Performance of density functionals for the structure and energetics of (M–O)0,± (M=Al, Si, Sc–Zn)

ABSTRACT We report the results of the performance of 20 exchange–correlation functionals of density functional theory (DFT) in the structure (Metal–Oxygen bond length) and energetical properties (bond dissociation energy, adiabatic ionisation energy, and adiabatic electron affinity) of twelve metal monoxides (M–O, M=Al, Si, Sc–Zn). The calculated results show that the selected DFT functionals have the ability to reproduce the M–O bond length with a mean deviation of 0.01–0.05 Å, the energy values are reproduced with a mean deviation of 0.20–1.00 eV. In general, the functionals with significant HF exchange show decent performance in the calculation of bond length and harmonic vibrational frequency. These functionals show poor performance in energetics. Our calculated results show that the M06-L, B3LYP, and TPSSh functionals give good performance in both structure and energetical properties of metal monoxides. These functionals are recommended for the studies of structure and energetics in metal oxide systems. Further, our studies indicate that M06-L can be used for the studies in larger molecular systems. Among the 20 DFT functionals, the recently developed N12 functional gives poor performance in the studies of metal monoxides. Hence this functional is not recommended for the studies of structure and energetics in metal oxide systems.


Introduction
'Metal Oxides' can act as a catalyst and catalytic support materials in many chemical reactions because of its acid-base and redox nature [1]. The examples of such chemical reactions are oxidation over alkanes, alkenes, hazardous carbon monoxide (CO), and reduction of nitrogen oxides (NO x ) [2,3]. Further, a few metal oxides possessing magnetic behaviour, are used as high dielectric materials in the semiconductor industry [4,5]. Due to these wide range of applications, a large number of experimental [6][7][8][9][10][11][12][13][14][15] and theoretical investigations [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] were carried out to understand the structural, electronic, vibrational, and magnetic properties of metal oxides. For instance, Yu et al. [6] have performed the collision induced dissociation (CID) investigation on titanium oxide cluster ions (Ti n O 2n-1/2 ) + . They suggested that the (Ti n O 2n-1/2 ) + were dissociated into Ti or TiO or TiO 2 subunits based on their stability. Desai et al. [7] have studied the structure and bonding of aluminium oxide clusters (Al x O y ) − using anion photoelectron spectroscopy. They found that 2 ∑ + state to be the ground state of AlO and 2 ∏ g state to be the ground state of AlO 2 . Citra et al. [8] have used both experimental (Infra-red spectroscopy) and density functional theory (DFT) techniques to investigate the reactivity of nickel oxide clusters. They have studied the formation mechanism of nickel oxide clusters from nickel and molecular oxygen using these techniques. In theory, Bauschlicher et al. [30] have used highly correlated coupled-cluster (CCSD(T)), internally contracted multireference configuration (IC-MRCI), and DFT methods to investigate the ground states of NiO 2 0/-. From the investigation, they found that, 1 A 1 state to be the ground state of NiO 2 and 4 B 1 state to be the ground state of NiO 2 -. Recently, Miliordos et al. [22,23] and Sakellaris et al. [24][25][26][27] also used highly correlated ab initio methods to investigate the structure and energetical properties of 3d transition metal monoxides and their ions. They have obtained ground state and low-lying excited states of 3d transition metal monoxides and compared their results with the available experimental results.
In general, the problems behind the study of transition metal oxides using theoretical methods is, the presence of low-lying partially filled orbital, strong electron correlation effects, and spin contamination in open shell systems. Therefore, the accurate prediction of structure and energetics is a difficult task in these systems. In order to get accurate theoretical results, earlier studies were performed with highly correlated coupled-cluster or Multireference methods. The results are highly accurate, but their computational cost is high. Hence, the above treatments are not possible for the larger molecular systems. Alternatively, DFT methods are used to study the structure and energetics of molecular systems with reasonable accuracy and lower computational cost. For instance, Gutsev et al. [17] have investigated the geometrical and electronic structure of 3d transition metal monoxides in neutral and anion form. They found that the pure DFT functionals give better performance than the B3LYP method in the calculation of spectroscopic constants and electron affinity. Similarly, Nakao et al. [18] have studied the bond length and dissociation energy of the first-row transition metal oxide cations by ab initio and DFT methods. They suggested that the multireference perturbation method and the B3LYP method gives the smallest deviation for the bond length and dissociation energy.
