Calculating theoretical octane numbers of various petroleum derivatives using quantum mechanical DFT methods

This research included calculating the octane number of a group of chemical compounds using the (DFT) method by using multiple regression analysis of the relationship between the practical octane number values with the theoretically calculated physical variables, where a number of variables were calculated including the calculation of energy functions, including the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LOMO), hardness (η), chemical potential (µ), and spherical electrophilic index (W), and the thermodynamic functions (∆G, ∆H, ∆S, ∆E) and dipole moment were calculated. They were evaluated from the correlation coefficient relationship (R2). Through the four-way statistical analysis, we obtained the best correlation coefficient for the variables calculated using the DFT method, where (R2) showed the highest values for the following variables: (∆E+HOMO+∆G+µ) (R2=0.793) (∆S+ LOMO + ∆G +W) (R2=0.753). Through the single and multiple statistical analysis, the most influential variable on the octane number values is (HOMO).

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R2: HOMO + LOMO + η + W vs HOMO + η + W + µ vs HOMO + η+ ∆S + ∆E vs HOMO + η + ∆E +LOMO vs ∆E + HOMO + ∆G+ µClose ×

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Theoretical octane number and True octane number: 1. vs 2. vs 3. vs 4. vs 5.Close ×

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Introduction Octane Number

It is a measure of the ignition quality or ignition ability of gasoline. Research Octane Number (RON) refers to normal road performance under moderate inlet temperature (65.6 °C) and rpm (600 rpm). Engine octane number (MON) refers to high-speed performance under severe inlet temperatures (148.9 °C) and 900 rpm. Research Octane Number (RON) and Engine Octane Number (MON) are evaluated using ASTM D2699 and ASTM D2700, respectively [1,2] .

Computational chemistry

Computational chemistry is one of the branches of physical chemistry, including quantum mechanics and molecular mechanics. Its purpose is to find the most important properties of chemical compounds and compare them with practical values. It is one of the main branches of research used in the analysis, diagnosis, and support of scientific research [3] . Computational chemistry depends on the use of computers instead of chemicals and the creation of effective mathematical methods to solve theoretical problems through advanced computational methods. John People is considered the first to introduce computers into quantum chemistry in the fifties of the last century by creating new integrated algorithms, highly efficient in calculations, and then marketing computational programs such as the Gaussian program [4] . Computational chemistry is characterized by the fact that its calculations are easy to perform compared to In experiments that are often difficult to perform and involve the risks of chemicals as well as less expensive because they do not require chemicals, and can be performed on any system even if it is not present, while experiments are limited to relatively stable molecules, however, computational chemistry is not an alternative to experimental methods, but it plays an important role in knowing the properties of chemical compounds [5] .

The term computational chemistry is used when the mathematical method has been developed sufficiently to be applied using computer programs [6] .

Density Function Theory (DFT)

The first to use and develop this theory were the researchers (Hohenberge-Kohn) in 1964 AD, and it later became one of the important components related to the basic analysis methods that were used extensively in theoretical studies, especially after the development in the field of electronic computers. This theory is based on finding the electronic properties of the molecule in the stable state, which are found using the electronic density. The advantage of this theory is that it includes finding the approximate electronic correlations with the same The accuracy calculated by the well-known method (Hartrey-Fock) (HF) is usually faster than the experimental method and is considered the most accurate method in calculations, so it can be used as one of the basic calculation methods (Ab-initio). It is similar to the basic calculations and semi-empirical calculations in terms of its reliance on the Schrodinger equation, but it differs from the two methods in that it does not calculate a wave function but derives the distribution of electrons (electron density function) directly, where the electron density is expressed as the square of the function. Wave, the basis of this method is to simulate the actual distribution of the electron density of the system, so the larger the set of functions used to model the electrons, the more accurate the calculations become, and the results are much better, as they have been developed by introducing electronic exchange and correlation functions that increase the accuracy of the calculations. There are many approximations used in this method, including:  LDAM: Local Density Approximation.  LSDA: Local spin Density Approximation.  GCM: Gradient Corrected Method.

