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On the Relative Contribution of Combined Kinetic and Exchange Energy Terms vs. Electronic Component of Molecular Electrostatic Potential in Hardness Potential Derivatives Rituparna Bhattacharjee, and Ram Kinkar Roy J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 01 Oct 2013 Downloaded from http://pubs.acs.org on October 2, 2013

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The Journal of Physical Chemistry

On the Relative Contribution of Combined Kinetic and Exchange energy terms vs. Electronic Component of Molecular Electrostatic Potential in Hardness Potential Derivatives Rituparna Bhattacharjee and Ram Kinkar Roy* Department of Chemistry, Birla Institute of Technology and Science, Pilani-333031, Rajasthan, India.

Abstract:

The relative contribution of the sum of kinetic [

10 4 C F ρ (r ) 2 / 3 ] and exchange energy [ C X ρ (r )1 / 3 ] 9 9

terms to that of the electronic part of the molecular electrostatic potential [ Vel (r ) ] in the variants of Hardness Potential is investigated to assess the proposed definition of ∆+ h( k ) = −[VelN +1 ( k ) − VelN ( k )] and ∆− h( k ) = −[VelN (k ) − VelN −1 ( k )] (Saha et al. J. Comp. Chem. 2013, 34, 662). Some substituted benzenes and polycyclic aromatic hydrocarbons (PAHs) (undergoing electrophilic aromatic substitution), carboxylic acids and their derivatives are chosen to carry out the theoretical investigation as stated above. Intra and intermolecular reactivity trends generated by ∆+ h(k ) and ∆− h(k ) are found to be satisfactory and are correlated reasonably well with experimental results.

Keywords: Hardness Potential Derivatives, Reactivity Descriptor, TFD Functional, Kohn–Sham potential, Electrophilic Aromatic Substitution *

Corresponding author. E-mail: [email protected]

ACS Paragon Plus Environment


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1. Introduction: Parr and his collaborators proposed several systematic and simple methods to gain a deep insight into the chemical reactivities of various electronic systems by exploiting mathematical formulation of DFT in the late 1970s and early 1980s.1-9 These led to the foundation of ‘Conceptual Density Functional Theory’ or ‘Chemical Reactivity Theory’ or ‘Density Functional Reactivity Theory (DFRT).1-17 It offers a series of chemical reactivity indices for the description of chemical phenomena which are governed by some essential principles—the electronegativity equalization principle,18 the hard/soft acid/base principle,19-25 and the maximum hardness principle.20,26-31 Reactivity indices are classified as global [chemical potential µ,1 global hardness ( η ),2 global softness( S ),9 electrophilicity index ω32-35], local [Fukui function f ( r ) ,5,6,36,37 local softness s (r ) ,9 local hardness η (r ) ,8,38-40 relative nucleophilicity and relative electrophilicity,41,42 philicity ω (r ) 43 etc] and non-local [softness kernel44 and hardness kernel44,45]. In literature, analysis of several local indices revealed that Fukui function remains a suitable descriptor for regioselectivity,41,46-56 but it cannot be applied to intermolecular cases due to its normalization condition or more precisely, ‘intensive’57 nature. It is worth mentioning here that local descriptors from the grand canonical ensemble should be able to predict intra as well as intermolecular reactivity because they generally possess a combination of a local and a global part [e.g., local softness: s( r ) = f ( r )S and philicity ω α (r ) = ω f α (r ) = ( µ 2 2η ) f α (r ) = µ 2 Sf α (r ) ]. But ‘intensive’ nature of Fukui function f ( r ) has made these two descriptors ‘intensive’ (in spite of the fact that the number of electrons and energies associated with evaluating them for intermolecular comparison are extensive) and so constrained the applicability58-62 of these two indices as intermolecular reactivity descriptors. Also, the global softness part in these two descriptors is 2


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‘extensive’, but true (unambiguously) for a conductor. For most of the commonly used chemical systems this may not always be true. Hard-hard interaction is charge-controlled63 and hence suitable to account for long range interactions, i.e., suitable for intermolecular studies. Local hardness which is defined by, η ( r ) =

1 ∫ η ( r , r ′)λ[ ρ ( r ′)]d r ′ , is such a descriptor that takes care of hard-hard N

interaction and so is expected to be suitable for studying intermolecular reactivity. But in a study by Saha et al.64 it is clearly shown that it cannot always be used as a reliable tool to predict intermolecular reactivity trends between systems of different sizes but having common reactive centers. The

1 factor in the definition of η (r ) , as shown above, alters the expected reactivity N

trends. In another recent article, Saha et al.65 have shown that N-dependence problem of local hardness η (r ) can be addressed formally by hardness potential h( r ) .66 The approximated analytical form of Hardness potential [ h( r ) ] is, h( r ) =

10 4 C F ρ ( r )2 / 3 − V el ( r ) − C X ρ ( r )1 / 3 9 9

(1)

and then it was approximated further as h( r ) = −Vel ( r ) , where Vel (r ) is the electronic contribution to the molecular electrostatic potential. It is worth mentioning here that complete mathematical definition of h( r ) requires some pretty strong approximations. The Hohenberg–Kohn functional F [ ρ ] 3,4 is approximated on the basis of Thomas–Fermi–Dirac (TFD)67-69 approach (which has its

own limitations70) plus the Weizsäcker71 term, to DFT. So, basically, the universal functional of Hohenberg and Kohn is approximated by Thomas-Fermi-Dirac-Weizsäcker approach while obtaining mathematical definition of h( r ) . One can systematically improve TFD functional by 3