Nowadays DFT is one of the most preferred methods for the studies in transition metal containing compounds [31]. Since DFT has a wide range of exchange-correlation functionals; it is a tedious task to select which functional gives an accurate result for a particular metal system. Hence benchmarking of DFT functionals is necessary to find out the suitable DFT functional for studying the metal containing compounds. Previous DFT benchmark calculations on metal containing complexes were carried out by Truhlar et al. [32][33][34] and Dixon et al. [19,35,36]. Recently, Bao et al. [32] have assessed 50 DFT functionals for the bond length and bond energies of 3d and 4d bimetallic diatomic molecules. They concluded that M06-L, OreLYP, and N12-SX functionals perform well for both metal-metal bond lengths (mean unsigned error less than 0.07 Å) and bond energies (mean unsigned error less than 10 kcal/mol). Fang et al. [37] have investigated the performance of 55 DFT functionals for energetical properties of group IV and VI transition metal oxide (TMO) nanoclusters. They found that B1B95 and PBE0 functionals provide better results for the energetical properties of TMO nanoclusters. Recently, Paranthaman et al. [38] have also studied the performance of DFT functionals for the structure and energetical properties of Ruthenium containing organo-metallic complexes. They have concluded that the recently developed Minnesota functionals MN12-SX and MN12-L show good performance for the structure and energetics of Ru-containing organo-metallic complexes.
Even though DFT has now become the preferred method for the studies of transition metal containing complexes, it has difficulty in determining the bond length and bond dissociation energies that involve metal-oxygen bonds [39]. A prominent example of this problem is dissociative adsorption of molecular oxygen on Aluminium surface [40]. Therefore, it is necessary to find out the suitable DFT functional with the accuracy of highly correlated ab initio methods to study the metal-oxygen bond, which is involved in many catalytic systems. Hence, in the present study, the performance of 20 DFT functionals for the structure and energetics of metal monoxides (MOs) is assessed. Twelve MOs (AlO, SiO, and ScO-ZnO) are selected in the neutral, anion, and cation forms. The performance of the selected DFT functionals are evaluated by calculating their structural property, i.e. M-O bond length, spectroscopic property, i.e. harmonic vibrational frequency, and energetical properties such as bond dissociation energy, adiabatic ionisation energy, and adiabatic electron affinity. To the best of our knowledge, this is the first systematic DFT assessment study in the metal oxide systems. Because earlier benchmark studies mainly considered training set which includes metal dimers, monohydrides, monocarbides, mononitrides, and monoxides. The major problem in using the training set is, we do not know which functional performs well for which MO system and why. Our study includes traditional as well as recently developed functionals along with Minnesota functionals. This work will be very much helpful for the selection of DFT functional for studying larger MO systems. Further, this work will also helpful in the development of new exchange-correlational functionals for the studies in metal oxide systems.
The adiabatic ionisation energy (IE ad ) is the energy difference between the total energy of the neutral system and its cation, i.e.
The adiabatic electron affinity (EA ad ) is the energy difference between the total energy of neutral system and its anion, i.e.
All the DFT and CCSD(T) calculations are performed using Gaussian09W program [63].

Results and discussion
Twelve MOs (AlO, SiO, and ScO-ZnO) in the neutral, anion, and cation forms are optimised using 20 exchange-correlation functionals and CCSD(T) method. The selected DFT functionals along with their Hartree-Fock (HF) exchange percentage are listed in Table 1. The DFT functionals that are taken in this study are classified as generalised gradient approximation (GGA), meta-GGA, hybrid GGA, and hybrid meta-GGA along with nonseparable gradient approximation (NGA) methods. The key components in these functionals are density, gradient of density, kinetic energy density, and HF exchange. The difference between GGA and NGA functional is, in the former case density functional depends on the up and down spin densities and their reduced gradient, in the latter case, density functional depends on the up and down spin densities and their reduced gradient and also adopts a nonseparable form. The calculated bond length, harmonic vibrational frequency, bond dissociation energy, adiabatic ionisation energy, and adiabatic electron affinity of MOs are given in Tables 2-12. The available experimental and previous theoretical results are also listed in Tables 2-12 [65]. This indicates that our CCSD(T) results can be used as an alternate to experimental data, to assess the performance of DFT functionals. This is because, as mentioned earlier, only very limited experimental and highly correlated (MRCI) theoretical data are available for some MOs.