Hybrid functional function

In this, the functional function theory (DFT) is combined with the Hartree-Fock theory (HF) to describe the functional function of the exchange energy, including:  B3LYP: Becke's 3 parameter exchange correlation functional, which uses 3 Parameters and LYP, the Lee, Yang and Parr correlation functional.

This function is considered the most used in calculations due to the accuracy of the results that can be obtained. However, new functions are constantly being developed to increase the accuracy of calculations [7,8] .

Theoretical calculation

Quantum mechanics methods were used in this study, represented by the DFT method, by using these theoretical methods to find the values of the octane number by the (Chem. Office) program [9] . The calculations were performed using the following methods 1. Ab-initio methods (DFT).

Gaussian 09 program

Steps of theoretical calculations for the compounds under study using Gaussian 09: 1. Drawing the molecular formula of the compounds using the (Chem. Office) program. 2. Transferring the compound drawn in the (Chem.

Office) program to the (Chem. 3D) program. 3. Save the compound in the (Chem. 3D) program in the file format (Input File). 4. Open the program (Gaussian 0.6) then open the file saved in (Input File) format. 5. Select the (Calculate) button, then select (Gaussian Calculation Setup). 6. From the Job Type button, select (Optimization), then select (Minimum) to obtain the lowest energy, which will be more stable and have the best spatial shape. 7. After that, from the (Method) button, select the theoretical method and determine the basis set. 8. After that, select (Submit) to perform the calculations in the (Gaussian 09W) program. 9. After finishing and completing the results, they will be saved via (Notepad). 10. The statistical analysis is performed using the SPSS program.

Statistical analysis

The statistical analysis was conducted between the selected and calculated variables (HOMO, LOMO, dipole moment, hardness, electronic potential, spherical electrophilic index, ∆H, ∆G, ∆S, ∆Ε) using theoretical methods with the practical Octane Number (ON) values calculated using SPSS

Program

The correlation coefficient (R 2 ) values were adopted as a reference for this purpose to know the strength and type of relationship between the calculated variables and the practical Octane Number (ON) values, as the statistical analysis was conducted between the variables with each other and with the practical Octane Number (ON) values using the methods used, where each method was taken separately and with the following steps:

1. The calculated variables of the studied compounds (HOMO, LOMO, dipole moment, Hardness, electron potential, spherical electrophilic index, ∆H, ∆G, ∆S, ∆Ε) were entered as non-dependent variables with the practical octane number values that are dependent variables.

2. The variables were analyzed through a series of steps in the SPSS program using the following steps: (Linear + regression + analysis). 3. The values of the correlation coefficient (R 2 ) are then obtained, since knowing the values of (R 2 ) determines the variables that have a significant effect on the Octane Number (ON) in the simple regression analysis. 4. We then conduct the process of multiple statistical analysis to determine the strength of the relationship between the variables that were used with each other and with the scientific octane number values obtained from the literature.

Theoretical calculations

In this research, the compounds were studied theoretically and the Octane Number (ON) values were extracted theoretically, as a set of variables were calculated, including the thermodynamic functions (∆H, ∆G, ∆S, ∆Ε), dipole moment and orbital energy (HOMO, LOMO), and energy functions (hardness η, electron-chemical potential µ, spherical electrophilic index W). The octane number values were calculated for a number of compounds to obtain the best relationship between the variables and the octane number values. (Table 4) shows the descriptive statistics of the variables studied using the DFT method. The table shows that the average octane number of the studied samples is 127, and the average polar moment of the studied materials is about 0.3024. The enthalpy (∆H), entropy (∆S), and free energy (∆G) changes are around the average values of 87.3512, 0.1724, and 0.1309 respectively, while the average energy change (∆E) of the studied materials is about 107.6117. For the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LOMO), the average values are -0.2238 and 0.0270, respectively. The average transport efficiency (η) of the studied materials is 0.1132, and the average polarity (µ) is -0.0984. It is noted from the attached (Table 9) and from the results of the regression analysis, that there is a significant correlation between the practical octane number values obtained from the literature with the theoretically calculated octane number values, with some differences in some compounds for the four methods used. This is evidence of the strength of the proposed variables for calculating the octane number values for some gasoline derivatives [10] . This study opens up broad horizons in the field of software chemistry (computational chemistry).