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considering

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1 of the Weizsäcker functional (for gradient expansion).71 Thus, formally, the presence 9

of the Weizsäcker term in the kinetic energy description compensates for several of the deficiencies of the Thomas–Fermi–Dirac functional. However, it was also shown that the contribution of the Weizsäcker term to the local hardness, η (r ) [and hence to the hardness potential, h( r ) ) when density, ρ (r ) , is used as composite function, λ[ ρ ( r ′)] 39 is equal to zero. Theoretical investigations are done to overcome the limitations of TFD for years. It is shown that TFD somewhat works well when the exact electron density is used to evaluate it. Burke et al.72 have shown that the exact electronic ground-state density and external potential can be used to improve the accuracy of approximate density functional by exploiting the fact that exact exchange-correlation energy functional is more local for full-coupling strength than for the couplingconstant average and that exact virial can be used to reduce the exchange energy error by a factor of 2. To apply his ‘local model’ to the hardness kernel and related reactivity parameters in DFT, Fuentealba73 used TFD approach to approximate any local functional F [ ρ ] and also used the electron density ( ρ ). Using this method, all the calculated reactivity indices are positive definite since ρ is so. Senet74 has emphasized that in spite of several loopholes, the TF model has been ‘extensively implemented by using an electronic density arising from an independent ab initio calculation.75,76 But it is warned that TF model only yields fruitful results for a system having a relatively large number of electrons. Some desirable properties are also obtained by using TFDW Euler–Lagrange equation3 for the ground-state electronic densities of atoms and jellium surfaces.77-79 Thomas-Fermi like theories supports the existence of the electrostatic potential component of the hardness kernel. But from the first derivative of Kohn-Sham kinetic energy with respect to the electron density (which emerges out to be the difference between chemical potential 4


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and effective Kohn-Sham potential), it can be demonstrated that the electrostatic term as well as exchange correlation term cancel out. Liu and Ayers80 discussed it while showing that the functional derivative of the non-interacting kinetic energy density functional can unambiguously be represented as the negative of the Kohn–Sham effective potential, arbitrary only to an additive orbitalindependent constant. However, while discussing second functional derivative of Kohn-Sham kinetic energy, Ayers has shown81 that the derivative of the Kohn–Sham potential with respect to the electron density contains a contribution from a coulomb term (eq 61 of ref. 81). Here, the explicit coulomb contribution vanishes, but an implicit dependence (which is embedded in the Kohn–Sham potential derivative i. e., the first term in eqn 61 of ref. 81) exists. It may also be noticed that hardness potential belongs to the class of derived (arising out of more fundamental response functions and not derived from first principles) chemical reactivity descriptors. There are three types of derived chemical reactivity descriptors according to the classification by Johnson et al.16 Some of them are derived by rigorous mathematical methods (electrophilicity,32-35 nucleofugality,82 electrofugality,82 point charge response83,84 etc), some are proposed for their easy-to-compute nature and efficiency to analyze chemical phenomena (relative nucleophilicity and electrophilicty41,42 and leaving group indicators85-87 etc) and some lie within these two extremes (which are mentioned as ‘intermediate’ such as local hardness,8,38-40 multiphilic descriptor,88,89 indicators associated with CDASE90 etc). Hardness potential is also one of the “intermediate” derived chemical reactivity indicators. A close analogy to hardness potential may be found in the recent work by Ayers et. al.,91 where they showed that local hardness emerges ‘naturally’ while using variational approach to define local stability conditions for electronic states. Variational approach is also exploited to optimize the electron density or the electron number in a fixed external potential previously.29 5



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The present authors proposed two variants of hardness potential65 (evaluated at the nucleus) [ ∆+ h(k )] and [ ∆− h(k ) ] to explain both intermolecular and intramolecular reactivity trends of systems having variation of atom types of the reactive centres, different sizes (i.e., number of electrons) and characteristics. These two derivatives (or variants), generated from the approximated form of h(r ) [i.e., h( r ) = −Vel ( r ) ], have emerged out to be identical to the left and right derivatives of Fukui potential.56,92-97 Thus, without going into the ambiguity of adopting any particular approach to DFT (i.e., either TFD or Kohn-Sham) if we define h(r ) as in equation 1 or its approximated form [i.e., h( r ) = −Vel ( r ) ] then these two variants [i.e., ∆ + h(k ) and ∆− h(k ) ] emerge out to be two useful reactivity descriptors in their own merit. Because, then these two variants of h(k ) [i.e., when r → 0 in h(r ) ] can represent the reactivity of the concerned reactive atom towards an approaching nucleophile and an electrophile, respectively. Use of the difference of electrostatic potential to explain reactivity is not new in theoretical chemistry but only the new things in ∆ + h(k ) and ∆− h(k ) is that here the differences of only electronic components of electrostatic potential is used (the details to be discussed in Section 2.B.). Also, it is important to indicate that these two variants of hardness potential are originated (with some approximation) from hardness functional, H [ρ ] ,66,98 which in turn belongs to grand canonical ensemble (i.e., − H [ ρ ] = E − µN = Ω ). Hence, in principle, they have the ability to take care of both intramolecular (i.e., site selectivity) as well as intermolecular reactivity trends, which also have been illustrated with some model systems and real examples.65

6


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In the present article, an effort is made to rationalize relative contribution of the sum of kinetic energy [

10 4 C F ρ (r ) 2 / 3 ] and exchange energy [ C X ρ (r )1 / 3 ] terms to that of electronic 9 9

contribution to the molecular electrostatic potential [ Vel (r ) ] while defining the working equations of [ ∆+ h(k )] and [ ∆− h(k ) ]. Keeping in mind the broad readership discussion will also be there for considering only the electronic component, instead of taking the total (i.e., nuclear + electronic) contribution, of molecular electrostatic potential which was exploited in several other approaches.84,99-102 Several arenes (undergoing electrophilic chlorination, nitration and benzylation) and polycyclic aromatic hydrocarbons (undergoing electrophilic bromination) as well as carboxylic acids and their derivatives have been chosen to carry out the theoretical investigations as mentioned above. Arenes and polycyclic aromatic hydrocarbons have been exploited further to study intramolecular (regioselectivity) as well as intermolecular reactivity (relative reaction rates compared tothat of benzene) using nucleophilic variant of the hardness potential [ ∆− h(k ) ] (evaluated at the nucleus).

