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Quantitative Structure-Activity/Property/Toxicity Relationships
ties like molecular weight, atomic number, number of OH groups, number of non-hydrogenic atoms and so on correspond to 1-D descriptors.2-D descriptors are usually obtained by applying algorithms to the topology based representation. 3-D descriptors, also called geometrical descriptors are generated from the spatial coordinates (x,y,z) of a molecular system whereas the interaction energy values between an embedded molecule in a grid and a probe will provide 4-D descriptors. In general, molecular descrip­tors should satisfy some basic requirements as suggested by Randić (1991). Different other articles also showed the role of QSAR in the generation of different algorithms (Karcher & Devillers, 1990; Roy & Mitra, 2009; Selassie et al., 2002).
Molecular descriptors are mathematical representations of various properties of a molecule which are generally derived after the application of a reliable algorithm to the molecular representation or are determined from experiment. Being related to the molecular Hamiltonian, the molecular descriptors like electron density, electrostatic potential, etc. would represent all the properties of a molecule (Bonaccorsi et al., 1970; Murray & Politzer, 2003; Pathak & Gadre, 1990; Politzer, 1988). These descriptors help in the development of models for chemistry, toxicology, health research, pharmaceutical sciences, etc. They are generated by the application of different principles from different fields and theory, such as organic chemistry, graph theory, quantum-chemistry, information theory etc. The electron localization and delocalization functions obtained from the quantum theory of atoms in molecules (QTAIM) are also very effective in modeling QSAR (Buttingsrud et al., 2007a, 2007b, 2007c; Chaudry & Popelier, 2003, 2004; O’Brien & Popelier, 1999; Popelier et al., 1999, 2004; Smith & Popelier, 2005). Popelier et al. (1999) showed that a QTAIM based approach called quantum topological molecular similarity (QTMS) could predict the toxicity of health-hazardous lipid-soluble environmental pollutants as efficient as that of comparative molecular field analysis (CoMFA). Very recently, Matta et al. (2014) reviewed the ap­plicability of the electron density, electron localization and delocalization functions, and the electrostatic potential in developing effective QSAR models.
Bohari et al. (2011) adopted so-called analogue-based approach to predict the anti-cancer activity of 266 different compounds against 29 various cancer cell lines. Various models were constructed through 1 to 10 different descriptors. They found that in general three-parameters regression models could predict the anti-cancer property satisfactorily. Filho et al. (2012) performed QSAR studies on 32 Morita–Bay­lis–Hillman adducts against Leishmania amazonensis. Energy of the lowest unoccupied Kohn–Sham orbital, number of o−NO
group, number of –CH2− group and an increment in log P parameters were
2
used to build the regression model. A reasonably good predictivity was found for this model. QSAR studies were also performed in two sets of antitumour drugs 2-(4-aminophenyl)benzothiazoles by using different indices encompassing constitutional, geometrical, topological, and electronic descriptors (Hilal
2
& Elroby, 2011). Good correlation having a range of r
as 0.867-0.954 was attained between computed and experimental activities with these models. A number of molecular descriptors like polar surface area, Randic shape index, number of phenolic groups and energy of the highest occupied molecular orbital were used to predict the estrogenic activity of different terpenoid ester derivatives obtained from Ferula
2
plants (Rasulev et al., 2007). An r
value of 0.892 implied the reliability of the regression model to predict the activity of these compounds. Li et al. (2010) carried out the QSAR studies on 27 diacyl-hydrazine derivatives by using DFT based method and semi-empirical AM1-MOPAC method. They found higher predictivity at the DFT level than that at the semi-empirical level. 3D-QSAR models were also generated based on the geometries obtained at the DFT level and electrostatic potential fitting charges. The results of the QSAR and 3D-QSAR were found to be comparable. For these compounds, it was argued that the electrostatic and hydrophobic properties are important in deciding their overall biological activities.