Compound	𝚫 𝐄	𝐇𝐎𝐌𝐎	𝚫𝐆	µ	Theoretical octane number	True octane number	𝚫
1.	66.511	-0.25191	0.074269	-0.123225	102	102	0
2.	84.979	-0.23900	0.098629	-0.116600	132	122	10
3.	110.61	-0.18242	0.136583	-0.042935	122	130	8
4.	122.926	-0.23793	0.153158	-0.115160	110	110	0
5.	151.617	-0.18145	0.195365	-0.042025	101	92	9
6.	103.524	-0.23202	0.126019	-0.111590	125	145	20
7.	103.486	-0.23357	0.122993	-0.113790	148	140	8
8.	102.881	-0.22756	0.123732	-0.110865	147	136	11
9.	121.971	-0.22873	0.147628	-0.109540	157	166	9

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Theoretical octane number values for DFT method

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DFT	ON	Dipole moment	∆H	∆S	∆G	∆E	HOMO	LOMO	η	µ	W
Mean	127	0.3024	0.1724	87.3512	0.1309	107.6117	-0.2238	0.0270	0.1132	-0.0984	0.1160
Std. Error of Mean	8	0.0715	0.0128	3.2623	0.0114	8.0310	0.0083	0.0132	0.0129	0.0107	0.0683
Median	130	0.3936	0.1659	86.6370	0.1260	103.5240	-0.2320	0.0076	0.1204	-0.1116	0.0540
Std.div.	23	0.2146	0.0384	9.7870	0.0342	24.0930	0.0248	0.0397	0.0388	0.0320	0.2049
Variance	534	0.0460	0.0010	95.7850	0.0010	580.4740	0.0010	0.0020	0.0020	0.0010	0.0420
Range	74	0.5825	0.1356	31.6180	0.1211	85.1060	0.0705	0.0919	0.1272	0.0812	0.6545
Minimum	92	0.0000	0.1069	68.7530	0.0743	66.5110	-0.2519	0.0055	0.0122	-0.1232	0.0063
Maximum	166	0.5825	0.2426	100.371	0.1954	151.6170	-0.1815	0.0974	0.1395	-0.0420	0.6608

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Descriptive statistics of the study variables calculated using the DFT method

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Figures and tables from the paper

No.	ON	Nameclature	Structure
1.	102	Benzene
2.	122	Toluen
3.	130	Ethyl benzene
4.	110	Propyl benzene
5.	92	Butyl benzene
6.	145	O- Xylene
7.	140	M- Xylene
8.	136	P- Xylene
9.	166	Mesitylene

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Structures of the compounds under study and their names with octane number (ON) values

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No	Dipole moment	∆H Hartee/Particle	∆ S Cal.K-1.mol-1	∆G Hartee/Particle	E∆ Cal.K-1.mol-1
1.	0.0000	0.106936	68.753	0.074269	66.511
2.	0.3305	0.136366	79.424	0.098629	84.979
3.	0.4188	0.177212	85.512	0.136583	110.610
4.	0.4375	0.196839	91.934	0.153158	122.926
5.	0.4596	0.242561	99.332	0.195365	151.617
6.	0.5825	0.165920	83.979	0.126019	103.524
7.	0.3936	0.165859	90.219	0.122993	103.486
8.	0.0004	0.164896	86.637	0.123732	102.881
9.	0.0983	0.195317	100.371	0.147628	121.971

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Calculating thermodynamic functions and dipole moment using DFT method

