DFT calculations revealed Si(OH)4 to react with [Zn(OH)4]2− via a barrierless transition state, evidencing a kinetic ground for the retardation effect. Additionally, Raman spectroscopy corroborated the DFT results by showing that Q3 species in the proposed mechanism are formed more rapidly in the presence of zinc ions than in its absence.Geopolymers are the most promising alternative to Ordinary Portland Cement for oil-well cementing and well abandonment. To that end, the slurry needs a required pumping time ensured by the addition of retarders. Although zinc has been widely known to prolong the setting time of geopolymers, its mechanism of action has yet to be fully elucidated. It is herein hypothesized that zinc ions impede the first stages of silicate oligomerization (Si–O–Al), culminating in longer setting times. Pumping time measurements showed that Zn(NO However, to apply such material in cementing and well abandonment operations, chemical admixtures such as retarders must be used to delay setting and guaranty a safe period for displacement into wellbores4. Geopolymer formation from solid materials is a complex, multistep process roughly comprising i) alkaline depolymerization of the poly(siloxo) framework and dissolution of aluminum ii) formation of monomers and oligomers from ortho-sialate (OH)3− Si–O–Al–(OH)3, and iii) polycondensation into higher oligomers and polymeric 3D networks6. Advantageously, it has been demonstrated that the degree of polymerization/depolymerization of glasses and geopolymers can be determined via Raman spectroscopy7. In essence, SiO4 species in a silica network differ from each other spectroscopically according to the number of sharing oxygen atoms. An isolated SiO4, for instance, is referred to as Q0 due to its lack of sharing oxygen. A Q1 entity denotes, in turn, a SiO4 with one sharing oxygen in the network. The reasoning extends then to Q2, Q3, and Q4 meaning two, three, and four sharing oxygen atoms, respectively. Upon contacting glass or a silica-rich mineral with an alkaline environment it is expected that the amount of Q0–Q3 species increase with time due to silica depolymerization, a phenomenon that can be tracked since each Qn species appear at distinct frequencies in the Raman spectrum9.Geopolymers are an alternative cementitious material with the potential of replacing Ordinary Portland Cement (OPC) in both construction and oil & gas applications. The applicability of this material into oil and gas has been under research in the past period since its production has a lower carbon footprint and maintains superior properties over OPC specifically in long term periods2+) species, as a retarder, have been under study where its mechanistic and kinetic aspects have been taken into consideration13. Zinc oxide (ZnO), for instance, is thought to dissolve into Zn2+ which prolongs the setting time by sequestering calcium ions (Ca2+) and forming calcium zincate [Ca(Zn(OH)3)2.2H2O]11. This is also the conclusion reached by Cong et al.14, who only speculated that Zn2+ could have had an effect on the condensation polymerization. The possibility of Zn2+ playing a role in the early stages of the geopolymerization should not be overlooked. Zeng et al.15 demonstrated the synthesis of a coagulant based on poly-zinc-silicate to yield a complex compound with mainly zinc-silicon polymeric species rather than a simple mixture of raw materials. Upon studying the impact of Zn2+ and lead (Pb+2) ions on OPC, Oretgo et al.16 discovered Zn2+ ions to retard the silicate polymerization. The authors demonstrated, via NMR, a high proportion of Q0 and Q1 species after curing OCP with Zn2+ ions, implying a low degree of polymerization of SiO4 units.Zinc 4]2− due to the high pH, as a reactant that is inserted into monomers and oligomers to hamper momentarily the progression of the reaction. Density Functional Theory (DFT) calculations revealed that SiO4 reacts more quickly with [Zn(OH)4]2− via a step with a barrierless transition state, meaning that the reaction between Si(OH)4 and [Zn(OH)4]2− is kinetically more feasible in comparison to that between Si(OH)4 and Al[(OH)4]−. Raman results revealed that the presence of Zn2+ led to a higher rate of initial depolymerization of the poly(siloxo) framework, which supports the DFT proposition.The present study intends to shed light on the mechanism via which Zn.17. The retardation phenomenon can be observed through consistency measurements of slurries that show the behavior of material under different conditions. In this case, pumpability into an oil & gas well. The pumping time of the granite-based geopolymer slurries Geo-0%Zn and Geo-1%Zn is presented in Fig. 2+ species (Geo-1%Zn) delays the pumping time by around 5 h in comparison to the neat sample (Geo-0%Zn), which reached its maximum after around 1 h. It can be perceived that the oligomerization phase was highly extended due to the presence of Zn2+ species. A poisoning phenomenon of the reaction can be concluded from Fig. 2+ in silica-rich geopolymer systems. Wang et al.10 analyzed the role of ZnO on metakaolin based geopolymer material. They concluded that the presence of ZnO have created metastable “Na/K-Zn” phase materials , which have a prolonging effect on setting time. Their study also included the investigation of Q4(3Al) and Q4(4Al) sites using NMR spectra. A decrease in the Si/Al ratio was observed due to the increase in number of Q4(4Al) sites. It has been foreseen that not only Q4 sites to be affected, but rather also the possibility of having a deviation in Q3 sites’ development, which can as well be sites to investigate while the reaction is developing18.The impact of retardation phenomena in geopolymer material can be initially observed through pumping time measurements as described in previous studies by Chamssine et alGibbs free energy of reaction (ΔrG) for fifteen dehydration reactions involving the Si(OH)4, [Al(OH4)]−, [Zn(OH3)]−, and [Zn(OH4)]2− monomers  are presented in Table 4 and [Al(OH4)]− monomers is exergonic by 11.0 kcal mol−1. Besides, condensation reactions to yield the (OH)2Al–(O–Si(OH)3)2, (OH)Al–(O–Si(OH)3)3, and Al–(O–Si(OH)3)4 oligomers are exergonic by 20.4, 30.5, and 38.7 kcal mol−1, respectively (reactions 2–4 in Table 3Al–O–Si(OH)3]−  and oligomers species, which is the driving force of the reaction, is in line with the explanation previously given that the geopolymerization mechanism must occur through the condensation of oligomers19.The calculated Gibbs energy. Condensation reactions from the [Zn(OH3)]− and [Zn(OH4)]2− species are all exergonic (reactions 5–9), suggesting that Zn–O–Si units can be incorporated into the polymeric framework. However, the dehydration reactions without the condensation of the reactant species are only favorable for reactions involving the [Zn(OH4)]2− specie (reactions 10–15). Indeed, the reactions 12 and 15 are computed to be exergonic by 13.3 and 3.8 kcal mol−1. Figure Gibbs energy profiles and the relevant optimized structures calculated for the [Al(OH4)] −  + Si(OH)4, [Zn(OH3)] −  + Si(OH)4, and [Zn(OH4)]2−  + Si(OH)4 reactions. In our model, the condensation reactions from the [Al(OH4)]− and [Zn(OH3)]− anionic species are described by mechanism analogous. The reactants are linked to each other by the attraction between hydrogens from the Si(OH)4 specie and OH groups from the ionic specie, [Al(OH4)]− or [Zn(OH3)]−. Then, the two reactants condense to form the [(OH)3Al–O–Si(OH)3]− and [(OH)2Zn–O–Si(OH)3]− species, respectively, with concomitant to release the H2O molecule. Based on Fig. 4)]−  + Si(OH)4 and [Zn(OH3)]−  + Si(OH)4 reactions proceed through a transition state (TS) located at 16.4 and 2.1 kcal mol−1 above the reactants’ energy, respectively. The origin of the difference between these reaction barriers can be attributed to steric hindrance at the TS structures. As shown in Fig. 4)]− specie involves a greater geometrical deformation when compared to the TS of the reaction with [Zn(OH3)]−.The calculations also predict the formation of the Zn–O–Si linkage which is favorable in terms of the 4 + [Al(OH4)]−/[Zn(OH3)]− reactions, the mechanism of the reaction between [Zn(OH4)]2− and Si(OH)4 species is predicted to proceed by the initial formation of the [Zn(OH3)(OH2)…OSi(OH)3]2− intermediate (Int). The formation of the Int occurs virtually barrierless, where the Si(OH)4 specie undergoes a hydrogen abstraction process by the OH group from the [Zn(OH4)]2− 3]2−  + H2O products or fragment to yield the [Zn(OH)3]−  + [SiO(OH)3]−  + H2O products. According to our calculations, the reactions involving the [Zn(OH4)]2− anions are kinetically and thermodynamically more feasible than that reaction with other anions.On the contrary to what is computed for the Si(OH)9. An advantageous feature is that their Raman spectra can be compared with those of SiO4 glasses7. SiO4 glass creation relies on the condensation of isolated SiO4 tetrahedra (referred to as Q0) by linking to each other via sharing one to four oxygen atoms (Q1–Q4). Since differently bonded tetrahedra have distinct Raman signatures in the range of 1000 cm−1–1100 cm−1, variations in that region may be used to track the SiO4 depolymerization required for geopolymer formation7. Interestingly, the computational chemistry results in Fig. 3 species via a barrierless TS that appears only in the presence of Zn2+. It is then conceivable that Zn2+ would lead to a higher rate of Q3 formation in geopolymers when compared to their no Zn counterparts (CNT-0%Zn), which would support the mechanism determined by computational chemistry. This hypothesis has