Semi-empirical quantum chemical method is the computational method to calculate the electronic and molecular orbital properties using self-consistent field molecular orbital theory. The integration method employs parametric method 6 (PM6). This quantum molecular descriptor was calculated using semi-empirical MOPAC2016, Version: 21.002 James J. P. Stewart software [12]. The input of the structure and geometry optimization was generated using Avogadro version 1.2.0. The geometry optimization used force field method MMFF94s with step per update 4. The molecular structure was generated using 3-dimension Avogadro interface, and the molecular descriptor involved three coordinates. Therefore, this method employed the 3D-QSAR approach. The calculation was done using PM6 parameters.

α_ij: the dynamic linear polarizability constant at ω = 0.25 eV. The value of i and j represented tensor vector inx, y or z.

β_i: the average value of the hyperpolarizability at ω = 0.25 eV. The quantity of interest is defined as β_i = [β_iii + β_ijj + β_ikk] [13].

γ_ijkl: the second order hyperpolarizability at ω = 0.25 eV. The value i, j, k and l represented tensor vector in x, y and z.

ε_HOMO: the energy of the highest occupied molecular orbital.

ε_LUMO: the energy of the lowest unoccupied molecular orbital.

d_p: the molecular dipole moment contributed from the net charge density [14].

d_hyb: the molecular dipole moment contributed from a hybridization of molecular orbital [14].

COS: the Conductor like screening model (COSMO) surface area.

μ: the chemical potential. The chemical potential can defined as, μ = (ε_HOMO – ε_LUMO)/2.

η: the molecular chemical hardness. The η can be defined as, η = (ε_HOMO + ε_LUMO)/2.

W: the electrophilicity index. The electrophilicity is defined by Parr et al. that is W = μ²/2η [15].

W^–: the electrodonating power. This index explain chemical response to the donation of electron which given by (3ε_HOMO – ε_LUMO)²/16(ε_HOMO + ε_LUMO) [16].

GE¹: the graph energy index. The graph energy index is the sum of eigen value. The index is defined as G E 1 = ∑ | − λ i | where λ is the negative eigen value.

GE²: the graph energy index with positive eigen value G E 2 = ∑ | + λ i | where λ is the positive eigen value.

Est: the Estrada index. The Estrada energy index is defined as E s t = ∑ exp ( λ i ) [17].

The chemical data was collected using the Reaxys database. Reaxys is a well-known curated database for finding and comparing relevant chemical information from different sources that attempt to bring together chemical, biological activity and toxicogenomic into their center location [18] [19]. In this study, all biological activity and toxicology values were extracted from a single reference. This was done to ensure a compatible relationship among them. The data used in this study are presented in the appendix section [20] - [26]. Inferential statistical analyses were performed. Multiple linear regression was applied to generate the prediction models. The multiple linear regression is a relatively simple approach pursuit to model the linear relationship between both independent and explanatory variables by fitting the linear equation. Variable selection in a block was entered and calculated using a single step. The researchers performed regression using SPSS statistical software. The chosen regression was based on a good fitting with the regression equation’s standard of deviation not less than 0.0001. The observed plot vs calculated toxicity with a 95% degree of confidence was plotted using Minitab. The descriptors had very small standard deviations, being inadequate to represent the changes in electronic structure from molecule to molecule [27]. Principal Component Analysis (PCA) was used to determine the inherent dimensionality of groups of molecular descriptor properties. The orthogonal PROMAX rotation was used during the PCA analysis.

Alkanes are hydrocarbons derived from petroleum processing streams. These alkanes contain only carbon and hydrogen atoms. It contains carbon numbers ranging from approximately C5 - C20 with three-type constituents: normal paraffins, isoparaffins, and cycloparaffins. These constituents result in different physical and chemical properties [28]. Tables 1-3 show the electronic and molecular orbital for alkanes generated from the semi-empirical calculation. To understand the factor contribution of each descriptor, the researchers plotted the component analysis based on ProMax rotation as shown in Figure 1. Five principal component factors are classified. The first component is the dynamic linear polarizability constant (α_xx, α_yy and α_zz), conductor-like screening model (COSMO), Ge¹ Ge² second order hyperpolarizability of γ_xxxx and γ_xxyy. The second component is the average value of the hyperpolarizability of β_z, hybrid and the net charge dipole moment. The third component is the second order hyperpolarizability of γ_yyyy and γ_zzzz. The fourth component is the average value of hyperpolarizability of β_x and β_y. The last component is the electrophilicity index, second order hyperpolarizability of γ_yyyy and γ_yyzz.