In the case of anion MOs, the previous MRCI calculation predicted 1 ∑ + state to be the ground state of ScO − with the bond length of 1.729 Å [22]. Our CCSD(T) calculation also predicts 1 ∑ + state to be the ground state of ScO − with the bond length of 1.739 Å ( Table 7). Further our CCSD(T) calculation predicts 6 Σ + to be the ground state of FeO − . Earlier Hendrickx et al. [66] also predicted 6 ∑ + state to be the ground state of FeO − using CASPT2 + DKH method. Similarly, the ground state of NiO − is predicted to be 4 ∑ − with the bond length of 1.665 Å by previous MRCI calculation [26]. Our CCSD(T) calculation also predicts the same ground electronic state with the bond length of 1.669 Å. Our CCSD(T) results are in agreement with the previous MRCI data in the case of cations also. For instance, the previous MRCI calculation predicted 3 ∑ − state to be the ground state of CrO + with the bond length of 1.614 Å [22]. The CCSD(T) method in this study predicts 3 ∑ − state to be the ground state of CrO + with 1.609 Å as the bond length (Table 10). Similarly, previous MRCI calculation predicted 5 ∏ state to be the ground state of MnO + [22]. The same result is observed by the CCSD(T) method in our study. The MRCI calculation gives 1.757 Å as the bond length of MnO + , whereas the CCSD(T) method predicts 1.764 Å. Even though CCSD(T) predicts accurate results in anions and cations, in a few cases, the CCSD(T) results are deviated from previous high-cost theoretical results. For instance, the (Cr-O) − bond length predicted to be 1.712 Å by previous MRCI calculation [22], while the CCSD(T) in this study gives it as 1.678 Å (Table 7). Similarly, the (Ni-O) + bond length predicted to be 1.611 Å by previous MRCI method [26], and it is obtained to be 1.655 Å by CCSD (T) method in this study (Table 10). However, the difference between CCSD(T) and previous MRCI bond length is minimum. This indicates that the CCSD(T) results can be used for the benchmarking study i.e. for assessing the performance of DFT functionals.
The maximum, minimum, and mean deviation values of the bond length, harmonic vibrational frequency, and bond dissociation energies of each MO (neutral, anion, and cation) along with their reference values (experimental or CCSD(T)) are listed in Tables S1-S3 in the supporting information file. The best-performed DFT functionals in the calculation of  [74]. b [75]. c [65]. d [76]. e [12].
bond length, harmonic vibrational frequency, and energetics of each MO are given in Tables S4-S6. The mean unsigned deviation (MUSD) values are calculated and are listed in Tables S7-S17. The performance of DFT functionals in the calculation of structure and energetics of neutral, anion, and cation MOs are discussed in the main text.
1.724 Å [12], and the M11 functional gives 1.822 Å (Table 2). However, the difference between the experimental and M11 functional bond length is minimum. While considering the performance of functional as a function of HF exchange, if the HF exchange percentage increases, the deviation also increases. From Figure 1 and Table S7, it can be seen that the B3LYP (20% HF exchange) gives MUSD value of 0.010 Å, PBE0 (25% HF exchange) gives 0.016 Å, and M11 (43% HF exchange) gives 0.018 Å. However, this deviation is small. In summary, all the DFT functionals considered in this study give good performance in the calculation of the bond length of neutral MOs.
