On the Validity of Minimum Magnetizability Principle in Chemical Reactions

Tandon et al.: On the Validity of Minimum Magnetizability Principle ... Abstract A new principle known as Minimum Magnetizability Principle has recently been introduced in the context of Density Functional Theory. In order to validate this principle, changes in the magnetizability (Δ ξ ) and its cube-root (Δ ξ 1/3 ) are computed at B3LYP/LanL2DZ level of theory for some elementary chemical reactions. The principle is found to be valid for 77% of reactions under study. It is observed that the molecules with the lowest sum of ξ or ξ 1/3 are generally the most stable. The principle fails to work in the presence of hard species. A comparative study is also made with change in hardness (Δ η ), electrophilicity index (Δ ω ), polarizability (Δ α ) and their cube-roots (Δ η 1/3 , Δ ω 1/3 , Δ α 1/3 ). It is observed that the Minimum Magnetizability Principle is nearly as reliable as Minimum Electrophilicity Principle. It appears that this principle could be helpful in predicting the direction of diverse reactions as well as stable geometrical arrangements.


Introduction
Theoretical Chemistry aims at unearthing novel concepts and principles to explain a broad range of chemical reactions. The most common questions that arise for any kind of reaction are about the pace and extent of the reaction. It is logical that thermodynamic data is required for providing an answer to the latter. Constructively, numerous reactivity descriptors have been established within the context of Conceptual Density Functional Theory (CDFT). 1 These descriptors play a significant role in studying the changes taking place in a reacting system. This ultimately helps in understanding the reactivity and stability patterns of the reactants and products in a chemical reaction. Some of these descriptors are hardness (η), 2 electrophilicity index (ω), [3][4][5][6] and polarizability (α) 7 whose definitions are given as follows: (1) Here E refers to the system's energy with respect to the number of electrons N at fixed external potential v(r). E L and E H stand for the frontier orbital energies while μ is the chemical potential. [ represents the external electric field. Another reactivity descriptor is magnetizability (ξ) which is defined as the linear response of an atom, molecule or ion's electron cloud towards an external magnetic field. 8 It is expressed as: (4) where B signifies the external magnetic field. It is an important descriptor to study chemical reactivity, stability and aromaticity of different atoms and molecules. [9][10][11][12][13][14] Dynamics of reactions have also been studied for molecules in confinement using magnetizability. 15 This property is of multidisciplinary interest and has extensive applications in the realm of physical, biological, engineering and materials science such as organic electronics, 16 magnetically labelled cells, drugs and therapeutic agents, 17 magnetic flux concentrators, 18 magnetizable elastomers, 19 bionanocomposites, 20 magnetic immunoadsorbents, 21 magnetic nanoparticles/ nanofibres 17 and so on besides its use in general chemical science. Looking at the widespread relevance of this property, it is believed that further exploration on its behaviour and characteristics will assist in strengthening the fundamentals of the concept, thereby supporting its efficient and appropriate use in various applications.
Based on above mentioned concepts and descriptors, some principles have been formulated in the framework of Density Functional Theory (DFT). The Maximum Hardness Principle (MHP) states that "there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible". 22 Another principle known as Minimum Polarizability Principle (MPP) proposes that "the natural direction of evolution of any system is toward a state of mimimum polarizability". 23 A Minimum Electrophilicity Principle (MEP) was also suggested according to which "the natural direction of a chemical reaction is toward a state of minimum electrophilicity". 24 Numerous efforts have been made to verify the reliability of these principles in diverse reactions/species. 25,26 However, a breakdown of these principles was also observed in many cases. [27][28][29][30] Recently, one more principle, known as the Minimum Magnetizability Principle (MMP), has been introduced in the context of DFT based on the descriptor magnetizability. According to this principle, "a stable configuration/conformation of a molecule or a favourable chemical process is associated with a minimum value of the magnetizability". 31 In case of MMP, unlike other principles, no study has been performed yet to test the validity of this principle for chemical reactions. As understandable, operational efficiency of a principle can only be determined through performing a validation procedure. Accordingly, in order to employ MMP for practical purposes, it is necessary to carry out its verification based on some criteria. Thus, in the present study we have tried to assess the potential and accuracy of this principle for the first time with respect to some chemical reactions.