2. Theoretical Background: A. Working Equations of ∆+ h(r ) and ∆− h(r ) : The analytical equations of ∆+ h(r ) and ∆− h(r ) can be written as, ∆+ h(r ) = N∆η + (r )

(2)

∆− h(r ) = N∆η − (r )

(3) 7



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The ∆+ h(r ) and ∆− h(r ) , when evaluated at the nucleus (i.e., ∆+ h(r )

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r→0

and ∆− h(r )

r→0

) of

atom ‘k’, and denoted as ∆+ h(k ) and ∆− h(k ) , respectively, can be formulated (after neglecting the kinetic and exchange energy terms in equation 1) as65 : ∆+ h( k ) = −[VelN +1 ( k ) − VelN ( k )]

(4)

∆− h( k ) = −[VelN (k ) − VelN −1 ( k )]

(5)

and,

Incidentally, these two approximated forms of ∆+ h(r ) and ∆− h(r ) lead, respectively, to,65

∆ h(r ) = ∫

f + ( r ′) dr ′ r − r′

(6)

∆− h( r ) = ∫

f − ( r ′) dr ′ r − r′

(7)

+

and,

where,

f (r ′)

∫ r − r ′ dr ′ is Fukui potential.

56,92-97

The fact that ∆+ h(r ) and ∆− h(r ) are related to the electronic contribution of the molecular electrostatic potential (MEP) [i. e., Vel (r ) ] directly (eqns 4 and 5) and hence to electron density [i. e., ρ (r ) ] indirectly (eqns 6 and 7), they should be able to interpret reactions where both frontier molecular orbital and electrostatic control play important roles.

8


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Normally, electronic contribution to the molecular electrostatic potential, Vel (r ) , is the dominant one when compared to the other two terms (i.e., kinetic and exchange energy terms). Also, the negative sign in the exchange energy term cancels out, to some extent, the effect of the kinetic energy term38,51,52,55,103-107 (see eqn 1). Thus, eqns 4 and 5 seem to provide reasonably reliable working formula of ∆+ h(k ) and ∆− h(k ) , respectively. Some similar kind of idea (i.e., change of kinetic energy functional equals to the negative of the sum of changes of exchange and correlation energy functionals at isodensity contours of 0.00872) was first invoked by Politzer et al.108 when they tried to establish the relationships between atomic chemical potentials, electrostatic potentials, and covalent radii. A similar protocol was used by Liu et al.105 while establishing a simplified model for

hardness

η (r , r ′) =

kernel,

1 r−r

'

η ( r , r ′) .

They

stressed

that

it

must

be

of

the

form,

+ R ( r , r ' ) where the first term on the right-hand side of the above equation results

from the classical Coulomb repulsion term and the second term includes contributions from the kinetic, exchange, and correlation energy functionals. Liu et al.105 assumed that the contribution from the second term is typically small and hence η ( r , r ′) got reduced to η ( r , r ′) ≈

1 r − r'

.

Contrary to all these, Jin et al.109 have argued that the equation η (r ) =

− Vel (r ) N

or

h( r ) = −Vel ( r ) cannot be accepted. Because most of their observations were not satisfactory at a

surface of 0.001 electrons/Bohr3 (based on the cancellation of first and third terms in the right hand side of eqn 1). In our earlier study,65 we conjectured that the definition of ∆+ h( k ) and ∆− h( k ) as projected by eqns 4 and 5, respectively, seem to be acceptable because electron density differences 9



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(of neutral systems and its ions) at the atomic nuclei may be very small (even if not 0.001 a. u.) and thus net contribution of the first and third terms will be almost negligible. Analytically, this can be shown from the original equations65 of ∆+ h( k ) and ∆− h( k ) taking into account of all three terms as below: 2 1  10 2 1   4 4 10 ∆+ h(k ) =  CF ρ (k ) 3 − C X ρ (k ) 3  −  CF ρ (k ) 3 − C X ρ (k ) 3   9 9 N   N +1  9  9

N  N +1  - Vel ( k ) − Vel (k )  

(8) 2 1   10 2 1  4 4 10 ∆− h(k ) =  CF ρ (k ) 3 − C X ρ (k ) 3  −  CF ρ (k ) 3 − C X ρ (k ) 3   9 9 N  9  N −1   9

N −1  N  - Vel ( k ) − Vel (k )  

(9) Here, the net contribution of the first four terms (i.e., the differences of kinetic and exchange energy terms) is expected to be negligible when compared to the net contribution of the last two terms (i.e., difference of Vel (k ) terms) in both the equations. The quantitative scenario will be presented in Section 4 (Results and Discussion) based on generated numerical values of ∆+ h(k ) and ∆− h(k ) of the concerned reactive centers in the chosen systems.

B. Total (i. e., VT = Vnu (r ) + Vel ( r ) ] vs electronic [i. e., Vel ( r ) ] contribution of Molecular electrostatic potential within the context of ∆+ h(r ) and ∆− h(r ) : In an earlier work,66 Parr and Gázquez obtained an important relation,

H (ρ ) =

1 ∫ h(r )ρ ( r ) d r + higher order terms 2 10


(10)

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Where, at large r, h( r ) equalizes to the classical electrostatic potential generated by the electron density ρ. Here, ‘higher-order’ means terms involving third and higher functional derivatives of F [ ρ ] , the so-called Hohenberg-Kohn functional3,4 (which is the sum of kinetic energy functional T [ ρ ] and electron-electron interaction energy functional Vee [ ρ ] ). Molecular electrostatic

potential (MEP) has been exploited over years in the formulation of DFT-based reactivity indicators. Ayers and collaborators84,94,95,99 showed that the effective external potential felt by an electrophile due to the presence of a nucleophile can, in principle, be calculated exactly (when the reagents are far apart, electrostatic control dominates) by involving a correction to MEP with a term which involves the nucleophilic Fukui function of the nucleophile (similar case was obtained for nucleophile in presence of an electrophile). Gadre and co-workers exploited MEP for various organic molecules.110-116 They have developed three dimensional characterization and topographical approaches towards MEP to explore the molecular reactivities, weak intermolecular interactions, and a variety of chemical phenomena.110,112,113,115 Langenaeker et al51 argued that MEP is the best reactivity indicator for comparing reactivity of different mono-substituted benzenes. But according to them, application of MEP is limited to only charge-controlled processes and it cannot be a good indicator of hardness as well. Politzer et al117,118 showed that MEP of substituted arenes can be used to indicate the activation or deactivation of aromatic rings towards electrophilic attack. MEP is labeled as liable and rigorous descriptor to predict both Lewis basicity and Brønsted-Lowry basicity119 in the study of zeolites. Several other use of MEP together with the atomic charges can be found elsewhere.119-121 Some studies show that electrostatic potential at the nucleus is in good agreement with acidity/pKa.122-124 More recently, it is presented that electrostatic potential at nuclei (EPN)125-132 can correlate well to the experimental trends.