In the case of harmonic vibrational frequency, Figure 1 clearly shows the M06-L and TPSSh functionals perform well Table 8. The calculated harmonic vibrational frequency (in cm −1 ) of anion MOs.
i [27]. about 104 and 129 cm −1 respectively. This indicates that the above mentioned local Minnesota functionals show poor performance for the vibrational frequency of the MO systems. However, the RSH meta-NGA functionals such as M11 and MN12-SX yield significantly improved the result. This indicates that the functionals with HF exchange (hybrids) show good performance in this MO system. In this case also, if the fraction of HF exchange percentage increases, the deviation from the experimental vibrational frequency also increases ( Figure 1 and Table S8). The functionals with out HF exchange (M06-L) or with small amount (TPSSh) give good performance. The calculated bond dissociation energy of neutral MOs are given in Table 4 and their MUSDs are shown in Figure 1. From Figure 1, it can be seen that the M06-L and B3LYP functionals give good performance. The mean deviation value given by the M06-L and B3LYP functional is about 0.2 eV for neutral MOs. The pure functionals such as BLYP, BP86, and BPW91, which show good performance in M-O bond length and vibrational frequency, do not show good performance for the energetics of neutral MOs. These functionals give MUSD of 0.65-0.88 eV in the calculation of bond dissociation energy of neutral MOs. Generally, the GGA functionals give large deviations in the energetics of metal containing complexes. These functionals are unable to predict metal atom energy accurately [67], because metals possess near degenerate energy levels. Among the 20 DFT functionals, PBE has the maximum deviation value (0.91 eV) in the calculation of bond dissociation energy of neutral MOs. However, PBE0 (hybrid version of PBE) gives an improved result (∼0.50 eV). This indicates that the mixing of exact exchange improves the performance of the functionals. That is, hybrid functionals show good performance in the calculation of bond dissociation energy. The functionals with out HF exchange (M06-L) or with small amount (B3LYP) give good performance (Figure 1 and Table S9). Figure 2 shows the comparison between CCSD(T) and experimental results in bond length, harmonic vibrational frequency, and bond dissociation energy of neutral MOs. From Figure 2, it is found that the CCSD(T) results are in good agreement with the experimental result except in bond dissociation energy of VO and MnO. The experimental bond dissociation energy of VO and MnO are 6.44 (±0.20) eV and 3.83 (±0.08) eV, respectively [12]. The CCSD(T) calculation in this study gives 4.92 and 5.47 eV for VO and MnO respectively (Table 4). This indicates that, these metal oxides may require larger basis set such as correlation consistent, to get the accurate results. The calculated adiabatic ionisation energy of MOs are given in Table 5. Figure 3 shows the MUSD error of adiabatic ionisation energy and adiabatic electron affinity of neutral MOs. From Figure 3, it is seen that, TPSSh, B3LYP, B3PW91, PBE0, and M06 functionals perform well with the mean deviation value less than 0.25 eV. The  B3P86 hybrid functional gives large deviation (0.58 eV) in the calculation of adiabatic ionisation energy of neutral MOs. The calculated adiabatic electron affinity of MOs are given in Table 6. From Table 6, it is evident that the CCSD(T) results in this study are in good agreement with the previous experimental results. The B3LYP, M06, ωB97X, and ωB97XD functionals show good performance. These functionals give the mean deviation value less than 0.30 eV. Among the 20 DFT functionals, N12 functional does not perform well for the adiabatic electron affinity of neutral MOs, because it gives large MUSD value (0.64 eV). The above results indicate that the hybrid functionals give good performance in the calculation of adiabatic ionisation energy and electron affinity of MOs. In summary, all the functionals considered in this study give good performance in the calculation of bond length and vibrational frequency. In the case of energetics, hybrid functionals give better performance than pure functionals. The pure functionals yield larger error in this case. The reason is, as mentioned above, these functionals have difficulty in determining the metal atom energy accurately.

Anion MOs
The optimised M-O bond length of anion MOs are given in Table 7. The MUSD errors of bond length of anion MOs are shown in Figure 1. Due to the lack of experimental results in anion MOs, high-cost CCSD(T) calculations are performed. These CCSD(T) results are used to validate our DFT results. From Figure 1, it is evident that the BP86, BPW91, PBE, M06-L, TPSS, TPSSh, and B3LYP functionals perform well for the bond length of anion MOs. The deviation is about 0.01 Å. This indicates that most of GGA functionals (without HF exchange) perform well in this case. Among the hybrid functionals, TPSSh (10% HF exchange), B3LYP (20% HF exchange), and PBE0 (25% HF exchange) have MUSD of 0.011, 0.012, and 0.016 Å respectively ( Figure 1 and Table  S12). This shows that if the HF exchange percentage increases in the functionals, deviation in bond length also increases. Among the 20 DFT functionals, the M11-L functional gives the maximum mean deviation value (0.06 Å) in the anion M-O bond length (Figure 1). However this difference is small. This indicates that all the DFT functionals considered in this study show decent performance in the calculation of bond length in anion MO systems.
