In this work we discuss certain consideration for quantum chemical and chemometric assessing in the crystallographic polymorphism. It is aimed primarily at researchers in ‘Analytical chemistry’ and is designed to help readers’ understanding of the implications of the different quantum chemical theories in the chemical crystallography.
The term ‘polymorphism’ reflects molecular ability to crystalize in more than one structure. Since, properties of polymorphs can vary, their quantification becomes an important task to manufacturing pharmaceutics. It affects packing properties via molar volume and crystal density; optical properties and refractive index; electrical and thermal properties; conductivity; hydroscopicity; other differences associated with thermodynamics; kinetics; surface and mechanical properties, and more.
In order to make a complex prediction of correlation among ‘molecular structure’–‘electronic structure’–‘energetics’ using crystallographic and quantum chemical data it should be taken into consideration various polymorph modifications.
TABLE OF CONTENT
Acknowledgments
Keywords
Polymorphism; Pharmaceutics; Quantum chemistry; Crystallography; Chemometrics Abstract
Abbreviations and acronyms
Introduction
2. Crystallographic and theoretical analyses of crystals of 5–hydroxy thryptophan
3. Crystallographic and quantum chemical analyses of crystals of biogenic amines
4. Crystallographic and quantum chemical analyses of crystals of 2’,3’- o -isopropylideneadenosine
Conclusion
References
PREFACE
In this work we discuss certain consideration for quantum chemical and chemometric assessing in the crystallographic polymorphism. It is aimed primarily at researchers in ‘Analytical chemistry’ and is designed to help readers’ understanding of the implications of the different quantum chemical theories in the chemical crystallography. The focal part of the discussion sets out the polymorphism of aspirin. The other parts of the paper extend the scope of the contribution considering crystallographic, quantum chemical and chemometric data about pharmaceutics. In the corresponding results–sections we present discussion concentrating on subtle electronic effects in polymorphs, thus illustrating the great capability of the computational quantum chemistry to distinguish between modifications showing perturbations of the atomic positions in the crystals.
Acknowledgments
The authors thank the Deutscher Akademischer Austausch Dienst for a grant within the priority program “Stability Pact South-Eastern Europe” and for an Evolution 300 UV–VIS–NIS spectrometer ; the Deutsche Forschungsgemeinschaft; the Alexander von Humboldt Stiftung (Germany) for the donation, a single crystal X–ray diffractometer; the central instrumental laboratories for structural analysis at Dortmund University (Federal State Nordrhein-Westfalen, Germany) and analytical and computational laboratory clusters at the Institute of Environmental Research at the same University. Conflict of interests: Michael Spiteller has received research grants (Deutsche Forschungsgemeinschaft, 255/22–1 and 255/21–1); Bojidarka Ivanova has received research grants (Deutsche Forschungsgemeinschaft, 255/22–1; Alexander von Humboldt Foundation, research fellowship).
This work was carefully carried out. Nevertheless, authors and publisher do not warrant the information therein to be free of errors. The work is being published in English aiming a widest access to the scientific contributions. English is not native language of the authors. Therefore, stylistic rough edges may occur. The authors hope of the understanding of the reader.
