Corrected zero-point relative energies (kcal/mol) of ABIOD isomers.
In the present work, molar fractions were obtained as a function of temperature with different levels of theory for the most representative isomers of three systems belonging to the family of pseudosaccharins. The choice of those three systems was due to the fact that it is known in the scientific literature that these systems present very small differences in their relative energies which make a complicated experimental characterization, in addition these compounds are of interest in the biological area. These systems represent challenges not only from an experimental point of view but also from a theoretical point of view. From the theoretical perspective, this study is also complicated since several possible isomers with very similar energies are presented. The diagrams of species distribution (molar fractions) provide information that cannot be accessed through the electronic structure calculations at T = 0. Here, this tool was useful to identify the most probable isomer from several quasi-degenerate isomers and to discern if thermal effects favor any of them, as well as to find trends despite the different results of each level of theory. Additionally, an analysis was performed on vibrational, rotational and electronic data in order to know the reason of the behavior of molar fractions as function of temperature.
- molar fractions
- DFT calculations
- statistical mechanics
Having the same chemical composition, a molecule can present different geometrical arrangements of atoms, leading to distinct chemical and physical properties. Here lies the importance of knowing the arrangement of the atoms inside molecules. The different atomic arrangements in a molecule are known as isomers and their study gives rise to isomerization or isomerism. Computational chemistry is one of the most used tools in this branch of research. Through the analysis of stabilities by means of the electronic energy, the isomers that could be found in an experiment can be known. However, when the energetic differences between isomers are very small, it is difficult to assure that the lower energy isomer is the only one present in the experiment. Existing models, both in the framework of methods of electronic structure or molecular dynamics, have contributed greatly to the study of the different types of isomerization, and have been improved taking into account, for example, the inclusion of solvents. Despite advances in theory and technology, no current theoretical methodology is 100% accurate. This leads to the fact that it is not sufficient to consider only the results yielded by these methods, mainly in molecules whose isomers have very similar energies, which could be considered as part of the error associated to the level of theory.
For this reason, given the importance of elucidating the isomerism of a molecular system with very small energy differences, it is important to make use of other tools. In this chapter, we will obtain diagrams of species predominance (molar fractions) as a function of the temperature (assuming that the equilibrium has been reached) for some systems of the pseudosaccharin family that have isomeric structures with small energy differences and are of great importance in various fields of applied sciences.
Pseudosaccharins, also known as benzisothiazoles, are molecular systems with a wide variety of properties. Their structures are derived from saccharin, which is the first synthetic sweetener that is still used with that aim, but also has many industrial applications. Its photosensitive properties have aroused interest in the derivatives of this system for applications of synthesis of coordination compounds and bioorganic synthesis . Its general structure can be seen in Figure 1, where
Pseudosaccharins are compounds of interest in several fields of research, specifically in biologically active systems . Their usage has been reported in the treatment of cardiac and hepatic diseases [5, 6], and even for the treatment of breast cancer, as antibiotic agents and herbicides [7, 8, 9]. Adding to these applications, Hlasta et al.  proposed a series of compounds of the pseudosaccharins family as inhibitors of the human leukocyte elastase protein, which causes the destruction of the connective tissue of the lungs in people with pulmonary emphysema. They also found how this series of compounds have therapeutic usage in people with this condition [10, 11].
Due to the importance of this type of compounds, studies have been carried out to characterize experimentally and theoretically both their structure and properties. A high reactivity can be found mainly in the junction of the heterocyclic ring and
In previous studies [4, 13, 14], the importance of isomerization in these systems has been demonstrated. As the
Due to all the previously mentioned, in the present chapter a systematic study of the isomerization of three pseudosaccharins, which have very small energy differences between them and that have also been previously studied experimentally and theoretically, is performed. For this study, a variation of functional and BS is carried out, however, considering that a better comparison can be made between experimental and theoretical results if thermal effects are taken into account, molar fractions are obtained for the different isomers of pseudosaccharins. It is important to mention that, unlike previous studies on these compounds, populations are obtained as a function of temperature, taking into account different contributions through the vibrational, rotational and electronic partition functions. From this, it is possible to observe the effects of methodology in the molar fractions, as well as to find the reason of behavior of molar fractions plots.
2. Theoretical background
All calculations were performed with Gaussian09 suite of programs , while the construction of graphics of molar fractions as a function of temperature was made with a program designed by Professor Slanina et al. . Relaxation of molecular structures, vibrational frequencies, rotational constants and excited states were obtained through B3LYP [17, 18], M06-2X  and PBE  exchange-correlation (XC) functionals, along with 6-311++G(3df,3pd), 6-31++G(3df,3pd), 6-311+G(d,p) and 6-31+G(d,p) [21, 22] basis. The choice of the methodology was made taking as a starting point the level of theory used in previous studies [4, 13, 14]. Additionally, the other functionals and BSs were used in order to be able to compare the results and analyze the importance of the methodology. It is important to mention that, for selected cases, total electronic energies were refined through the second-order Møller-Plesset perturbation theory, MP2, along with 6-311++G(3df,3pd) and 6-31+G(d,p) level of calculation.