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Quantitative Structure-Activity/Property/Toxicity Relationships
Zhang et al. (2009) combined DFT-based conformation analysis with QSAR to develop a simple new DFT/QSAR approach to predict the nature of bioactive conformation in the side chain of cyclic imide derivatives as protoporphyrinogen oxidase inhibitors. The potential energy surface scan, molecular docking and molecular dynamic simulation supported the reliability of this new approach. Damme and Bultinck (2010) employed conceptual DFT bsed quantum chemical molecular field in order to construct 3D-QSAR models to describe the antituberculotic activity of salicylamide derivatives. QSAR studies on the interactions between the terpenoid mosquito repellents and lactic acid were made by Song et al. (2013). They found that in addition to the structures of the repellents, the repellent–lactic acid complexes also have a role in deciding their activities. Use of ordered predictors selection algorithm and partial least squares regression was also made to select variables and to construct QSAR models of various 4,5-dihydroxypyrimidine carboxamide derivatives as HIV-1 integrase inhibitors (de Melo et al., 2009). The electronic distribution of the studied compounds appeared to have deciding role in their inhibition activities. The antifungal activity of N-nitrosopiperidone semicarbazone derivatives against the plant pathogens viz., Fusarium oxysporum and Rhizoctonia solani can also be predicted by the regression models devepoled by the descriptors like molecular electrostatic potential, charge on N center, dipole moment, minium surface area and log P (Hemalatha et al., 2012). The QSAR studies on the imidazole and benz­imidazole derivatives as corrosion inhibitors revealed that HOMO energy, polarizability, dipole, frontier orbital charge density, the interaction mode between inhibitors and metal surface and steric hindrance of molecules are deciding descriptors to represent their inhibiting potency (Zhang et al., 2005). Various DFT based descriptors like HOMO-LUMO energy gap, HOMO and LUMO energies, mean molecular polarizability, atomic charge, weighted nucleophilic atomic frontier electron density were employed in order to build regression models for the various cyclic imide derivatives as protoporphyrinogen oxidase (Protox) inhibitors (Wan et al., 2004). The DFT-based QSAR and 3D-QSAR approaches on the activity of aurone derivatives against chloroquine resistant parasite by Adhikari et al. (2013) are also notworthy. A review summarizing various quantum-chemical descriptors in developing QSAR and QSPR written by Karelson and Lobanov (1996) is highly recommended to get more details about these approaches.
Geometrical descriptors were designed and included in the QSAR during the development of 3D­QSAR models in mid 1980s. These included gravitational indices (Katritzky et al., 1996), shadow indices (Rohrbaugh & Jurs, 1987), EVA descriptors (Ferguson et al., 1997), GETAWAY descriptors (Consonni et al., 2002), 3D-Morse descriptors (Schuur et al., 1996), charged partial surface area descriptors (Stanton & Jurs, 1990), and WHIM descriptors (Todeschini et al., 1994).
The descriptors such as topological descriptors utilize molecular graph. Different topological indices such as those developed by Weiner (1947), Randić (1975), Balaban (1998), Kier and Hall (1995) were reported in literature to generate QSAR models. Other descriptors such as steric (Camenisch et al., 1998; Mezey, 1992) viz., volume, shape and surface, were found to be useful in representing the global structural characteristics of a molecule and different software such as VolSurf software (Cruciani et al.,
2000) were employed to create effective QSAR models. In order to generate 2D and 3D descriptors for the prediction of physicochemical properties, the methods provided by Rekker (1992) were used.