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However, in almost all the studies in literature, it is a common practice to consider the overall electrostatic potential [i.e., Vnu (r ) + Vel ( r ) ]84,94,95,99-102,117-121,125-132 to account for different types of interactions that govern chemical reactions. Hence, it raises a question about the working formulation of ∆+ h( k ) and ∆− h( k ) either by using only the difference of electronic contribution to MEP (i.e., eqns 4 and 5) or the difference of all the three terms (eqns 8 and 9). It is worthy of mentioning here that the hardness potential arises from the second derivative of the universal Hohenberg-Kohn functional, which by definition does not contain the nuclei repulsion [i.e., the external potential is held constant in the response function, the hardness kernel η ( r , r ' ) , itself]. Hence, only electronic part of the molecular electrostatic potential exists in the definition of hardness potential. It would be interesting to note the effect of the absence of the nuclear contribution of the electrostatic potential term in ∆+ h( k ) or ∆− h( k ) in the generated trends of inter and intramolecular reactivity. Incidentally, in a recent paper,94 Ca ′rdenas et al. have also favored the idea that the Fukui potential at the nucleus can rationally be computed from differences of the electronic contribution to the molecular electrostatic potential of the neutral atoms and their corresponding ions.

3. Computational Details: To verify quantitatively the approximation [i.e., h( r ) = −Vel ( r ) ] used in the formulation of ∆+ h(k ) and ∆− h(k ) 65 and to elucidate their potency for studying some intra as well as intermolecular reactivity studies, we have chosen several model systems and classified them as Group I, Group II and Group III. Group I consists of several substituted benzenes (i.e., alkylbenznes 12


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and halobenznes) (see Table 1, 2 and 3) which undergo electrophilic substitutions (chlorination, nitration and benzylation are discussed here for brevity). Group II contains some polycyclic aromatic hydrocarbons (bromination is discussed in detail, see Table 4 and Figure 1). Experimental partial rate factors (of different positions of the Ph ring and fused rings) and relative reaction rates (of substituted benzenes and polyarenes with respect to unsubstituted benzene) were verified by

∆− h( k ) values. It is a test of the reliability of ∆− h( k ) to study inter as well as intra-molecular reactivity trends. Some electrophiles such as carboxylic acids and their derivatives (Table 5) are listed in Group III. Incidentally, these are part of the systems chosen in one of our earlier studies.65 The three series generated from these systems are given below: (i) HCOOH, CH3COOH, CH3CH2COOH (ii) HCOF, CH3COF, CH3CH2COF (iii) HCONH2, CH3CONH2, CH3CH2CONH2 Optimizations (neutral systems) and for all the systems of the three Groups are carried out at B3LYP133-135/6-31G** level while the corresponding single point calculations for cations and anions are carried out at UB3LYP/6-31G** level. Also, calculations for all systems are performed in gas phase and the SCF density is used to evaluate Vel ( k ) and ρ ( k ) . All computations are carried out using Gamess136 software package.

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4. Results and Discussion: A. Group I: Alkylbenzenes and Halobenzenes: (i) Intramolecular reactivity trends by ∆− h(k ) : To test the reliability of eqn 5 as a proposed definition of ∆− h(k ) , positional selectivity and relative rates (compared to that of benzene) of nitration, benzylation and chlorination on halobenzenes and alkylbenzenes are studied here. There are a number of earlier attempts where electrophilic substitutions on arenes are studied using reactivity indices based on density functional reactivity theory (DFRT) and other theoretical approaches.9,51,117,118,137-149 Normally, higher the ∆− h(k ) value higher should be the reactivity of that site (here atom) (similar intramolecular studies were done earlier on indolynes and unsymmetrical arynes).65 From Table 1 (third column), it is evident that for chlorination of methylbenzenes, ∆− h(k ) of individual arenes correlated satisfactorily with experimental partial rate factors132 in all cases. Schaefer and Galabov132 have recently demonstrated successful use of electrophile affinity and electrostatic potential at nuclei (EPN, denoted by Vc ,150,151 reﬂecting the electron densities at different ring positions of arenes), which can function as an excellent local reactivity index for charge-controlled chemical interactions.125-132 But EPN failed to produce exact intramolecular reactivity trends in some cases132 (shown in bold in Table 1). The ortho position of toluene has the higher Vc (electrostatic potential at nuclei) but the reactivity ( ln f value) for the para position is greater. The same irregularities exist between the 3 and 4 positions of 1,2-dimethylbenzene and the 2- and 4-positions of 1,3-dimethylbenzene (Table 1). Cause of these irregularities in Vc values are attributed to the ortho steric effects.132 Because in Ph-ring all reactive centers (i.e., atoms) are same 14


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(i.e., carbon), Vc values reflect the variations of electron densities at particular sites and more negative the Vc value greater is the electron density, making the site more susceptible to electrophilic attack. As Vc values fail to produce expected intramolecular reactivity trends in the system as mentioned above, it is expected that ∆− h(k ) values will also fail. But surprisingly ∆− h(k ) values produce correct trends in all these systems. Naturally, the question arises why ∆− h(k ) is successful where Vc fails? This may be due to the fact that though ∆− h(k ) also takes care of only electronic effects, it does so in a unique way. It reflects how that particular reactive site responds in the changing electron density scenario in the event of an approaching electrophile, whereas Vc is all about the neutral system only. This might be the reason why ∆− h(k ) values show superior intramolecular reactivity trends than those by Vc values of the atoms of the neutral system. It is worth to note that similar expected intramolecular reactivity trends are obtained from ∆− h(k ) values for nitration of halobenzenes152,153 (Table 2, third column) and benzylation of halobenzenes154,155 (Table 3, third column). From the Vc values it is obvious that although for nitration intramolecular reactivity trends are as expected (Table 2, sixth column), for benzylation it does not match with experimental observation for chlorobenzenes and bromobenzenes (Table 3, sixth column).