The calculated harmonic vibrational frequency of anion MOs are given in Table 8. The MUSD errors of vibrational frequency of anion MOs are shown in Figure 1. Limited experimental results are available in the case of vibrational frequency of anion MOs also. For instance, the experimental vibrational frequency of NiO − is 760 ± 40 cm −1 [68]. The CCSD(T) method in our study gives 837 cm −1 . The vibrational frequency calculated by the DFT functionals varies from 742 to 854 cm −1 . Further, the calculated results of the CCSD(T) method are in good agreement with previous MRCI results (Table 8). For example, the previous MRCI calculation gave 786 cm −1 for MnO − [22] and 827 cm −1 for CoO − [25]. The CCSD(T) method in this study gives 757 and 833 cm −1 for MnO − and CoO − respectively. Among the 20 DFT functionals considered in this study, BP86, BPW91, PBE, TPSS, TPSSh, B3LYP, and B3PW91 functionals perform well in the calculation of vibrational frequency of anion MOs. The MUSD value is less than 30 cm −1 . In this case also, if the HF exchange percentage increases in the functionals, the deviation also increases in the vibrational frequency ( Figure 1 and Table  S13). Similar to the neutral MOs, the recently developed Minnesota functionals M11-L and MN12-L fail to show its accuracy in vibrational frequency of anion MOs. These two functionals give the mean deviation value of 67 and 60 cm −1 respectively.
The calculated bond dissociation energy of anion MOs are given in Table 9 and MUSD errors are given in Figure 1. From Table 9, it is noted that the CCSD(T) results coincide very well with the previous MRCI results. For instance, the previous MRCI calculation showed the bond dissociation energy of ZnO − to be 2.24 eV [27], and our CCSD(T) study gives 2.45 eV. However, in some cases significant deviations are noted. For instance, our CCSD(T) results are deviated from the previous theoretical results in MnO − and FeO − systems. The previous MRCI calculation gave 3.07 eV [22] and 3.92 eV [24] for MnO − and FeO − respectively. The CCSD(T) calculations in this study overestimate both the values to 5.90 and 6.53 eV for MnO − and FeO − respectively. The long range corrected functionals such as ωB97X and ωB97XD show good performance because the difference is less than 0.30 eV. The B3LYP, B3PW91, and PBE0 functionals provide some reasonable results. The MUSD value is less than 0.40 eV (Figure 1 and Table S14). These functionals have small amount of HF exchange percentage (20-25%). In general, the pure DFT functionals (no HF exchange) give poor performance in this case. In particular, BP86 and PBE functionals give large MUSD value (> 0.80 eV). This result suggests that the above said functionals do not yield valuable results for the energetical properties of anion MOs. Over all, the hybrid functionals show good performance in the calculation of bond dissociation energy of anion MOs.

Cation MOs
The optimised M-O bond length of cation MOs are given in Table 10. The MUSD errors of the bond length of cationic MOs are shown in Figure 1. From Figure 1, it is seen that the M06-L, B3LYP, PBE0, M06, ωB97X, and ωB97XD functionals give the minimum deviation (0.03 Å). The pure DFT functionals BLYP, BP86, BPW91, and PBE give MUSD of ∼0.04 Å for the bond length of cation MOs. Similar to neutral and anion M-O bond lengths, the M11-L functional gives small deviation (0.05 Å) in the calculation of cation M-O bond length. In this case, the functionals without HF exchange (M06-L) or small amount (B3LYP) gives good performance ( Figure 1 and Table S15). This indicates that all the DFT functionals considered in this study show decent performance in the case of bond length of cation MOs.