CoMFA (Cramer et al., 1988) is an effective method in building effective 3D-QSAR, which was successfully applied in drug designing and other related applications (Aboye et al., 2004; Akamatsu, 2002; Amin & Welsh, 2006; Avery et al., 2002; Bhongade & Gadad, 2004; Hannongbua etal., 2001; Jayatilleke et al., 2000; Sippl & Holtje, 2000). History behind the idea of CoMFA in modeling QSAR is that in the classical QSAR, the different biological acitivies of drugs or other compounds of interest are correlated with various physico-chemical properties or quantum-chemical descriptors, provided
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Quantitative Structure-Activity/Property/Toxicity Relationships
they could represent certain structural characteristics. Therefore, they did not take care of the informa­tion regarding their 3-D geometries and chirality. On the other hand, at longer distance an active site of a drug can only experience the electrostatic field of a target molecule and at shorter distance its hard body including the charge distribution (Kubinyi, 1998). Other two important observations were that the most of the molecular interactions which generate biological activity, are of non-covalent type and the most of the molecular mechanics force fields, wich describe the non-covalent interaction as steric and electrostatic forces, could explain a large variety of molecular properties. Therefore, it appeared that an appropriate sampling of the steric and electrostatic forces surrounding the considered molecules could give all the necessary information regarding their observed biological properties. Cramer and Milne (1979) compared the molecules for the first time by arranging them in space and by molecular field mapping in a 3-D grid. The same approach was then implemented in dynamic lattice-oriented molecular modeling system (DYLOMMS) method (Kubinyi, 1993). However, to obtain success in broader applications of this approach, certain factors like the use of partial least square analysis instead of principal component analysis and use of appropriate software are important (Kubinyi, 1998). In 1988, Cramer et al. (1988) termed this approach as CoMFA. The major precondition for this approach is that all the selected mol­ecules under study must interact with the receptors of same kind and in same mechanistic way. Then in order to develop a CoMFA model, a traning set is chosen from a certain subgroup of the considered molecules and the remaining part is treated as a test set. This test set is used to verify the efficacy of this model to describe certain properties. The final output of this analysis provides a regression equation having several coefficients and they are very often depicted as a set of contour plots. From these plots, the favorable and unfavorable steric regions and favorable and unfavorable regions for electropositive or electronegative substituents could be identified. Then one could predict the desired properties of the test set or other related samples by qualitative examination of these plots or by computing the fields of these molecules quantitatively. Cramer et al. (1988) first employed this method to predict the binding
2
affinities of steroids towards the human corticosteroid. The squared correlation coefficient (r
) for the affinity values was found to be very high (close to 0.9). Since 1988, a lot of articles including reviews are reported in the literature on this topic (Adhikari et al., 2014; Blankley, 1996; Kim, 1995; Martin et al., 1996; Martin & Lin, 1996).
In addition to the development of the QSAR, quantitative structure-property relationship (QSPR) was also used to predict a certain property of the molecules. The hypothesis, that structural alteration changes the observed macroscopic properties of systems is the base of QSPR methods. Several research groups contributed significantly in the development of QSPR models (Abraham, 1994; Abraham et al., 1995; Balaban, 1997; Hilal et al., 1994; Katritzky et al., 1995; Kier & Hall, 1986; Lucic & Trinajstic, 1999; Murray & Politzer, 1994; Randic & Razinger, 1996; Stuper et al., 1979). The efficacy of QSPR as a useful tool in connecting molecular structure with the various physical, chemical, biological and technological properties was further demonstrated by Katritzky et al. (2000). The successful implication of QSPR in medicinal chemistry to predict different property was also available in the literature (Katritzky et al., 2002). A fair correletion between struc­ture and property was obtained by using QSPR models made on an empirical basis even for large data sets. The importance of data reduction methods like principal component analysis (PCA) of a matrix generated by the different properties of the systems was also found in providing insight into these properties quantitatively (Leo & Hansch, 1999). The reviews summarizing the construction of QSPR models in predicting diverse properties of materials are highly recommended. Variuos approachs were also made to model effective QSPR for anti HIV drugs. The applicability of QSPR in pharmaceutical arena including different pharmacokinetic aspects towards drug designing was reviewed by Grover et al. (2000a, 2000b).
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Quantitative Structure-Activity/Property/Toxicity Relationships
Further, quantitative structure-toxicity relationship (QSTR) is essential in order to have proper knowl­edge about the toxicity of various essential systems; hence it is an issue of environmental safety and security. The approach of QSTR modeling even becomes more popular since experimental determina­tion of toxicity of each and every essential compound is hard with respect to time, cost, and the avail­ability of resources. Different theoretical approaches have been made to connect the structural features (electronic, geometric, topological, quantum chemical, etc.) with the toxicity. An effective QSTR model could predict the relative toxicity/safety of different systems in a reliable way saving the experimental effort. Cronin and Schultz (1998) developed useful QSTR model for vibrio pscheri toxicity including all probable mechanisms. Schultz et al. (1999, 2002) evaluated toxicity of numerous aliphatic chemicals and substituted benzenes towards Tetrahymena pyriformis.