Keywords
Polymorphism; Pharmaceutics; Quantum chemistry; Crystallography; Chemometrics
Abstract
That paper deals extensively with relations between ‘ molecular structure ’–‘ electronic structure ’–‘ energy ’ of pharmaceutics such as salicylic acid (1), polymorphs (I–III) of aspirin (2), 5–hydroxy thryptophan crystalizing as crystallohydrate (3), 5-hydrohy-L-tryptophan barbituric acid co–crystal (4) and 5-hydroxy-L-tryptophan 1,3-dimethylbarbituric acid dihydratemolecule co–crystal (5); tyrammonium iodide (6), 3,5–diiodotyrosine (7), tyrammonium 5–sulfosalicylate (8), dopammonium 5–sulfosalicylate dihydrate (9)) and 2’,3’- o -isopropylideneadenosine (10), using crystallographic, quantum chemical ab initio and DFT molecular dynamics – adiabatic and diabatic computations – and chemometrics. Structures (1) and I have been redeterminated ((1): Monoclinic P21/c; a = 4.9293(8), b = 11.232(2), c = 11.602(2) Å, b = 90.648(6)o; V = 642.314 Å3, Z = 4; (2): Monoclinic P21/c; a = 11.4511(18), b = 6.6028(9), c = 11.4182(18) Å, b = 95.690(5)o, V = 859.069 Å3; Z = 4). The theoretical analyses have accounted for whether high resolution crystallographic measurements of disordered systems can be treated theoretically, producing distinguishable quantitatively trajectory of energetics accounting for subtle electronic effects. The results have contributed insights into the followings: (i) The total energy appears sensitive parameter towards subtle electronic effects; (ii) As far as the total energy difference between two independent crystallographic solutions of I of aspirin is D(ETOT) = |0.24701| a.u. The latter is lower than D(ETOT) of modifications I and II (ETOT Î 0.4780–0.5975 kcal.mol-1), but in parallel is higher than D(ETOT) = |0.1573| a.u. of disordered II, the evaluation of total energy parameters as only a quantity prevents a reliable study of subtle electronic effects; but (iii) the chemometrics of the trajectory profile of the total energy provides meaningful statistical information allowing us to distinguish between the electronic effects due to perturbations and atomic positions; disorders; plane stacking effect, and more.
Abbreviations and acronyms
ADMP – Atom-Centered Density Matrix Propagation (quantum chemical method); AIM – Atoms in molecules; ANOVA – Analysis of variance; BDE – Bond dissociation energy; BOMD – Born Oppenheim Molecular Dynamics (quantum chemical method); BOs – Bond orders; BCP – Bond critical point (analysis); BVM - Bond valence model; CT – Charge transfer effects; DFT – Density functional theory; ED – Electron density; ESPs – Electrostatic potentials; FBO – Fuzzy bond order; IR – Infrared (spectroscopy); LBO – Laplacian bond order; LPA – Lowdin population analysis; MD – Molecular dynamics; MM – Molecular mechanics; MOs – Molecular orbitals; MS – Mean square; NBO – Natural bond orbital (analysis); NCE – Natural Columbic energy (potential); NLMO – Natural local molecular orbitals; NEC – Natural electronic configuration; PCM – Polarizable continuum model; SS – Sum of squares; SD – standard deviations; TSH – Trajectory surface hopping; VNCE – Columbic potential energy term; UV–VIS–NIR – Ultraviolet-visible-Near infrared (spectroscopy); XRD – X-ray diffraction (in context measurements of powders).
Introduction
The term ‘ polymorphism’ reflects molecular ability to crystalize in more than one structure (Bond et al; 2007; Higashi et al. 2017). Since, properties of polymorphs can vary, their quantification becomes an important task to manufacturing pharmaceutics. It affects packing properties via molar volume and crystal density; optical properties and refractive index; electrical and thermal properties; conductivity; hydroscopicity; other differences associated with thermodynamics; kinetics; surface and mechanical properties, and more (Datta and Grant, 2004). In order to make a complex prediction of correlation among ‘ molecular structure ’–‘ electronic structure ’–‘ energetics ’ using crystallographic and quantum chemical data it should be taken into consideration various polymorph modifications. On this view, we have struggled with an analytical chemical problem to define quantitatively borders of perturbation of electronic structure and energetics of polymorphs, using as molecular templates crystals of (1) and (2) (Scheme 1), respectively. Compound (1) appears a structural analogous of aspirin, but is a conformational blocked, due to presence of intramolecular (OH…O=C) hydrogen bond.