been verified by means of Raman spectroscopy with results shown in Fig. 2+ at this point): 207 cm−1 (Si–O-Si bond twisting), 355 cm−1 (SiO4 bending), and 456 cm−1 (bending of O-Si–O)20, which tend to vanish due to quartz consumption. Vibrations at 712 cm−1, 1345 cm−1, and 1361 cm−1 refer to CO3−2 in calcite as a consequence of natural carbonation21, which is known to take place in cementitious materials such as OPC22. Regarding geopolymers, the reaction between CO2 from the air and OH− produces Ca and sodium (Na) carbonates24. It is important to notice that the bands assigned to carbonate display no clear tendency over time, which is expected given that the variable amount of CO2 in the atmosphere leads to an uncontrolled carbonation process. The most important feature of Fig. −1 related to a Si–O- vibration with O− denoting a non-bridging oxygen within a Q3 species25. Another important characteristic is that the Q3 band initially increases with time (up to 150 min), meaning that OH− attacks the Si network to produce increasing amounts of unbounded oxygen. It is important to emphasize that the increase in intensity of the peak at 1051 cm−1 is not a result of any artifact since; i) the time of laser illumination and laser potency were the same for all samples (ruling out intensity variation due to heating effects), ii) the illuminated spot of the sample was always the same (same area probed), and iii) there is no contribution of carbonation in the wavenumber region of interest25. Upon addition of 1% of Zn to the mixture  as a consequence of weakened Si–O force due to charge transfer from Zn(OH)42− to SiO4. The negative charge transferred to tetrahedra accumulates preferentially on Si atoms leading to a decrease in the Si–O coulombic interaction26. When comparing Q3 peaks, Fig. 3 formation. The slopes of the straight lines from t = 0 to t = 150 min are 23.75 and 37.97 for mixtures without and with Zn2+, respectively. Both curves reach the steady state at 150 min, a situation in which the rate of Q3 formation is counterbalanced by the emergence of oligomers and subsequently geopolymer.Raman spectroscopy has proven to be a valuable tool in the study of geopolymers2+. The precursor was designed with a low Ca content composition (< 10 wt.%) as presented in Table 3)20.6H2O) was used as the source of Zn2+ species. The mix design of slurries is presented in Table A granite-based aluminosilicate-rich solid precursor has been used to exemplify the retardation effect of Zn2+ species, in the form of Zn(NO3)20.6H2O, were also used to mimic retardation with a similar 1 wt% as in the original mix design. In order to accurately mimic the designated components’ behavior, the number of molecules of Si, Al, Fe, Ca, Mg, Na, and K was calculated and implemented with the proposed chemical replacements27 . The solid-to-liquid ratio was around 2.0 Table . Initial29. This test was performed only on granite-based geopolymer slurry.An OFITE HPHT consistometer (Model 2040) was used to examine the pumping and setting time at a BHCT of 50 °C and pressure of 2000 psi. The standard for pumping time was set from the starting point until 40 BC while setting time was from 40 to 100 Bc following API RP 10-B2 recommendations−1 were examined using a LabRAM HR Evolution using a 532 nm diode laser operating at 25 mW. Spectra were collected at 30 min time interval. Freshly mixed geopolymer paste was analyzed at time (t) 0 and transported to a nearby oven operating at 50 °C. Thus, maintaining the reaction temperature like the one in the consistometer.Geopolymer paste, of the controlled system samples (CNT), was analyzed using Raman Spectroscopy. Spectra from 400 to 4000 cm2+ in the geopolymerization reaction following the method presented by Yang et al.30. For such, dehydration reactions between the Si(OH)4 and [Al(OH)4]−, [Zn(OH)3]− and [Zn(OH)4]2− species were used to model the formation of the Al–O–Si and Zn–O–Si bonds31. In our model, Si, Al, and Zn are coordinated by hydroxides, which is consistent with the high alkaline condition (pH around 13.0 to 13.5) used in the experimental setup32. In this range of pH, there is an equilibrium between [Zn(OH)3]− and [Zn(OH)4]2− anionic species. Figure Density functional theory (DFT) calculations were performed to support the experimental evidence of the retardation effect of Zn34. To globally explore conformers, the GFN2-xTB method was used in the framework of meta-dynamics36. Secondly, the minimum-energy conformers were chosen as guess structures for calculations of geometry optimizations with the ωB97X-D/6-311 + G(3df.2p)//6-31 + G(d.p) level of theory. The structures were optimized at the ωB97X-D/6-31 + G(d.p) level of theory, and single-point calculations were performed at the ωB97X-D/6-311 + G(3df.2p) on these structures. All calculations were performed with an implicit solvent (water) using the polarizable continuum model. The ωB97X-D functional has presented a good performance in the description of structural kinetics and thermochemical properties38. The Gaussian 16 software package was used for all DFT calculations39.First, structure predictions of the reactants and products were performed with the conformer-rotamer ensemble sampling tool of the xtb software


P14AS competitively binds to CBX7 and prevents the binding of CBX7 to ANRIL and P14ARF, P15INK4B, P16INK4A promoters” are shown at the end of the article.In the published article, there was an error in Discussion, Paragraph 1.In the published article, there was also another error: “and” was lost in the first sentence in the discussion section. A correction has been made to ANRIL promoters of P14ARF, P15INK4B, P16INK4A can directly bind to the PcG proteins including CBX7”This sentence previously stated: “It has been reported that both lncRNA The corrected sentence is as follows:ANRIL and promoters of P14ARF, P15INK4B, P16INK4A can directly bind to the PcG proteins including CBX7”“It has been reported that both lncRNA The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.


The performance of predictive methods for three carcass traits including liveweight (LW), dressing percentage (DP), and longissimus dorsi muscle weight (LDMW) was evaluated using three approaches . In this study, we found the accuracies of LD‐based and fixed‐SNP haplotype Bayesian methods outperformed the Bayesian models . GHBLUP showed a high improvement (up to 11.29 ± 9.87%) compared with GBLUP. The Bayesian models have higher accuracies than BLUP models in most scenarios. The average computing time of the BayesBH+BayesB model can reduce by 29.3% compared with the BayesB model. The prediction accuracies using the LD‐based haplotype method showed higher improvements than the fixed‐SNP haplotype method. In addition, to avoid the influence of rare haplotypes generated from haplotype construction, we compared the performance of GP by filtering four types of minor haplotype allele frequency (MHAF)  under different conditions . We found the optimal MHAF threshold for LW was 0.01, and the optimal MHAF threshold for DP and LDMW was 0.025.Genomic prediction (GP) based on haplotype alleles can capture quantitative trait loci (QTL) effects and increase predictive ability because the haplotypes are expected to be in linkage disequilibrium (LD) with QTL. In this study, we constructed haploblocks using LD‐based and the fixed number of single nucleotide polymorphisms (fixed‐SNP) methods with Illumina BovineHD chip in beef cattle. To evaluate the performance of different haplotype block partitioning methods, we constructed haploblocks based on LD thresholds (from  Single nucleotide polymorphisms (SNP) are routinely applied to increase the prediction accuracy at a young age compared with the approach using pedigree information in farm animals  assess the performance of GP for carcass traits and computation time based on different haploblock methods (the fixed‐SNPs and the LD‐based) using GBLUP and BayesB model; (2) compare the prediction accuracies under different LD thresholds and different number of SNPs; (3) evaluate the prediction accuracy with different MHAF thresholds for three traits.22.1The phenotypic data were generated from 1478 Chinese Simmental cattle, which were born between 2008 and 2020 from Ulgai, Xilingol League, and Inner Mongolia, China. After weaning, all individuals were moved to Jinweifuren Co., Ltd. for fattening under the same feeding and management conditions. All individuals were slaughtered at an average age of 20 ± 2.2 months. More detailed description of the management processes was reported in previous studies , and individual call rate (<0.90). Missing genotypes were imputed using BEAGLE (v4.1)  were used for genotyping. Before phasing the quality control of the original SNP dataset was carried out using PLINK (v1.07) .Multivariate variance analysis was used to determine the fixed effects of sex, year, fattening farm, and slaughterhouse and the covariates (body weight upon entering the fattening farm and the number of fattening days). The phenotypes were adjusted by the significant factors using glm function in R programming. Variance components were estimated using the following univariate animal model in ASREML (v4.1).2.42.4.1The LD information was used to determine the boundary of haploblocks for each chromosome. Due to the variation in local recombination rates, the breakdown of LD is often discontinuous and presents a haploblock‐like structure . There were no overlapping SNPs between adjacent haploblock. For two bi‐allelic loci (A1/A2 and B1/B2), r2 was calculated asD=pA1B1pA2B2−pA1B2pA2B1. D is the degree of LD