Inside each cluster, the molecular properties differed for their dimension, polarizability, and isomerization. The occurrence of various descriptor components was clustered using a different classification of molecular electronic property

descriptors, based on the calculation of the molecular spherical coordinate system, the localization of electron, oscillating electrical field, and electron-electron interaction. The cumulative total variance of all five components is 85.136%, by which most of the contributions came from component 1 (40.434%). This contribution might be classified according to the position of carbon atoms to the molecule in the spherical coordinate system based on self-consistency molecular orbital [29]. The second principal contribution is 19.363% which is plausibly related to the orientation of dipole moment in the molecule [30]. The third contribution for the cumulative total variance is 13.115%. The fourth and fifth contributions are 6.242% and 5.982% respectively.

The biological activity is normally related with the molecular interaction that goes through the compound during the transport through biological membranes, or in the reaction with the active site. This interaction is strongly related to the molecular chemical structure [20]. A change in the structure can result in a change in biological response. One of the techniques to measure the biological response is by manifesting a change in the chromatographic retention data. The prediction of chromatography retention indexes can also be explained using quantum chemical descriptors or parameters. The volatile component of Scorzonera hispanica L has been demonstrated by a wide variety of molecules [21]. The statistical analysis yields the following results in which n is the number of samples, r and r² are the correlation coefficients, and s is the standard deviation. In this study, the researchers only discussed three best regression Equations (with the highest value of r and r²) that are given in Equations (1a)-(1c). The chromatography retention indexes for alkanes in this series were computed. The three best statistical analyses yield the following results:

log R t 1 = 0.006333 C O S ( ± 0.000455 ) + 0.630636 d p ( ± 4.127 )                           − 0.477 d h ( ± 2.15171 ) n = 25 , r = 0.9742 , r 2 = 0.948 , s = 0.311 (1a)

log R t 1 = 0.000307 α x x ( ± 0.017201 ) − 0.001759 α y y ( ± 0.041598 )                         + 0.014028 α z z ( ± 0.052832 ) n = 25 , r = 0.970 , r 2 = 0.942 , s = 0.329 (1b)

log R t 1 = 0.000044 γ x x x x ( ± 0.000014 ) + 0.000856 γ z z z z ( ± 0.00034 )                           + 0.000515 γ x x y y ( ± 0.000210 ) − 0.000083 γ x x z z ( ± 0.000392 ) n = 25 , r = 0.9996 , r 2 = 0.99916 , s = 0.047 (1c)

Linear alkanes were also used as a reference to calculate the retention index for the volatile components from marjoram oil sample [22]. The computed isotherm retention time for alkanes in this series used statistical analysis, yielding the following results:

log R t 2 = 0.00375 C O S ( ± 0.00002 ) + 189.6037 d p ( ± 3.33786 )                           − 58.47219 d h ( ± 1.01753 ) n = 5 , r = 0.9999 , r 2 = 0.9999 , s = 0.0047 (2a)

log R t 2 = 0.072612 α x x ( ± 0.00806 ) − 0.075058 α z z ( ± 0.009266 ) n = 5 , r = 0.9998 , r 2 = 0.9996 , s = 0.027780 (2b)

log R t 2 = 0.000041 γ x x x x ( ± 0.00001 ) + 0.000137 γ z z z z ( ± 0.0001 )                           + 0.000356 γ x x y y ( ± 0.000064 ) n = 5 , r = 0.9996 , r 2 = 0.99916 , s = 0.047 (2c)

The researchers computed the series of correlation with chromatography retention index to produce branched alkanes by Cynobaterium Microcoleus vigantus. [20] The correlation with quantum molecular descriptor was found, yielding the following results:

log R t 3 = 0.173 W ( ± 0.051 ) + 0.168 η ( ± 0.006 ) n = 24 , r = 0.997 , r 2 = 0.994 , s = 0.106 (3a)

log R t 3 = − 0.03275 α x x ( ± 0.00517 ) + 0.00518 α y y ( ± 0.01265 )                           + 0.04186 α z z ( ± 0.01503 ) n = 24 , r = 0.997 , r 2 = 0.994 , s = 0.108 (3b)

log R t 3 = 0.000082 E s t + ( ± 0.000002 ) n = 24 , r = 0.994 , r 2 = 0.989 , s = 0.149 (3c)

The obtained results show that the best regression r² descriptors are different between experiments. This might be due to the number of samples being different from each other. To determine the experiment modeling with the quantum chemical descriptor, the molecular structure plays an important role. Equations (1a) and (2a), and the retention index can fit with COSMO and dipole moment descriptor. They also showed the best r and r² values in both results. In chromatography, the retention data are proportional to the free-energy change of solute-stationary phase interactions changed by the mobile phase. This shows that the molecular interaction with the molecules affect the regression model. The molecules such as butane, 3,4-Dimethylhexane and hexane (the total dipole moment is zero) tend to induce London dispersion forces. While the molecules with the total dipole moment is not zero, influenced by van der Waals forces. The linear polarizability constant (α) and second hyperpolarizability (γ) are the physical quantities derived from dipole moment interaction with static and fluctuation of electrical field. Both quantities show a good correlation with the retention index. Therefore, the effects of charge screening in molecule and dipole moment are correlated with the retention index.