Aspirin is a remarkable example of a bestselling pharmaceutics, which is broadly used to treat cardiovascular diseases, in addition to reduce angiogenesis of cancer (Xie et al. 2021). It is already implemented in the practice, however, shows complex polymorphism, including few known polymorphs (Bond et al. 2007a,b, 2011; Vishweshwar et al. 2005; Bag and Reddy, 2012; Wen and Beran, 2012; Shtukenberg et al. 2017; Arputharaj et al. 2012; LeBlanc et al. 2016); Price et al. 2009; Ouvrard and Price, 2004; Brela et al. 2016). The employment of crystallographic data for purposes of computational chemistry needs a detail assessment of variations of experimental crystallographic parameters, in order to, determine the error contribution to energy – the major parameter assessing bonding properties of molecules and thus their biological activity – and/or molecular properties, in parallel to, error contributions from theoretical methods. Despite, numerous efforts devoted to quantify accuracy of computational quantum chemical methods (Price, 2004; Duarte and Kamerlin, 2017), there are few contributions, stressing importance to quality of crystallographic data, talking about quantum chemical prediction of energetics of polymorphs (Ivanova and Spiteller, 2017; Zheng et al. 2014). The validation of crystallographic data is still not well understood; and, thus, even high-resolution measurements of a virtually perfect as quality single crystal could produce not straightforward 3D crystallographic coordinates of a molecule from the perspective of statistical uncertainty, which should affect on input 3D coordinates. There is of importance to understand the statistical meaning of these, so-called variations of experimental crystallographic data and how these variations affect on the computational results, in order to assess the reliability of computed free Gibbs energy parameter depending on quantum chemical methods. Notice, that there are standard set of validation procedures for structural deposition, however, a high statistical significance of crystallographic structural solution (or reliability at about 95.5 %) can be obtained only by means of minimal validation to a large set parameter (Zheng et al. 2014; Spek, 2017). Therefore, with increasing in complexity of molecular models, it becomes increasingly difficult to assess accurately comparing among sets of structural parameters; are these groups of data sufficiently different or not? The lack of precise determination of error contributions to theoretical energetics, has led to a conclusion that such data could only partially be true.
As an additional point of great interest, we will discuss the relation between electronic structures of conformational polymorphs; the energy of chemical bonds obtained on the base on experimental crystallographic and theoretical data about electron densities; and bond orders, respectively. With above concerns in mind, there can be deduce that since aspirin is an attractive molecular template (LeBlanc et al. 2016; Ouvrard and Price, 2004; Li, 2006, 2008), considerable previous work has been focused on analyes of conformational preference to different modifications; evaluation of phase transitions among polymorphs; and examining of energy contribution of vibrational motion of acetylating group, respectively (Bond et al. 2007a,b; Vishweshwar et al. 2005; Bond et al. 2011; Bag and Reddy, 2012; Wen and Beran 2012; Shtukenberg et al. 2017; Arputharaj et al. 2012, LeBlanc et al. 2016; Price, 2009; Brela et al. 2016). However, the largest part of these contributions mainly include static computations. It is known that solid-state 3D conformations of molecules in their crystals could be non-equilibrium ones, as well (Ivanova and Spiteller, 2017). So, in this work we investigate polymorphism of aspirin using ab initio and DFT molecular dynamics, involving adiabatic and diabatic approaches. On this base we provide a complete picture on boundaries of variation of energetics of all known polymorphs. The experimental topological refinement of EDs has been reported (Arputharaj at al. 2012). Nevertheless, it is not shown and discussed energies of bond, so that we have provided this new information associated with electronic structural redistribution and its effect on energetics of polymorphs. MD methods have been used studying hydrogen bonding interactions in dimers of I and II (Brela at al. 2016), but there is a lack of data about computations of unit cell content and conformations along with data about III, reported more recently (Shtukenberg et al. 2017).
