/Solvent Interface[END_REF] between the electrode surface and the boundary of the implicit solvent region in determining the interface capacitance for both models.

Theory and Methods

Grand potential, electrode potential, and capacitance

We begin with a brief review of the general thermodynamic equations for an electrochemical interface [START_REF] Melander | Grand-canonical approach to density functional theory of electrocatalytic systems: Thermodynamics of solid-liquid interfaces at constant ion and electrode potentials[END_REF][START_REF] Schmickler | Interfacial Electrochemistry[END_REF] , which hold, in particular, for the PBM method. Their validity for the HBM, however, must be carefully assessed as discussed in the subsequent section. Electrochemical processes involve the exchange of electrons and ions between the interface region and the bulk of the electrode and electrolyte, respectively. The interface therefore corresponds to an open system coupled to an electron reservoir, the electrode, and an ion reservoir, the electrolyte. The corresponding grand canonical ensemble is characterized by a certain electron chemical potential µ e and ion chemical potentials µ i for each of the ion species present in the electrolyte. [START_REF] Adib | E-Mobility -Advancements and Challenges[END_REF] The equilibrium state of the electrochemical interface corresponds to the minimum of the grand potential

Ω = A -µ e N e - i µ i N i ( 1
)
where A is the Helmholtz free energy, and N e and N i are the excess particle numbers of electrons and ions, respectively, which quantify the difference with respect to the uncharged interface.

An aspect of central importance is the overall charge neutrality. The electronic excess charge at the electrode surface must, on average, be balanced by the ionic excess charge of the electrolyte to avoid energy divergence. Therefore, the electron and ion numbers N e and N i cannot vary independently, but they must fulfill the charge neutrality condition N e = i z i N i with the charge numbers z i of the ion species. Splitting the ion chemical potentials as µ i = µ 0 i + ez i φ elyte , where φ elyte is the inner electrostatic potential in the bulk electrolyte, we obtain the grand potential in the form

Ω = A -(µ e + e φ elyte )N e - i µ 0 i N i (2)
where we used the charge neutrality condition.

The electrode potential Φ can be defined as a "work function" in the electrolyte environ-ment [START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF] , e Φ = (-e)φ elyte -µ e

where the electron chemical potential µ e is referenced to the electrostatic potential energy (-e) φ elyte in the bulk electrolyte. The (differential) interface capacitance

1 C = ∂Φ ∂q T,V (4)
relates changes in the interfacial charge q to changes in the electrode potential Φ. Since we consider fixed temperature T and volume V throughout the following, we omit the indication of the T, V -subscript. Due to the charge neutrality requirement discussed above, the electron and ion numbers cannot be varied independently. We therefore consider variations in electron number N e , and thus interfacial charge q, under fixed ion chemical potentials µ i . In this partial grand canonical setting [START_REF] Binninger | Piecewise nonlinearity and capacitance in the joint density functional theory of extended interfaces[END_REF] , a variation in N e implicitly includes a compensating variation in the ion numbers N i to maintain charge neutrality. The fixed ion chemical potentials µ i = µ 0 i + ez i φ elyte also fix the electrostatic potential φ elyte in the electrolyte. Therefore, with the electrode potential given in Eq. ( 3), the interface capacitance of Eq. ( 4) fulfills

e 2 C = ∂µ e ∂N e {µ i } (5)
where we used the relation q = -eN e for the interfacial charge, which holds in the PBM, but not in the HBM, as discussed below. From now on, we consider all partial derivatives under constant ion chemical potentials {µ i }, and, for simplicity, we omit the indication of the {µ i }-subscript. From the well-known relation

∂Ω ∂µ e = -N e (6)
we find

∂ 2 Ω ∂µ 2 e = - ∂N e ∂µ e = - C e 2 (7)
where we used Eq. ( 5) and (∂N e /∂µ e ) = (∂µ e /∂N e ) -1 . Due to the fixed electrolyte potential φ elyte , we have ∂/∂Φ = (-e)∂/∂µ e for the derivative with respect to the electrode potential Φ according to Eq. (3). Consequently,

∂ 2 Ω ∂Φ 2 = -C (8)
which is the well-known relation that the curvature of the grand potential as a function of the electrode potential is equal to the negative of the capacitance.

Grand potential and capacitance in HBM

The original HBM [START_REF] Filhol | Elucidation of the Electrochemical Activation of Water over Pd by First Principles[END_REF] adopts an electronic perspective on the definition of electrode potential and grand potential. Accordingly, a number of corrections are typically applied to correct for the unphysical influence of the homogeneous background charge [START_REF] Filhol | Elucidation of the Electrochemical Activation of Water over Pd by First Principles[END_REF][START_REF] Mamatkulov | An abinitio study of electrochemical vs. electromechanical properties: the case of CO adsorbed on a Pt(111) surface[END_REF][START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] . Motivated by the general form of the grand potential (1), we explore the consequences of a novel perspective, including the homogeneous background charge in the definition of the grand potential within the HBM. In this way, we regard the homogeneous background charge not only as a numerical tool for avoiding the energy divergence of infinite charged systems, but we consider it as the simplest possible model for a physical compensating counter-charge distribution.

