Chemical abundances in Seyfert galaxies – VII. Direct abundance determination of neon based on optical and infrared emission lines

Although the objects in our sample have been classified as AGNs by the authors from which the data were compiled, we produced an additional test based on standard Baldwin-Phillips-Terlevich (BPT) diagrams (Baldwin et al., 1981; Veilleux & Osterbrock, 1987). These diagnostic diagrams, based on optical emission-line ratios, have been used to distinguish objects whose main ionization mechanisms are massive stars from those that are mainly ionized by AGNs and/or gas shocks (see also Kewley et al. 2001, 2013; Kauffmann et al. 2003; Pérez-Montero et al. 2013; Ji & Yan 2020). We adopted the criteria proposed by Kewley et al. (2001) where all objects with

and

We performed the reddening correction to the optical emission lines by considering the expression

In comparison with the Case B recombination value of 2.86, Halpern (1982) and Halpern & Steiner (1983), adopting photoionization models, found that $I$(H$\alpha$/H$\beta$) is close to 3.10 in AGNs with high and low ionization degree. This contradicts Heckman (1980) preposition of an anomalously high Balmer decrement in these objects. Therefore, $I$(H$\alpha$/H$\beta$) = 2.86 and 3.10 intrinsic ratios are usually considered to be estimations for H ii regions and AGNs, respectively (Ferland & Netzer, 1983; Gaskell, 1982, 1984; Gaskell & Ferland, 1984; Veilleux & Osterbrock, 1987; Wysota & Gaskell, 1988). Particularly, in AGNs, there is a large transition zone, or partly ionized region, in which H⁰ coexists with H⁺ and free electrons. In this zone, collisional excitation is also important in addition to recombination (Ferland & Netzer, 1983; Halpern & Steiner, 1983). The main effect of collisional excitation is to enhance H$\alpha$. The higher Balmer lines are less affected because of their large re-excitation energies and smaller excitation cross-sections.

In order to check the H$\alpha$/H$\beta$ value assumed in our reddening correction, we consider results from AGNs photoionization models built with the Cloudy code (Ferland et al., 2013) by Carvalho et al. (2020). This grid of models assume a wide range of nebular parameters, i.e. a Spectral Energy Distribution with power law $\alpha_{ox}=-0.8,-1.1,-1.4$, oxygen abundances in the range of $\rm 8.0\>\lid\>12+\log(O/H)\>\lid\>9.0$, logarithm of the ionization parameter ($U)$ in the range of $-4.0\>\lid\>\log U\>\lid\>-0.5$, and electron density $N_{\rm e}$= 100, 500 and 3000 $\rm cm^{-3}$. The AGN parameters considered in the models built by Carvalho et al. (2020) cover practically all the range of physical properties of a large sample of Seyfert 2 nuclei. We excluded models with $\alpha_{ox}=-1.4$ and $\log U=-4.0$ because they predicted emission lines which are not consistent with observational data (see Pérez-Montero et al. 2019; Carvalho et al. 2020). The Carvalho et al. (2020) models assume constant electron density along the nebular radius while spatially resolved studies of AGNs have found $N_{\rm e}$ variations from $\sim 100$ to $\sim 3000\>\rm cm^{-3}$ along the NLRs of some AGNs (e.g. Freitas et al. 2018; Kakkad et al. 2018; Mingozzi et al. 2019; Riffel et al. 2021a). However, to provide a simple test for this problem, Dors et al. (2019) built AGN photoionization models assuming a profile density similar to observational estimations by Revalski et al. (2018a) in the Seyfert 2 Mrk 573, i.e. with a central electron density peak at $\rm\sim 3000\>cm^{-3}$ and a decrease in this value following a shallow power law. Dors et al. (2019) found that predicted emission lines assuming this density profile are very similar to those considering a constant electron density along the AGN radius. Therefore, $N_{\rm e}$ variations have almost a negligible effect on emission lines and abundances predicted by photoionization models, at least for the low electron density limit ($\la 10^{4}\>\rm cm^{-3}$).