for bi‐allelic loci (A1/A2 and B1/B2); pA1, pA2, pB1, and pB2 are the alleles  frequencies; pA1B1, pA2B2, pA1B2, and pA2B1 is the genotype  frequencies  were set as the thresholds in this study. To reduce the dimensionality of LD‐based haplotype analyses, haplotype alleles with a minor allele frequency of less than 0.01 were discarded.Seven different LD levels . Three different levels  were set to construct the haploblock. There were no overlapping SNPs between adjacent haploblocks. Consistent with the LD method, haplotype alleles with a minor allele frequency of less than 0.01 were discarded in the current study.2.4.3r2 > 0.3 and the fixed number of SNPs was 5, while the quality control of haplotype alleles was set at four different levels .To compare the haplotype structure distribution and performance of GP with different MHAF thresholds, the LD levels were set at 2.5HBLUP and BayesBH model (only based on haploblock), and GHBLUP+GBLUP and BayesBH+BayesB model (contain both the haploblock and the nonblocked SNPs) were considered for predictions. Seven different r2 thresholds and three levels of a fixed number of SNPs were used for haploblock construction. The Bayesian algorithm was performed using GCTB (v2.0)  was used for LW, DP, and LDMW as described in Equation .G=M−PM−P′m2pi1−pi, where M means the ‐encoded genotype matrix, pi is the MAF of marker i, m is the number of markers, and P is a matrix with columns equal to 2pi  was performed for all SNPs. The haplotype‐based genomic relationship matrix GH was expressed as GH=MH−PHMH−PH′mH2pi1−pi, where MH is the pseudo‐markers matrix with entries 0, 1, and 2 representing the number of copies of each haplotype allele in a haploblock, and pi is the MAF of haplotype alleles i, mH is the number of haplotype alleles of the whole genome, and PH is a matrix with columns equal to 2pi. In the GHBLUP+GBLUP model:a∼N0,σa2GH is a vector of random additive genetic effect, where GH is the additive genetic relationship matrix constructed using haplotype alleles and σa2 is the additive genetic variance based on the haplotype alleles, Z is the incidence matrix associating a; au∼N0,σau2G is a vector of random additive genetic effect, where G is the additive genetic relationship matrix constructed using nonblocked SNPs and σau2 is the additive genetic variance based on the nonblocked SNPs, Zu is incidence matrix associating au; a is composed of haplotype alleles effects and au is composed of SNP effects estimated from outside the haploblocks.The haplotype‐based genomic best linear unbiased prediction  or haplotype alleles (BayesBH model) or SNPs and haplotype alleles (BayesBH+BayesB model), Zj is a vector of alleles counts (0/1/2) at SNP j or haplotype allele j, BayesB assumes that some of the αj have no effect. This is defined by the binary variable δj that represents whether covariate j was fitted in the model according to hyperparameter π, such that δ=1 with probability 1−π, or δ=0 with probability π. In this study, π was set to 0.96, 0.995, and 0.995 for LW, DP, and LDMW, respectively, according to the previous studies y=u+2.6y is the vector of the adjusted phenotypes, and gebv is the vector of the GEBVs.Both the accuracy and bias of GP were assessed based on 5‐fold cross‐validation (CV). The CV procedure is that the whole population was randomly divided into 5 groups, one of which was taken as the validation group each time, and the remaining 4 groups were used as the reference group. This procedure was randomly repeated 5 times for GBLUP and Bayesian models, respectively. The prediction accuracy is the average Pearson correlation coefficient between the adjusted phenotypic values and genomic estimated breeding values (GEBVs) in the validation groups , the average number of SNPs within each haploblock ranged from 3.53 to 5.17, while the maximum number of SNPs within each haploblock was from 53 to 99.As shown in Table r2, and the number of fixed‐SNPs) were presented in Table As for the fixed‐SNP method, the number of haploblocks was from 33,603 to 134,412, while the number of fixed‐SNPs was between 5 and 20. Details about haploblocks  decreased from 605,908 to 403,841, while the number of fixed‐SNPs were 5 (5‐SNPs) decreased from 755,697 to 382,073. The number of alleles based on the two methods decrease by 33.35% and 49.44%. Our results revealed that 5‐SNP methods contain more alleles with low MHAF. Details about haplotype structure at different MHAF thresholds were shown in Table Moreover, under the strict MHAF thresholds (from 0.01 to 0.1), the number of haplotype alleles based on the haplotype‐based method with 3.23.2.1HBLUP, GHBLUP+GBLUP, BayesB, BayesBH, and BayesBH+BayesB) using different haplotype construction methods. In this study, our results revealed the accuracies of GP decreased with the trait heritability. The accuracies of haplotype methods showed great improvement for all three traits under some LD compared with the GBLUP model. Among these traits, we observed the highest accuracies for LW, followed by DP and LDMW . Comparing with BayesB model, BayesBH+BayesB model with r2 > 0.3 for LW (h2 = 0.40) and DP (h2 = 0.22) increased ~6.53 ± 3.02% and 0.72 ± 5.64%, while this model increase ~8.54 ± 7.44% for LDMW (h2 = 0.16) under r2 > 0.4  were used to investigate the performance of GP. We found that the 5‐SNPs GHBLUP model can significantly increase (p < 0.05) the accuracies of GP for all three traits compared with the GBLUP model, while the bias between GHBLUP based on fixed‐SNPs and GBLUP model significantly increased with the number of fixed‐SNPs , except for LW with 5‐SNPs BayesBH model (p < 0.001). Comparing with the BayesB model, the difference in accuracies of the 5‐SNPs BayesBH model was 5.74 ± 2.95%, 0.2 ± 5.49%, and 3.61 ± 6.41% for LW, LDMW, and DP, respectively  based on the number of fixed‐SNPs. Four models  and the increase of the number of fixed‐SNPs (from 5 to 20). The computation time of GP using the LD‐based BayesBH+BayesB model was similar to that of the fixed‐SNPs BayesBH model. The computation time of the GHBLUP+GBLUP model was slightly larger than GHBLUP, and GBLUP model due to long time was required to calculate the covariance of the two G matrixes in the GHBLUP+GBLUP model. Because the MCMC sampling often takes at least an order of magnitude, the Bayesian models cost much more time to fit covariates compared with the GBLUP models  based on the different haplotype construction methods in beef cattle. We found the haplotype‐based GP generated better performance compared with SNPs, which was consistent with previous studies  and the number of fixed‐SNPs (from 5‐SNPs to 20‐SNPs). These results were similar to previous studies  based on diverse LD levels and different numbers of fixed‐SNPs. We found that the haplotype methods can produce high accuracies across the three traits in most scenarios when LD thresholds were  Figures . As showve Table  reportednd Janss  also revnd Janss  found thIn this study, the LD‐based methods showed slightly higher accuracy under some thresholds compared with the accuracies of GP using fixed‐SNP methods. One possible explanation is that LD‐based haploblocks can reflect the true block structure of the population compared with fixed‐SNPs haploblocks. However, LD‐based methods did not consistently show improvement over SNPs or fixed‐SNPs haplotype as expected for all three traits Figure , which m4.3HBLUP model has a high improvement for GP compared with the GBLUP model  were used to construct the haploblock, the LD‐based methods produce a high accuracy of GP compared with the fixed‐SNP methods Tables  and S2. 4.6In this study, the optimal MHAF threshold for LW was set at 0.01, and the optimal MHAF threshold for DP and LDMW was set at 0.025. Under the optimal MHAF threshold, we found the accuracies of GP increased using the haplotype‐based method compared with the SNP method, while the computation time was reduced. Hess et al.  suggeste5r2 > 0.3 or r2 > 0.4 and the number of fixed‐SNPs was five, the Bayesian haplotype methods (BayesBH+BayesB) showed high accuracies across the three traits in most scenarios. Meanwhile, the average computation time of the LD‐based BayesBH+BayesB method can be reduced by 29.3% compared with the BayesB SNP method. LD‐based GHBLUP+GBLUP and fixed‐SNPs‐based GHBLUP methods using the 5‐SNPs can increase the accuracy of GP compared with GBLUP for all three traits. The optimal MHAF threshold for LW was 0.01, and the optimal MHAF threshold for DP and LDMW was 0.025. Further studies were required to evaluate the relationships within and between training and validation sets and further improve the accuracy of GP.Our study showed that haplotype‐based Bayesian and GBLUP methods can improve the prediction accuracy up to 8.54% and 9.93% compared with the individual SNP approach. The LD‐based haplotype approach has larger improvements than that of fixed‐SNPs. When LD thresholds were This work was supported by the National Natural Science Foundation of China , the Chinese Academy of Agricultural Sciences of Technology Innovation Project , the Beijing Natural Science Foundation (6154032), the National Beef Cattle Industrial Technology System (CARS‐37), the China Agriculture Research System of MOF and MARA, and the Science and Technology Project of Inner Mongolia Autonomous Region (2020GG0210).The authors declare that they have no competing interests.Table S1Click here for additional data file.Table S2Click here for additional data file.Table S3Click here for additional data file.Table S4Click here for additional data file.Table S5Click here for additional data file.