Since the homogeneous background extends across both the electrolyte region and the electrode slab, the HBM essentially combines two different models: First, the homogeneous background charge in the electrolyte region models the ionic counter charge of the electrochemical double-layer. And second, the homogeneous background charge in the electrode slab region can be considered as an entirely delocalized model for a dopant of the electrode material. Therefore, charging the electrode-electrolyte system with the HBM consists of both charging the electrochemical double-layer and changing the dopant concentration of the electrode at the same time. We will see below that, in fact, the latter dopant contribution to the grand potential is negligible for the case of metal electrodes, making the HBM an easy-to-use method for such systems.

The chemical potential of the HBM counter-charge (i.e the uniform background) is given by its electrostatic interaction with the rest of the system as µ bg = e φ with the mean electrostatic potential φ of the simulation cell [START_REF] Mamatkulov | An abinitio study of electrochemical vs. electromechanical properties: the case of CO adsorbed on a Pt(111) surface[END_REF][START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] , and the corresponding "background particle number" N bg = q bg /e is simply derived from the background charge q bg . Including this contribution in the Legendre transformation, analogous to Eq. ( 1), the HBM grand potential reads

Ω = A -µ e N e -µ bg N bg = A -(µ e + e φ)N e (9)
where we used the charge neutrality condition N bg = N e (note that N e only refers to the excess electron number with respect to the uncharged electrode). Because the global system is neutral, this expression does not depend on the choice of a certain potential reference.

The definition of the electrode potential Φ according to Eq. (3) remains valid also within the HBM. The definition of the interface capacitance within HBM, however, is less straightforward. In principle, we still assume that we can define a "true" interface capacitance according to Eq. ( 4). However, the interfacial charge q appearing in this expression is not anymore equal to -eN e with the total excess electron number N e of the simulation cell.

Because the homogeneous background charge extends across the bulk region of the metal electrode, a certain part N b of the excess electrons will screen the background charge therein and therefore not contribute to the electrode surface charging. Only the remaining part N s = N e -N b of the excess electrons results in a charging of the electrode-electrolyte interface. The interfacial charge q = -eN s in the definition of the interface capacitance according to Eq. ( 4) is therefore not clearly defined within the HBM a priori.

Nevertheless, we can still define a capacitance analogous to Eq. ( 4) simply using the charge q e = -eN e of the total number of excess electrons,

1 C ΦN = - 1 e ∂Φ ∂N e (10)
where we have to keep in mind that this capacitance will not be equal to the "true" interface capacitance, as discussed above. Also, in analogy to Eq. ( 8), we define the capacitance

C ΩΦ = - ∂ 2 Ω ∂Φ 2 (11)
where Ω refers to the HBM grand potential of Eq. ( 9).

We saw in the previous section that the capacitance definitions according to Eqs. (10) and ( 11) are precisely equivalent in the PBM and simply equal to the interface capacitance of Eq. ( 4). In the HBM, however, C ΦN and C ΩΦ correspond to different quantities. As discussed above, C ΦN is expected to be different from the "true" HBM interface capacitance as it is including the contribution from the excess electrons in the electrode bulk region. The central question arises whether at least C ΩΦ can provide a representation of the latter, to be addressed in detail in the following.

Computational Details

Calculations were performed within the density-functional theory (DFT) framework, using the Vienna ab initio simulation package (VASP) [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF] . The electronic wave functions were expanded in a plane-wave basis set with a kinetic energy cutoff up to 450 eV for Li and Computing the grand potentials. We highlight a subtle point regarding the PBM implementation in the VASPsol package [START_REF] Mathew | Implicit self-consistent electrolyte model in plane-wave density-functional theory[END_REF] , which we used in the present study. As the VASPsol authors emphasize [START_REF] Mathew | Implicit self-consistent electrolyte model in plane-wave density-functional theory[END_REF] , an energy contribution ∆E = e φ elyte N e must be added to the free energy A VASP printed by the VASP core package in order to obtain the total free energy A = A VASP + ∆E of the system. Furthermore, in the Kohn-Sham-Mermin DFT approach, the electron chemical potential µ e = E F is equal to the Fermi energy of the Fermi-Dirac distribution on the Kohn-Sham eigenvalues, which is directly printed by the VASP core package. Finally, the chemical part µ 0 i of the ion chemical potentials is simply set to zero. Therefore, the PBM grand potential according to Eq. ( 2) is obtained from the VASPsol implementation as

Ω PBM = A VASP + e φ elyte N e -(µ e + e φ elyte )N e = A VASP -E F N e (12)
To compute the HBM grand potential according to Eq. ( 9), we note that the mean electrostatic potential is set to zero within VASP, φ = 0 , so

Ω HBM = A VASP -E F N e (13)
and we obtain the same form as in Eq. ( 12). We provide our processing scripts for the PBM and HBM calculations with VASPsol in an online repository 38 . 111) and (c) Li(100) electrodes in contact with an implicit solvent with b = 78.4 for Pt and Au and b = 89.9 for Li, corresponding to water and ethylene carbonate, respectively. Grey regions highlight the potential ranges (PZC ± 0.25 V) on which the fitted values for the capacitances in Table 1 have been computed.