Thus, the methods of computational quantum chemistry underlying both static and dynamic computations appear robust approaches providing highly accurate information about electronic effects, electronic interactions in the molecular systems, energetics, and more (Kroger et al. 2013 ; Rega et al. 2004; Schlegel et al. 2002; Iyengar et al. 2005; Iyengar, 2005; Thompson, 1998). In particular, ab initio and DFT molecular dynamics provide bridge between static thermochemical properties and dynamic simulations (Kroger et al. 2013 ; Rega et al. 2004; Schlegel et al. 2002; Iyengar et al. 2005; Iyengar, 2005; Thompson, 1998). The latter methods allow us to describe the evolution of electronic system of a molecular structure as function of the trajectory, furthermore accounting for electronic properties of systems at each of trajectory steps (Kroger et al. 2013 ; Rega et al. 2004; Schlegel et al. 2002; Iyengar et al. 2005; Iyengar, 2005; Thompson, 1998). Thus, a detail examination of chemical bond or electronic interactions between atoms in crystals can be carried out at various electronic states at every step studying the energy behaviour of systems depending on dynamics of electrons and nuclei in molecular crystals. As we could expect, the individual dynamics of systems at the trajectory steps should be different depending on electronic interactions. The latter ones should be distinguishable within the framework of different 3D molecular and electronic structures as well as intermolecular interactions of molecules in crystals, along with a difference of energy of interacting ensemble of polymorphs of crystals. It is important to emphasise that analysis could be applicable studying subtle bonding properties and electronic effects using high accuracy ab initio or DFT methods, or both of these. In this study, in particular, we have employed mainly atom-centred density matrix propagation, amongst others MD methods, which has been proven to provide an accurate description of bond breaking and formation along with a commonly highlighted advantages of method consisting on: (a) Treatment of all electrons in systems and/or employment of pseudopotentials, so that the accuracy can be balanced within a broad spectrum of methods; (b) employment of large time step for trajectory analysis; (c) capability of study charged species; (d) to account for effect of temperature and pressure on average electronic structure of a molecule; (e) accounting for quantum nuclear effects; et cetera (Kroger et al. 2013 ; Rega et al. 2004; Schlegel et al. 2002; Iyengar et al. 2005; Iyengar, 2005; Thompson, 1998). Given that, these approaches are able to obtain high accuracy information about electronic effects and electronic structures of crystals accounting for fluctuations of atomic positions, obtaining information from trajectory analysis as the so-called ‘ energy spectrum ’. More detail can be found (Kroger et al. 2013 ; Rega et al. 2004; Schlegel et al. 2002; Iyengar et al. 2005; Iyengar, 2005; Thompson, 1998). Abbildung in dieser Leseprobe nicht enthalten
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Scheme 1. Chemical diagram of crystals (1)–(10) and atom labelling; Dio denotes bond dissociation energies (i = 1–5).
The diabatic computations are based on trajectory surface hopping approach. There is employed method by Tully and Preston (1971), among others, where nonadiabatic transitions are described as vertical transitions between adiabatic potential energy surfaces at given concrete localized regions. The nuclear trajectory is propagated adiabatically using Born-Oppenheim approximation. In the surface hopping region, the electronic forces were treated using stochastic hopping. The conservation of the total electron–nuclear energy after the hope is achieved using rescaling of the nuclear velocity (More detail can be found in the section Theory/Computations).