In Fig. 3, bottom panel, we show the model predictions of the gas ionization degree parameterized by the [O iii]$\lambda 5007$/[O ii]$\lambda 3727$ line ratio versus H$\alpha$/H$\beta$ ratio. In this figure, the expected values for the H$\alpha$/H$\beta$ ratio, considering the theoretical values by Storey & Hummer (1995) for different temperature values of 5000 K, 10 000 K and 20 000 K are indicated by the solid black lines representing 3.10, 2.86 and 2.69, respectively. We notice that most of the models ($\sim$95 %) predict H$\alpha$/H$\beta$ values in the range from 2.69 to 3.10. In Fig. 3, top panel, the distribution of H$\alpha$/H$\beta$ values predicted by the models is shown, where it can be seen that, the most representative value is around (H$\alpha$/H$\beta$)=2.90 with an average value of $2.89\pm 0.22$. Therefore, for the intrinsic ratio of Seyfert 2 nuclei, we adopted the theoretical value given by (H$\alpha$/H$\beta$)=2.86.

The wavelength dependence in the optical domain, $f(\lambda)$, is the reddening value for the line derived from the curve given by Whitford (1958), which is defined such that $f(\infty)=-1$ and $f({\rm H}\beta)=0$. An analytical expression for the estimation of $f(\lambda)$ following the proposal by Kaler (1976) was used in the derivation of the extinction curve, which is given by:

Since several measurements for the emission lines compiled from the literature do not have their errors listed in the original papers where the data were compiled, we adopted a typical error of 10% for strong emission-lines (e.g. [O iii]$\lambda$5007) and 20 % for weak emission lines, in the case of [O iii]$\lambda$4363 (see, for instance, Kraemer et al. 1994; Hägele et al. 2008). These errors were used to calculate the uncertainties in the derived values of electron temperatures (in order of 800 K) and abundances (in order of 0.1 dex).

In determining ionic abundances using the $T_{\rm e}$-method, basically, it is necessary to obtain measurements of the intensity of the emission lines emitted by the ions under consideration and the representative values of the electron temperature ($T_{\rm e}$) and electron density ($N_{\rm e}$) of the gas region where these ions are located (Osterbrock, 1989).

Hägele et al. (2008) obtained, from the task temden of iraf¹¹1Image Reduction and Analysis Facility (iraf) is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. (De Robertis et al., 1987; Shaw & Dufour, 1995), functions to determine electron temperatures. It is considered that $t_{3}$ and $t_{2}$ are the electron temperatures (in units of $10^{4}$ K) for the electrons that are exciting the O²⁺ and O⁺ ions in the high and low ionization zones, respectively. The expressions obtained by Hägele et al. (2008) were assumed to calculate $t_{3}$, $\rm O^{2+}/H^{+}$, $\rm O^{+}/H^{+}$ and $\rm Ne^{2+}/H^{+}$.

First, for each object in our sample, the electron temperature in the high-ionization zone ($t_{3}$) was obtained by using the expression

Consequently, it is not possible to explicitly estimate $t_{2}$ in the AGN spectra of our sample where the [O ii]$\lambda$3727Å/$\lambda$7325Å emission line ratio can not be measured. Thus, we assumed the $t_{2}$-$t_{3}$ relation derived by Dors et al. (2020c) from a grid of photoionization models built using the Cloudy code (Ferland et al., 2013). The theoretical resulting relation is given by

where $\rm a=0.17$, $\rm b=-1.07$, $\rm c=2.07$ and $\rm d=-0.33$.