Results and Discussion

despite the "unphysical" extension of the homogeneous background charge across the bulk of the electrode slab. This is a remarkable finding, because the definition of the HBM grand potential according to Eq. ( 9) neither involves any homogeneous background corrections, nor any specification of the actual surface electron number. Table 1 presents the fitted values of the capacitances defined in Eqs. (10) and (11) for HBM and PBM at the potential of zero charge (PZC), indicated by the "0" superscript. As discussed in the Theory and Methods section above, the capacitance definitions are precisely equivalent in the PBM, which is confirmed by their fitted values for all systems. In the HBM, however, they correspond to different quantities, resulting in entirely different fitted values. Interestingly, the fitted values of the capacitance C 0 ΩΦ obtained for the HBM agree within less than 10% with those obtained for the PBM. Since C 0 ΩΦ corresponds to the curvature of the Ω(Φ) curve, this agreement is directly correlated with the agreement of the Ω(Φ) curves shown in Figure (1).

Thus, the grand potentials Ω(Φ) and the capacitances C 0 ΩΦ at the PZC, agree remarkably well between the HBM and PBM, despite their very different counter-charge distributions. Noting that, within the PBM, C ΩΦ is precisely equal to the interface capacitance of Eq. ( 4), we conclude that C 0 ΩΦ for the HBM provides a satisfactory representation of the "true" interface capacitance at the PZC, at least within the common limitations of HBM and PBM to be discussed below. [START_REF] Fan | Low-carbon production of iron and steel: Technology options, economic assessment, and policy[END_REF] We furthermore see that the straightforward definition of the HBM grand potential according to Eqs. ( 9) and ( 13), which avoids any a posteriori energy corrections, provides results in excellent agreement with the PBM. This finding will be further rationalized in the following.

Separating bulk electrode and electrolyte contributions in HBM

To explore the reason why the capacitance C 0 ΩΦ in the HBM provides a good description of the interface capacitance at the PZC, we investigate a simplified model where the HBM contributions within the bulk electrode and the electrolyte regions are treated separately.

A schematic simulation cell of a periodic electrode-slab model is shown in Figure 2(a).

Here, t b denotes the total electrode slab thickness (b: bulk ) and t s corresponds to the width of the remaining interspace region (s: space or surface), which contains the implicit solvent/electrolyte. A sub-partitioning of t s into an actual solvent-region width t sol and twice a gap width t g will be treated later on. For now, we continue with the coarse partitioning into t b and t s . Accordingly, the total width of the simulation cell in the z-direction perpendicular to the slab surface is T = t b + t s , and the volume fractions of the bulk-electrode and interspace regions are α b = t b /T and α s = t s /T , respectively.

Imagine that we could independently change the homogeneous background charge within both the bulk electrode and the electrolyte regions (while, at the same time, changing the electron number accordingly to preserve charge neutrality). We denote the respective excess electron numbers by N b and N s , and the Helmholtz free energy A(N b , N s ) becomes a function of both. We next perform a second-order expansion around the state with zero excess electrons,

A(N b , N s ) ≈ A 0 + µ 0 b N b + µ 0 s N s + 1 2 η 0 b N 2 b + 1 2 η 0 s N 2 s + η 0 bs N b N s (14)
and we neglect higher-order terms, which do not contribute to the capacitance at the potential of zero charge, see Eq. (11). Here,

µ σ = ∂A ∂N σ (15)
and

η σ = ∂ 2 A ∂N 2 σ ( 16
)
correspond to the chemical potential and chemical hardness, respectively, of the bulk (σ = b)

and surface/interspace (σ = s) sub-systems, and the "0" superscript denotes the state with zero excess electrons. Because the derivatives are defined by charge-neutral variations in both electron number and homogeneous background (within the respective region), we note that the chemical potentials µ σ = µ e,σ + µ bg,σ are the sum of the electron and background chemical potentials, where µ bg,σ = e φσ is simply determined by the average value of the electrostatic potential φσ within each of the regions [START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] . The last term in Eq. ( 14) with

η bs = ∂ 2 A ∂N b ∂N s (17)
quantifies the interactions between the bulk/surface excess-electron and background charges.

+ µ s N s ≈ µ 0 b N b + η 0 b N 2 b + µ 0 s N s + η 0 s N 2 s + 2η 0 bs N b N s ( 18
)
where we used N b = α b N e , N s = α s N e , and N e = N b + N s . In the last step, the bulk and surface chemical potentials were again expanded around the zero-excess state, µ b ≈

µ 0 b + η 0 b N b + η 0 bs N s and µ s ≈ µ 0 s + η 0 s N s + η 0
bs N b , respectively, which follows from Eqs. ( 15), (16), and ( 17). Using Eqs. ( 14) and ( 18), the HBM grand potential reads to second order