X-ray crystallographic analysis demonstrated C–H hydrogen bonds and bidentate conformations with triflate and iodide anions, yet the steric bulk of the central functional group seems to impact the expected trends in halogen bond distance. In solution, anion titration data indicated elevated performance from the receptors that utilize C–H Hydrogen Bond enhanced Halogen Bonds (HBeXBs). Collectively, the results suggest that even modest hydrogen bonds between C–H donors and iodine acceptors can influence molecular structure and improve receptor performance.C–H hydrogen bonds have remarkable impacts on various chemical systems. Here we consider the influence of C–H hydrogen bonds to iodine atoms. Positioning a methyl group between two iodine halogen bond donors of the receptor engendered intramolecular C–H hydrogen bonding (HBing) to the electron-rich belt of both halogen bond donors. When coupled with control molecules, the role of the C–H hydrogen bond was evaluated. Gas-phase density functional theory studies indicated that methyl C–H hydrogen bonds help bias a bidentate binding conformation. Interaction energy analysis suggested that the charged C–H donors augment the halogen bond interaction—producing a >10 kcal mol C–H hydrogen bonds to iodine halogen bond donors are shown to improve halogen bonding and molecular preorganization. Rev. 2010, 110, 10, 6049–6076.Historically, evaluations of CH HBing to organic halogens have focused on various conformational analysis studies highlighting a preference for gauche conformations over anti in alkyl halides. For a relevant review see:  Considering over half of organohalogen drugs launched contain heavier halogens with the faculty for halogen bonding ,17 this deficiency should be addressed. Herein, we evaluate the capacity of C–H hydrogen donors to hydrogen bond with iodine atoms by measuring the performance of halogen bonding anion receptors  justifies the attractive interaction between the halogen and Lewis bases. The electron rich region is largely responsible for the linearity of the interaction  and various attractive “side-on” interactions with electrophilic species such as metals.20The halogen bond is a noncovalent interaction between an electrophilic halogen and a Lewis basea hydrogen bond-enhanced halogen bond (HBeXB).21 A hydrogen bond to the electronegative belt of a halogen bond donor further polarizes and strengthens the halogen bond donor. HBeXBs have influenced macromolecule stability,22 small molecule anion binding,23 organocatalysis,24,25 and have been quantified in a fundamental solution study.16 Each of these studies employ “traditional” –OH or –NH donors, prompting us to consider the efficacy of C–H hydrogen bond donors to operate in a similar manner. To test this, we have constructed a series of charge-assisted bidentate halogen bond receptors to evaluate C–H HBeXBing in solution, the solid-state, and in silico.More recently, this electronegative region of a terminal organohalogen has been utilized as a hydrogen bond acceptor while simultaneously donating a halogen bond—23 The meta-bis-ethynyl core enables three planar conformations—bidentate, S, and W  control (4) . In scaffolds 2 and 3, the pyridinium methyl functionality was placed on the backside of the receptor, removed from the binding pocket. Receptor 2 featured a methyl group para to the pyridinium nitrogen, directed into the binding pocket to evaluate C–H hydrogen bonding to the halogen bond donors. Scaffold 3 is a control molecule of 2 where the methyl group was replaced with a hydrogen atom. Scaffold 4 is structurally like 2 and 3 with the pyridinium methyl group directed away from the pocket but has an internally directed –NH2 group—included to benchmark the C–H HBeXBing.To evaluate this hypothesis, four bidentate halogen bond receptors with a bis-ethynyl pyridinium core and flanking benzene arms were constructed (trol (4) . The pyrTo evaluate the influence of C–H hydrogen bonding on scaffolds 1–4 computational investigations were carried out using Gaussian 09 at the M06-2X/def2-TZVPP level of theory and the small-core energy-consistent relativistic effective core potential (def2-ECP) applied to iodine . However, the W conformation of 2 is nearly 0.5 kcal mol−1 higher in energy than the S conformation. The relative preference for 1 to adopt the bidentate conformation highlights the impact of the stronger C–H hydrogen bond and favourable molecular dipoles. Control molecule 3, lacking the C–H hydrogen bond donor, favors the W conformation by 0.34 kcal mol−1 over the S conformation. Notably, the bidentate conformation of 3 is 0.68 kcal mol−1 less stable than the favored W conformation. 4 was evaluated to compare the conformational preference for a receptor containing a more traditional N–H hydrogen bond donor. 4 favors the bidentate conformation by 2.34 and 4.82 kcal mol−1 over the S and W conformation, respectively. The data indicate the stronger N–H hydrogen bonds provide more preorganization within this system. Overall, the conformational analysis highlights that C–H hydrogen bonding to the iodine halogen bond donors can stabilize the convergent bidentate conformation.Scaffold 1 clearly prefers the bidentate conformation over the S and W conformation by 0.93 and 2.29 kcal molVs,max) of the iodine donors when adopting the bidentate conformation  maps of 1–4 provided an estimate of halogen bond strength by assessing the σ-hole . If C–H HBeXBing was enhancing the halogen bond donor in 2, we would expect 2 to have a greater Vs,max value. One possible explanation for this observation is that the electron donating effects of the methyl group nullified any polarization afforded by the C–H hydrogen bonds. Although a recent paper discussing distance and substituent effects indicates this may be negligible.26 The similar Vs,max values of 2 & 3 may suggest a potential limit to the C–H HBeXB and that receptor differences in solution between these two could be dictated by preorganization effects. Molecule 4 has the second greatest Vs,max  of the receptors evaluated, nearly 7 kcal mol−1 greater than 2 and 3, confirming that a stronger hydrogen bond donor will elicit greater σ-hole augmentation.1 had the greatest −1). Receptor 4 had the second strongest interaction energy which was 0.66 kcal mol−1 less than 1. The results of 1 and 4 was expected considering the electronics of 1 and the strengths of the hydrogen bond donors in 1 and 4.Halogen bond interaction energies  analysis. Receptor 2 had a 3.26 kcal mol−1 greater interaction energy than 3, suggesting that the methyl C–H hydrogen bond donors strengthen the halogen bond.§§− and each time the receptor adopted a distorted receptor conformation resulting in two halogen bonds and a C–H hydrogen bond to the iodine  bond critical points (BCPs) and bond paths (BPs) between the methyl group and the iodine atoms, suggesting a bonding relationship. Scaffolds 1 and 2 have two unique BPs—due to the geometry of the receptor and the methyl C–H hydrogen bond donors. In one case, there is a BP running directly from the hydrogen atom to the iodine in the same plane suggesting the presence of a monodentate C–H⋯I–C hydrogen bond. In the other case the BP splits the two hydrogen atoms and runs from the parent carbon atom to the iodine  and what is more commonly asserted as a bifurcated (or trifurcated) hydrogen bond.30,31 These reports when compared to our BP findings suggest the BP may be considered a carbon tetrel bond, yet the angle between the donor and acceptor doesn't fit the tight criteria suggested for identifying these tetrel bonds in the solid-state.30 Furthermore it has been noted with an oxygen acceptor that the C–H⋯O hydrogen bond is comparable in binding energy to the C⋯O tetrel bond.32 The geometry presented here represents an opportunity for further carbon tetrel bond investigation and classification. Regardless, the BPs identified here provide additional evidence of C–H⋯I–C hydrogen bonds.Facilitated by Multiwfn, Fig. 36S. Previou33 Overall, these theoretical data indicate that C–H hydrogen bonding to iodine atoms is occurring which would aid in receptor preorganization. The in silico data also suggests that the halogen bond is enhanced by the hydrogen bond but further physical studies in the solid and solution state are required.In contrast, 3 shows no BCP or BP between the aryl CH proton and the iodine donors. As expected, 4 has BCPs and BPs between the amine protons and the iodine atoms, aligning with a previous AIM HBeXB study evaluating intramolecular amide hydrogen bond donors.−) salts of 1–4 were synthesized and crystallized to provide an initial assessment of preorganization and C–H HBeXB in the solid state   and 3·OTf−  have halogen bond contacts that are longer than 1·OTf− which agrees with the computational evaluations on interaction strength. Between these two receptors 2·OTf− exhibits both the longest and shortest halogen bond contacts whereas 3·OTf− without C–H HBeXB has contacts that are intermediate  —highlighrmediate . The OTf− crystallizes in the tetragonal space group P̄42c1 and adopts an S conformation with a Z′ = 1  Å, 174.72(11)° and 3.6590(4) Å, 176.96(6)°, respectively. The ≈0.04 Å shorter halogen bond contact of 3·I−, without any C–H hydrogen bond donors directed to the iodine rich belt of the halogen bond donor, potentially suggests the lower limit of C–H HBeXB or that electron donating effects of the methyl group are influencing the contact distance. Alternatively, the steric bulk of the methyl group of 2 might be preventing the alkynes from bending as much, thereby inhibiting shorter halogen bond contacts. For example, the iodine-to-iodine distance in 2·I− is 6.3882(5) Å while in 3·I− the distance is 6.1792(5) Å. The alkyne distortion is further demonstrated by measuring the angle formed by the centroids of each ring (arm-core-arm angle) of the receptor—a smaller angle would indicate the alkynes clamping down on the anion. In 2·I− this angle is 124.559(3)° whereas 3·I− is reduced to an angle of 123.025(4)°. So, it is possible that the C–H hydrogen bond donors from the methyl group enhance the halogen of 2 yet also introduce steric hindrance.To limit the influences of the polyatomic anion we crystalized structures 1–4 with monoatomic iodide . Paralleand 0.88 . The str− offered another opportunity to compare the intramolecular hydrogen bond as this structure was also isomorphous with 2·I− and 3·I−  and 175.46(11). The halogen bonds of 4·I− were shorter than both the 2 and 3 iodide structures which correlates with both the MEP and interaction energy analysis. The iodine-to-iodine distance was 6.2764(10)Å which represents a midpoint between 2·I− and 3·I−, further suggesting that a central hydrogen bond donating group, whether that is a NH2 or a CH3, may limit the ability of the receptor to distort in this system.