The databases regarding toxicity of a number of fish species like the guppy (Poecilia reticulata) and zebrafish (Brachydanio rerio) are also available. The fish data sets for the fathead minnow (Pimephales promelas) to predict the acute toxicity and related modes of toxic action was built by studying the es­tablishment of toxicodynamic profiles, and behavioral and dose-response interpretation of 96-h LC
50
tests (Russom et al., 1997). Good correlation between fish toxicity and toxic potency measures in T. pyriformis was also found in few studies (Bearden & Schultz, 1998; Schultz, 1997). McFarland (1970) proposed quite simple toxicokinetic- and toxicodynamic-based approach in which chemical toxicity was expressed as a combination of uptake into or through biological membranes and the interaction of the toxicant with the site of action to predict toxicity. Different approaches in modeling QSTR were also reported in the literature (Chakraborty et al., 2013; Cronin & Schultz, 1996; Karabunarliev et al., 1996; Veith & Mekenyan, 1993). QSTR could also be applied to measure toxicity of disinfection by-products (DBPs) generated during disinfection of water (Moudgal et al., 2000).
Cronin et al. (1998) developed QSTR model by using LUMO energy and maximum acceptor super­delocalizability as descriptors to predict toxicity of various alkyl- and halogen-substituted nitro- and dinitrobenzenes towards T. Pyriformis. Mishra et al. (2014) employed various quantum-chemical descrip- tors like HOMO energy, electronegativity, electron affinity (EA), ionization potential (IP), total energy and log P to predict the toxicity of nitrobenzene derivatives towards T. Pyriformis. The topological descriptors were also found to be effective to describe the toxicity of nitrobenzene derivatives towards the same (Tripathi & Mishra, 2014). Further based on several energy descriptors, heat of formation, atomic charge, dipole moment and polarizability values, Bao et al. (2012) predicted the toxicity of several nitrobenzene derivatives towards the Scenedesmus obliguus. Very recently, Tawari et al. (2014) attempted to build QSTR model to describe the mutagenicity of nitroaromatic compounds. Pandith and Islam (2013) further built QSTR models by using log P, electrophilicity (ω) and LUMO energy to predict the acute toxicity of aliphatic compounds towards V. fischeri. HOMO energy, molecular weight, LUMO energy, chemical potential, zero point energy, and hardness were also used to model QSTR in order to predict 50% effective inhibition concentration of various substituted benzene derivatives towards the algae Scenedesmus obliquus (Haghdadi & Fatemi, 2010). Quite high correlation (r = 0.91) revealed that such properties can be properly described by using such molecular indices. Eroglu et al. (2007) used quantum-chemical descriptors like translational entropy, principal moment of inertia, the highest and the lowest vibrational wave numbers to predict the acute toxicity of several organic compounds towards Fathead Minnow. Further, toxicity of organophosphorus compounds was assessed using the regression models mady by different quantum-chemical and topological descriptors (Senior et al., 2011). Wiener index, Harary index, modified Randic index, Kupchik modified connectivity index (Todeschini & Con­sonni, 2000), electronegative Randic index, heteroatom corrected extended connectivity Randic index
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Quantitative Structure-Activity/Property/Toxicity Relationships
as different topological descriptors and HOMO-LUMO gap, maximum positive charge, total negative charge, charge polarization, local dipole index, electronic topological index, etc. as quantum-chemical descriptors were used for this purpose. Attempt to predict the lethal toxicity of organophosphorous pesticides occurred in the tadpoles was also reported in the literature (Yan et al., 2008). Antifungal and insecticidal activities of perfluorophenyl antimony(III) and antimony(V) chlorides were predicted by several energy and CDFT based descriptors by Singhal et al. (2014). Regression model based on total energy and LUMO energy was also found effective in describing the toxicity of several substituted benzenes against freshwater photobacteria (Mo et al., 2011). The hardness, electronegativity and total energy descriptors were found to be useful descriptors for QSTR modeling on a series of aliphatic alcohol derivatives (Kumar et al., 2011). Trohalaki and Pachter (2003) showed that the use of recipro­cal of LUMO energy could improve the correlation in predicting the toxicity of halogenated aliphatic hydrocarbons than that of LUMO energy. The acute fish toxicity of organic solvents was also described by using the octanol-water partition coefficient, LUMO energy, dielectric constant and surface tension as descriptors (Levet et al., 2013). Gironés and Carbó-Dorca (2006) reported the molecular quantum similarity measures approach in develping QSTR.