(ii) Intermolecular reactivity trends by ∆− h(k ) : The grand canonical ensemble origin of ∆− h(k ) 65 approves that it should, in principle, explain the intermolecular reactivity trends also. However, depending on the nature of substituents 15



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(e.g., methyl groups are electron-pushing, while halides are electron-withdrawing) interpretation of intermolecular reactivity trends will vary when ∆− h(k ) [or ∆+ h(k ) ] are used as descriptors. Situations are further complicated when the concerned systems have two or more sites of comparatively high reactivity and also if the reactions are not single step concerted ones. In the next few paragraphs, systematic analysis of these trends are carried out. While studying relative reaction rates132 of alkylbenzenes compared to that of benzene by using ∆− h(k ) ), the existence of electron pushing +I effect of several methyl groups (in alkylbenzenes) makes the interpretation different from intramolecular cases. It is argued in our earlier study, that for intermolecular reactivity comparison of this type of systems (where rest part of the system is electron pushing) lower the positive value of ∆− h(k ) higher is the nucleophilicity of the corresponding nucleophilic atom.65 The reason is that, with increasing number of − CH 3 groups, the value of both VelN ( k ) and VelN −1 ( k ) of the nucleophilic atom on the Ph-ring will increase. But what exactly happens is that relative increase of VelN −1 ( k ) is more than that of VelN ( k ) , which in turn, leads to lowering of ∆− h(k ) value. If eqn 5 (or eqn 9) are examined

carefully, it is revealed that the argument is justified. In the course of studying the relative reaction rates of halobenzenes155,156 compared to that of benzene, the presence of two opposing forces, i.e., electron-withdrawing –I effect and electron pushing +R effect of halo groups make the interpretation different from that of alkylbenzenes. If again we look at eqn 5 (or eqn 9), it is evident that if the rest part of the system is not electron pushing (i.e. not compensating for the loss of electron to an incoming electrophile),

16


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N −1  N  then higher the absolute value of Vel ( k ) − Vel (k ) or ∆− h(k ) , higher is the reactivity towards  

an electrophile. However, if after the attack of the electrophile the + R effect dominates over the − I effect then the interpretation will be as outlined in the previous paragraph (i.e., lower the

positive value of ∆− h(k ) , higher is the reactivity or vice versa). In several earlier studies,58-60 it was argued that these type of descriptors [e.g., local softness s( r ) , philicity ω ( r ) etc, which are originated from grand canonical ensemble] can be used for intermolecular reactivity studies only in limited cases where the systems have only one predominant reactive site or sites of equal reactivity (i.e., chemically equivalent sites). Systems, in which there are more than one sites of comparable reactivity, these descriptors may not be suitable. Thus, from Table 1, third column, we get expected reactivity trends for almost all systems of the former type (first, fifth, sixth, seventh and eighth rows) by using eqn 5. There is only one irregularity with 1,3,5-trimethylbenzenewhich is resolved when eqn 9 is used instead of eqn 5 (Table 1, fourth column). Here, it should be noted that discrepancy is also observed in Vc values for 1,3,5-trimethylbenzene and 1,2,4,5-tetramethylbenzene (observed value of Vc is higher for the latter which is actually less reactive, Table 1, sixth column).132 For systems with multiple reactive sites such as toluene, 1,2-dimethylbenzene and 1,3dimethylbenzene(Table 1) expected intermolecular reactivity trends are obtained when the ∆− h(k ) values of the most reactive sites of individual systems are compared (however, trend of reactivity of these three systems cannot be compared with those having only one reactive site, as discussed in the preceding paragraph). If we consider the ∆− h(k ) values of position 4 (most reactive site on the basis of intramolecular study) for toluene, 1,2-dimethylbenzene and 1,3-dimethylbenzene (Table 1, third 17



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and fourth columns) it is observed that lower is the value, higher is the reactivity (as per discussion in paragraph 2 above in this Section). Here, it is important to note that 1,3-dimethylbenzene (mxylene) is the highest reactive among all xylenes. Kobe et al.157 suggested that in m-xylene, the two methyl groups strengthen one another in their activation of the ring. But in 1,2-dimethylbenzene (oxylene) and 1,4-dimethylbenzene (p-xylene), the methyl groups cancel the ring activation to some extent. If we look at position 4 (most reactive site on the basis of intramolecular study) of halobenzenes, (Table 2 and 3, third and fourth columns) expected results are obtained (as per discussion in paragraph 3 above) except for bromobenzene. However, as Br is a heavy atom, use of higher basis set may correct the anomaly. It is worth mentioning here that due to the presence of two opposing forces, i.e., –I effect and +R effect, reactivity trends of halobenzenes are ambiguous and also depend on the type of reactions.158 However, what apparently seems from the correlation of ∆− h(k ) values with the experimental relative reaction rate data is that +R effect is a stronger operating force than –I effect in the process of an electrophilic attack on halobenzenes. That might be the reason why ∆− h(k ) values (of the strongest reactive sites) increase (again except for bromobenzene) as the relative rate of nitration and benzylation decrease. This argument is more justified because electrophilic aromatic substitution is a two-step process and passes through a positively charged σ -complex (see the mechanism below). It should be stressed that ∆− h(r ) is based on electrostatic potential (in spite of the fact that only electronic component is present here) and so, hopefully, can take care of long range interactions (intermolecular) more accurately. This may be the probable reason why ∆− h(k ) can predict intermolecular reactivity trends of the systems those are discussed so far. 18


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The general mechanism of electrophilic aromatic substitution is as shown below158 :

1.

E+ E

B

E

2.

E

H

+ BH

where, E+ is the electrophile involved and B is a nucleophile / base. Electrophilic aromatic substitution is a two-step reaction. In the first step aπ-complex and then a σ complex formation takes place. As this step involves the loss of aromaticity, it is the slowest, i. e, the rate-determining one. The second step involves the elimination of H+ from the intermediate σ complex by a base / nucleophile. In an earlier study, Bagaria and Roy62 demonstrated that in multistep reactions electrophilicity / nucleophilcity of the starting substrate may not always provide reliable information on the overall rate of the reaction. Similar arguments will hold for intermolecular comparison of reaction rates on the basis of most reactive site of the individual species (when systems have multiple sites of comparable reactivity). However, for the reaction chosen in the present study, the first step is the rate-determining one. Hence higher nucleophilicity of the substrate will favor the first step as well as the overall rate of the reaction. That might be the probable reason why ∆− h(k ) is able to produce expected trends for reactivity for most of the chosen systems though electrophilic aromatic substitution is not a single-step concerted reaction.