The calculated harmonic vibrational frequency of cation MOs are given in Table 11. The MUSD errors of harmonic vibrational frequency of cation MOs are shown in Figure 1. The CCSD(T) and DFT results are different in this case. For instance, the CCSD(T) vibrational frequency of VO + is 1006 cm −1 , all the DFT functionals overestimate this value by 79-195 cm −1 . The CCSD(T) vibrational frequency of ZnO + is 766 cm −1 , all the DFT functionals underestimate this value by 63-188 cm −1 . Among the 20 DFT functionals, the M06-L and B3LYP functionals provide some reasonable results. However, these two functionals give 77 and 79 cm −1 as the mean deviation value and this difference is large. The maximum deviation value is observed by the N12-SX and MN12-SX functionals. These two functionals yield the mean deviation value greater than 100 cm −1 . In general, all the DFT functionals give large error in the calculation of vibrational frequency of cation MOs. The reason for the large discrepancy between our DFT and CCSD(T) results is not known.
The calculated bond dissociation energy of cation MOs are given in Table 12 and MUSD errors are shown in Figure 1. A few DFT functionals in this study reproduce the experimental bond dissociation energy. For instance, the experimental bond dissociation energy of VO + is 5.81 (±0.17) eV [69]. The M06-L, TPSSh, B3P86, M06, and N12-SX functionals reproduce this value (Table 12). Similarly, the experimental bond dissociation energy of ZnO + is 1.65 (±0.12) eV [69]. The bond dissociation energy calculated by the functionals M06-L, TPSSh, B3LYP, B3P86, ωB97X, and ωB97XD is in agreement with the experimental bond dissociation energy. From Figure 1, it is evident that all the pure functionals show poor performance in the calculation of bond dissociation energy of cation MOs (MUSD value ranges from 0.70-0.96 eV). In general, hybrid functionals give decent performance and in particular, B3LYP and PBE0 functionals perform well. Because these functionals give MUSD value less than 0.40 eV. In this case also, if the HF exchange percentage increases, the deviation from the CCSD(T) value is also increases ( Figure 1 and Table S17). In general, the functionals with small amount of HF exchange (20-25%) yield good performance.

Performance of pure and hybrid functionals
The performances of pure (BLYP, BP86, BPW91, PBE, TPSS, M06-L, M11-L, N12, and MN12-L) functionals are compared with their hybrid versions in our study. Figure 4 shows the comparison between the performance of pure and hybrid functionals in the structure and energetics of MOs. In Figure 4, we also give a comparison between the long-range corrected functionals such as ωB97X and ωB97XD. Both these functionals are hybrid functionals. In order to understand the effect of dispersion term in these functionals, these functionals are given in Figure 4. From the figure, it is seen that both pure and hybrid functionals show good performance in bond length and harmonic vibrational frequency of MOs. The only exception is M11-L functional, which gives slightly higher MUSD value (0.05 Å for bond length and 70 cm −1 for vibrational frequency). Earlier Peverati et al. [49] have mentioned that the M11-L (local meta-GGA functional) is good for transition metals, inorganic, organometallic complexes, and noncovalent interactions. They have used a training set (BC338) to assess the performance of DFT functionals. This training set includes the energetics of main-group elements, transition metals, inorganic, organometallic complexes, and solid state systems. They found that M11-L functional gives good performance in these systems. However, it must be noted that the errors are averaged over large set of data in this training set. In our study, we have assessed the performance of 20 DFT functionals in 12 MOs. This is the reason for the poor performance of M11-L functional in our study. Further, the hybrid functionals give good performance in the case of energetics. In particular, B3LYP functional gives small deviation (0.25 eV). In comparison with BLYP functional, the MUSD value significantly reduced in B3LYP functional. Among the hybrid functionals, the maximum deviation (∼0.45 eV) is obtained by MN12-SX functional. This functional is screened exchange (SX) functional with 25% short range HF exchange and 0% in long range. It is designed for energetics and structural problems in chemistry and solid-state physics. Earlier Peverati et al. [59] have assessed the performance of MN12-SX functional on 22 datasets. These datasets include main-group elements, transition metals, solid state systems, etc. They concluded that, MN12-SX gives good performance in all chemistry and solid-state physics systems. However, it must be noted that they did not assess the performance of MN12-SX in MOs in their study. In general, the functional designed for a particular system (main-group element or transition metals), gives good performance in the similar system. This is the reason for the poor performance of this functional. In our study, we have selected GGA, meta-GGA, hybrid GGA, and hybrid meta-GGA along with NGA functionals for the assessment study. Generally, the exchange and correlation holes in GGA and meta-GGA functional are short ranged, whereas hybrid functionals have long ranged exchange hole. This treats strong nondynamical correlation in transition metal containing complexes.