3. CONCEPTUAL DENSITY FUNCTIONAL THEORY
The conceptual density functional theory (CDFT) (Chattaraj, 2009; Chattaraj & Giri, 2009; Chakraborty et al., 2010; Geerlings et al., 2003) has become a powerful theoretical tool to shed light into the chemical reactivity of a molecule. Within the framework of the CDFT, several response functions were introduced in order to describe chemical reactivity, which are also called as reactivity descrip­tors. CDFT along with its different global reactivity descriptors viz., hardness (η) (Parr & Pearson, 1983; Pearson, 1997), electronegativity (χ) (Chattaraj, 1992; Parr et al., 1978) and electrophilicity (ω) (Parr et al., 1999; Chattaraj et al., 2006, 2007, 2011) and different local reactivity descriptors like Fukui functions (Parr & Yang, 1984)
1986) (f
), local softness (s(r)) (Wang & Parr, 1985; Berkowitz & Parr, 1988), local hardness (η(r))
k
(Wang & Parr, 1985; Ghosh & Berkowitz, 1985) and philicity (ω(r)) (Chattaraj et al., 2003; Roy et al., 2006a) predict the reactivity behavior of a molecule as a whole and / or at a particular site.
The concept of electronegativity (χ) was first introduced by Pauling (1932, 1960), which represents the “power” of an atom in a molecule to attract the bonded electrons towards itself. Later, Mulliken (1934, 1935) described χ as the average of the first ionization potential (IP) and the electron affinity (EA), two experimentally obtainable parameters of a molecule. He showed that χ could be determined by an arithmetic mean of the corresponding IP and EA values as given in Equation 1.
χ =
+IP EA
. (1)
2
f ( )r and their condensed-to-atom variants (Yang & Mortier,
The chemical potential (μ) (Parr & Yang, 1989; Hohenberg & Kohn, 1964) of an atom/molecule may be quantitatively assessed by the theory of statistical ensembles. Suppose, an atom or a molecule is a part of a grand canonical ensemble in which energy (E) and the number of electrons (N) are two independent parameters and continuous functions as well, then μ of the ensemble can be expressed as follows:
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∂EN
Quantitative Structure-Activity/Property/Toxicity Relationships
µ =
, at constant entropy (2)
Further, Gyftopoulos and Hatsopoulos (1968) put forward a definition of χ of a system where the
electronic chemical potential (μ) was considered as the negative of χ. Now, following Mulliken’s (1934,
1935) prescription for χ, the Mulliken chemical potential can also be expressed as a finite difference approximation of the derivative of E with respect to N as:
µ χ
= − = −
Mulliken Mulliken
+
IP EA
2
=
 
E N
( )
N
 
. (3)
 
N N
=
0
Pearson (1963, 1966, 1968a, 1968b, 1987) introduced the concept of “hardness” or “softness” in
chemistry during the study of Lewis acid-base reactions
A +:B → A:B
where A, an acid, acts as an electron acceptor whereas B, a base is the electron-pair donor. Global hard­ness (η) implies the extent of compactness of the electron cloud encompassing the nucleus/nuclei of an atomic/molecular system whereas global softness (S) is the reciprocal of η and hence represents the extent to which the electronic environment surrounding the nucleus/nuclei of an atomic/molecular spe­cies tends to loosen itself. Parr and Pearson (1983) expressed the qualitative concept of η quantitatively in the form of a mathematical descriptor. It is defined as the first derivative of μ or the second derivative of E as a function of N at a fixed external potential v(r) as follows:
2
E
N
. (4)
2
( )rr
ηµ=
N
=
vv( )
The convexity in the E vs. N curve signifies that the value of η would be always positive and from the finite difference approach the corresponding curvature equals to difference between IP and EA, which actually represents hardness. Hence
2
E
2
N
=
( )rr
IP EA
. (5)
ηµ=
N
=
vv( )
Therefore, η of a given system describes the extent of its reluctance towards the electron shift to another system in a general charge-transfer phenomenon in between a donor-acceptor couple.