Ω ≈ A 0 - 1 2 α 2 b η 0 b + α 2 s η 0 s + 2α b α s η 0 bs N 2 e ( 19
)
To define the electrode potential, we introduce a further approximation, restricting our model to high-dielectric-constant solvents with r 1, e.g. water with r ≈ 80. Then, the electrostatic potential in the electrolyte region assumes an approximately constant plateau value φ elyte , and the curvature due to the homogeneous background charge can be neglected, see Figure 2(b). Consequently, the mean value φs across the interspace region becomes approximately equal to the plateau value φ elyte . The electrode potential, defined according to Eq. ( 3), is then simply given by -e Φ ≈ µ e,s + e φs = µ s (20) with the chemical potential µ s = µ e,s + µ bg,s , as discussed above. We develop the electrode potential to first order around the potential of zero charge (PZC) Φ 0 as Φ ≈ Φ 0 -(α s η 0 s + α b η 0 bs )N e /e, which follows from inserting the first-order expansion of µ s , see above, into Eq. (20). Resolving for N e and inserting into Eq. ( 19) leads to the grand potential as a function of the electrode potential up to second order,

Ω ≈ A 0 - 1 2 α 2 b η 0 b + α 2 s η 0 s + 2α b α s η 0 bs (α s η 0 s + α b η 0 bs ) 2 e 2 (Φ -Φ 0 ) 2 (21)
We now assess the capacitance definitions for the present model. Firstly, according to Eq. ( 4), we define the "true" interface capacitance as

1 C s = ∂Φ ∂q s = 1 e 2 ∂µ s ∂N s = η s e 2 (22)
where q s = -eN s is the excess charge at the electrode surface, and we used Eqs. ( 20), (15), and (16). Note that the surface excess electron number N s cannot be directly specified as an input parameter of HBM-DFT calculations. Instead, the total excess electron number N e of the simulation cell is fixed as an input, which corresponds to the capacitance C ΦN defined in Eq. ( 10). Using N s = α s N e , we find

1 C ΦN = α s C s = α s η s e 2 (23)
The capacitance C 0 ΩΦ , defined by Eq. ( 11), at the PZC is obtained from Eq. ( 21),

C 0 ΩΦ = C 0 s 1 + [(α 2 b η 0 b + 2α b α s η 0 bs )/(α 2 s η 0 s )] (1 + [α b η 0 bs /(α s η 0 s )]) 2 ( 24
)
where we used Eq. ( 22) for the "true" interface capacitance C 0 s at the PZC.

For metal electrodes, the delocalized electrons effectively screen the homogeneous background charge in the bulk electrode region, resulting in a large "bulk capacitance", or small bulk chemical hardness η 0 b in comparison to η 0 s of the surface, so η 0 b η 0 s . This is, for instance, verified for the Pt(111) slab, for which we obtain η 0 b = 0.052 eV η 0 s ≈ 2.1 eV. [START_REF] Greeley | Alloys of platinum and early transition metals as oxygen reduction electrocatalysts[END_REF] The metallic screening makes the bulk electrode region charge neutral, and, due to overall charge neutrality, the same must hold for the surface/interspace part. Therefore, the bulksurface interactions quantified by η 0 bs result from higher-order multipole interactions between both regions, which are expected to be significantly weaker than the intra-region interactions quantified by η 0 b and η 0 s , so η 0 bs η 0 b and η 0 bs η 0 s , which is confirmed by our computational results discussed below.

Under these conditions, the terms in square brackets in Eq. ( 24) are negligible and it follows that C 0 ΩΦ ≈ C 0 s . Thus, for metal electrodes, the HBM capacitance C 0 ΩΦ at the PZC is in good agreement with the "true" HBM interface capacitance C 0 s . This provides a first part of the explanation for the good agreement between the HBM C 0 ΩΦ and the PBM capacitance, as established by the results in Table 1. The second part of the explanation, i.e. why C 0 s of the HBM, in turn, is approximately equal to the PBM capacitance, will be discussed further below.

Advancement of the Homogeneous Background Method

Our finding C 0 ΩΦ ≈ C 0 s enables an advancement of the HBM, both facilitating and extending the applicability of the method. The advanced version is simply represented by the calculation of the grand potential Ω according to Eqs. ( 9) or ( 13), which can be readily performed with the knowledge of the total electron excess N e of the simulation cell that is specified as an input parameter for the HBM-DFT calculations. The interface capacitance can then be derived in good approximation from the curvature of the computed Ω(Φ) curve, see Eq. (11). Unlike the original approach [START_REF] Filhol | Elucidation of the Electrochemical Activation of Water over Pd by First Principles[END_REF][START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] , the calculation of the Ω(Φ) curve according to the advanced HBM does not require an estimation of the number N s of excess electrons at the electrode surface. The consistency between the advanced and original approaches is discussed in the following section and demonstrated by the results in Figure 3.

η 0 b , an independent HBM calculation of the corresponding Pt-bulk cell was performed, as described in the Computational Details section, and η 0 b was obtained from a second-order polynomial fit of the resulting A b (N b ) curve. The meaningfulness of the obtained Ω(Φ) curve is further demonstrated by the good agreement with the Ω(Φ) curve from PBM, shown in Figure 1 and discussed above. We can now explain this agreement. At the PZC, there is no counter charge present, so the HBM and PBM grand potentials are identical. Furthermore, the grand potential has a maximum at the PZC, so the first derivative vanishes, cf. Eq. (6) with N e = 0. Finally, as shown below, the "true" interface capacitance C s of the HBM is approximately equal to the capacitance of the PBM. Therefore, our result C 0 ΩΦ ≈ C 0 s means that also the curvature of the Ω(Φ) curve at the PZC is approximately equal in HBM and PBM, cf. Eqs. ( 8) and (11). At least up to second order in Φ around the PZC, a good agreement is thus established between the Ω(Φ) curves of the advanced HBM and PBM.