1. Crystallographic and theoretical analyses of polymorphs of aspirin and salicylic acid
1.1. Crystallographic data of aspirin
In the literature, researchers have been urged to study aspirin’s polymorphism (Bond et al. 2007a,b, 2011; Vishweshwar et al. 2005; Bag and Reddy, 2012; Wen and Beran, 2012; Shtukenberg et al. 2017; Arputharaj et al. 2012; LeBlanc et al. 2016); Price et al. 2009; Ouvrard and Price, 2004; Brela et al. 2016). The polymorph I (hereafter referred to (2)–I or I) has been discussed (Bond et al. 2007a,b,2011). Table 1 contains unit cell parameters of known polymorphs, including crystallographic solution reported in this work (CCDC 1557027). The assessment of a second polymorph (hereafter referred to (2)–II* or II*) may be made (Vishweshwar et al. 2005) and (Bond et al. 2007a,b). A second polymorph (hereafter referred to (2)–II) has been discussed in (Bond et al. 2011) and (Bag and Reddy, 2012), as well. A third polymorph (hereafter referred to (2)–III or III) is known (Shtukenberg et al. 2017). A more critical chemometric analysis of crystallographic parameters of I–III shows that our structure corresponds to polymorph I. We see that chemometric data of both group of variables (unit cell parameters from independent measurements of I in this work and in (Bond et al. 2011)) and data about first determination of structure of aspirin dating since 1964 indicate an insignificant difference (Tables 1–3). We also see that method performance shows a high reproducibility of data. The chemometrics is based on single - factor analysis of variance; Kruskal - Wallis one-way analysis of variance; Tukey test and F-ratio. Pairs (or groups) variables of different populations are compared. The null hypothesis (H0) is that all mean values are equal, while the alternative hypothesis is that at least one mean value is different. The discussion considers mean values; standard deviations; maximum and maximum values, respectively. By understanding of these statistical parameters, we have assumed that: (i) The populations have been regarded continuous (not discrete), which in case of a comparative evaluations of unit cell parameters means that we have described data blocks quantities per polymorph (Table 1). Any polymorph is characterized by a concrete set of values a, b, c, a, b, g and V; and (ii) different populations are independent, which is ensured within the framework of independent data collections and crystallographic solutions. Figure 1 illustrates statistical difference among populations of I.
Table 1. Crystallographic data of polymorphs of (2); references are given in brackets
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Table 2. Chemometric analysis of total MD energetics of unit cell content of polymorphs (2)–II* (Vishweshwar et al. 2005), (2)–I (This work and work Bond et al. 2011); SS – Sum of square; MS – Mean square.
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Figure 1. Crystal structures of (1) and polymorph I of (2) reported in this work; CCDC 1557027 and 1557025; PLUTON plots of the unit cell contents and hydrogen bond networks; Chemometric analysis of the unit cell and geometry parameters of polymorph I obtained in this work and data from (Bond et al. 2011)
Table 3. Chemometric analysis of simulated powder XRD patterns of (1) (this work) from measurements in triplicate; SS – Sum of squares; MS – Mean square; α=0.0001.
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Table 2 shows F value 7.10-4 and p value 0.9795, thus, indicating an insignificant difference in quantities, as well. The populations of parameters of I: 11.27762 (a), 6.55171 (b), 11.27412 (c), 95.8371 (b) and 828.702 (V) (Bond et al. 2011); and 11.45112 (a), 6.60289 (b), 11.418218 (c), 95.6905 (b), 859.069 (V) are equal statistically. An absolute correspondence among groups surely requires a F-parameter equal to (‘0’) and p -parameter equal to ‘1’. The number of discrete values per population should be equal (or higher) than five (See Fig. 1). The linear regression and correlation operate with correlation coefficient (r), which should be +1 or –1 when a perfect linear relationship occurs. The obtained linear regression equations describe relations among groups of quantities, but do not indicate any exact relations between corresponding variables. The parameter ‘goodness–of–fit’ (r 2) of a regression model also reflects linearity. The r 2 varies Î [0–1]. The r 2 value is determined by a sum of squares of model divided by total sum of square of groups of variables shown a “y” – axis in Fig. 1. From this brief comment can be seen that r = 0.99999 examining the crystallographic unit cell parameters of I. But the geometry parameters are different (Scheme 2, Fig. 2). The statistical parameter is r = 0.99359. One does not need much crystallographic experience to see that despite identical unit cell parameters there are different electronic structures reflecting different energies of interactions.
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Scheme 2. Selected experimental geometry parameters of (2)–I (this work), (2)–II* (Vishweshwar et al. 2005), (2)–II (Bond et al. 2011) and (2)–III Shtukenberg et al. 2017; Bond lengths [Å] and dihedral angle [o]; Dimer of aspirin II as reported in Vishweshwar et al. 2005 (Molecules A and B with relating atom labelling scheme); D – Difference between values.
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Figure 2. Chemometric analysis of DFT–MD total and potential energies of polymorphs (2)–I and (2)–II*.