Global softness (S) (Pearson, 1987), a reciprocal of the global hardness (η) of a system, is expressed as:
N
1
S
= =
η µ
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. (6)
v
( )r
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Quantitative Structure-Activity/Property/Toxicity Relationships
Electrophilicity (ω) (Parr et al., 1999; Chattaraj et al., 2006, 2007, 2011) and its recently reported
±
mathematical refinement, net electrophilicity (Δω
) (Chattaraj et al., 2009) are also important descrip-
tors from the perspective of constructing useful SAR based models
In chemistry electrophiles represent those species, which have a special affinity towards accepting electrons. In general, electrophiles are electron-deficient species, which prefer to combine with electron­rich systems, nucleophiles. In an experiment of the human immunodeficiency virus type 1 (HIV-1) nucleocapsid protein p7 (NCp7) with various electrophiles, Maynard et al. (1998) found that the decay
2
rates of fluorescence varied almost linearly with the ratio of the square of χ and η, χ
. This ratio could
further be rationalized as the ability of an electrophile to attract electrons from a nucleophile in forming a covalent bond. This observation of Maynard et al. (1998) led Parr et al. (1999) to describe electron­attracting power of a system in a more precise way.
Consequently, a new reactivity descriptor namely electrophilicity index (ω) was introduced in chemistry. Different from the approach of Maynard et al., (1998) Parr’s postulate of ω was based on a thermodynamic background and thus ω becomes a measure of the favorable change in energy upon saturation of a system with electrons. ω as prescribed by Parr et al. (1999) is:
2 2
µ
η
2 2
χ
. (7)
η
ω
= =
2
Eqution 7 resembles W = V
/R, where W, V and R are electrical power, voltage and electrical resis-
tance, respectively. Hence, ω describes the electrophilic power of a system. Few reviews summarizing the different application of ω in understanding the chemical reactivity of different systems are also available in literature (Chattaraj et al., 2006, 2007, 2011).
In order to correlate the associated energy changes between the interacting acceptors and donors in a charge-transfer process, based on the second-order Taylor series energy expansion formula as a function of the number of electrons (N) in the intervals between N - 1 and N, and N and N + 1, Gazquez et al.
(2007) showed that the electrodonating (ω
2
µ
( )
ω
=
+
ω
=
where μ whereas η proved that μ
+
become equivalent with ω and thereafter ω, ω− or ω+ may be represented in terms of μ and η as ω− =
ω
+
= ω = μ2/2η. Further, Gazquez et al. (2007) proposed two different expressions for ω− or ω+ depend-
ω
(8a)
2
η
2
+
µ
( )
(8b)
+
2
η
and μ+ are the chemical potential for electron donation and electron acceptance, respectively,
and η+ are the hardness for electron donation and electron acceptance, respectively. It was also
and μ+ could be equalized to μ and η− and η+ could be equalized to η. Therefore, ω− and
) and the electroaccepting (ω+) powers might be defined as:
ing on two different approaches; one following the original formula as above and can be expressed as:
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{
}
Quantitative Structure-Activity/Property/Toxicity Relationships
2
ω+=
ω−=
EA
IP EA
2( )
2
IP
IP EA
2( )
(9)
(10)
and other by exploiting an alternative expression for energy as (Gazquez et al., 2007):
3
2
(11)
2
. (12)
or ω+ provided better correlations than the
+
( )
ω+=
ω−=
IP EA
( )
IP EA
16
IP EA
+
( )
3
IP EA
( )
16
It was also reported that the alternative expression of ω
original one (Gazquez et al., 2007).
±
Chattaraj et al. (2009) recently put forward the concept of net electrophilicity (Δω
) to describe the resultant
electron-accepting power of a molecule arising from the combined attractive and repulsive effects due to the
±
presence of both electrons and nuclei within a molecule. This dual descriptor (Δω
) assesses the electrophilicity
of a molecule relative to its own nucleophilicity, therefore is more meaningful descriptor in representing the
±
“electrophilic power” of a system. Thus, net electrophilicity (Δω
± + +
ω ω ω ω ω
= − −
( )
= +
( )
(13)
) in terms of ω+ and ω− can be expressed as:
or,
± +
ω ω
=
 
1
.