The $t_{3}$-$t_{2}$ relation for SFs has issue of some uncertainties due to the large scatter between these temperatures, around 900 K (e.g. Berg et al. 2020) and this problem has been addressed in several chemical abundance studies. For example, Hägele et al. (2008) pointed out that the scatter in the $t_{3}$-$t_{2}$ relation can be due to electron density effects because the [O ii] temperature is somewhat dependent on the density. Curti et al. (2017) pointed out that the [O iii]$\lambda$4363 can be contaminated by the neighboring [Fe ii]$\lambda$4360 line, mainly for objects with high metallicity ($\rm 12+\log(O/H)\>\ga\>8.4$). Recently, Arellano-Córdova & Rodríguez (2020) showed that the $t_{3}$ and $t_{\rm e}$(N ii) ($\approx t_{2})$ relation depends on the ionization degree of the gas phase in SFs. In fact, the model results adopted to derive Eq. 8 by Dors et al. (2020c) also present a large scatter, which is not explained by electron density effects. However, Riffel et al. (2021a) showed that the relation given by Eq. 8 is in consonance with direct estimates of temperature for AGNs when no clear gas outflows are present in these objects. Unfortunately, direct estimates of $t_{2}$ for AGNs are rare in the literature and we stress that the use of Eq. 8 can yield somewhat uncertain in $\rm O^{+}$ temperature estimates.

We make use of the relations to estimate ionic abundance of the singly and doubly ionized oxygen originally derived by Pagel et al. (1992) and in its current form given by Hägele et al. (2008) as:

where $n_{\rm e}=10^{-4}\>\times\>N_{\rm e}$.

The electron density $N_{\rm e}$ for each object was derived through the relation of this parameter with the line ratio [S ii]$\lambda 6717$/[S ii]$\lambda 6731$ by using the iraf code (Tody, 1986; De Robertis et al., 1987; Shaw & Dufour, 1995) and assuming the $t_{2}$ value obtained for each object. We derived electron density values in the range of $300\>\la\>N_{\rm e}\rm(cm^{-3})\>\la 3\,500$, with an average value of $\rm\sim 1\,000\>cm^{-3}$. In Dors et al. (2020c), a detailed analysis of the effect of the electron density on the direct abundance determination was presented and it is not repeated here. We only point out to the fact that, despite high $N_{\rm e}$ values in order of 13 000 - 80 000 ${\rm cm^{-3}}$ derived when optical lines emitted by ions with higher ionization potential than the ${\rm S^{+}}$ are used to derive the electron density, e.g. [Ar iv]$\lambda 4711$/$\lambda 4740$ line ratio (see Congiu et al. 2017; Riffel et al. 2021a), these values are much lower than the critical densities (e.g. see Vaona et al. 2012) for the optical lines used here. Additionally, the electron density determined from the line ratio [S ii]$\lambda 6717$/[S ii]$\lambda 6731$ is much lower than that obtained using auroral and transauroral lines, as well as ionization parameter based approach (Davies et al., 2020).

Generally, in H ii regions studies, the same temperature $t_{3}$ is used to estimate the ${\rm O^{2+}}$ and ${\rm Ne^{2+}}$ ionic abundances. This approach is based on the similarity of ${\rm O^{+}}$ and ${\rm Ne^{+}}$ ionization potentials, i.e. 35.12 and 40.96 eV, respectively, which indicates that both ions coexist in similar nebular regions. The same assumption is considered for ${\rm O^{+}}$ and ${\rm N^{+}}$, which is to assume $t_{2}$ for both cases whenever it is not possible to directly derive the $T_{\rm e}$ from the [N ii]$\mathrm{\lambda 6584/\lambda 5755}$. However, Dors et al. (2020c) found for AGNs (see also Riffel et al. 2021a) a slight deviation from the equality of the temperature for the $\rm O^{+}$ ($t_{2}$) and $\rm N^{+}$ [$t_{\rm e}({\rm\text{N\,{ii}}})$]. Therefore, in order to test if the temperature for ${\rm Ne^{2+}}$, defined for $t_{\rm e}({\rm\text{Ne\,{iii}}})$, can be considered to be the same as $t_{3}$, we used results from the grid of AGN photoionization models built with the Cloudy code by Carvalho et al. (2020). In Fig. 4, the model predicted values for $t_{\rm e}({\rm\text{Ne\,{iii}}})$ versus $t_{3}$ are shown. In each panel of Fig. 4, the model results are discriminated in accordance with the parameters $\alpha_{ox}$ (botton panel), $N_{\rm e}$ (midle panel) and $\log U$ (top panel) assumed in the models. It can be seen that for $t_{3}\>\ga\>1.0$, the models predict $t_{\rm e}({\rm\text{Ne\,{iii}}})$ lower than $t_{3}$ and the outlier of a point can not be explained by the variation in the nebular parameter assumed in the models. It is worth mentioning that the variations in the nebular parameters produce temperatures, in most part, within the uncertainty of $\pm 800$ K derived in direct estimates (e.g. Kennicutt et al. 2003; Hägele et al. 2008). In the top panel of Fig. 4, we can note that for objects with higher ionization parameter a high difference between the temperatures closer to the 1 to 1 relation is derived. The fit to the estimations considering all the points in Fig. 4 produces the relation