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Group II: Polycyclic Aromatic Hydrocarbons (PAHs): The reactivity of alkylbenzenes and halobenzenes may be solely explained on the basis of resonance theory due to their structural simplicity. To test the efficacy of eqn 5 as a proposed definition of ∆− h(k ) , positional selectivity and relative rates (compared to that of benzene) of bromination of polycyclic aromatic hydrocarbons (which are structurally as well as mechanistically more complex systems compared to simple arenes) are studied here. Some of the earlier theoretical approaches which are exploited to study electrophilic substitution on polycyclic aromatic hydrocarbons, are based on MO reactivity parameters159-162 and electrophile affinity.163 The chosen polycyclic aromatic hydrocarbons have only one predominant reactive site (experimentally found for bromination) except for naphthalene which has two reactive sites (of comparable strength) and reactivity of position 1 is higher than that of position 2. In section 4. A. (i) it is discussed that higher the ∆− h(k ) value higher should be the reactivity of that site (here atom) within a molecule (intramolecular). Here, in case of naphthalene also it is observed that the ∆− h(k ) value of position 1 is higher than that of position 2 (Table 4, third column) i. e., position 1 is more reactive compared to position 2. Also, as per discussion in section 4. A. (ii) (fourth paragraph), ∆− h(k ) can be safely applied to systems with single predominant reactive sites to study relative reaction (in this case bromination) rates (i.e., intermolecular reactivity). But the interpretation will be different from intramolecular cases. Here, five or six membered fused rings adjacent to the reactive sites in polycyclic aromatic hydrocarbons are neither electron-withdrawing nor electron releasing. However, variation of the electron pushing power (after the electrophilic attack) of the rest part of 20


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the molecule in different PAHs will be reflected in the generated ∆− h(k ) values. From Table 4, third column, we get expected reactivity trends for almost all systems, except for fluorene, pyrene and acenaphthene (for naphthalene, the value of most reactive site i. e., position 1 is considered). The bromination of PAHs are carried out experimentally in different concentrations of aqueous acetic acid for different systems. In the present study, theoretical calculations are performed in gas phase. This may be one of the probable reasons for the irregular reactivity trends in some cases generated by ∆− h(k ) values. Also, some of the PAHs undergo substantial amount of addition reaction apart from substitution ones. Both the types of reaction involve the same transition state162 and the ratios of substitution vs. addition reaction also vary from one system to the other. While the experimental values of relative rates are for substitution only the calculated values of ∆− h(k ) represent the overall reactivity. This also makes the comparison of ∆− h(k ) values with experimental relative rates of bromination more complicated and inconclusive.

B. Relative contributions of the sum of kinetic and exchange energy terms to that of the electronic component of the molecular electrostatic potential in ∆− h(k ) : Upto this point, intra and intermolecular reactivity trends of the chosen systems towards electrophilic attacks on them are analyzed. However, the prime objective of the present study is to test the reliability of the approximated forms of ∆+ h(k ) and ∆− h(k ) (eqns 4 and 5). So, the relative 10  4  contributions of the sum of kinetic  C F ρ (r ) 2 / 3  and exchange energy  C X ρ (r )1 / 3  terms and 9  9 

electronic part of the molecular electrostatic potential [ Vel ( r ) ], as indicated in eqns 9 and discussed in section 2.A., are shown separately [Table 1, 2, 3 and 4]. If the numerical values in the third and 21



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fifth columns of these tables are compared, it is evident that the net contribution of the first four terms (i.e., kinetic and exchange energy terms) is negligible when compared to the net contribution of the last two terms in eqn 9. Moreover, if we consider all the three terms (i.e., kinetic energy, Vel ( r ) and exchange energy, eqn 9)or only the Vel ( r ) (eqn 5) while generating ∆− h(k ) values, in

both the cases predicted reactivity trends do not vary much from each other (Table 1, 2, 3 and 4, third and fourth columns). It encourages us to argue that eqn 5 is a reasonable approximation to eqn 9 in the formulation of variants of hardness potential.

C. Relative contributions of the sum of kinetic and exchange energy terms to that of the electronic part of the molecular electrostatic potential in ∆+ h(k ) : To validate the ability of the approximated form (eqn 4) as a working eqn of ∆+ h(k ) , carboxylic acids and their derivatives have been chosen as electrophilic systems (i.e., systems 10  belonging to group III). The relative contributions of the sum of kinetic  C F ρ (r ) 2 / 3  and 9  4  exchange energy  C X ρ (r )1 / 3  terms and the electronic part of the molecular electrostatic 9 

potential [ Vel ( r ) ], as formulated in eqn 8 and discussed in section 2.A. are shown in Table 5, (second and fourth columns). If the numerical values are compared, it is clear that the net contribution of the first four terms (i.e., kinetic and exchange energy terms) is not always negligible when compared to the net contribution of the last two terms in eqn 8. However, it is gratifying to note that intermolecular reactivity trends generated by ∆+ h(k ) values using eqns 4 and 8 do not vary much from each other (Table 5, second and third columns). Here, the strategy adopted to interpret

22


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intermolecular reactivity trends using ∆+ h(k ) is that, higher is the value of ∆+ h(k ) , higher is the reactivity.65

Concluding Remarks: The primary theme of the present article is to assess formally, as well as computationally, the relative contribution of the sum of kinetic and exchange energy terms to that of the electronic part of molecular electrostatic potential in the working equations of the variants of hardness potential [i.e., ∆+ h(k ) and ∆− h( k ) ]. The reason behind considering only the electronic part of molecular electrostatic potential, unlike many other approaches where total electrostatic potential i.e., both electronic and nuclear components are used, is also addressed. It is worth mentioning here that reactivity trends generated by ∆+ h(k ) and ∆− h( k ) are in good agreement with experimental ones, even though Vnu (r ) part is absent in the definition (i.e., eqns 4 and 5 or eqns 8 and 9) of ∆+ h(k ) and ∆− h( k ) .