||||1·I has a iodine-to-iodine distance of 6.0546(4) and the angle of 121.002(19). These parameters do not follow the pattern suggested. We believe there are several possibilities for why 1·I does not follow the trend. First is that the structure crystalized in a different space group resulting in different packing. Second is that the location of the pyridinium is different. Thus, the halogen bond is more potent and possibly overcomes some of the steric clashing penalties imposed by the central methyl group. The angle formed by the centroids of each ring of the receptor was 123.94(4)° and is a midpoint between 2·I− and 3·I− following the trends of steric size (i.e. H < NH2 < Me).34The crystal structure of 4·I− Fig. S1. 4·I− hameta-bis-ethynyl core enables 1 and 2 to display C–H hydrogen bonds to the iodine atoms. In the presented structures of 1 and 2, the methyl groups maintain a single “strong to moderate” C–H hydrogen bond with the iodine as well as two “weak” CH hydrogen bonds—based on the parameters outlined by Johnson, Haley, Pluth et al. in a study focused mainly on CH hydrogen bonding to sulfur species.7 In their study the authors also expanded their analysis to other acceptors including organic iodine in a Cambridge Structural Database (CSD) analysis. The distances and angles of the strong to moderate C–H hydrogen bonds of 1 and 2 fall within regions that have the greatest number of observations in this reported CSD study. In contrast, the weak Hydrogen bonds fall in a region where C–H contacts trend, albeit fewer observations are noted. These geometries, coupled with the findings of the aforementioned CSD search is further indication that C–H⋯I–C are operating within this system.The geometry and the rigid directional nature of the 1H NMR titrations were carried out to obtain association constants. However, minimal shifting of 1H resonances upon introduction of anions as tetrabutylammonium salts dictated the use of other spectroscopic methods.**d6 was the media investigated with various tetra-n-butylammonium anion sources. Unfortunately, minimal shifting of the 1H NMR resonances necessitated modifications. An initial consideration was that the OTf− anion was potentially outcompeting the iodide anion, leading to minimal changes in chemical shift. However, the PF6− and the BArF− complexes 1 also showed minimal shifting upon addition of different anions. We then considered that a derivative with a –CF3 group para to the iodine donors might enhance binding  and potentially improve solubility. Unfortunately, CF3–1·OTf− displayed the same minimal shifting and exhibited minimal solubility enhancements when evaluating other solvents. Presumably the DMSO solvent leads to weaker binding vs. the ultimate solvent choice of THF/DMSO/H2O (90/9.9/0.1) in the UV-vis studies. During all these studies there was minimal shifting of all the proton resonances, including the CH3 methyl groups. This suggests no hydrogen bonding between the anion and the CH3 groups.Early efforts used the triflate salts of 1–4 as the starting host species. To ensure solubility, DMSO- As such, we employed UV-Vis spectroscopic titrations. UV-vis titrations of 1–4·OTf− were conducted with tetra-n-butylammonium bromide (TBABr) at 20 °C in a THF/DMSO/H2O (90/9.9/0.1) mixture to ensure that all compounds were soluble and that there was a constant amount of water present  by fitting the change in the absorbance to a 1 : 1 binding model using Bindfit.36,37 The Ka values measured followed the trend of 1·OTf > 4·OTf > 2·OTf > 3·OTf (−1). When comparing the values for 2·OTf and 3·OTf there is a slight difference in receptor performance. 2·OTf has a slightly larger association constant than 3·OTf (15 000 M−1vs. 12 000 M−1) indicating that C–H HBeXB is improving binding. As alluded to in the theoretical evaluations (as well as previous HBeXB papers), this is likely due to a combination of both preorganization as well as halogen bond enhancement. For comparison, 4·OTf (with the amine hydrogen bond donor) exhibited stronger binding (18 000 M−1) than 2·OTf and 3·OTf.Our theoretical analysis and the solid-state investigations suggested that C–H HBeXB may improve anion receptor performance. To test this hypothesis in solution  > 3·OTf  and coreIn conclusion, we have synthesized a series of halogen bond anion receptors to systematically evaluate C–H HBeXB. Collectively, the computational, solid-state, and solution phase evaluations suggest that C–H hydrogen bonds to iodine atoms can improve halogen bond receptor performance. The location of the charge is important—having the charge located near the binding pocket can produce C–H HBeXB receptors that outperform traditional HB donors where the charge is further away. In contrast, weaker C–H donors may be the limit to this polarization enhanced noncovalent interaction. The data also provide rare evidence of a hydrogen bond between C–H donors and iodine acceptors and suggest that preorganization may be a dominate factor, while the augmentation of the halogen bond seems to be subtle based on the solid-state and theoretical data. The identification of C–H HBeXB has broad implications for supramolecular designs as traditional hydrogen bonds are often used to preorganize molecular structure. These common N and O hydrogen bond donors are pH sensitive which can, in some circumstances, limit their utility. In contrast, non-traditional C–H donors offer pH insensitivity and, when coupled with the pH insensitive halogen bond, highlights one functional potential for this unique supramolecular design. Future studies are intended to target solvent effects and preorganization studies to better understand the influence of C–H hydrogen bonds to iodine atoms.See ESI.†DAD and OBB conceptualized the project. DAD, JS, and MRB conducted synthesis and characterization. DAD conducted computational and solid-state studies. DAD and JS conducted the solution studies. DAD wrote the paper. OBB supervised the investigation and provided editorial assistance during manuscript preparation. All authors examined the data, results, and conclusions presented here.There are no conflicts to declare.SC-013-D2SC03792K-s001SC-013-D2SC03792K-s002


The boosting of ionic permittivity is attributed to local defect-induced strain, and maximum Ti–O bond length is proposed as a descriptor. The Ti–O phonon mode that is responsible for the large dielectric constant can be tuned by a local strain and symmetry lowering from substitutions. Our findings help explain the recently observed colossal permittivity in co-doped rutile, attributing its intrinsic permittivity boosting solely to the lattice polarization mechanism, without the need to invoke other mechanisms. Finally, we identify new perovskite- and rutile-based systems that can potentially display colossal permittivity.We examine the effect of isovalent substitutions and co-doping on the ionic dielectric constant of paraelectric titanates  using density functional perturbation theory. Substitutions increase the ionic dielectric constant of the prototype structures, and new dynamically stable structures with  Due to their temperature and electric field stability, paraelectric materials are preferred for CP over ferroelectrics whose dielectric constant is high but also unstable near the phase transition temperature. With recent experimental reports on colossal permittivity in paraelectric TiO2 rutile via co-doping3, microscopic mechanisms for permittivity boosting need to be well understood. The apparent CP due to polarization of charge depletion layers at contacts, grain boundaries, interfaces, and other spatial heterogeneities5 , is classified as “extrinsic”. Other extrinsic mechanisms include thermally activated conductivity by carrier hopping. A notable example is CaCu3Ti4O12, having a colossal permittivity close to 105 by the Maxwell–Wagner effect8. On the other hand, possible bulk mechanisms including ionic polarization, metal–insulator transition, and charge density wave formation8 can be classified into “intrinsic”, i.e. bulk mechanisms that are active at all temperatures. Of these the ionic polarization is of greatest practical interest, as other mechanisms incur high dielectric losses, limiting applications. Since the early reports of large, temperature-dependent permittivity in insulating tetragonal YBa2Cu3Ox6+10, the overlap of intrinsic and extrinsic processes has complicated pinpointing the microscopic origins of the apparent CP. In co-doped rutile, a thermally activated process, most likely the transport of hopping carriers, was found to be responsible for the largest part of the apparent colossal permittivity3. At the same time, a nonactivated, intrinsic mechanism of permittivity boosting coexists in Nb + In co-doped rutile. The mechanism does not switch off at low temperatures; in fact, the enhanced permittivity increases with decreasing temperature11. The temperature trend follows that of pure rutile, suggesting a common microscopic origin. While theory accounts well for the observed dielectric constant of pure rutile, and attributes it largely to the ionic mechanism, no theoretical explanation exists for permittivity boosting in co-doped rutile. Originally, the “electron-pinned defect-dipoles” mechanism has been proposed in In + Nb co-doped TiO2 rutile2, suggesting the electronic nature of the large response. Further studies of In + Ta and Al, Ga, In + Nb co-doped TiO213 also attributed the observed colossal permittivity to the electronic mechanism. On the other hand, the studies of Nb + In co-doped TiO2 single crystals at sufficiently low temperature where thermally excited carriers are frozen suggested that the enhancement of the permittivity should stem from intrinsic effects11.Achieving materials with a very large  dielectric permittivity (CP) is crucial for various technological applications, including miniaturization of on-chip capacitive elements in transistors, high density memory, sensors, as well as high power density energy storage2, perovskite CaTiO3, and Ruddlesden–Popper Sr2TiO4 and and Sr3Ti2O7, modified via co-doping or isovalent replacement. From the analysis of the results, the permittivity increase in these materials is attributed to impurity-induced local strain, which enhances the dielectric response by softening the Ti–O phonon mode. We find that the frequency of the mode dominating the dielectric response can be changed over a wide range by substitutions, offering a possibility to precisely engineer the ionic dielectric constant.Here we show that the ionic mechanism can indeed account for boosting the permittivity in co-doped rutile and other substituted metal oxides. To this end, we perform a computational study of the ionic part of the dielectric