In order to produce an additional test for ascertaining the $t_{\rm e}({\rm\text{Ne\,{iii}}})$ and $t_{3}$ relations between AGNs and H ii regions, we analyse the electron temperature ($T_{\rm e}$) as well as the $\rm O^{2+}/O$ and $\rm Ne^{2+}/Ne$ ionic abundance structures along the nebular radius. In view of this, we consider photoionization models built with the Cloudy code in order to represent both kind of objects. For both models we adopt the same nebular parameters, i.e. electron density $N_{\rm e}=500\>\rm cm^{-3}$, solar metallicity $(Z/\rm Z_{\odot})=1.0$, and logarithm of the ionization parameter $\log U=-2.5$. The outer radius in both AGN and H ii region models was considered to be the radius at which the electron temperature of the gas reaches 4 000 K, i.e. the default lowest allowed kinetic temperature by the Cloudy code. It is worth noting that gases cooler than $\sim$4 000 K practically do not emit the optical and infrared emission lines considered in this work. Despite the fact that AGNs have slightly larger $N_{\rm e}$ values (by a factor of $\sim 2$) than H ii regions (see, e.g. Copetti et al. 2000; Vaona et al. 2012), the same value for this parameter was used in both models in order to maintain consistency. For the lower electron density regime, the $N_{\rm e}$ value does not change the temperature and ionization structure predicted by the photoionization models. For the AGN model, we adopt the SED as being a power law described by the slope $\alpha_{ox}=-1.1$ (for a detailed description of this SED see Krabbe et al. 2021). The SED for the H ii region model was taken from starburst99 code (Leitherer et al., 1999) and it assumes a stellar cluster formed instantaneously with the age of 2.5 Myr, which is a typical age of normal star-forming regions (e.g. Dors et al. 2008). For detailed description of the AGN and H ii region models see Dors et al. (2018) and Carvalho et al. (2020). The model results from AGN and H ii region are compared with each other in Fig. 5. In the bottom panel of this figure, it can be seen that the AGN model presents a very distinct temperature distribution over the nebular radius as compared to the H ii region one, implying that the former has a stronger decrease with the radius than the latter. Also, the $\rm O^{2+}/O$ and $\rm Ne^{2+}/Ne$ ionization structures are very distinct for both kind of objects. Similar ionic abundance distributions for both ionic ratios are derived for the H ii region, confirming the assumption of $T_{\rm e}(\text{O\,{iii}})\approx T_{\rm e}(\text{Ne\,{iii}})$. However, for the AGN model, the $\rm Ne^{2+}/Ne$ ionic abundance extends to an outer nebular radius (lower temperature) in comparison with $\rm O^{2+}/O$, implying that the approach $T_{\rm e}(\text{O\,{iii}})\approx T_{\rm e}(\text{Ne\,{iii}})$ is not valid for this object class. Moreover, the neon and oxygen ionic abundance structures for the AGN clearly indicate that the supposition $\rm(Ne^{2+}/O^{2+})=(Ne/O)$, usually assumed to derive the total neon abundance in H ii region studies (e.g. Kennicutt et al. 2003), can not be applied to AGNs. The result shown in Fig 4 is further supported by this simulation.

To estimate the ${\rm Ne^{2+}/H^{+}}$ abundances we use the relation given by Hägele et al. (2008):

where $t_{\rm e}$ is the electron temperature. We considered both $t_{3}$ and $t_{\rm e}({\rm\text{Ne\,{iii}}})$ in the estimations for ${\rm Ne^{2+}/H^{+}}$.