In context McCrone’s definition of polymorphism (Bond et al 2007): “… a polymorph is a solid crystalline phase of a given compound resulting from the possibility of at least two different arrangements of the molecules of that compound in the solid state …”, to an ensemble of arranged molecules are different electronic structures with different energies. However, when differences in geometry parameters of polymorphs are as slight as in case of aspirin, then it is of importance to account for uncertainty of quantum chemical methods studying perturbations of atomic positions. Moreover, according to a description of a known definition, showing that ‘polymorphism’ can considers “each different crystal structure to be a different polymorph” (Bond et al. 2007) (next section).
1.2. Quantum chemical treatment of aspirin
1.2.1. Molecular and electronic structure of aspirin in its polymorphs
A conformational analysis has shown that total energy (ETOT) at T = 0K provides meaningful information about correlation between ‘molecular structure’–‘energy’ (Ouvrard and Price, 2004). However, we explore correlation among ‘molecular structure’–‘electronic structure’–‘energy’, or we may comment that a quantitatie correlation is carried out not only accounting for crystallographic 3D geometry parameters, but also for electronic redistribution within the framework of each molecular conformation. There seems that in gas-phase a conformation of I close to crystallographic structure occurs. But is a less stable form comparing with conformation at global potential energy minimum (DETOT = 3.5 kJ.mol-1 or 0.8365 kcal.mol-1) (Ouvrard and Price, 2004). The study devoted to MD analysis of hydrogen bonded dimers of (2) (Brela at al. 2016) reports deviation from 1.4 kcal.mol-1 of bonding energy, which has been regarded as close to the ‘ chemical accuracy ’ (Ehrlich et al. 2017). The error is 0.4 kcal.mol-1 (6.37.10-4 a.u.). There is more than clear, in this respect, that evaluations of absolute total energy as only quantitative parameter prevent a reliable assignment of polymorphs. Moreover in (2) (Bond at all 2007) the structural could be so slight and consisting of almost identical rearrangements with differences between I and II consisting of stacked domains (Chan et al. 2010). This similarity can be assigned to twinning, which might be described using ‘order-disorder’ theory (Chan et al. 2010). The energy of interaction should be affected significantly by perturbation of position of disordered atoms (Massobrio et al. 2015). Within one and the same polymorph when disorder occurs different energies are obtained. Conversely, one and same molecular and electronic structures yield to identical energies. Apparently, ab initio and DFT static and MD data of I (this paper), II* (Vishweshwar et al. 2005), III (Shtukenberg et al. 2017), I (Bond et al. 2011) and II (Bond et al. 2011) show that an understanding of polymorphism is surely more comprehensive when we are able to consider MD energy trajectories of different polymorphs statistically. Identical electronic structures should produce identical energy-trajecory relations, keeping in mind that MD computations show accuracy at about 1.2 kcal.mol-1 of bonding free energy (Ehrlich et al. 2017). We see that MD simulations reveal a difference of total energy between I and II* DETOT = |0.14065| a.u. (88.259 kcal.mol-1, Fig.3). Form I is less stable comparing with II* modification. Apparently, these results confirm conclusions in (Wen and Beran, 2012). The MD data (Fig. 3) agree well with the obtained great difference of geometry parameters of (2) in both crystals as mentioned before (Scheme 2). Also, a known DFT study of I and II has shown that the former form is less stable (DETOT = 2–2.5 kJ.mol-1 (0.4780–0.5975 kcal.mol-1)). However, the range is larger than sub kJ.mol-1 energies in disordered crystals or both modifications have been classified as “virtually isoenergetic” (Wen and Beran, 2012). Despite, the employment of ab initio random structure searching method shows that both polymorphs I and II* are distinguishable in energy terms (Mathew et al. 2020.) The latter results agree excellent with our computations presented, herein. Tables 4 and 5 show that static DFT computations also indicate a higher stability of II. The DETOT is |0.009026| a.u. (|5.66390526| kcal.mol-1). Looking on the data in Tables 1,4 and 5 and Fig.3 it may seem to be a contradiction that crystallographic solutions of II* in (Ouvrard and Price, 2004) highlight the structure in (Bond et al. 2007a,b) as an inaccurate one.