Method of Computation
In order to verify the validity of Minimum Magnetizability Principle along with comparing it to the other principles, viz. MHP, MPP and MEP, 30 elementary reactions are selected. Geometry optimizations for each of the reactant and product molecules have been performed at B3LYP/LanL2DZ level of theory. This is a very common and powerful method employed in computations and incorporates the correlation effects. Hence, the energy values of the HOMO (Highest Occupied Molecular Orbital) (E H ) and the LUMO (Lowest Unoccupied Molecular Orbital) (E L ) as well as molecular polarizability are obtained for the molecules. The canonical molecular orbitals differ from Kohn-Sham orbitals, nevertheless their values for orbital energy are found to be comparable. 32 As a result, the operational forms of hardness and electrophilicity presented in Eq. (1) and Eq. (2) are used to calculate the value for hardness and electrophilicity. Further, magnetizabilities are computed using the Keith and Bader's method 33 at the same level of theory. All computations have been performed on the computational software Gaussian 03. 34 Next, in view of the MMP, the reaction is favoured only when the products to be formed are of lower magnetizability than the reactants. Thus, for a chemical reaction (A j ), represented as Σa j A j = 0 in a condensed form, where a j corresponds to the stoichiometry of the j th atom/molecule, the corresponding deviation in a parameter X through the reaction can be clearly expressed as: (5) In Eq. (5), X j refers to the value of X for the j th atom/ molecule in the given chemical reaction. It follows that the direction in which the chemical reaction will progress may be indicated by the sign of ΔX. Thus in accordance with MMP, it appears that the sign of Δξ provides an evidence of the direction in which any reaction proceeds. Hence, we have calculated the change in parameter X, viz. hardness (Δη), electrophilicity (Δω), polarizability (Δα), and magnetizability (Δξ), for the selected reactions using Eq. (5). Change in the cube-roots of hardness (Δη 1/3 ), electrophilicity (Δω 1/3 ), polarizability (Δα 1/3 ) and magnetizability (Δξ 1/3 ) have also been calculated. Enthalpy change (Δ f H 0 ) for each molecule is also calculated using the atomization energy data from reference [35].

Results and Discussion
The study presents validity of Minimum Magnetizability Principle using chosen 30 elementary reactions. A comparison of the results is also presented with respect to the other principles, viz. MHP, MPP and MEP. It must be noted that several of the chosen reactions are hypothetical. All the reactions are exothermic signifying that the products are thermodynamically stable. Also, the variation of chosen parameters is not considered along the reaction path, but merely the overall change in these parameters in the chemical reaction is evaluated, which can be simply computed as long the geometry can be. Further, it should be noted that the reactants in reaction 14 are the same as in reaction 19, however the type of products formed in both are different.
The computed values for hardness (η), electrophilicity index (ω), polarizability (α), magnetizability (ξ) and Tandon  HOMO (E H ) and LUMO (E L ) energy, along with magnetizability (ξ T ) data from the literature, 36 for all the reactants and products of the selected reactions are presented in Table 1. A comparison of computed magnetizabilities (ξ) with tabulated magnetizabilities 36 (ξ T ) reveals a good similarity between the two datasets (R 2 = 0.997) indicating the efficacy of our computed data. The changes in enthalpy (Δ f H 0 ) and in parameter values with their reaction are provided in Table 2. As mentioned above, the formation of products is only favoured when the magnetizability of reactant is more than that of the products to be formed according to the MMP. Moreover the sign of Δξ gives an idea of the direction in which any reaction proceeds. A close look at Table 2 reveals that Minimum Magnetizability Principle is valid for chemical reactions since Δξ < 0 for various reactions. Further, Δξ is negative in numerous cases which undoubtedly demonstrates that Δξ provides a sign for the most stable species. Thus, the favoured direction of a chemical reaction is towards less magnetizability. When the stability of products is more than the reactants, change in enthalpy is less than zero (Δ f H 0 < 0). This condition is also clearly met by the reactions in the study. However, MMP is not valid always. It is reliable for approximately 77% of the selected reactions and fails in case of the remaining reactions. It is observed that the majority of the reactions where the principle fails contain a hard base such as OH, F, Cl, N, etc. It is further noted that MEP is as convincing as MMP and proves to be valid for 77% of the reactions tested. MPP performs slightly better than MHP; however it is less suitable than MEP and MMP. MPP works for 50% of the reactions while MHP is applicable for 47% of the considered reactions. It is observed that the minimum magnetizability Table 1. Computed frontier orbital energies (E H and E L ), hardness (η), electrophilicity index (ω), polarizability (α) and magnetizability (ξ) values of the reactant and product molecules of the selected reactions using B3LYP/LanL2DZ method and their tabulated magnetizabilities (ξ T ) (in au) principle is almost as valuable as minimum electrophilicity principle in predicting the direction of a reaction. The sign of Δξ can be used to provide an indication for higher stability of products thermodynamically. In order to understand the significance of the results, a brief statistical analysis is performed for each parameter with respect to Δ f H 0 . Our study is based on the stability criterion, i.e. minimum energy condition, thus it is important to validate the correlation of Δ f H 0 with these descriptors. Regression analysis has been performed for this purpose. An attempt to explore the relationship of Δα with Δ f H 0 is futile (R 2 = 0.4871, R = +0.70, p = 0.001), although Δη is found to follow a satisfactory linear relation with Δ f H 0 (R 2 = 0.6266, R = −0.79, p = 0.001). Lower values of R 2 denote inferior correlation of Δα and Δη with Δ f H 0 whereas p-values indicate that the result is significant. It can be inferred that MPP and MHP may not always follow the minimum energy criterion and may become invalid in several cases. An analysis of Δξ and Δω with Δ f H 0 presents an excellent relationship. For Δξ and Δ f H 0 , R 2 = 0.9792, R = +0.99 and p = 0.001 while R 2 = 0.9679, R = +0.98 and p = 0.001 for Δω and Δ f H 0 . Such high values of R 2 clearly signify the outstanding correlation between the parameters. Further, a perfect positive correlation is presented for Δξ and Δ f H 0 . Significance of the correlation is highlighted by the p-values. It is concluded that MMP as well as MEP are highly related to Δ f H 0 and thus follow the minimum energy and high stability condition. As the validity of minimum energy principle increases, MMP and MEP become valid as well.