The infrared${-}$lines method (hereafter, IR-method) is based on determining the abundance of a given element using intensities of emission lines in the infrared spectral region (for a review, see Fernández-Ontiveros et al. 2017). Infrared emission lines have the advantage over optical lines for being less dependent on the electron temperature and on reddening correction, however, they have lower critical density ($10^{4-6}~{}{\rm cm^{-3}}$; e.g. Förster Schreiber et al. 2001) than the others ($10^{4-8}~{}{\rm cm^{-3}}$; e.g. Vaona et al. 2012).

Regarding the IR lines involved in our study, the critical electron density $N_{\rm c}$ for the [Ne ii]$12.81\micron$ and [Ne iii]$15.56\micron$ emission lines are $7.1\times 10^{5}$ and $2.1\times 10^{5}$ cm^-3 (Osterbrock & Ferland, 2006), respectively. The electron densities for the NLRs of our sample (see Sect. 3.1) derived from the $\rm S^{+}$ line ratio $(300\>\la\>N_{\rm e}(\rm cm^{-3})\>\la\>3500)$ are much lower than the $N_{\rm c}$ values. However, $N_{\rm e}$ derived from line ratios emitted by ions with different ionization potential (IP) other than the $\rm S^{+}$ (IP=10.36 eV) can reveal gas regions with higher $N_{\rm e}$ values than the ones derived for our objects and, consequently, indicate an influence on physical properties based on IR lines. In fact, $N_{\rm e}$ estimates from the [O iii]52$\micron$/88$\micron$ line ratio [PI($\rm O^{2+})=35.12$ eV] carried out by Vermeij & van der Hulst (2002) for H ii regions showed electron densities lower than 2 000 $\rm cm^{-3}$. Also, Storchi-Bergmann et al. (2009), who built $N_{\rm e}$ map based on the [Fe ii]1.533$\micron$/1.644$\micron$ [PI($\rm Fe^{+})=7.90$ eV] for the NLR of NGC 4151, found values between 1 000 and 10 000 $\rm cm^{-3}$. Finally, $N_{\rm e}$ determinations based on [S ii]$\lambda 6716/\lambda 6731$ and [Ar iv$\lambda 4711/\lambda 4740$ [IP($\rm A^{3+})=40.74$ eV] line ratios in two Seyfert 2 (IC 5063 and NGC 7212) by Congiu et al. (2017) show $N_{\rm e}$ values ranging from $\sim$200 to $\sim$13 000 $\rm cm^{-3}$. Although studies indicate the existence of an electron density stratification in NLRs of AGNs with values higher than the ones for our sample (see also, e.g. Kakkad et al. 2018; Freitas et al. 2018), effects of collisional de-excitation are probably negligible in our IR abundance estimates. Furthermore, we selected emissivity ratio values considering lower electron density compared to the aforementioned $N_{\rm c}$ values (see Table 1).

The Ne⁺ and Ne²⁺ ionic abundances can be determined using the intensities of the [Ne ii]12.81$\micron$ and [Ne iii]15.56$\micron$ emission lines following a similar methodology presented by Dors et al. (2013). Considering two ions Xⁱ⁺ and H⁺, the ratio of their ionic abundances is determined by

Using this method, any error in the determination of these emissivities directly translates into a systematic shift to the derived neon ionic abundance. To obtain the Ne⁺ and Ne²⁺ ionic abundances with respect to hydrogen (H⁺), near to mid-infrared H i recombination lines must be preferably used as reference line, such as P$\alpha$, P$\beta$, P$\gamma$, P$\delta$, Br$\alpha$, Br$\beta$, Br$\gamma$, Br$\delta$ and Br11, which are detected in most of the sources under consideration.