In this work, calculations of molar fractions (
3. Results and discussion
From previous studies on pseudosaccharins, three isomers were reported for 2-allyl-1,2-benzisothiazol-3(2
The molecular structure of the ABIOD pseudosaccharin can be considered to be mainly formed by two components; a heterocyclic ring and a flexible allyl substituent. In ABIOD, the molecule can be rotated two dihedral angles and they give rise to different isomers (see Figure 2(a)), nevertheless, according to Ref. , the dihedral angle formed by atoms 1, 2, 3 and 4 is irrelevant in the formation of isomers because of the sterical hindrance between the allyl group and the rest of the molecule. Consequently, only the dihedral angle formed by atoms 2, 3, 4 and 5 originate different isomers. In the same study, there were found a total of three isomers of ABIOD in a minimum of the potential-energy surface (PES) and they were represented using the abbreviations
In a first step, molecular optimization was performed through B3LYP functional; after that, reoptimizations were performed with M06-2X and PBE functionals.
The corrected zero-point relative energies of the three ABIOD isomers are presented in Table 1. For B3LYP XC-functional, it can be observed that the most stable isomer is
As can be observed from DFT calculations at T = 0, the analysis of isomerism is complicated, especially when small energy differences are presented between isomers. Since none of the three functionals (B3LYP, M06-2X and PBE) obtain the same most stable isomer, this quest appears to be quite dependent on the method used. However, it is also worth commenting that the energy difference between the three isomers is very small, regardless of the XC-functional used. Then, it could be considered that the energy differences are within the anticipated error of calculation and therefore, it would be expected that molar fractions as a function of temperature can provide relevant information.
Considering the previous discussion, Figure 3 shows the molar fractions as a function of temperature for the three isomers of ABIOD pseudosaccharine.
For B3LYP XC-functional it can be observed that the molar fractions do not vary significantly as the temperature increases, in fact, the line corresponding to the
For M06-2X functional, results show that for the first two BSs the graphs are similar and display the same trend in general; however, with 6-31++G(3df,3pd) it is observed that distribution of isomer populations are narrower than in 6-311++G(3df,3pd). Nonetheless, 6-311+G(d,p) and 6-31+G(d,p) present a different trend since the populations change, this behavior could be understood noticing that the energy difference between the three isomers obtained with these methodologies is very close (see Table 1), then, the vibrational frequencies, rotational constants and energy levels become important in the construction of the molar fractions. A deeper analysis on vibration frequencies, rotational constants and energy levels is presented below.
As mentioned in a previous paragraph, the relative energies obtained with PBE XC-functional place the
From Figure 3, it can be observed that B3LYP and PBE XC-functionals obtain the same trends: the less stable structure (
Due to the complicated comparison between the different results obtained, mainly with M06-2X, both at T = 0 and varying temperature, it was decided to refine the electronic energy through the MP2 method. As a first step, a comparison of energies of the three isomers MP2 optimized and non-optimized was performed.
Firstly, the molecular structures of the three isomers were optimized with MP2/6-311++G(3df,3pd) methodology resulting in the following relative energies:
In a second step and according to the results discussed in the previous paragraph, single point calculations were performed with MP2 along with 6-311++G(3df,3pd) and 6-31+G(d,p) BSs on previously relaxed structures with B3LYP, M06-2X and PBE in order to obtain relative stabilities. For B3LYP optimization and single point calculations through MP2, the relative energies were
As it can be seen, the
Molar fractions as a function of temperature were re-built using the electronic energy data from MP2 calculations on optimized structures with B3LYP, M06-2X and PBE (see Figure 4). As expected, the most populated isomer in all the range of temperature was
Having noticed that the vibrational partition function influences in an important way, the vibration frequencies obtained through M06-2X/6-311++G(3df,3pd), were observed for each isomer. Interestingly, it was found that the first frequencies for each isomer are 32, 35 and 14 cm−1, for
3.2. Pseudosaccharyl allyl ether 3-(allyloxy)-1,2-benzisothiazole1,1-dioxide
The pseudosaccharyl allyl ether 3-(allyloxy)-1,2-benzisothiazole1,1-dioxide (ABID) is a system that has been studied from an experimental and theoretical point of view . In that study, five structures were found in a minimum in the PES and whose energetic differences are small. The differences between the five isomers are particularly due to the values of two dihedral angles in the structure. In Figure 2(b) atoms that conform these angles are shown, the first of them is formed by the atoms numbered as 1, 2, 3, 4 and the second by the atoms 2, 3, 4, 5. From Ref.  the five isomers were named according to the values of the dihedral angles in question, the value of the angle formed by the atoms 1, 2, 3, 4 was represented using the abbreviations
Molar fractions as a function of temperature of the five isomers with the XC-functionals and the four BSs used in previous subsection were obtained. For each level of theory, the necessary frequency vibrations were obtained in order to obtain, on one hand, the molar fractions and, on the other hand, to corroborate that molecular structure is in a minimum of PES; rotational constants for building the rotational partition function; and 10 excited states for obtaining the electronic partition function. In Table 2, the relative energies obtained with B3LYP, M06-2X and PBE XC-functional and the proposed BS for the five isomers are shown. It can be observed that B3LYP and PBE got the same order of stability (
With M06-2X functional a change in order of stability is shown, in contrast to B3LYP and PBE functional,
The temperature of interest is around 350 K, since these compounds are synthesized at this temperature. Remember that for B3LYP XC-functional and the four BSs, the
Regarding M06-2X, there is an interesting event with the lowest potential-energy species, in spite of
From these results, it can be concluded that, in one hand, both B3LYP and PBE XC-functional remark the same trends and, on the other hand, in spite than M06-2X obtained a more complicated distribution of relative energies at T = 0, when thermal effects are taken into account, the same trends in populations of the five isomers compared to B3LYP and PBE functionals are obtained.