While considering the performance of hybrid functionals as a function of HF exchange percentage, if the HF exchange percentage increases, the MUSD values also increases ( Figure 4). This indicates that the functionals with significant HF exchange give poor result. Further, the functionals with small amount of HF exchange yield good result. For example, B3LYP which contains 20% HF exchange show good performance in the calculation of bond length, harmonic vibrational frequency and bond dissociation energy. However, the range separated functionals (M11, N12-SX, and MN12-SX) yield poor performance. These functionals may provide good results for specific properties (thermochemical properties and kinetics) that require significant HF exchange. Our calculated result coincides with earlier result. Earlier studies have mentioned that the functionals with the significant amount of HF exchange, affect the accuracy of the calculated atomisation energies [54,70]. Further, the functionals such as B3LYP, B3P86, and B3PW91 have same amount of HF exchange, yield different results (0.20 eV for B3LYP, 0.35 eV for B3P86, and 0.30 eV for B3PW91). The reason is, the contribution of correlation energy in exchange-correlation energy is generally smaller than (less than 10%) the exchange energy. Therefore, the exchange energy must be more precise than the correlation energy. In addition, the exchange functional show good performance when it is combined with the same correlation functional for example, the exchange functional PBE yield better results when it is combined with its correlation part, i.e. PBEPBE. However, some exceptional cases like B3LYP are also available. In addition to this, the long-range corrected functionals (ωB97X and ωB97XD) yield some reasonable results for the MO systems. The mean deviation value given by these functionals is 0.02 Å in bond length, 53-55 cm −1 in vibrational frequency, and 0.37-0.38 eV in the energetical properties of MOs. The above results show that, there is no significant difference in the performance of ωB97X and ωB97XD functionals in the studies in MO systems. This indicates that the dispersion effect does not change the results much in the case of MOs. In summary, the hybrid DFT functionals perform well when compared with the pure DFT functionals for the MOs considered in this study. This is due to the mixing of HF exchange in the hybrid functionals. The inclusion of HF exchange in the hybrid functionals improves the result. However, the functionals with significant HF exchange give poor performance.

Performance of Minnesota functionals
Totally eight Minnesota functionals (M06-L, M11-L, N12, MN12-L, M06, M11, N12-SX, and MN12-SX) are considered in this study. Figure 5 shows the percentage of error contribution by the Minnesota functionals in the bond length, harmonic vibrational frequency, and bond dissociation energy of MOs. Among the eight Minnesota functionals, M06-L (meta-GGA) functional performs well for the MO systems. It shows the best overall performance for the structure, harmonic vibrational frequency, and energetics of MOs ( Figure 5). This implies that, it has the ability to provide reliable results for neutral, anion, and cation MO systems. The mean deviation value obtained by M06-L functional varies from 0.01-0.03 Å for bond length, from 24 to77 cm −1 for vibrational frequency, and from 0.19-0.62 eV for bond dissociation energy ( Figure 1). The M06 functional shows the second best performance in the structure, harmonic vibrational frequency, and energetics of MOs ( Figure 5). The mean deviation error varies from 0.01-0.03 Å for bond length, from 32 to 78 cm −1 for vibrational frequency, and from 0.25-0.41 eV for bond dissociation energy (Figure 1). Our calculated result is consistent with previous Zhao et al. study [48].  (Figure 1). It must be noted that apart from M06-L, all other best performed Minnesota functionals are hybrid functionals. This indicates that the inclusion of HF exchange in these functionals yield better result in the calculation of both structure and energetics of MOs. However, among the Minnesota hybrid functionals, if the HF exchange percentage increases, the performance of the functional decreases. For example, M06 gives better performance than M11 functional. Figure 6 shows the percentage of error contribution by the selected DFT functionals in each MOs. In ScO (particularly in anion), the maximum error is obtained while calculating the bond length. In MnO, the maximum error is obtained while calculating the bond length and energies. In FeO