Table 4. Chemometric analysis of unit cell parameters of polymorphs (2)–I (this work) and (2)–II* (Vishweshwar et al. 2005); SS – Sum of squares; MS – Mean square; a = 0.0001.
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Figure 3. DFT–MD computations of conformational polymorphs I, II* and III; Total and potential energies [Hartree] versus time in trajectory [fs]; DE – Energy difference [a.u.].
Table 5. Gas–phase thermochemistry (T = 273.15K, P = 1 atm) and BDEs (Dio) of the studied compounds in Scheme 1; e0 – Total electronic energy [a.u.(particle)-1]; EZPVE – Zero-point vibrational energy [kcal.mol-1]; e0 – Zero-point correction [a.u.(particle)-1]; Ecorr – Thermal correction to energy [a.u.(particle)-1]; Hcorr – Thermal correction to enthalpy [a.u.(particle)-1]; Gcorr – Thermal correction to free energy [a.u.(particle)-1]; E – Sum of electronic and thermal energies [a.u.(particle)-1]; E – Sum of electronic and thermal enthalpies [a.u.(particle)-1]; G – Sum of electronic and thermal free energies [a.u.(particle)-1].
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On the contrary our computations of energies of I and II* indicate a significant difference (Table 6). The potential MD energy trajectories of I and II indicate a relative mutual correspondence.
Table 6. Bond orders of conformers of (2); Atom labelling is in Scheme 1.
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The r is 96.12 % (Fig. 2). The same computational approaches are applied studying energetics of polymorph I and using our crystallographic solution along with data from work (Bond et al 2011) (See Figs. 4 and 5). As can be seen the results about the interacting ensemble of the molecules in the unit cell indicate that this form is more favorable from a thermodynamic point of view comparing with total energy value of I. The D(ETOT) is |0.24701| a.u. Importantly, this value is lower than energy differences between I and II (ETOT Î 0.4780–0.5975 kcal.mol-1) reported in (Wen and Beran, 2012). However it is higher than D(ETOT) = |0.1573| a.u. of II and II*, obtained in our study. As far as the later value, should correspond to a disordered systems according to the crystallographic description of the structure of II (Chan et al 2010) or to a polymorph II with a planar like defect (or stacking faults). This fact explains the great similarity between I and II as far as the DETOT is |0.009026| a.u. However, the higher energy difference between II and II* (D(ETOT) = |0.1573| a.u.) along with a D(ETOT) = |0.24701| a.u. value corresponding to different crystallographic solutions of identical structures of I clearly illustrates that the employment of only the total energy trajectory could describe unambiguously not only variations of geometry parameter within one and the same polymorph, but phenomena like crystallographic disorders, other stating effects, etc. Despite the available methods there is still a lack of a consolidated approach which to provide highly accurate and unambiguous description of such as systems (Balbuena et al. 1999; Frenkel and Smit, 2002). But the show, here, 2D and 3D energy–trajectory analyses of polymorphs are very distinguishable. The ANOVA test shows that I and II* cannot be treated as equal (Fig. 2, Table 7), Nevertheless from the perspective of crystallographic data (Bond et al 2007) they are described as slightly different. An r 2 value 96.12 % is obtained. This value itself is lower than a corresponding correlation coefficient obtained studying the energy profile of I (r = 0.99232–0.99176). Figures 4 and 5 show a significantly more sensitive approach studying subtle electronic effects in crystals. Table 7 and Fig. 2 illustrate a large F parameter. The column ‘Reject Equal Means?’ indicates whether equality of mean values can be rejected or not? When the answer to that question is ‘not’, then this means that means are the same. The answer ‘Not’ can be obtained when sample size is too small as well. However, in our case N = 51 and an answer ‘Not’ is associated with equality of datasets of values. The H–value in tables reflects uncorrected Kruskal–Wallis test data. As is expected at F–ratio = 0, H = 0 and probability level ‘1’ there is a complete coincidence between populations.