The third system of the pseudosaccharine family studied in this work is tetrazole-saccharyl (see Figure 2(c) and (d)). This system, as well as the previous ones, has been studied from an experimental and theoretical approach, for this molecule there are two tautomers and for each one of them several isomers can be generated by the rotation of two dihedral angles. In Figure 2(c) and (d) the involved atoms in the dihedral angles are shown, again the structures were named according to the letters
Tautomers are called as 1H tautomer and 2H tautomer, the first one is obtained when H is bonded to atom number 5 (Figure 2(c)) and the second one is obtained when H is bonded to the nitrogen atom beside to atom number 5 (Figure 2(d)) . Relative energies were obtained for the same levels of theory as previous subsections.
In Table 3, corrected zero-point relative energies are shown. As it can be seen, B3LYP and PBE are consistent in the sense that they present the
Here it is worth to mention that in ABIOD and ABID molecules, M06-2X reduced the energy differences, while tetrazole-saccharyl M06-2X does not reduce the energy differences in comparison with the results of B3LYP and PBE. The molar fractions as a function of temperature are analyzed below.
Observing Figure 7, it can be seen that the trend of B3LYP and PBE with the four BSs is maintained, there are two predominant isomers in all the graphs at room temperature, with the first two BSs the isomer of greatest abundance is the
Analyzing the molar fractions as a function of temperature for M06-2X functional, it can be observed that the first two BSs found three isomers with significant populations and the rest remained with a very low population, therefore, it is possible to say that the three isomers whose populations are considered important are the same ones that are predicted as the most stable. For the third and fourth BSs, we can observe that the
An important difference between the results obtained by B3LYP and M06-2X is the change of the order in stability of isomers since M06-2X always brought the
As it can be observed, M06-2X predicts a different isomer as the most populated compared with B3LYP and PBE, however, it seems to be that the
In this work, an analysis of molar fractions as a function of temperature on three pseudosaccharyl systems (ABIOD, ABID and tetrazole-saccharyl) was performed with three XC-functionals and four BSs. In general, for ABIOD and ABID systems, M06-2X functional reduces the energy differences between the isomers, while B3LYP and PBE are in agreement with each other, but different from M06-2X. It is important to observe that in ABID pseudosaccharine, even if the order of stability changes with the M06-2X functional, the molar fractions maintain the same trend, placing the
Even though the molar fractions show a dependence on the enthalpies of the isomers, the most stable structure is not necessarily the one with the greatest population. By taking into account the vibrational, rotational and electronic partition functions in the calculation of the molar fractions, several aspects are involved to obtain the most predominant isomers at T ≠ 0. It was found that vibrational contribution is the most important factor in obtaining the predominant isomer. Therefore, the use of methodologies that correctly describe the vibrational spectra in order to obtain molar fractions is recommended. Additionally, when a theoretical-experimental spectra comparison is carried out, calculated spectra could be weighted by molar fractions at the temperature of interest.
One of the main differences between the results obtained with the different methodologies is that herein calculations can predict a different order of stability and modify the relative energies. However, in order to be able to assume that an isomer will predominate, it is important to consider the contributions of the different partition functions instead of only considering the relative potential energies.
The authors would like to thank PRODEP (formerly PROMEP) for the support provided through the 103.5/13/6900 office. ZNCG would like to acknowledge the financial support from CONACyT (Mexico) (Grant No. 413573). FJTR would like to thank the University of Guadalajara for authorizing sabbatical leave. DAHV would like to thank CONACYT for the support provided by the program “Apoyos para la Incorporación de Investigadores Vinculada a la Consolidación Institucional de Grupos de Investigación y/o Fortalecimiento del Posgrado Nacional”. CONACYT through Project 52827 is also acknowledged.