While, the net contribution of the sum of kinetic [

10 C F ρ (r ) 2 / 3 ] and exchange energy 9

4 [ C X ρ (r )1 / 3 ] terms is found to be negligible when compared to that of electronic contribution to the 9

molecular electrostatic potential [ Vel ( r ) ] [eqn 9 and Table 1, 2, 3 and 4] for systems like substituted benzenes and polycyclic aromatic hydrocarbons, this is not the case for systems like carboxylic acids and their derivatives (eqn 8 and Table 5). So, it may be broadly concluded that depending on the type of systems, the net contribution of the sum of the kinetic and exchange energy terms may or 23



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may not be negligible when compared to that of Vel ( r ) . Thus, to be on the safer side, it is better to use all the three terms i.e., eqns 8 and 9 in place of eqns 4 and 5, respectively. Incidentally the values of ∆+ h(k ) and ∆− h( k ) (wherever applicable), generated by the two approaches on the chosen systems maintain mostly identical trends which again are as per experimental observation in majority of cases. While ∆− h( k ) is able to reproduce experimental trends for site-selectivity for almost all alkylbenzenes, halobenzenes and polycyclic aromatic hydrocarbons in electrophilic aromatic substitution, it is observed that intramolecular reactivity trends for toluene, 1,2-dimethylbenzene and 1,3 dimethylbenzene (where ortho steric effects should be dominant) were also satisfactorily predicted by ∆− h(k ) . This latter observation is particularly encouraging because earlier study using total electrostatic potential at the nucleus (EPN) could not produce experimental trends.132 The success of ∆− h(k ) in generating expected reactivity trends is attributed to the fact that it can take care of the changing electron density scenario when the electrophile approaches towards the nucleophilic system of interest. In an earlier study,62 it was also argued that these type of descriptors can be reliably used only for reactions which are single step concerted ones and not always for multi-step ones. Similar logic can be applied to intermolecular comparison of reaction rates on the basis of reactivity of the most reactive site in individual species (when systems have multiple sites of comparable, but not equal reactivity). Although electrophilic aromatic substitutions are multistep processes, the first step itself is the rate-determining one. Hence, ∆− h(k ) can predict expected reaction rates.

24


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Finally, on the basis of ‘the most reactive site’ model of the individual species, ∆− h(k ) and ∆+ h(k ) can generate satisfactory intermolecular reactivity trends for most of the chosen systems having multiple sites of comparable (but not equal) reactivity. The probable reason of this success is that these descriptors are based primarily on electrostatic potential, which can take care of large distance effects and so the intermolecular reactivity. However, a thorough and detail probe is warranted to justify the argument as stated above. Some studies are under progress to explain different types of reactivity trends and phenomena by using ∆− h(k ) and ∆+ h(k ) .

Acknowledgments: Authors thank sincerely all the reviewers for their thoughtful comments which helped immensely in modifying the manuscript and enhancing its scientific significance. R. B is thankful to the Department of Science and Technology (DST), the Government of India, New Delhi, for financial support in the form of a Senior Research Fellowship (Project). R.K.R. acknowledges the financial support by DST, India (Project Ref. No. SR/S1/PC-41/2008) as well as Departmental Research Support under University Grants Commission (UGC) Special Assistance Programme (SAP) [Project Ref. No. F.540/14/DRS/2007 (SAP-I)], Government of India. Helpful discussion with Prof. Dmitri Fedorov, Research Institute for Computational Sciences (RICS), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan is gratefully acknowledged.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Values of ∆− h(k ) (using eqn 5), ∆− h(k ) (using eqn 9), sum of (kinetic and exchange) energy terms, Vc ,

ln f and rates of chlorination (relative to that of benzene) for Methyl Arenes at B3LYP/6-31G(D,P) level. Systems for Chlorination

Position of Attachment

∆− h(k ) ∆− h(k ) (a. u.) (a. u.) using,

using, three terms (eqn 9)

Vel ( r )

Vc a

(Kinetic + Exchange) energy terms (a. u.)

(volts)

ln f

b

RelativeRa tec

(eqn 5) 1. benzene

1

0.258

0.242

-0.016

-1.069

0.0

1

2. toluened

2

0.214

0.197

-0.017

-1.225

6.4

340

4

0.258

0.209

-0.049

-1.210

6.7

3

0.207

0.196

-0.011

-1.310

7.9

4

0.229

0.208

-0.021

-1.298

8.2

2

0.154

0.194

0.040

-1.372

12.6

4

0.199

0.205

0.006

-1.359

12.9

5. 1,4-dimethylbenzene

2

0.215

0.201

-0.014

-1.307

8.0

2000

6. 1,3,5-trimethylbenzene

2

0.238

0.188

-0.052

-1.500

17.9

30000000

7. 1,2,4,5tetramethylbenzene

3

0.194

0.195

0.001

-1.521

15.4

1580000

8. pentamethylbenzene

6

0.191

0.192

0.001

-1.686

20.5

134000000

3. 1,2-dimethylbenzened

4. 1,3-dimethylbenzened

2030

180000

a. Vc is the electrostatic potential at the reacting carbon atom132 b. Logarithm of partial rate factors ‘ f ’(i.e., ln f ) values were taken from ref. 132. The positive values indicate that the ring is activated (due to the presence of methyl groups) in alkylbenzenes w.r.t benzene. c. Taken from ref.132. d. Systems where EPN failed but ∆− h(k ) produced correct reactivity trends [as discussed in Section 4.A.i)] are shown in bold.

45



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Page 46 of 51

Table 2: Values of ∆− h(k ) (using eqn 5), ∆− h(k ) (using eqn 9), sum of (kinetic and exchange) energy terms, Vc , ln f and rates of nitration (relative to that of benzene) for halobenzenes at B3LYP/6-31G(D,P) level.

Systems for nitration

Position of Attachmentof Singlet Cl+

∆− h(k ) (

∆− h(k ) (

a. u.) using,

a. u.) using,thr eeterms(e qn 9)

Vel ( r ) (e qn 5)

Vc a

(Kinetic +Exchange ) energy terms(a. u.)