Table 7. Chemometrics of total and potential energy differences of polymorphs I (this work), II* (Vishweshwar et al. 2005) and III (Shtukenberg et al. 2017) obtained using crystallographic coordinates; α=0.0001.
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Figure 4. DFT-MD trajectory analyses of molecules A and B in 2–II and (2)–I; Chemometrics.
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Figure 5. Chemometrics of the DFT–MS trajectory analysis of total energy of II (Bond et al. 2011), II* (Vishweshwar et al. 2005); I (this work); I (Bond et al. 2011) and III (Shtukenberg et al. 2017), respectively.
Figure 3 illustrates MD data about III. The comparative chemometrics between III and II* (Table 8) indicate a difference of the energetics. In other words the MD computations provide sensitive information about the energy spectrum despite the close similarity of the geometry parameters of (2) in III and II* (Scheme 2). Accordingly, further, we have carried out quantum chemical computations of energetics of unit cell content of I, II* and III (Figures 6–8, Tables 2 and 8). The F–value is greater comparing with tabulated quantity according to the degree of freedom (Otto, 2017), which indicates that the patterns cannot be treated as different as random.
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Figure 6. DFT–MD computations (total energy [a.u.] versus time [fs]) of unit cell content of polymorphs of I and II* as well as hald unit cell content of III; Crystal structures of polymorphs (I (this work), II* (Vishweshwar et al. 2005), III (Shtukenberg et al. 2017); I (Bond et al 2011, CCDC 610952) and (II) (Bond et al. 2011, CCDC 617840).
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Figure 7. Chemometric analyses of the relations between total energies of forms II* and III studying both conformational situations of the molecules in the crystals and the unit cell content.
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Figure 8. Chemometrics of energetics (DFT–MD total and potential energies [a.u.] versus time in trajectory [fs]) of unit cell contents of polymorphs (2)–I (This work) and (2)–II* (Vishweshwar et al. 2005); DFT–MD data about the potential energy [a.u.] versus the time in trajectory [fs] of polymorph I using the crystallographic solution in this work and the data from reference (Bond et al. 2011), however, excluding from the protons in the computations
The conclusion of this line of analyses is that the results from energy trajectories of the polymorphs (Tables 7, 8, Figs. 2, 6, 8 and 9) show significant difference supporting works (Vishweshwar et al. 2005, Chan et al. 2010, Bond at al. 2011) about existing of a stable polymorph II of aspirin. A second avenue of our quantum chemical examination pursued by methods of NBO aims to gain energetics of molecular level interactions in the structures. The analyses of (2) shows NBO charges of (O=C)–OH/O=C(O)(CH3): 0.46764 (H)/-0.20948 (O) and 0.49362 (H)/-0.30412 a.u. (O), respectively. The interatomic distance 3.2938 and 3.90763 Å are taken from crystallographic data (this paper and work (Vishweshwar et al. 2005)). The difference of electrostatic potential energy DE(es) is –0.0086765 a.u. (or –5.4446 kcal.mol-1) accounting for that for II VNCE = - 0.03841707 a.u. and VNCE (I) > VNCE (II). In this context energetic advantages of II is underlined evaluating VNCE term. The latter consideration unambiguously confirms using different quantum chemical methodologies that II* appears favorable conformation from a thermodynamic point of view comparing with I one.
Table 8. Chemometrics of total energy differences of II* (Vishweshwar et al. 2005) and III (Shtukenberg et al. 2017); α=0.0001.
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- Citation du texte
- Prof. Dr. Bojidarka Ivanova (Auteur), Michael Spiteller (Auteur), 2021, The Crystallographic Polymorphism of Pharmaceutics. A Quantum Chemical and Chemometric Treatment, Munich, GRIN Verlag, https://www.grin.com/document/1014678
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X.