The emission coefficient for lines in the infrared has a weak dependence on the electronic temperature, which, in general, is disregarded. Therefore, abundance of a given ion can be obtained directly from the ratio between an emission line observed in the infrared and a hydrogen reference line. The calculation of Ne⁺ and Ne²⁺ ionic abundances can be obtained by the general relations with their emission lines:

Based on the above assumptions together with the values of the hydrogen line emissivities relative to $\mathrm{H\beta}$ listed in Table 1 and Eqs. 14 and 15, we deduce the following relations:

The Case B was assumed to derive the above equations because, as opposed to the broad-line region gas, much of the narrow-line region is believed to be optically thick to the ionizing radiation, even though studies of the He ii $\lambda 4686$ Å/H$\beta$ ratio in AGNs indicates the presence of some optically thin gas (Murdin, 2003). The [Ne iii] $\lambda 15.56$ µm line is always chosen over the [Ne iii] $\lambda 36.0$ µm when both are measured, because the spectrum is noisier at the long wavelength end of the long high-resolution (LH) module in the Infrared Spectrograph (IRS). Therefore, we preferred to use the [Ne iii] $\lambda 15.56$ µm line flux for the abundance determination of this ion, which also has larger transition probability and critical density.

In general, the total abundance of an element relative to hydrogen abundance is difficult to be calculated because not all emission line intensities emitted by the ions of this element are measured in the same spectral range. This fact, in principle, is circumvent by the use of ionization correction factor (ICF) proposed by Peimbert & Costero (1969). The ICF for the unobserved ionization stages of an element X is defined as

being N the abundance and $\rm X^{i+}$ the ion whose ionic abundance can be calculated from its observed emission lines. For instance, considering optical emission lines of oxygen, it is relatively easy to derive the $\rm O^{+}/H^{+}$ and $\rm O^{2+}/H^{+}$ abundances when $T_{\rm e}$ and $N_{\rm e}$ are derived. However, emission lines of oxygen ions with higher ionization states are observed in other spectral bands as, for instance, in X-rays (e.g. Cardaci et al. 2009; Bianchi et al. 2010; Bogdán et al. 2017). Recent studies (Flury & Moran, 2020; Dors et al., 2020c) indicate that the contribution of ions with ionization stage higher than $\rm O^{2+}$ in AGNs is in order of 20 per cent of the total O/H abundance. A smaller contribution of these ions, at least, in poor metal star-forming regions, is in order of only 1-5 per cent (Skillman & Kennicutt, 1993; Lee & Skillman, 2004).

To calculate the total oxygen abundance N(O/H) for our sample, we assumed

where ICF(O) is the Ionization Correction Factor for oxygen. We consider the ICF(O) expression proposed by Torres-Peimbert & Peimbert (1977)

(see also Izotov et al. 2006; Flury & Moran 2020). This ICF expression is based on the similarity between the $\rm He^{+}$ and $\rm O^{2+}$ ionization potential (about 54 eV).

To calculate the ionic helium abundance for each object taking into account the assumption that $t=t_{3}$, we use the relations proposed by Izotov et al. (1994) expressed as,

and

The total neon abundance determination in Seyfert 2 nuclei from either $T_{\rm e}$ or IR method can be realised by using an ICF taking into account the unobserved ionization stages ions of this element, such as Ne³⁺, whose emission lines are observed at 12$\mu$m and 24$\mu$m (e.g. Dudik et al. 2007). Peimbert & Costero (1969) and Peimbert & Peimbert (2009), based on the similarity between the ionization structures of neon and oxygen [$\mathrm{(Ne^{2+}/Ne)\approx(O^{2+}/O)}$], proposed

However, this approach does not seem to be valid for AGNs. For example, Komossa & Schulz (1997), by using multi-component photoionization models which permitted a successful match of a large set of line intensities from the UV to the NIR for Seyfert 2 nuclei, showed that the Ne²⁺ ion extends to a larger (where a lower temperature is expected) radius of the AGN than O²⁺. Similar result was found by Alexander & Balick (1997) for Planetary Nebulae (PNs), which also exhibited gas with high ionization. In fact, it can be seen from Fig. 4 that, generally, $t_{3}$ is higher than $t_{\rm e}({\rm\text{Ne\,{iii}}})$. Therefore, based on the results shown in Fig. 5, it is necessary to produce a new formalism to replace Eq. 25 for AGNs. We developed a semi-empirical neon ICF following a similar methodology assumed by Dors et al. (2013) for SFs.