(volts)

ln f

b

Relative Ratec

1. benzene

1

0.258

0.242

-0.016

-1.069

0.0

1

2. fluorobenzene

4

0.264

0.227

-0.037

-0.869

-0.3

0.45

3. chlorobenzene

3

0.193

0.203

0.010

-0.672

-7.1

0.14

4

0.269

0.227

-0.042

-0.768

-2.0

3

0.203

0.193

-0.010

-0.699

-6.9

4

0.230

0.197

-0.033

-0.817

-2.3

4. bromobenzene

0.12

a. Vc is the electrostatic potential at the reacting carbon atom132 b. Logarithm of partial rate factors ‘ f ’ (i.e., ln f ) values were taken from ref 132. The negative values in halobenzenes (due to the presence of halides) indicate that the ring is deactivated w.r.t. benzene. c. Taken from ref.156

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Table 3: Values of ∆− h(k ) (using eqn 5), ∆− h(k ) (using eqn 9), sum of (kinetic and exchange) energy terms, Vc , ln f and rates of benzylation (relative to that of benzene) for halobenzenes at B3LYP/6-31G(D,P) level.

Systems for benzylation

Position of Attachmento f Singlet Cl+

∆− h(k ) (

∆− h(k ) (

a. u.) using,

a. u.) using, three terms (eqn 9)

Vel ( r ) (eqn 5)

(Kinetic + Exchange) energy terms(a. u.)

Vc a(vol

ln f

ts)

b

Relative Ratec

1.benzene

1

0.258

0.219

-0.016

-1.069

0.0

1

2.fluorobenzene

2

0.237

0.219

-0.018

-0.767

-1.6

0.46

3

0.230

0.217

-0.013

-0.694

-5.9

4

0.264

0.227

-0.037

-0.869

0.9

2

0.217

0.217

-0.000

-0.653

-1.4

3

0.193

0.203

0.010

-0.672

-5.4

4

0.269

0.227

-0.042

-0.768

0.0

2

0.210

0.195

-0.015

-0.554

-1.7

3

0.203

0.193

-0.010

-0.699

-5.6

4

0.230

0.197

-0.033

-0.817

-0.3

3.chlorobenzene

4.bromobenzene

0.24

0.18

a. Vc is the electrostatic potential at the reacting carbon atom132 b. Logarithm of partial rate factors ‘ f ’(i.e., ln f ) values were taken from ref 132. The negative values indicate that the ring is deactivated (due to the presence of halides) in halobenzenes w.r.t. benzene. c. Taken from ref. 155

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Page 48 of 51

Table 4: Values of ∆− h(k ) (using eqn 5), ∆− h(k ) (using eqn 9), sum of (kinetic and exchange) energy terms, ln f and rates of bromination (relative to that of benzene) for polynuclear aromatic hydrocarbons at B3LYP/6-31G(D,P) level.

Position of reaction

∆− h(k ) (a.u.) using, Vel ( r ) (eqn 5)

1. benzene

1

0.258

2. naphthalene

1

Hydrocarbon

∆− h(k ) (

(Kinetic +Exchange) energy terms (a. u)

ln f b

0.242

-0.016

0.0

1

0.206

0.178

-0.027

5.3

1.24x105

2

0.193

0.179

-0.014

3.3

3. phenanthrene

9

0.189

0.166

-0.024

6.3

7.43x105

4. fluoranthene

3

0.170

0.151

-0.019

6.8

2.30x106

5.fluorenea

2

0.183

0.154

-0.029

7.0

3.53x106

6. chrysene

6

0.167

0.145

-0.022

7.6

1.25x107

7. 1,2 benzanthracene

7

0.152

0.140

-0.012

11.2

2.44x1010

8. pyrenea

1

0.167

0.155

-0.012

10.6

2.84x1010

9. acenaphthenea

5

0.195

0.168

-0.027

11.2

5.49x1010

10. anthracene

9

0.187

0.161

-0.026

12.4

7.87x1011

a. u.) using,th ree terms(e qn 9)

Relative Ratec

a. Systems where ∆− h(k ) produced irregular reactivity trends [as discussed in Section 4.B)] are shown in bold. b. Logarithm of partial rate factors ‘ f ’ (i.e., ln f ) values were taken from ref 163. c. Taken from ref. 162

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Table 5: Values of ∆+ h(k ) (using eqn 4), ∆+ h(k ) (using eqn 8) and sum of (kinetic and exchange) energy terms for carboxylic acid and its derivatives at B3LYP/6-31G(D,P) level. Electrophilic centres are shown in bold. ∆+ h(k ) (a. u.) using,

∆+ h(k ) (a. u.)

Vel ( r ) (eqn 4)

using, three terms (eqn 8)

1. HCOF

0.409

0.298

-0.112

1. HCOOH

0.393

0.286

-0.108

3. HCONH2

0.374

0.275

-0.099

4. CH3COOH

0.360

0.255

-0.106

5.CH3CH2COOH

0.329

0.237

-0.092

6.CH3CONH2

0.320

0.241

-0.079

7.CH3CH2CONH2

0.226

0.231

0.005

8.CH3COF

0.375

0.264

-0.110

9.CH3CH2COF

0.250

0.257

0.007

Systems for Nucleophilic Attack

49


(Kinetic +Exchange) energy terms(a. u.)


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Page 50 of 51

Figure 1: Chosen Polycyclic Aromatic Hydrocarbons: 9

1

3

2

3. Phenanthrene

2. Naphthalene

4. Fluoranthene

2

7

6

7. 1,2-benzanthracene

5. Fluorene 6. Chrysene 9

1

5

10.Anthracene

9. Acenaphthene 8. Pyrene Preferred Site of electrophilic attack

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For Table of Contents Only:

∆+ h (k ) ∆Vel (r ) ∆N

∆− h (k ) >>

r →0

∆ ∆N

10 2/3  - ∆  9 C F ρ (r )  ∆N r →0

4 1/ 3   9 C X ρ (r )  r →0

Reliable Approximation or Not?

51


Relative Contribution of Combined Kinetic and ... - ACS Publications

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