Intrinsic estimation of "active" electron fraction

The approximation C 0 ΩΦ ≈ C 0 s of Eq. ( 24), combined with Eq. ( 23), provides an estimation of the "active" fraction N s /N e = α s of excess electrons accumulated at the electrode-electrolyte interface,

α s = C 0 s C 0 ΦN ≈ C 0 ΩΦ C 0 ΦN (25)
without requiring any a priori specification of a boundary between the bulk electrode and interspace regions.

In contrast, the original HBM approach [START_REF] Filhol | Elucidation of the Electrochemical Activation of Water over Pd by First Principles[END_REF][START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] relies on an a priori definition of the interspace volume fraction α s . In fact, several reasonable choices for α s exist, according to the definition of the boundary between electrode and interspace region, as visualized in Fig-

ure 3(d).
First, the location of the center of the electrode-slab surface atoms can be used to define the boundary, yielding an interspace volume fraction α c s . Second, the atomic radius can be added to the center location of the surface atoms, resulting in α c+r s . And third, the interspace region can be defined simply as the implicit solvent region of VASPsol, leading to α sol s . As shown in Table 2, these definitions follow the order α sol s < α c+r s < α c s , with differences in value up to a factor of two for the given (typical) simulation cell dimension.

Interestingly, the "intrinsic" estimate of α s provided by the advanced HBM according to Eq. ( 25) lies in the middle of the value range and agrees rather well with the definition α c+r s .

Likewise, as shown in Figure 3(a)-(c), the Ω(Φ) curve directly computed according to Eq. ( 9) without a posteriori corrections falls in between the Ω(Φ) curves computed according to the original HBM approach using the three definitions of α s . These results demonstrate that the advanced HBM provides a method that is not only consistent with the original HBM approach, but also avoids the problem of defining α s , by providing an intrinsic "detection" of the "active" fraction of excess electrons. This will be highly useful for the application of the HBM in electrochemical-interface simulations with mixed explicit-implicit or fully explicit solvent models and in presence of surface adsorbate species that further complicate an a priori definition of the electrode-electrolyte boundary, and thus of α s .

To demonstrate this aspect, we performed HBM-DFT calculations for the Li(100) surface, where the first layer of ethylene carbonate (EC) solvent molecules was explicitly modelled, combined with implicit EC solvent in the remaining interspace region. To accommodate the explicit EC molecules, the simulation cell comprised a 2x2 surface cell of the Li(100) slab (with 9 atomic layers) and a 40 Å-wide interspace between the periodic slab images. For the Li(100) and Li(100)@EC systems without and with the explicit EC layer, respectively, Eq. ( 25

Comparison between "true" interface capacitance of HBM and PBM

We finally return to the question for the origin of the good agreement between the HBM and PBM grand potentials defined in Eqs. ( 9) and ( 2), respectively, as shown in Figure 1.

As discussed before, this is directly related to the agreement between the HBM and PBM capacitances C ΩΦ , presented in Table 1. Within the PBM, C ΩΦ of Eq. ( 8) is precisely equal to the interface capacitance C of Eq. ( 4). We further demonstrated, cf. Eq. ( 24), that the HBM capacitance C ΩΦ is approximately equal to the "true" HBM interface capacitance C s .

Therefore, the "true" interface capacitances of HBM and PBM seem to be approximately equal, which appears surprising given the very different counter-charge distributions in the electrolyte region for the two models.

This behavior can be understood by taking into account the existence of a "gap" between the electrode surface and the beginning of the implicit solvent region [START_REF] Hörmann | Grand canonical simulations of electrochemical interfaces in implicit solvation models[END_REF][START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF][START_REF] Lespes | Using Implicit Solvent in Ab Initio Electrochemical Modeling: Investigating Li+/Li Electrochemistry at a Li/Solvent Interface[END_REF] , see Figure 2(a).

In the present context, we consider the "gap" simply as a numerical aspect of the implicit solvent model, and we refer to other literature for a discussion of its relation to the Helmholtz capacitance of the physical system [START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF] . The width t g of this gap is numerically controlled within the VASPsol implementation by the parameter n c that represents the critical value of the electron density defining the location of the implicit solvent boundary [START_REF] Mathew | Implicit self-consistent electrolyte model in plane-wave density-functional theory[END_REF] . Similar parameters are also defined in other implementations of implicit solvent models for DFT codes [START_REF] Hörmann | Grand canonical simulations of electrochemical interfaces in implicit solvation models[END_REF] . Since the gap region does not contain implicit solvent, its dielectric properties only result from the small residual electronic density within this region, corresponding to a significantly reduced effective dielectric constant r,g and a gap capacitance 22 C g = (2A)( 0 r,g )/t g , where we take into account both electrode-electrolyte interfaces of the simulation cell with a total area of 2A.