permittivity of co-doped and substituted rutile and other Ti-containing dielectric oxides. We focus on substitutions in stable paraelectric prototypes, such as rutile TiO3 with Pnma structure, rutile TiO2, as well as Ruddlesden-Popper phases Sr2TiO4 and Sr3Ti2O7. Ca, Sr → Ba, Pb, Sr, Ca substitutions in perovskites, as well as III–V  and II-VI  co-doping in rutile are performed. A more detailed description of the parent compounds and substitutions is given in the Supplementary Information. A total of 150 rutile and 1178 perovskite and Ruddlesden-Popper structures were calculated. The goal is to analyze the structures where substitutions increase εion and identify relevant phonon modes. Among dynamically stable titanates (ν2 > 0), we found the maximum Ti–O bond length, max(dTi-O), reflecting maximum local strain, to be a good descriptor of the highest achievable ionic dielectric constant. The correlation between εion and max(dTi-O) is rationalized in terms of the softening of the phonon mode that corresponds to the Ti–O bond stretching. The descriptor was considered by analogy with Goldschmidt factor14, which is used as a descriptor for structural stability of perovskites. As high ionic dielectric constant is dominated by the Ti–O mode, lowering its frequency significantly increases the dielectric response.We examine the effects of isovalent substitutions and co-doping for titanates on phonons, Born effective charges, and ionic part of the dielectric constant. The electronic part of the dielectric constant is typically ~ 1–10 across most insulators and was not considered here in detail. The prototypes for substitution include mineral perovskite CaTiOion αβ is calculated as a sum over the gamma point phonon modes s according to16e is an elementary charge, M0 is a convenient mass reference, typically taken to be 1 amu16, V a unit cell volume, αth Cartesian component of the unnormalized effective charge vector for phonon mode s, and νs is mode frequency. ξs,κβ are the eigenvectors of the dynamical matrix, normalized according to Σκβξs,κβ ξs’,κβ = δss’, and Z*κ,αβ are the atomic Born effective charges, describing the proportionality17 between forces on atoms the and macroscopic electric field as Fκ,β is the βth component of the force acting on the κth atom, and αth component of the macroscopic electric field. As the mixed second derivatives of the total energy, Born effective charges can also be expressed as the derivatives of the components of the polarization vector τκ,β as: Z* and dynamical matrix eigenvector ξ need to have large components in that direction for the same atom.The ionic part of the dielectric tensor εdTi-O), with a visible peak around max(dTi-O) ≃ 2.02 Å. The average ionic dielectric constants of co-doped rutile TiO2 as a function of max(dTi-O) is also shown in Fig. ion>, as seen from Figs. dTi–O) near the peak of εion is higher than the typical bond lengths in titanates with moderate dielectric constants  = 1.98 Å), raising the possibility of mode softening by local strain from substitutions. The soft ferroelectric Ti–O mode has been studied extensively in titanates19, and its sensitivity to external perturbations such as volume changes or degree of Ti 3d–O 2p hybridization has been well recognized. In rutile, volume increase drives the Ti–O A2u mode softening20, with frequency becoming imaginary and structure developing instability and possibly becoming ferroelectric at a certain extensive strain. The instability is manifested across different titanates21, and stabilized by anharmonic effects22. In Fig. ion> with max(dTi–O) in Ti rutile under uniform strain.The dielectric constant has a pole where the A2u mode becomes unstable with its frequency becoming zero. The shaded area near the pole corresponds to the range of the maximum dielectric constants in Fig. ion in substituted/co-doped oxides is reached in the same range of max(dTi–O) where the softening of Ti–O mode occurs in rutile. By examining the microscopic origin of the large response, it is indeed confirmed that the same soft Ti–O mode is responsible for the large dielectric constant in these oxides. Note that we do not find similar mode softening in Si- or Sn-based rutile structures, as seen from Figs. 2, SiO2, and SnO2 rutile structures are compared.Figure 2Sr2Ti4O12, obtained from the orthorhombic CaTiO3 by Ca → Ba, Sr substitutions, Fig. z axis and is dynamically stable. It has an anisotropic ionic dielectric tensor with <εion>  = 3373, εzzion  = 9015, dominated by the lowest optical mode at ν4 = 0.521 THz, contributing 99.9% of εzzion . This is unlike CaTiO3, where many modes contribute to εion, as seen from the summary given in Table 2Sr2Ti4O12, max(dTi–O) = 2.01 Å, similar to that of the dynamically unstable cubic BaTiO3, but unlike that structure, shorter Ti–O bonds are also present. Apparently, the ferroelectric Ti–O mode of BaTiO3 is stabilized by insertion of Sr planes, preventing the formation of unstable Ti–O chains. This effect can also be viewed as the softening of the infrared-active modes of CaTiO3 by Ca → Ba and Ca → Sr substitutions. Just like the rutile, the infrared-active modes of CaTiO3 are softened by tension23. Projecting this low energy mode onto the modes of cubic BaTiO3 shows that it is mostly made up of the ferroelectric Ti–O mode of BaTiO3, with a small contribution from the displacement of Ba against the Ti–O cage. Similarly, a large εzzion  = 1228 is achieved in a Ba2Ca2Ti4O12(α) structure with alternating Ba and Ca planes, Fig. 2Sr2Ti4O12 and Ba2Ca2Ti4O12(α) structures, and the larger εzzion  in Ba2Sr2Ti4O12 is due to its larger 2Ca2Ti4O12(α) in Fig. 2Ca2Ti4O12(β), structure shown in Fig. 5 = 2.02 THz, with zzion  = 568 for that structure, as compared to ν4 = 0.53 THz, 2Ca2Ti4O12(α) in Fig. 4 and ν5 for the structures in Fig. z direction, which is prevented in the latter by staggering Ca and Ba as in Fig. As an illustration, consider Bazzion  = 252 in BaCa3Ti4O12 (not shown), a modest enhancement over the prototype CaTiO3 with εzzion  = 108. The dominant mode 5 in BaCa3Ti4O12 at 2.51 THz contributes ~ 184 to its εzzion . Projection onto the modes of Pnma BaTiO3 shows it to be a linear combination of the modes at 4.17i THz, 3.26 THz, and 5.26 THz of BaTiO3. The former is the ferroelectric Ti–O mode, whereas the latter two are Ba-only modes. Here, the stable A-site modes are mixed into the Ti–O mode, leading to its stabilization. Thus, lowering the symmetry of the structure via substitution is important for the stabilization of the Ti–O mode. This is to be expected, as the Ti–O chain structure instability is nonlocal, i.e., Ti and O atoms are coherently displaced along an infinite Ti–O chain, and there exists a minimum correlation length for the mode to become unstable24. The instability has been extensively analyzed24, with Ti 3d–O 2p hybridization and a partially covalent nature of the Ti–O bond being determined essential to the ferroelectricity19. The 1D chain-like nature of the instability was noted from the mode’s nearly flat zero-frequency isosurfaces in the Brillouin zone25. Real-space analysis has shown24 that the minimum length of the Ti–O chain for the instability to appear is ~ 10 atoms or 5 unit cells; thus any stabilizing perturbation must act on a shorter scale. In the Nb-In co-doped rutile, a soft mode with a large dielectric constant contribution appears at ν = 0.594 THz, as seen in Fig. 2u mode has a frequency of 5.28 THz.We find this mechanism of permittivity enhancement through softening/mixing Ti–O modes to act in other substituted titanates. We obtain ε2Sr2Ti4O12 and Ba2Ca2Ti4O12(α), shown in Fig. 3 mode, which is also confirmed by mode projections. In contrast, in the Nb-In co-doped rutile, the mode is localized on a single Ti atom in the xy plane, and delocalized along the z direction, as seen in Fig. The visualizations of phonon modes responsible for the large dielectric response are shown in Fig. zzion  = 9015 obtained for one of Ba2Sr2Ti4O12 structures may be close to a practically achievable maximum value for the ionic dielectric constant. From Eq. (εion ≃205 × (Z*/ν)2, assuming a cubic unit cell with a side of 6 Å, and mode frequency ν in THz. Thus, in order to have εion ~ 104, one needs Z*≃7 for ν = 1 THz. As larger Z* values to achieve even higher εion seem unrealistic, further increasing εion then requires decreasing the mode frequency below 1 THz, which may eventually result in instability and switching to a ferroelectric state.We also note that the value ε2Sr2Ti4O12, compared with its formation energy of ~ − 145 meV/atom. The closeness in energy among different configurations indicates that their mixing should occur at finite temperatures, and thermodynamic averaging, together with consideration of anharmonic effects, should be performed in order to estimate the dielectric constant at nonzero temperature.In order to assess the feasibility of synthesis of doped titanates, we calculate their formation energies by computing the energies of the reactions26. Our calculated average ionic dielectric constants as a function of the band gap are shown in Fig. 27. Discovering CP materials with both high dielectric constant and wide band gap remains a challenge that must be addressed in the future work.CP materials must possess sufficiently large band gaps in order to prevent dielectric breakdown, which is believed to be electronic in natureεion ~ 104 can be achieved in some compounds, establishing them as new candidates for colossal permittivity materials. The microscopic origin of the permittivity increase, namely the softening of the optically active phonon mode by a local defect-induced strain, may point to a new direction to search for colossal permittivity in other materials.Based on the results of first-principles calculations, we show that co-doping and substitutions offer a versatile and efficient pathway to increase the ionic dielectric constant in titanate paraelectrics. Most of the obtained compounds are found to be both dynamically and thermodynamically stable, offering a good prospect of experimental synthesis. Large ionic dielectric constants 28 with Hubbard U correction was used. U = 3 eV was applied to d electrons in transition metals, except for Ti and Sc, where U = 0 was used. The full list of the U values is given in the Supplementary Information. The k point grid density of 3000 k points·atom (approximately corresponding to a distance of ~ 0.2 Å−1 between the k points) was used for the integration over the Brillouin zone. Full cell shape and atomic position relaxations were carried out prior to DFPT calculations.The ionic dielectric response was calculated using density functional perturbation theory (DFPT) approach, as implemented in the VASP package. PBEsol functionalSupplementary Information.