The total neon abundance in relation to hydrogen is usually assumed to be

This approximation can be more reliable for SFs than AGNs, because it considers a null abundances of $\rm Ne^{i\>>\>2+}$. We use the photoionization model results by Carvalho et al. (2020) to ascertain the validity of Eq. 26 for AGNs. In the bottom panel of Fig. 6, the model results for $\rm y=1-[\rm(Ne^{+}+Ne^{2+})/H^{+}]$ versus $\rm x=[O^{2+}/(O^{+}+O^{2+})]$ is shown. It can be seen that, for $\rm x\>\la\>0.7$ or for $\log U\>\la\>-2.5$ the abundance sum $(\rm Ne^{+}+Ne^{2+})$ represents more than 80 % of the total Ne abundance. In the top panel of Fig. 6, a distribution of x values for the objects in our sample, calculated by using the $T_{\rm e}-$method (see Sect. 3.1), is shown. It can be seen that most of the objects ($\sim$90 %) have $\rm x\>\la\>0.6$ within the range $\rm 0.02\>\lid\>x\lid\>0.72$. Thus, a small correction factor is necessary in Eq. 26. A fit to the points in Fig. 6 produces

and Eq. 26 can be rewritten in the form

where

For the infrared abundance determinations, Eq. 28 was applied, where the $\rm Ne^{+}$ and $\rm Ne^{2+}$ estimates were based on Eqs. 16, 17, 18 and 19 and the $f$ factor was calculated from Eqs. 27 and 29 with x estimates obtained by using the $T_{\rm e}$-method (see Sect. 3.1).

For Ne/H estimates based on optical lines, it is only possible to calculate the $\rm Ne^{2+}/H^{+}$ abundance based on the Eq. 12 and assuming $t_{3}$ and $t_{\rm e}({\rm\text{Ne\,{iii}}})$. For that, the total neon abundance estimates via optical lines must be assumed

Using Eqs. 14, 15 and 28, we derive a semi-empirical neon ICF given by

The photoionization models and expressions employed to derive the ionic abundances, previously presented, probably use different set of atomic parameters which could introduce a small systematic uncertainty in the resulting abundances. However, Juan de Dios & Rodríguez (2017) found that atomic data variations introduce differences in the derived abundance ratios as low as $\sim$0.15 dex at low density ($N_{\rm e}\>\la\>10^{3}\>\rm cm^{-3}$). Since most NLRs of Seyfert 2 present $N_{\rm e}$ values lower than $10^{3}\>\rm cm^{-3}$ (e.g. Vaona et al. 2012; Dors et al. 2014; Dors et al. 2020b; Freitas et al. 2018; Revalski et al. 2018b; Kakkad et al. 2018; Revalski et al. 2021) the consideration of distinct atomic parameters in our calculations is expected to have a small effect in our abundance results.

Vermeij & van der Hulst (2002) obtained optical (by using the Boller & Chivens spectrograph on the ESO 1.52 meter telescope) and infrared (by using Short Wavelength Spectrometer - SWS and Long Wavelength Spectrometer- LWS on board the Infrared Space Observatory - ISO) spectra for 15 H ii regions located in the Magellanic Clouds. From these objects, it was possible to derive the Ne²⁺ ionic abundances via both IR and $T_{\mathrm{e}}{-}$method for 13 out of the 15 H ii regions. The differences (D) between these estimations ranges from $-0.6$ to +0.6 dex, thus, for some objects the $T_{\mathrm{e}}{-}$method resulted in higher abundances. The averaged value of D was about zero. The result obtained by Vermeij & van der Hulst (2002) is in disagreement with the findings by Dors et al. (2013), who found that the abundances obtained via infrared emission lines are higher than those obtained via optical lines in H ii regions, by a factor of $\sim 0.60$ dex.