and CoO, the maximum error is obtained in the case of energetics. In CuO (particularly in cation) the maximum error can be obtained in the calculation of bond length and harmonic vibrational frequency. This indicates that the selected functionals show poor performance in these systems. Therefore, further development of new exchange-correlation functional is required for the studies in these systems. Figure 7 shows the structure versus energetics MUSD errors in neutral, anion, and cation MO systems. Previous benchmark study [72] showed that B3LYP is not suitable for the studies in metal containing compounds. In our study B3LYP functional gives good performance in all the cases of MOs (Figure 7).  Further, previous DFT assessment study of the structure and energetics of aluminium clusters results show that, the B3LYP functional does not perform well for aluminium clusters [73]. Similarly in our study, the B3LYP functional shows poor performance in the case of structure and energetics of AlO 0,± (Tables S4-S6). This implies that, B3LYP functional is not suitable for the studies in aluminium oxide systems. The previous validation study by Zhao et al. showed that the M06-L functional performs well for the structure and energetics of metal containing compounds [48]. Similarly, our results show that the M06-L functional performs well for the structure and energetics of MOs. Even though the TPSSh functional has some inaccuracy in the cation MOs, it performs well in the case of neutral and anion MOs systems. Figure 7(a,b) do not include M11-L functional since its deviation value is large i.e. more than 0.04 Å. The above functionals show best average performance when the errors are averaged over a set of MOs in neutral, anion, and cation forms. The five best performed DFT functionals (based on the MUSD value) in each case of MOs i.e. M-O bond length, harmonic vibrational frequency, bond dissociation energy, adiabatic ionisation energy, and adiabatic electron affinity are listed in Table 13.

Conclusions
In this study, the performances of 20 exchange-correlation functionals are assessed for the structure and energetics of metal monoxides (in the form of neutral, anion, and cation). Our calculated results show that, all the selected DFT functionals show decent performance in the calculation of M-O bond length in MOs. In particular, M06-L, B3LYP, and TPSSh functionals show the best overall performance with the MUSD of 0.02 Å. Further, B3LYP and TPSSh functionals show good performance for the vibrational frequency of MOs. However, large MUSD errors were noted in the case of vibrational frequency of cation MOs. In general, the hybrid DFT functionals perform well for the energetical properties of MOs. The mean deviation values of most hybrid functionals lie below 0.30 eV, except the range-separated hybrids (M11, N12-SX, and MN12-SX). These functionals give MUSD value greater than 0.40 eV. This indicates that, functionals with the significant amount of HF exchange yield poor performance. Further, if the HF exchange percentage increases, the deviation from the experimental or CCSD(T) also increases. This trend is observed in most of the neutral, anion, and cation cases. The M11-L which is the improved version of M06-L, gives some reasonable results for the energetical properties of MOs. But its MUSD value in M-O bond length is large. The worst performance among the 20 DFT functionals is observed from N12 functional. This functional is not suitable for the studies in MO systems. Further, it is noted that the performance of the functional depends upon the MOs selected for the study. None of the functionals showed good performance in all the MOs and all three parameters considered in this study. Our study suggests that more attention should be given while selecting the functional for studying charged MO system. Over all, we recommend M06-L, B3LYP, and TPSSh functionals for the studies of MOs. In particular, out of the above three functionals, M06-L can be used for the studies of larger molecular systems, because of its lesser computational cost.
Our study provides a generalised overview of the performance of traditional as well as recently developed DFT functionals for MOs. We believe that this work can provide valuable guidance for further studies in these MOs.

Disclosure statement
No potential conflict of interest was reported by the authors. Table 13. The best DFT functionals for the bond length, harmonic vibrational frequency, bond dissociation energy, adiabatic ionisation energy and adiabatic electron affinity in MOs.