An accurate modelization of the physisorption of N2 and CO2 on the surface of carbon‐based porous materials is important to obtain accurate N2 and CO2 storage capacities and reliable PSDs. Physisorption depends on the dispersion interactions. High precision ab initio methods, such as CCSD(T), consider accurately the dispersion interactions, but they are computationally expensive. Double hybrid, hybrid and DFT‐based methods are much less expensive. In the case of graphene, there are experimental data of the adsorption of N2 and CO2 on graphite that can be used to build the Steele interaction potential of these gases on graphene. The goal is to find out hybrid and/or DFT methods that are as accurate as the CCSD(T) on benzene and as accurate as the experimental results on graphene. Calculations of the interaction energy curves of N2 and CO2 on benzene and graphene have been carried out using the CCSD(T) method and several double hybrid, hybrid, and DFT methods that consider the dispersion interactions. The energy curves on benzene have been compared to the CCSD(T) and the energy curves on graphene have been compared with the Steele energy curves. The comparisons indicate that double hybrids with dispersion corrections and ωB97 based DFT methods are accurate enough for benzene. For graphene, only the PBE‐XDM functional has a good agreement with the Steele energy curves.Experimental isotherms of N 2 and CO2 physisorption on benzene and graphene.N Obtaining the pore size distributions (PSDs), of these materials is an important part of the characterization. An accurate PSD is relevant for optimal gas storage and separation applications. Unfortunately, this is a complicated task due to the complex structure of these materials and to the indirect methods used to obtain the PSD.NMost of the indirect methods consist on using experimental and theoretical isotherms of a gas inside slit‐shaped pores of different widths, to solve an adsorption integral equation. The PSD is the solution of the equation. The indirect methods of this type differ on (a) the procedure to obtain the theoretical isotherms and/or (b) the procedure and details to solve the adsorption integral equation. The first method of this type was proposed by Seaton et al.Some indirect methods of this typeThe PSD obtained by means of any indirect method depends on the interaction potential or on the density functional used to obtain the isotherms. An interaction potential more accurate than the LJ potentials and than the non‐local functional based on hard spheres of the NLDFT model could yield more accurate (a) theoretical isotherms and (b) PSDs of solid nanoporous materials.2 and CO2 on carbon‐based solid porous materials. More specifically, the goal is to find the DFT or hybrid method that yields the most accurate interaction energy curves of N2 and CO2 on benzene, compared with the CCSD(T) energy curves, and on graphene, compared with the Steele potential energy curves of N2 and CO2 on graphene, obtained from experimental data. The most accurate DFT and hybrid methods will be used to run first principles calculations of the interaction of N2 and CO2 on carbon‐based nanoporous materials, saving memory and computer time. The 3d interaction potential energy obtained in those calculations will be used to run GCMC simulations of the N2 and/or CO2 isotherms of the carbon‐based nanoporous materials. These accurate theoretical isotherms, in turn, will be used to obtain more accurate PSDs.The goal of this paper is to find accurate DFT or hybrid methods that can be used to calculate the 3d‐interaction potential energy of N2 and CO2 on graphene is, due to its experimental origin and to experimental data available, the best option to study accurately the interactions of these molecules on graphene or in slit‐shaped pores based on graphene layers. However, for realistic carbon‐based solid porous materials, the Steele potential cannot be used. The Coupled Cluster Single and Double (Perturbative Triple), CCSD(T), method takes into account accurately the weak interactions of molecules, H2, N2, CO2, and so forth, on a surface, while the density functional theory, DFT, methods do not include those interactions or include them with different degrees of complexity. Hybrid and double‐hybrid DFT functionals include with some degree of accuracy the weak interactions. The computational cost of the CCSD(T), double‐hybrid, hybrid, and DFT methods scales as O(N7), O(N5), O(N4), and O(N3), respectively, where N is proportional to the system size. The amount of atoms of many carbon‐based solid porous materials makes unfeasible to run CCSD(T) calculations, while DFT and hybrid calculations of the interaction of a molecule with those materials are feasible.Graphene and benzene have been selected as carbon‐based surfaces, because they have well geometrically defined surfaces and because slit‐shaped pores are made of graphene layers. The Steele interaction energy curve2 on benzene, obtained with the CCSD(T) method and several DFT methods. The present research is, somehow, a continuation of the H2 on benzene research. Of particular relevance to this study is a previous work by Witte and co‐workers2‐benzene complex. The present study assesses a larger set (28 vs. 15) of DFAs, including double‐hybrids, the interaction of N2 in addition to the interaction of CO2, and extends the analysis to the physisorption of both moieties on graphene. Up to our knowledge, no assessment of such a large set of functionals has been performed so far on these systems.In a former publication,2, benzene‐N2, graphene‐CO2, and graphene‐N2. Finally, Section The structure of this paper is as follows. The methodology used in this study is described in Section 22.12 and CO2 were placed on top of the three main sites of the substrates (benzene or graphene): The center of an hexagon, H, the center of a C—C bond, B, and a carbon atom, A. There are three main relative orientations of the molecular axis of the molecule: The axis perpendicular to the substrate plane, ⊥, the axis parallel to the substrate plane and parallel to two C—C bonds of the hexagon, ∥∥, and the axis parallel to the substrate plane and perpendicular to two C—C bonds of the hexagon, ∥⊥. The combination of site and orientation is a configuration. There are nine main configurations. The following notation has been used for the configurations: The letter indicates the site where the molecule was placed and the set of symbols after the letter, denotes the relative orientation of the molecular axis. An additional possible configuration in which the main axis of N2 or CO2 orients itself parallel to the benzene surface on top of a C—H bond has not been considered since this particular configuration is not relevant for graphene. The nine configurations considered are depicted in Figures 2 on benzene and graphene, respectively, but they are also valid for CO2.The molecules N2.22 and CO2 on benzene and graphene: Gaussian 16 (G16RevA.03), Orca 4.2.1, Molpro 2018.2, and Quantum Espresso (QE) 6.3. DFT and DFT‐based methods that include dispersion interactions are implemented in these codes. The same procedures have been used with the four codes: For each of the methods used, the geometries of N2, CO2, benzene, and graphene were optimized. The geometry optimization was performed until the forces acting on the atoms were less than the threshold values depending on the code. We provide the specific values in the corresponding subsections in the Supporting Information.Four codes have been used to carry out the present calculations of N2, CO2, benzene, and graphene for each method, the most stable configuration of each molecule, N2 or CO2, on a substrate, benzene or graphene, was searched for, using one of the first principles method for all the configurations depicted in Figures After the optimization of the geometries of NE(d) between benzene or graphene and a N2 or CO2 molecule placed at a distance d from the benzene or graphene plane (d is the distance between the center of mass of the molecule and the benzene or graphene plane) is defined, for all the computational codes, in terms of the energies of the interacting system and the separated components as:2 or CO2 and A can be benzene or graphene. The binding energy Eb is the interaction energy at the equilibrium molecule‐benzene or molecule‐graphene plane distance de, that is, Eb = E(de). A negative value of Eb means that the molecule is bonded to benzene or to graphene. The interaction energy curve is the set of pairs or points ).The interaction energy 2 and CO2 were kept frozen, with the optimized geometries obtained in the optimization process for the corresponding method, and only the distance between the center of mass of N2 or CO2 and the benzene or graphene plane was changed. That distance was changed around the interval 3–4 Å to obtain the most stable configuration and, to obtain the interaction energy curve, that distance was changed from 2 to 7 Å, with a step of 0.1 Å.To obtain the most stable configuration and also to obtain the interaction energy curve, the individual geometries of benzene or graphene, NTable The use of atom‐centered finite basis sets in DFT calculations in Gaussian, Orca, and Molpro gives rise to the well known Basis Set Superposition Error (BSSE). For this reason, the interaction energies have been calculated always using the counterpoise (CP) correction method of Boys and Bernardi.−3 eV/molecule in the region near the minimum, and it becomes negligible at larger N2‐benzene distances).The atom–centered all–electron aug‐cc‐pVTZ basis setA short description of the DFAs tested in this study along with technical details on the calculations performed with the four codes used in this work are provided as Supporting Information.2.32 on benzene and graphene.2 and CO2 on benzene. In that methodology, the amounts RMSEx, the root‐mean‐square error, and RMSPEx, the root‐mean‐square percentage error, (x = m or x = t) for the method F are defined as:x = m stands for the region around the minimum of the interaction energy and x = t stands for the tail region. The ranges of the molecule‐plane distances dix used to calculate RMSE and RMSPE of the interaction energies of N2 and CO2 on benzene are shown in Table In a previous publication we used a quantitative methodology to compare the interaction energy curves of H2 and CO2 on graphene,2 and C‐CO2 interactions used in the Steele potentials of N2 and CO2 on graphene, were obtained from experimental data of N2 and CO2 on graphite as follows.In the case of graphene, we have used as reference, the Steele potential energy curve of N2 on graphite are −0.104 ± 0.003 eV and 3.34 Å, respectively, and the binding energy and equilibrium distance of CO2 on graphite are −0.1784 eV and 3.2 ± 0.1 Å, respectively, according to the experiments reviewed by Vidali et al.