In Fig. 8, 12 + log($\mathrm{Ne^{2+}/H^{+}}$) abundances via $T_{\rm e}$-method assuming $t_{\rm e}$(Ne iii) (left panel) and $t_{3}$ (right panel) are compared with the results via IR${-}$method for our sample. In the top panels of Fig. 8, the differences between both estimates are plotted versus the IR estimates. As noted earlier in Fig. 8, the difference (D) is systematic in both cases, where D increases with $\rm Ne^{2+}/H^{+}$ from IR-method estimations. The average difference ($\rm<D>$) between ionic abundances values via $T_{\rm e}$-method assuming $t_{3}$ and IR estimates is obviously the same value ($\sim 0.60$ dex) as the average value found for H ii regions by Dors et al. (2013). Therefore, probably, any artificial effects attributed to the use of heterogeneous sample of data sets, aperture effects, different regions in the objects which are considered in optical and IR observations, can have influence on our results.

The origin of D was discussed in details by Dors et al. (2013) for H ii regions and it was attributed to be mainly the presence of abundance and/or electron temperature variations across the nebula rather than extinction effects in the area of the sky covered by the IR and optical observations, as proposed by Vermeij & van der Hulst (2002). An overview of the discrepancy derived from this work will be presented in a subsequent paper, even though we refer to few possible scenarios here. Recently, Dors et al. (2020a) by using the suma code (Viegas-Aldrovandi & Contini, 1989), which assumed that the gas ionization/heating is due to photoionization and shocks, found that Seyfert 2 nuclei have gas shock velocities in the range of 50-300 $\mathrm{km\>s^{-1}}$. These shocks can produce an extra gas heating source in the NLRs, which translates into underestimation of the elemental abundances via $T_{\rm e}$-method in relation with abundances derived from IR lines (less dependent on temperature). As an addition support to the presence of electron temperature fluctuations in AGNs, Riffel et al. (2021a) presented 2D electron temperature maps, based on Gemini GMOS-IFU observations at spatial resolutions ranging from 110 to 280 pc, in the central region of three luminous Seyfert galaxies, where a large variation of temperatures (from $\sim 8000$ to $\ga 30\>000$ K) were derived. This result indicates a large fluctuation of $t_{3}$.

The Pa$\alpha$ to Pa$\delta$ and Br$\alpha$ to Br$\delta$ emission lines are not only the strongest emission lines found in the NIR and MIR, they are also relatively free from blending features and dust attenuation. This makes them valuable tools to derive the chemical abundances of AGNs. The optical Balmer emission lines, although stronger, can suffer blending with other lines (i.e. H$\alpha$ normally blends with [N ii] $\lambda 6548$, $\lambda 6584$ Å), and at least to some degree, are expected to be more affected by dust absorption than Paschen and Brackett emission lines. Independent measurements of the narrow component fluxes can yield important constraints on the presence of dust within the line of sight which could also affect the emitter regions of IR lines. In fact, effects of dust on hydrogen emission line measurements are clearly observable only in the Balmer emission line ratios but they can not be detected at a significant level using the Paschen and Brackett emission lines alone (Landt et al., 2008).

For the NIR broad emission line region (BLR) of AGNs, Landt et al. (2008) obtained the dust extinction in the order of $A_{V}\sim 1$ to $\sim 2$ mag in consonance with other studies (e.g. Cohen 1983; Crenshaw et al. 2001; Crenshaw et al. 2002; Storchi-Bergmann et al. 2009). From these results, the effect of the dust causing the observed extinction of the narrow emission line region depends on the location of the dust, thus, being internal dust if it is mixed with the gas phase or if it is located outside the NLR, for instance, in the host galaxy. Since the covering factor of the narrow emission line clouds is assumed to be only a few per cent, the line of sight towards the BLR will not necessarily intercept the dust. However, dust external to the NLR will act as a screen to affect the smallest scale components such as the BLR and the continuum emitted by the accretion disk. However, reddening in Seyfert galaxies by means of NIR line ratios performed by Riffel et al. (2006), led to the fact that Sy2s tend to lie close to the locus of points of the reddening curve, with $E(B-V)$ in the interval $0.25{-}1.00$ mag.