The analysis of electrophilicity index and the molecular chemical hardness (in Equation (3a)) is related to molecular electron density distribution on HOMO and LUMO that are localized in the atoms. Chemical hardness is a property that measures the stability and reactivity of a molecule [15]. Therefore, the change of molecular structure, sightly changes the electron cloud deformation which affects the retention constant. The Estrada index also gives a good regression relation in the quantitative retention structure analysis. The Estrada index is based on the topological molecular descriptor based on eigenvalues of an adjacency matrix. The Estrada index is also suitable for molecular descriptor which is comparable with the molecular topology index using the graph theory approach [31]. This index is able to represent a mathematical numeric to character the molecules, especially the isomers structures which fit the retention constant.

Hydrocarbons include a vast number of existing chemicals in the environment and consumerism, and there is a need to study the toxicology impact on both environment and human. Hence, an alternative approach is needed to seek the generalities of “structure-activity relationships” from the presence of toxicological data. This can be used to predict another chemical isomer with similar effects. The in vitro oxidation by reconstituted enzyme system using alkane as substrate is demonstrated [24]. This report can provide additional information on the toxicity of substrate to the whole cells. The best three statistical analyses yield the following results:

log E a = 7.537 W ( ± 3.670 ) − 0.578 η ( ± 0.381 ) n = 24 , r = 0.997 , r 2 = 0.994 , s = 0.106 (4a)

log E a = − 0.063886 α x x ( ± 0.030311 ) + 0.086243 α z z ( ± 0.033889 ) n = 26 , r = 0.951 , r 2 = 0.905 , s = 0.897 (4b)

log E a = 0.00716 C O S ( ± 0.00089 ) − 1.9520 d p ( ± 10.1960 )                           + 2.0527 d h ( ± 4.9964 ) n = 26 , r = 0.948 , r 2 = 0.899 , s = 0.542203 (4c)

The acute toxicology of aqueous solutions of hydrocarbon to Daphnia magna is reported [23]. The researchers found the correlation with quantum molecular descriptors, yielding the following results. Figure 2 shows the correlation between the experimental and predicted LC50 values obtained from the dynamic linear polarizability descriptor with our Equation (5a):

log L C 50 = 0.037333 α x x ( ± 0.028125 ) − 0.002384 α y y ( ± 0.025083 )                                 − 0.088385 α z z ( ± 0.054480 ) + 5.380546 ( ± 0.186554 ) n = 7 , r = 0.9989 , r 2 = 0.9977 , s = 0.0694 (5a)

log L C 50 = − 0.006069 G e 1 ( ± 0.00286 ) − 0.01367 G e 2 ( ± 0.00825 )                                 + 4.926273 ( ± 0.13140 ) n = 7 , r = 0.9978 , r 2 = 0.9955 , s = 0.083796 (5b)

log L C 50 = − 0.023260 C O S ( ± 0.002623 ) − 39.969094 d p ( ± 24.824302 )                                 + 14.012292 d h ( ± 9.573416 ) + 5.053425 ( ± 0.419972 ) n = 7 , r = 0.9852 , r 2 = 0.9706 , s = 0.2485 (5c)

A number of aliphatic hydrocarbons after intravenous injections of emulsion formulations into Mice were examined [25]. The correlation with quantum molecular descriptor was found, yielding the following results. Figure 3 shows the correlation between the experimental and predicted LD100 values obtained from the second order hyperpolarizability descriptor with our Equation (6a).

log L D 100 = − 0.000079 γ x x x x ( ± 0.000047 ) + 0.00457 γ y y y y ( ± 0.007558 )                                   + 0.001743 γ z z z z ( ± 0.004965 ) − 0.004269 γ x x y y ( ± 0.003920 )                                   − 0.002150 γ x x z z ( ± 0.001030 ) + 0.008519 γ y y z z ( ± 0.006301 ) n = 10 , r = 0.9822 , r 2 = 0.9647 , s = 0.3100 (6a)

log L D 100 = − 0.012545 G e 1 ( ± 0.009357 ) + 0.041093 G e 2 ( ± 0.026479 ) n = 10 , r = 0.9664 , r 2 = 0.9340 , s = 0.3000 (6b)

log L D 100 = 0.00427 C O S ( ± 0.00065 ) − 18.51654 d p ( ± 23.85805 )                                   + 6.64021 d h ( ± 8.22676 ) n = 10 , r = 0.9659 , r 2 = 0.9329 , s = 0.32297 (6c)

The aquatic toxicity of hydrocarbon to aquatic organism was reported [26]. The correlation with quantum molecular descriptor was found, yielding the following results. Figure 4 shows the correlation between the experimental and predicted EC50 values obtained from the dynamic linear polarizability descriptor with our Equation (5a).