For the PBM, the overall interface capacitance C is given by the series of the gap capacitance C g and the Gouy-Chapman capacitance C GC = (2A)( 0 r,sol /λ D ) cosh ze∆φ sol 2k B T of the counter-charge layer in the solvent region [START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF] , so C -1 = C -1 g + C -1 GC . Even for a small gap width t g , the gap capacitance C g can become limiting and determining for the overall interface capacitance if C GC is sufficiently large. This is the case for small values of the Debye length λ D of few Å in a high-dielectric-constant solvent, as used in our calculations and consistent with typical experimental conditions, so C ≈ C g . For increasing Debye length, the influence of C GC causes a decrease of C, as confirmed by the computational results for the Pt(111) slab in contact with implicit water solvent shown in Figure 5.

For the HBM, the counter-charge in the implicit solvent (electrolyte) region is homogeneous, so its extension is directly controlled by the width t sol of the solvent region, indicated in Figure 2(a). Solving Poisson's equation for the homogeneous charge distribution in the solvent region, see Supporting Information, we readily obtain a corresponding capacitance C sol = (2A)( 0 r,sol )/(t sol /4). We note that t sol /4 represents the center-of-mass location of the homogeneous background charge in the implicit solvent region per half cell. Thus, for small t sol , the solvent-region capacitance is large, and the gap capacitance C g dominates the HBM interface capacitance, C s ≈ C g . At larger t sol , the influence of C sol causes a decrease of C s and, since C ΩΦ ≈ C s , also of C ΩΦ . This is confirmed by the computational results presented in Figure 5.

We thus find a direct correspondence between the Debye length λ D of the PBM and the quarter of the solvent-region width t sol /4 for the HBM, with good agreement between the corresponding capacitances. In particular, we note that for typically chosen computational parameters, i.e. λ D 5 Å for PBM and t sol 20 Å for HBM, both methods yield essentially equal results for the charging characteristics of the interface, which are dominated by the gap capacitance C g that is defined by the implicit solvent model, common to both methods. Also shown in Figure 5 is a fit of the HBM capacitance curve according to an electrostatic model including the presence of the gap, which is derived in the Supporting Information. The resulting Eq. (S13) for C s was inserted into Eq. ( 24), noting that η 0 s = e 2 /C 0 s , cf. Eq. ( 22), and α b /α s = t b /t s with t s = t sol + 2t g , to obtain C 0 ΩΦ expressed in terms of t sol (for fixed t g and t b ). The gap capacitance C g and the chemical hardness η 0 b of the bulk electrode region were fitted to the computed C 0 ΩΦ (t sol ) data4 , while neglecting the interaction term by setting η 0 bs = 0. The fitted curve reproduces the observed trend in the HBM-DFT results, demonstrating that the electrostatic model captures the essential features of the interface for the HBM. Importantly, the fitted value for the bulk chemical hardness of η 0 b = 0.047 eV is in good agreement with the computed hardness of 0.052 eV for the Pt bulk in absence of the interspace region, see above. This result corroborates the validity of the HBM model leading to Eq. ( 24), in general, and the assumption of negligible η 0 bs , in particular. For the gap capacitance, a fitted value of C g = 0.44 e V -1 (13.09 µF cm -2 ) was found, in good agreement with the overall PBM and HBM capacitance values in the range λ D 5 Å and t sol 20 Å, respectively, which further supports the conclusion that the gap capacitance is the decisive factor in both HBM and PBM. As discussed above, the gap capacitance is determined by the VASPsol parameter n c that controls the distance of the implicit solvent boundary from the electrode surface atoms, and thus the width of the gap. Beyond this numerical origin and in light of its decisive role, the question for the physical meaning of the gap capacitance is emphasized [START_REF] Letchworth-Weaver | Joint density functional theory of the electrodeelectrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[END_REF] .

Conclusion

By including the homogeneous background contribution in the definition of the grand potential, we developed an advanced homogeneous background method (HBM) that can be applied without any energy corrections and without requiring knowledge of the "active" fraction of excess electrons. This strongly facilitates the practical use of the HBM and extends its range of application for systems where the precise definition of the electrode-electrolyte boundary is difficult, e.g. in presence of surface adsorbates or explicit solvent molecules. We found that the advanced HBM and the Poisson-Boltzmann model (PBM) provide essentially equal results as long as the gap between the electrode surface atoms and the boundary of the implicit solvent region dominates the interface capacitance. This is fulfilled for practically chosen computational parameters, i.e. implicit-solvent-region width t sol 20 Å and λ D 5 Å (for PBM), in high-dielectric-constant solvents, such as water or ethylene carbonate. For the HBM, the chemical hardness of the bulk electrode region must be negligible, restricting its applicability to metal electrode materials. Under these prerequisites, the HBM provides a practical method for DFT simulations of charged electrochemical interfaces that does not require any particular implementation of a counter-ion model within the electrolyte region.

Supporting Information Available

Electrostatic model for the interface capacitance within the homogeneous background method (HBM).