S. No. Molecules
The cube-roots of exact polarizability have been proved to be more useful in comparison to Δα to predict the direction of a chemical reaction. 35 Hence, following the above notion, we have determined the cube--roots for hardness, electrophilicity index, polarizability as well as magnetizability and these are presented in Table 3. It is apparent from the outcomes that the validity of MHP as well as MPP increases when their cube-roots are considered. In fact, the soundness of MHP increases drastically as compared to MPP. As expected, these results indicate efficiency of cube-roots of hardness and polarizability in predicting the path followed by a reaction. Although in Table 2. Computed changes in the enthalpy (Δ f H 0 ) (at 298K in kJ mol -1 ), hardness (Δη) (in au), electrophilicity index (Δω) (in au), polarizability (Δα) (in au) and magnetizability (Δξ) (in au) along with MHP, MEP, MPP and MMP validity (+) or invalidity (-) for selected reactions  Table 3, a decrease in the validity of MMP takes place when the cube-roots of magnetizability are considered. However MMP is still valid for several reactions (Δξ 1/3 < 0) and performs remarkably in contrast to MHP and MPP. As a result, it appears that Δξ 1/3 is more or less as reliable as Δξ. It is further observed that MEP is the most the convincing of all principles considering its high validity in both the cases. Next, Δ f H 0 demonstrates an excellent linear relationship with Δξ 1/3 (R 2 = 0.9818) and Δω 1/3 (R 2 = 0.8916) but poor connections are found with Δη 1/3 (R 2 = 0.5530) or Δα 1/3 (R 2 = 0.5338).
We have tried to accommodate reactions with different types of molecules, viz. inorganic, aliphatic and aromatic, in the present study to consider as many chemical effects as possible. Although for further studies it is suggested that the validity of this principle should be assessed in case of other reaction classes as well.

Conclusion
We have tried to validate Minimum Magnetizability Principle employing 30 elementary chemical reactions. Change in the magnetizability (Δξ) and its cube-root (Δξ 1/3 ) is computed in order to check the applicability of the principle in determining the direction of the reaction as well as stability of the products. We have also calculated change in hardness (Δη), electrophilicity index (Δω), polarizability (Δα) and their cube-roots (Δη 1/3 , Δω 1/3 , Δα 1/3 ) in order to make a comparative study. It is observed that the Minimum Magnetizability Principle is valid for chemical reactions however with some limitations. The principle fails to work in the presence of hard species. Minimum Electrophilicity Principle and Minimum Magnetizability Principle are found to have comparable reliability followed by Minimum Polarizability Principle, however Maximum Hardness Principle appears to be less valid for the chosen reactions. In conclusion, Minimum Magnetizability Principle is found to be fairly valid for reactions and can be successfully employed solely or in Table 3. Computed changes in the enthalpy (Δ f H 0 ) (at 298K in kJ mol -1 ), cube-root of hardness (Δη 1/3 ) (in au), cube-root of electrophilicity index (Δω 1/3 ) (in au), cube-root of polarizability (Δα 1/3 ) (in au) and cube-root of magnetizability (Δξ 1/3 ) (in au) along with MHP, MEP, MPP and MMP validity (+) or invalidity (-) for selected reactions

Conflicts Of Interest
The authors declare no conflicts of interest.

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for--profit sectors.