ω
The polarizability (α) of an atom or a molecule is the lowest order response of its electron cloud with respect to an external weak electric field (Dalgarno, 1962; van Vleck, 1932). The static dipole polariz­ability (α) can be defined as the second derivative of the total electronic energy (E) with respect to the external homogeneous electric field as:
 
= −
∂ ∂
α
xy
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E
F F
x y
 
(14)
F
=20
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Quantitative Structure-Activity/Property/Toxicity Relationships
where Fx and Fy are the electric field components along x and y direction. Qualitatively, polarizability (α) varies inversely with η (Ghanty & Ghosh, 1993, 1996; Fuentealba & Reyes, 1993; Pal & Chandra, 1995: Pearson, 1986; Politzer, 1987) and is proportional to S. A linear correlation was found between α and the third power of S (Chattaraj & Poddar, 1998; Simon-Manso & Fuentealba, 1998; Vela & Gázquez, 1990).
The magnetizability (ξ) (Griffith, 1999) for a molecule is a measure of the linear response of its
electron cloud with respect to an externally applied magnetic field and is expressed as:
2
ξε= −
( )B
(15)
2
B
=
0
B
where B is the external magnetic field.
A softer molecule is also found to be more polarizable and hence more magnetizable (Chattaraj et
al., 2007).
In addition to the various CDFT based global reactivity descriptors, the local reactivity descriptors
like Fukui function (f(r)), philicity (ω(r)), local softness (s(r)), local hardness (η(r)) and electron density (ρ(r)) are also important in providing valuable information regarding the reactivity pattern at a local atomic site in a molecule.
Information about ρ(r) is important to get insights into site-selectivity for a molecule and can be
determined in terms of the first order variation in energy (E) as a function of the external potential (v(r)) (Parr & Yang, 1989, 1995).
( )rr=
ρ
 
 
δ
E
. (16)
δ
v
( )
N
The Fukui function (f(r)) (Ayers & Levy, 2004; Fukui et al., 1952, 1954; Fukui, 1975, 1982; Parr &
Yang, 1984; Yang et al., 1984) is very useful to evaluate quanlitatively the extent of local response at a particular atomic site of a molecule. It is defined as the differential change in ρ(r) due to an infinitesimal change in the number of electrons (N).
f
( )
 
r
=
r
ρ
( )
N v
v
( )
r
=
δµ
δ
. (17)
( )
r
N
Now, due to the discontinuity in the derivative in eq. 17, for integral values of N, by applying the finite difference and frozen core approximations three different types of Fukui functions can be defined as follows: (Ayers & Levy, 2004; Parr & Yang, 1984)
For nucleophilic attack:
+
f
=
r
+
r
ρ
( )
( ) ( ) ( ) ( )
N
v
( )
r
r r r
ρ ρ ρ
+
1
N N LUMO
. (18a)
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2
Quantitative Structure-Activity/Property/Toxicity Relationships
For electrophilic attack:
f
=
r
r
ρ
( )
N
v
( )
r
r r r
( ) ( ) ( ) ( )
ρ ρ ρ
N N HOMO
1
. (18b)
For radical attack:
0
f f f
1
+
( ) ( ) ( )r r r= +
 
. (18c)
Yang and Mortier (1986) further proposed a coarse-grained atom by atom representation of the Fukui function, called condensed-to-atom Fukui function. Based on a finite-difference approach, they can be represented as:
+
f q N q N
= + ( ) ( )1 for nucleophilic attack (19a)
k k k
f q N q N
= ( ) ( )1 for electrophilic attack (19b)
k k k
0
f q N q N
k k k
where q
1 1 2= + [ ( ) ( )] / for radical attack (19c)
is the electron population at a particular kth atomic site in a molecule.
k
Therefore, condensed-to-atom Fukui function can be easily computed through the data obtained from the population analysis.
Mathematically, global softness (S) can be expressed as an integral of the individual local softness (s(r)). A normalized condition between S and s(r) can be written as:
S s dr=∫( )r (20)
s(r) is connected to f(r) through a chain rule as:
s
( )
r
=
 
( ) ( )
r r
ρ
=
µ
v v
( ) ( )
r r
 
ρ
N
N

µ
f S
( ).rr (21)
=
v
( )
f(r) acts mainly as an intra-molecular reactivity descriptor by describing the reactivity of the atomic sites in a particular molecule whereas unlike f(r), s(r) acts as an inter-molecular reactivity descriptor by comparing and correlating the tendency of interaction between a pair of neighboring molecules during chemical response.
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