  550 eV for Pt and Au systems. Projector augmented wave (PAW) pseudopotentials as implemented in VASP were used35 . Exchange-correlation effects were accounted for by the generalized gradient approximation (GGA) using the functional of Perdew, Burke and Ernz-erhof (PBE)35 . The convergence criterion was set to 10 -6 eV for the electronic self-consistent iterations. Structural relaxation was performed until the maximum force on any atom was below 10 -2 eV Å-1 . Periodic interface calculations were performed on slabs comprising 7 atomic layers generated through the cleavage of the relaxed bulk crystals along the (100)surface orientation for Li and (111)-surface orientation for Pt and Au. Slab surfaces comprised 1 surface atom (11.84 Å2 surface area), 4 surface atoms (27.28 Å2 surface area) and 4 surface atoms (29.13 Å2 surface area) for Li (100), Pt (111) and Au (111), respectively. To prevent inhomogeneous charging of the two surfaces of the slab, cells were built symmetric in the z -direction and the central atomic layer was kept frozen while the other layers were allowed to relax. Except when stated otherwise, a 15 Å-wide interspace region separated the periodic images of the slab in the z -direction. Brillouin-zone sampling was done according to Monkhorst-Pack 36 on a 14 × 14 × 1 Γ-centered k-point grid for Pt and Li, and 11 × 11 × 1 for Au. The surrounding electrolyte within the interspace region was described by an implicit solvent using the polarizable continuum model (PCM) as implemented in VASPsol[START_REF] Mathew | Implicit solvation model for density-functional study of nanocrystal surfaces and reaction pathways[END_REF] . The dielectric constant of ethylene carbonate ( b = 89.9), which is a major solvent component of Li-battery electrolytes, was used for Li calculations, while the one of water ( b = 78.4) was used for Pt and Au. The effective surface tension parameter τ was set to 0 eV Å-2 31 and the critical density parameter n c was set to 5 × 10 -5 Å-3 for Li 37 and 2.5 × 10 -3 Å-3 for Pt and Au[START_REF] Mathew | Implicit self-consistent electrolyte model in plane-wave density-functional theory[END_REF] . The cavity-shape parameter σ was set to the default value of 0.6. Unless otherwise stated, for charged interface calculations with the Poisson-Boltzmann model (PBM) from VASPsol, a Debye length λ D of 1.5 Å (1/10 of the total interspace width) was used in order to ensure sufficient accommodation of the counter-charge distribution within the interspace region. To compute the bulk chemical hardness η 0 b of Pt, calculations were performed on a Pt-bulk cell that was constructed by removing the interspace region and one surface atomic layer from the Pt(111)-slab cell used for interface calculations, and the k-point grid was increased to 20 × 20 × 10.

Figure 1 5 ΦFigure 1 :

 151 Figure1shows a comparison between the HBM grand potential, Eq.(9), and the PBM grand potential, Eq. (2), as a function of the electrode potential computed for Pt(111), Au(111) and Li(100) electrodes in contact with implicit electrolyte. It is obvious that the grand potentials obtained from the two different counter-charge models are almost identical,

Figure 2 :

 2 Figure 2: (a) Scheme of an electrode slab simulation cell with electrode (grey), gap (yellow) and solvent (blue) regions. (b) Potential (averaged over xy-area) along the z-axis for a Pt(111) electrode in contact with an implicit solvent with b = 78.4 and b = 1.0 at different charge states N e = -1, 0, +1 using the HBM.

  Using the HBM, however, we cannot control N b and N s individually, but we fix the total excess electron number N e = N b + N s , compensated by homogeneous background charge across the entire simulation cell. The fraction of homogeneous background charge in the bulk electrode region is equal to the respective volume fraction α b . Because we only consider metallic electrodes, we assume that a corresponding fraction of N b = α b N e excess electrons accumulates within the bulk electrode region to screen the homogeneous background charge therein[START_REF] Kopač Lautar | Modeling interfacial electrochemistry: concepts and tools[END_REF] . Accordingly, the fraction of excess electrons accumulated at the electrode surface is given by N s = α s N e , corresponding to the homogeneous background charge in the electrolyte region (note that α s + α b = 1).In order to compute the HBM grand potential Ω = A -(µ e + e φ)N e according to Eq. (9), we use the free energy expansion of Eq. (14) and we also expand the second term (µ e + e φ)N e . The mean electrostatic potential φ = α b φb + α s φs of the entire simulation cell can be written as the weighted average of the mean values φb and φs across the bulk and interspace regions, respectively. Since the entire simulation cell is electronically equilibrated, the electron chemical potentials of the two regions are equal, µ e,b = µ e,s = µ e . We thus obtain (µ e + e φ)N e = µ e N e + e φb α b N e + e φs α s N e = µ e (N b + N s ) + e φb N b + e φs N s = µ b N b

α

  s : C = 14.62 μF cm -2 α s : C = 11.37 μF cm -2 α s : C = 14.01 μF cm -2 α s : C = 17.08 μF cm -2 sol c+r c α s : C = 5.21 μF cm -2 α s : C = 3.44 μF cm -2 α s : C = 6.88 μF cm -2 α s : C = 8.53 μF cm -2 sol c+r c α s : C = 14.05 μF cm -2 α s : C = 11.76 μF cm -2 α s : C = 14.51 μF cm -2 α s : C = 17.69 μF cm -2

Figure 3 :

 3 Figure 3: Comparison between the advanced HBM version and the original HBM approach: Grand potential Ω as a function of the electrode potential Φ computed for (a) Pt(111), (b) Au(111), and (c) Li(100) in contact with implicit electrolyte. The results of the original HBM are shown for the three different definitions of α s , see text. (d) Scheme of the interface between an electrode slab and (implicit) electrolyte with three possible definitions of the electrode-electrolyte boundary, and thus of α s , for the original HBM approach.