log E C 50 = − 0.000191 γ x x x x ( ± 0.000037 ) − 0.000503 γ y y y y ( ± 0.000373 )                                 + 0.000763 γ z z z z ( ± 0.001163 ) − 0.000364 γ x x y y ( ± 0.000527 )                                 − 0.000278 γ x x z z ( ± 0.000903 ) − 0.000955 γ y y z z ( ± 0.001035 )                                 + 2.0700 ( ± 0.310773 ) n = 12 , r = 0.98034 , r 2 = 0.961065 , s = 0.233098 (7a)

log E C 50 = − 0.034559 α x x ( ± 0.058434 ) + 0.099846 α y y ( ± 0.084750 )                                 – 0.094943 α z z ( ± 0.117111 ) + 3.086897 ( ± 0.516546 ) n = 12 , r = 0.9566 , r 2 = 0.9151 , s = 0.2980 (7b)

log E C 50 = − 0.019657 G e 1 ( ± 0.010376 ) + 0.031923 G e 2 ( ± 0.029880 )                                 + 3.251170 ( ± 0.336243 ) n = 12 , r = 0.9574 , r 2 = 0.9167 , s = 0.2784 (7c)

The process of toxicology analysis was done through computational techniques, and the relation of the molecular structure with the interaction subject can be represented by this relation:

Activity = f ( physiochemistryproperties ) (8)

The physiochemistry includes chemical properties, electronic properties, molecular topology, thermodynamic and optical properties [32]. The modification in molecular structure will change the biological activity. In linear form, the activity can be described by:

Activity = a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 + ⋯ + a n x n (9)

where x_n is the molecular descriptor and a_n is the constant. The investigation of quantitative relationships between chemical structures characterized by electronic properties is one of the most important tools in biological activity such as toxicology analysis. The regression Equations (4)-(7) show that the electronic properties such as the electrophilicity index, chemical hardness, dynamic linear polarizability, second order hyperpolarizability, graph energy, dipole moment and conductor like screening model show a good relationship with the biological activity.

The electrophilicity index and chemical hardness show the best regression for in vitro oxidation by the reconstituted enzyme system. The electrophilicity index shows the stabilization energy when the molecules receive the charges from the environment. The electrophilicity is also related to the chemical potential, which can be related to the binding of environment in the biological system. The chemical hardness describes the transfer of charge in the system with high charge density. These two descriptors are useful indicators to estimate enzyme activities on alkane substrates. The regression result shows that both descriptors are best fitting to relate with enzyme activity system. This is also agreeable with the result by Grillo and his coworkers [33].

The GE index is a new molecular descriptor related to the characteristic value of eigen function. The eigen function is assigned to Linear Combination Atomic Orbital Self-Consistent Filed (LCOASCF) molecular orbital. This index corresponds to orbital energy of the molecular structure. GE index is capable of describing the binding of energy interaction, the total electron exchange energy, electrostatic that exists in the molecule and the resonance energy [34] [35]. Therefore, GE index can explain the chemical interpretation based on the electronic properties of molecular structure involved in the biochemical interaction. Concerning the GE index as molecular quantum description in the biological activity and toxicity, it is a good fitting with the regression equation.

The non-linear polarizability of the molecule is an important physical property that draws our attention in chemical-biological interaction. The non-linear polarizability is a measurement of distortion of molecules in an electric field. This property measures the strength of molecular interactions such as long-range intermolecular induction, dispersion forces, scattering and electrons interaction [30]. The linear polarizability (α) shows a good regression fitting for most biological activities and toxicities. Hansch and Kurup also reported that the linear polarizability shows a good regression result with their toxicity study [36]. The hyperpolarizability descriptor is related with non-linear polarizability constant. This descriptor shows a moderate regression fitting (with the value of r between 0.901 - 0.788). The second order hyperpolarizability shows the best regression fitting in most biological activities and toxicities. This might be due to the induced non-linear dipole moment interaction with the applied field, giving a fine electronic property in the molecular structure. The electron-correlation effect in the microscopic polarizability calculation produced a good correlation in predicting the qualitative trends for structure-property relationships [37].

The conductor like screening model (COSMO) area with molecular dipole moment is surprisingly a good fitting for enzyme activity and toxicity. COSMO area is related with the surface charge densities on the neighboring segments. The COSMO area is an effective area of the screening surface. The screening surface is related to the perturbation Coulomb interaction in the molecule. The screening depends on the localization of charge, and the molecular polarizability. The molecular polarization is contributed by electronic, vibration and rotation. This molecular polarization contribution is also related with molecular dipole moment. In the researchers’ work, the COSMO area and molecular dipole moment are important molecular properties in relating the biological and toxicology activities.