*2.2.2 Global descriptive parameters*

Global descriptive parameters are parameters that give information's about the reactivity of coumestrol derivatives and also give the relation between the reactivity of derivatives and responses to the changes in external conditions. So, by calculating these parameters, we can compare the reactivity of coumestrol with its derivatives. It is an attractive method for understanding the reactive nature of all the products [11]. Global parameters include ionisation potential (I), electron affinity (A), hardness (η), softness (S), electronegativity (χ), chemical potential (μ) and electrophilicity index (Ω) [12]. These parameters depend upon the number of electrons and electron density due to the external changes [13]. Global descriptive parameters can be calculated by two methods; they are according to Koopman's theorem and the Energy vertical method. These methods have particular relevance in the comparison of different molecules. Low ionisation potential, high electron affinity and high electronegativity contribute to high reactivity. So, by analysing the values of these parameters' reactivity can be studied.

According to energy vertical, difference in total electronic energy of the neutral molecule and its corresponding anion and cation were considered. The equations for finding ionisation potential (I) and electron affinity (A) are given below;

$$\mathbf{I} = \mathbf{E}\_{\text{cation}} - \mathbf{E}\_{\text{neutral}} \tag{1}$$

$$\mathbf{A} = \mathbf{E}\_{\text{neutral}} - \mathbf{E}\_{\text{anion}} \tag{2}$$

According to Koopman's theorem of closed shell compounds;

$$\mathbf{I} = -\mathbf{E}\_{\text{HOMO}}\tag{3}$$

$$\mathbf{A} = -\mathbf{E}\_{\text{LUMO}} \tag{4}$$

Where EHOMO is the energy of the highest occupied molecular orbital (HOMO) and ELUMO is the energy of the lowest unoccupied molecular orbital (LUMO). The global properties were computed by using the equations given below;

$$\mathbf{Hardness \(\eta\)} = (\mathbf{I} - \mathbf{A})\mathbf{1}/2\tag{5}$$

$$\text{Electromagnetic} \left( \mathbf{y} \right) = (\mathbf{I} + \mathbf{A})\mathbf{1}/2 \tag{6}$$

$$\text{Softness } (\mathbf{s}) = \mathbf{1}/(2\eta) \tag{7}$$

$$\text{Chemicalpotential } (\mu) = -\chi \tag{8}$$

$$\text{Electropibilityindex } (\mathfrak{o}) = \mathfrak{\mu}^2/2 \tag{9}$$

#### *2.2.3 Donor acceptor map (DAM)*

A donator-acceptor map is a useful tool for a qualitative comparison among substances. DAM can be used for classifying molecules in terms of their electron accepting and donating capacity (with respect to coumestrol). Graphical representation of DAM plot is shown in **Figure 1**. DAM also provides information's regarding anti-radical capability of molecules and also gave us a base for antioxidant studies. Single-point calculations (Energy vertical) were used to compute ionisation potential (I) and electron affinity (A). Ionisation potential was calculated as the difference between the energy of the cation and that of the neutral molecule.

**Figure 1.** *Graphical representation of DAM.*

And electron affinity was calculated as the energy difference between the neutral and the anion, and both were assumed to have ground state nuclear configuration of the neutral molecule.

According to J.J. Gázquez's approximation, the tendency to donate charge, or electron donating power, maybe defined as;

$$\mathbf{a} - \mathbf{a} = (\mathbf{3I} + \mathbf{A})^2 / 1\mathbf{6} (\mathbf{I} - \mathbf{A}) \tag{10}$$

whereas, the tendency to accept charge, or electron accepting power, maybe defined as;

$$\mathbf{a} + = (\mathbf{I} + \mathbf{3A})^2 / \mathbf{16(I - A)}\tag{11}$$

I and A donate or accept a single electron whereas, ω- and ω + refer to fractional charges. Lower values of electron donating power indicate the greater capacity for donating charge and higher values of electron accepting power indicate the greater capacity for accepting charge. So, it is a simple charge transfer model expressed in terms of chemical potential and hardness. Chemical potential gives more importance for ionisation potential in the context of charge donation and give more importance on electron affinity in the context of charge acceptance.

#### *2.2.4 Full electron donor acceptor map (FEDAM)*

FEDAM is a plot of electron donation index (RI) vs. electron acceptance index (RA), which gives information about the radical scavenging activity of different molecules. The ionisation enthalpy (I) and electron affinity (A) were obtained through DFT-B3LYP/6–31 + G(2d,2p) using energy vertical method. The electron donating and accepting indexes of the coumestrol derivatives were calculated with respect to the parent molecule, coumestrol, by using the equations given below;

*Theoretical Studies on Anti-Oxidant Activity of the Phytochemical, Coumestrol and Its… DOI: http://dx.doi.org/10.5772/intechopen.96967*


**Figure 2.** *Graphical representation of FEDAM.*

$$\mathbf{RI} = \mathbf{I}\_{\mathbf{L}} / \mathbf{I}\_{\mathbf{Cou}} \tag{12}$$

$$\mathbf{RA} = \mathbf{A}\_{\mathrm{L}} / \mathbf{A}\_{\mathrm{Coul}} \tag{13}$$

Where, L = Ligand (Derivatives).

Cou = Coumestrol.

The graphical representation of FEDAM is shown in **Figure 2**. It is used for evaluating the single electron (SET) transfer processes. Generally, the electron transfer takes place from region-3 (good donor) to region-1 (good acceptor). From this graph, it's vivid that the molecules with low I value and high A value exhibits the best scavenging activity.

#### *2.2.5 Antiradical activity*

To clarify the radical scavenging potential of phenolic anti-oxidants, three main mechanisms have been proposed. Consequently, antioxidants can deactivate free radicals according to the following mechanisms [14, 15].

#### **3. HAT (***hydrogen atom transfer***) mechanism**

$$\text{ArOH} + \text{X} \cdot \text{ArO} + \text{XH} \tag{14}$$

The phenolic anti-radical interacts directly with a free radical that is neutralised, according to this mechanism, and a radical form of phenolic antiradical develops. The hydrogen atom is transfered (HAT, Eq. (14)) from antioxidant molecules (ArOH) to radicals. Bond dissociation energy(BDE) is a numerical parameter connected to this mechanism. A better anti-radical property is defined by the lower BDE parameter.

$$\text{BDE} = \text{H}(\text{ArO}) + \text{H}(\text{H}) \text{-H}(\text{ArOH}) \tag{15}$$