Figure 4 :

 4 Figure 4: Effective location of the electrode-electrolyte boundary (dashed vertical lines) within the HBM, computed from the α s values for the Li(100) and Li(100)@EC systems without and with an explicit layer of EC solvent molecules, respectively. The normalized Fukui functions from the PBM, averaged over the surface-parallel xy-coordinates of the two systems, are shown for comparison, with the dotted vertical lines indicating the center of mass (c.o.m) of the Fukui functions. The shifts in the effective boundary locations due to the explicit EC molecules are indicated by arrows. The relaxed Li(100)@EC structure is shown in the background for orientation. The Li surface atoms of the relaxed Li(100) structure are indicated by dashed circles.

Figure 4

 4 Figure 4 has been computed using the PBM, in order to avoid the contribution of the bulk excess electrons within the HBM that would bias the Fukui center of mass. In presence of the explicit EC molecule layer, the Fukui function gets split into two dominant peaks, one centered at the surface Li atoms, and the other at the EC molecules. This means that excess electrons spill over onto the latter, and the EC molecules participate in the electronic charging of the interface[START_REF] Hagopian | Ab initio modelling of interfacial electrochemical properties: beyond implicit solvation limitations[END_REF] . Consequently, this first layer of EC solvent cannot be uniquely ascribed to either the electrode or the electrolyte side of the interface, making an a priori definition of the electrode-electrolyte boundary difficult. The intrinsic detection of the boundary location according to the α s values, however, correctly reflects the outwards shift of the Fukui function in presence of the explicit EC solvent layer in comparison with the purely implicitly solvated Li(100) surface. Even quantitatively, a consistent behavior is found: Whereas the boundary computed from α s is shifted by 1.2 Å, the center of mass (c.o.m) of the Fukui function in the z-direction is shifted by 1.9 Å. Given the fact that both represent different definitions of the boundary location, the quantitative agreement between these values is satisfying and corroborates the validity of the approximation in Eq.(25).

Figure 5 :

 5 Figure 5: HBM and PBM capacitance C 0 ΩΦ at the PZC computed for the Pt(111) slab in contact with implicit water solvent as a function of t sol /4 and λ D , respectively, together with the fitted curve according to the electrostatic model for the HBM, see Supporting Information.

Table 1 :

 1 Comparison between the PBM and HBM methods of the fitted values for the capacitances C 0 ΦN and C 0 ΩΦ at the PZC obtained from the computed Φ(N e ) and Ω(Φ) curves, respectively, for Li(100), Pt(111), and Au(111) electrodes in contact with implicit electrolyte. Capacitance values are given in µF cm -2 .

		Electrode	PBM ΦN C 0 C 0 ΩΦ	HBM ΦN C 0 C 0 ΩΦ
		Li(100)	5.12 5.07 14.13 5.21
		Pt(111) 13.98 13.98 33.60 14.05
		Au(111) 13.71 13.59 32.84 14.62
	t b /2	t s		t b /2
				Potential [eV]
	t g	t sol	t g
	Electrode	Gap	Solvent		Distance from electrode center [Å]

Table 2 :

 2 Comparison of the different definitions for α s , see text, for Li(100), Pt(111), and Au(111) electrodes in contact with implicit electrolyte. The last column contains the "intrinsic" estimate of α s provided by the advanced HBM according to Eq. (25).

	System	α c s	α c+r s	α sol s	α s =	C 0 ΩΦ C 0 ΦN
	Li(100) 0.59 0.48 0.24	0.37
	Pt(111) 0.52 0.43 0.35	0.42
	Au(111) 0.51 0.42 0.34	0.44

In a complete description[START_REF] Schmickler | Interfacial Electrochemistry[END_REF] , also the solvent molecules must be included. However, they do not influence the present discussion, so we neglect this aspect for the sake of clarity.

We emphasize that, in the present context, we understand the "true" interface capacitance as the one defined by Eq. (4) for the given model where q refers to the interfacial charge, only. The question whether the value of this "true" model interface capacitance agrees with the real interface capacitance of the real physical system is a different one and lies outside of the scope of the present work, where we exclusively focus on a comparison between the computational HBM and PBM methods.

Here, η 0 s was computed from C 0 ΩΦ using the approximation C 0 s ≈ C 0 ΩΦ . For the bulk chemical hardness

The solvent-region width was computed from t sol = α sol s T , where T is the total width of the simulation cell, see Figure2(a). The electrode slab thickness t b was computed and fixed as the distance between the Pt surface atoms (including atomic radius) of the two slab surfaces. The gap width was computed from 2t g = T -t b -t sol , yielding a fixed value of t g = 1.17 Å.

Acknowledgement T.B. acknowledges financial support in the form of a research fellowship grant funded by the SNSF (Swiss National Science Foundation). A.H and J.-S.F. thank the French National Research Agency for its support through the Labex STORE-EX Project (ANR-10LABX-76-01). This work was performed using HPC ressources from GENCI-CINES (Grant 2021-A0100910369).