ϵ and σ of the C‐molecule interaction.Vidali et al.2 and CO2 on graphite, simulating the graphite as a slab made of 101 graphene layers, separated the experimental distance between layers on graphite, 3.35 Å, for different values of the parameters ϵ and σ of the C‐N2 and C‐CO2 interaction, searching for the values that yielded a Steele potential of N2 on graphite with a binding energy and an equilibrium distance that matched their respective experimental values. The obtained values for C‐N2 were ϵ = 65.3kB and σ = 3.36 Å. The parameters obtained in the present work were used later to calculate the Steele potential energy curve of N2 on graphene, a single layer. The minimum of the Steele potential energy curve of N2 on graphene is at 3.4 Å and −0.0912 eV/mol.We ran calculations of the Steele potential of N2 interaction used in the Steele potential of CO2 on graphene, were obtained using the same procedure as in the case of N2. The values of the parameters of the C‐CO2 interaction of the Steele potential obtained in the procedure were ϵ = 123.2kB and σ = 3.22 Å, as mentioned before. The equilibrium CO2‐graphene distance and binding energy of the Steele energy curve of CO2 on graphene are 3.2 Å and − 0.1584 eV/mol, respectively.The values of the parameters of the C‐COϵ and σ parameters for the Steele potential of N2 and CO2 on graphite obtained in the present work, with those published in the scientific literature. We compare also the equilibrium distance and minimum of the Steele potential corresponding to those parameters with the experimental values of the interaction of N2 and CO2 on graphite. The C‐N2 and C‐CO2 parameters are obtained using the Lorentz‐Berthelot rules.2 and C‐CO2 parameters are reported. The parameters for N2 are similar to the obtained with present procedure. There is some disagreement with the parameters for CO2. However, the present obtained parameters agree better with the experiments of N2 and CO2 on graphite published by Vidali et al.In Table 33.12, CO2, benzene and graphene using CCSD(T) and the DFAs collected in Table 2 and CO2 on benzene and graphene.We have optimized the geometries of N3.22 on benzene consisted on finding the most stable or optimal configuration of the N2 molecule on benzene. The equilibrium and binding energies for the nine configurations of N2 on benzene obtained in CCSD(T) calculations are reported in Table The first step of the calculations of N2 on benzene in the configuration H⊥ using the aug‐cc‐pVTZ basis set and the counterpoise method, yielded a binding energy of −0.0311 eV at an equilibrium molecule‐plane distance of 3.9 Å. The binding energies of the configuration H∥∥ and H∥⊥ are −0.0608 at the same equilibrium molecule‐plane distance, 3.45 Å, according to CCSD(T) calculations, using the counterpoise method and the aug‐cc‐pVTZ basis set. The difference is very small, 2.72 × 10−5 eV, and identical to the energy converge tolerance, 2.72 × 10−5 eV. We have selected the configuration H∥∥ is the most stable configuration and all the DFT and hybrid calculations of N2 on benzene have been carried out on this configuration.CCSD(T) calculations of N∥∥ and H∥⊥ configurations.A previous study on this complex by Jaeger et al.2 molecule on the most stable configuration on benzene, H∥∥. The N2‐benzene plane equilibrium distances and the binding energies obtained are given in Table The second step on this research consisted on calculations to obtain the equilibrium distance and binding energy with DFT and hybrid methods, with the N2 and A = benzene. The interaction energy curves of N2 on benzene obtained in the calculations are plotted in Figures 2 molecule was in the H∥∥ configuration. Some horizontal lines corresponding to 0, −0.01, −0.02, −0.03, −0.04 eV, and so forth have been plotted in those figures, to guide the eye in the comparison of the different curves.The last step consisted on obtaining not only the location and value of the binding energy, but also the whole interaction energy curve. The interaction energy is defined in Equation , with mo2 and collected in Table The values of the RMSE and RMSPE quantities, defined in Equations  and 3),,3), have2 storage and adsorption on large carbon pores, the best performer is PW91, but still with a large RMSE of 0.0026 eV (37%). Clearly, LDA and GGA interaction energy curves of N2 on benzene are very different from the CCSD(T) curve  energy curve than the curves obtained with the other methods, as can be noticed in Figure The DFT‐D or semiempirical method curves  curve in both regions: the binding and the tail region. The vdW‐DF curves are very different from the CCSD(T) curve  energy curve in the tail region: The RMSEs of this group of methods are between 0.0012 and 0.0026 eV and the RMSPEs between 6% and 24%. In the binding region, however, there is not a general trend of this family of methods. The rev‐vdW‐DF2 curve coincides very well with the CCSD(T) curve (RMSE of 0.0028 eV and RMSPE of 6%), the vdW‐DF2‐C09 curve has high values of RMSE and RMSPE  and the vdW‐DF2 and RVV10 have a regular performance at the binding region, with RMSEs and RMSPEs around 0.011% and 24%, respectively. Finally, the DFSAPT curve is very similar to the CCSD(T) curve, with very low RMSE and RMSPE values in the binding and tail regions.2 on benzene, are ωB97X‐D, rev‐vdW‐DF2, B2PLYP‐D3, and B97D, in this order. The comparisons also indicate that all the hybrids, except MN15 and DSD‐BLYP, perform very well. The methods PBE‐D2, PBE‐XDM, vdW‐DF2, and RVV10 perform worse, but their RMSE and RMSPE values are reasonable.The qualitative comparisons in Figures 3.32 on benzene, on the nine configurations depicted in Figure 2 on benzene is the B∥⊥ one (see Table We have performed CCSD(T) calculations of CO2 on benzene, in the configuration B∥⊥, using different first principles methods, to obtain the interaction energy curve. The corresponding CO2‐benzene plane equilibrium distances and the binding energies are given in Table 2 on benzene is about −0.1 eV.We ran calculations of CO2‐benzene system. The results are similar to those discussed previously for the N2‐benzene system. Thus, the energy curves of the LDA and GGA methods are very different from the CCSD(T) energy curve  energy curve  curve in the binding region, but their RMSE and RMSPE values are reasonable. In the tail region the ωB97 energy curves are very similar to the CCSD(T) curve: Very low RMSE values, between 0.0004 and 0.0010 eV, and also very low RMSPE values, between 3% and 6% in the tail region. This shows that these DFAs describe correctly the long tail of the potential energy. The DSD‐BLYP curve has a regular performance (RMSPEs of 21% in both regions). The MN15 curve has a reasonable performance in the binding region (a RMSPE of 17%) and a poor performance in the tail region, with a high relative error: A RMSPE of 50%.The The agreement of the DFT‐D energy curves with the CCSD(T) curve in the binding and tail regions is similar  analysis. Table In order to shed some light on the nature of the bonding between benzene and both NEelst1+Eind2+Edisp2) for the N2 and CO2 complexes, respectively. Indeed, first‐order interactions (E1=Eelst1+Eexch1) are positive, meaning that electrostatic forces alone cannot hold these systems bound. Induction contributions (Eind2+Eexch−ind2) are small and also positive for both complexes. Thus, DFSAPT results show that, as expected, these systems are mainly bound by dispersion forces.Table E(1) + E(2) yield interaction energies of −0.0587 and −0.0936 eV for N2 and CO2, respectively, a little smaller than the reference CCSD(T) values, −0.0608 and −0.0990 eV, respectively. As pointed out in the methods sections, higher that second‐order contributions can be estimated by means of the δ(HF) term. In these systems, δ(HF) represents a small percentage of the total interaction energy  suggesting that first‐ and second‐order terms are clearly the most important contributors. Including higher‐order terms, the DFSAPT interaction energy is −0.0667 and −0.0999 eV, for N2 and CO2, respectively, slightly larger than the reference values. For N2 the uncorrected energy is closer to the reference; in the case of CO2, the corrected one is better. In any case, both of them are within 0.007 eV of the CCSD(T) value, showing that DFSAPT provides very accurate interaction energies for these systems.The total first‐ and second‐order components 3.52 on graphene, plotted in Figure ∥⊥. The interaction energy and the binding energy are defined in Equation . That potential is based on experimental data, as it was explained in a former section. The Steele potential energy curve is the reference curve for N2 on graphene and all the DFT energy curves will be compared with that.Calculations of the interaction energy of N2 on graphene. The errors, equilibrium distances and binding energies obtained with several DFAs are in Table 2 on graphene, used to calculate the errors, are in Table The plot of the interaction energy curves serves as a qualitative comparison. A quantitative comparison with the Steele potential energy has been also made, by means of the calculation of the RMS and RMSP errors, using as pivot or reference the Steele potential energy curve of NThe comparison of the DFT interaction energy curves with the Steele interaction energy curve Figures  reveals The semiempirical methods perform very well in the tail region: Their RMSE and RMSPE values are low, between 0.0020 and 0.0042 eV and between 10% and 22%, respectively  and the other functionals have a regular or poor performance . The vdW‐DF2‐C09 underestimates a little the binding region and the other vdW‐DF2 functionals overestimates by a large factor the binding region, as can be noticed in Figure 2 on graphene, the PBE‐XDM and the vdW‐DF2‐C09 functionals have a good or reasonable agreement with the Steele interaction energy curve in both regions and the PBE‐D2, PBE‐D3 and rev‐vdW‐DF2 have a regular agreement, according to the results plotted in Figures Among the DFAs studied for N3.62 on graphene, plotted in Figure 2 on graphene is the configuration B∥⊥. The interaction energy and the binding energy are defined in Equation  and the performance of the other functionals is regular or very poor in the binding region (RMSEs in the range 0.0237–0.0369 eV and RMSPEs in the range 17%–28%). The vdW‐DF2‐C09 functional underestimates the binding energy and the binding region by a large factor, the RVV10 functional overestimates the binding region also by a large factor (RMSPE of 26%) and the vdW‐DF2 overestimates the binding region by a moderate factor (RMSPE of 17%).2 on graphene, the functionals with a good or reasonable performance  for CO2 on graphene are: PBE‐D2, PBE‐D3, PBE‐XDM, vdW‐DF2 and rev‐vdW‐DF2.Taking into account the results in both regions of the energy curve of CO42 and CO2 on benzene and graphene have been calculated by means of different methods: CCSD(T), double hybrid, hybrid and DFAs. The energy curves on benzene have been compared with the CCSD(T) energy curve, and the energy curves on graphene with the Steele potential energy curve obtained from experimental data.The interaction energy curves of N2‐benzene, CO2‐benzene, N2‐graphene, and CO2‐graphene) can be noticed. The overestimation of the binding energy and region by the LDA functionals and the underestimation by the GGA functionals observed in the present calculations of the four systems, have been also observed in many physisorbed systems. The group of vdW‐DF functionals underestimates the binding energy and the energies of the binding and tail region of the interaction energy curves of the four systems studied in this work. A similar underestimation of the binding and/or the interaction energies, but not identical, was also reported for another systems. Kocman et al.2 on coronene and coronene modified with boron and lithium using different DFAs and the CCSD(T) method. They also obtained that vdW‐DF functionals overestimate the binding energy and the energies in the binding and tail regions of the interaction energy curve. The vdW‐DF functionals also underestimate part of the binding region and all the tail region of the interaction energy curve of H2 on benzene, when compared with the CCSD(T) interaction energy curve.2 on benzene. These functionals underestimate by a large factor the adsorption energies of the lowest level of H2 on graphene, compared to the experimental value of the adsorption energy.Some general trends of the DFAs used for the four systems  calculations, we find that the best agreement is found among double hybrids with dispersion corrections and ωB97 based DFAs. In the short or binding region of the interaction energy curve of these molecules on benzene, double hybrids with dispersion corrections slightly outperform ωB97 functionals for benzene‐CO2, whereas for benzene‐N2 both set of functionals perform equally well. On the other hand, in the long or tail region of the interaction energy curve, both double hybrids with dispersion corrections and ωB97 functionals provide similar accuracy. It is worth to point out that, according to the present results of N2 and CO2 on benzene, the dispersion corrections must be added to the double hybrids, in order to reproduce the CCSD(T) curves accurately. More specifically, DSD‐BLYP (without dispersion corrections) performs poorly in the long range region for both systems and must be used with caution.When comparing atom centered basis set results for both systems, benzene‐N2‐benzene and CO2‐benzene). Nevertheless, we would like to highlight the PBE‐D2 and rev‐vdW‐DF2 functionals, which efficiently reproduce long range curves and can be recommended for studies focused on this region, like physisorption processes. DFSAPT, on the other hand, provides very accurate results compared to CCSD(T) and can be recommended both in the binding and long range regions. It must be pointed out that DFSAPT has a similar computational cost as a GGA functional.Regarding the rest of tested DFAs, none of them achieve the same consistent accuracy for both systems (N2 and CO2 on graphene obtained using several DFAs, with the Steele interaction energy curves, based on experimental data, indicate that only the PBE‐XDM functional has a good or reasonable agreement for both systems in the binding and tail regions. The PBE‐D2, PBE‐D3, and rev‐vdW‐DF2 functionals have a mediocre performance for N2 on graphene, but perform reasonably well for CO2 on graphene.The comparison of the interaction energy curves of NAppendix S1 Supporting Information.Click here for additional data file.