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No	HOMO e.v	LOMO e.v	Η e.v	µ e.v	W e.v
1.	0.25191-	0.00546	0.128685	-0.123225	0.059022273
2.	0.23900-	0.00580	0.012240	-0.11660	0.055537418
3.	0.18242-	0.09655	0.139485	-0.042935	0.660792997
4.	0.23793-	0.00761	0.122770	-0.11516	0.054010853
5.	0.18145-	0.09740	0.139425	-0.042025	0.00633514793
6.	0.23202-	0.00884	0.120430	-0.11159	0.051699444
7.	0.23357-	0.00599	0.119780	-0.11379	0.054049775
8.	0.22756-	0.00583	0.116695	-0.110865	0.052663086
9.	0.22873-	0.00965	0.119190	-0.10954	0.050335647

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Calculating the energy values of the studied compounds using the DFT method

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DFT	—	ON	Dipole moment	∆H	∆S	∆G	∆E	HOMO	LOMO	η	µ	W
ON	R	1	-0.113	-0.059	0.265	-0.102	-0.059	-0.183	-0.375	-0.056	-0.304	0.083
	p		0.772	0.881	0.49	0.794	0.881	0.637	0.32	0.886	0.427	0.831
Dipole moment	R	-0.113	1	0.438	0.272	0.455	0.438	0.389	0.373	0.02	0.383	0.177
	p	0.772		0.239	0.479	0.219	0.239	0.3	0.323	0.96	0.31	0.649
∆H	R	-0.059	0.438	1	0.916**	0.999**	1.000**	0.688*	0.573	0.407	0.623	-0.014
	p	0.881	0.239		0.001	0	0	0.04	0.107	0.278	0.073	0.971
∆S	R	0.265	0.272	0.916**	1	0.893**	0.916**	0.488	0.317	0.294	0.386	-0.114
	p	0.49	0.479	0.001		0.001	0.001	0.183	0.406	0.443	0.304	0.77
∆G	R	-0.102	0.455	0.999**	0.893**	1	0.999**	0.706*	0.601	0.417	0.647	0
	p	0.794	0.219	0	0.001		0	0.033	0.087	0.265	0.059	0.999
∆E	R	-0.059	0.438	1.000**	0.916**	0.999**	1	0.688*	0.573	0.407	0.623	-0.014
	p	0.881	0.239	0	0.001	0		0.04	0.107	0.278	0.073	0.971
HOMO	R	-0.183	0.389	0.688*	0.488	.706*	0.688*	1	0.962**	0.389	0.985**	0.565
	p	0.637	0.3	0.04	0.183	0.033	0.04		0	0.301	0	0.113
LOMO	R	-0.375	0.373	0.573	0.317	0.601	0.573	0.962**	1	0.393	0.994**	0.597
	p	0.32	0.323	0.107	0.406	0.087	0.107	0		0.295	0	0.09
η	R	-0.056	0.02	0.407	0.294	0.417	0.407	0.389	0.393	1	0.395	0.229
	p	0.886	0.96	0.278	0.443	0.265	0.278	0.301	0.295		0.292	0.553
µ	R	-0.304	0.383	0.623	0.386	0.647	0.623	0.985**	0.994**	0.395	1	0.59
	p	0.427	0.31	0.073	0.304	0.059	0.073	0	0	0.292		0.094
W	R	0.083	0.177	-0.014	-0.114	0	-0.014	0.565	0.597	0.229	0.59	1
	p	0.831	0.649	0.971	0.77	0.999	0.971	0.113	0.09	0.553	0.094

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Pearson correlation coefficient between variables calculated by DFT method

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Model No.	Variables	R2
1.	HOMO + LOMO + η + W	0.734
2.	HOMO + η + W + µ	0.734
3.	HOMO + η+ ∆S + ∆E	0.699
4.	HOMO + η + ∆E +LOMO	0.649
5.	∆E + HOMO + ∆G+ µ	0.793
6.	∆E+ LOMO+µ +W	0.729
7.	∆E + W+HOMO+∆G	0.722
8.	LOMO + ∆S G ∆ +W+	0.753
9.	W+ LOMO+ ∆S+∆H	0.753
10.	W+ ∆S + ∆H +HOMO	0.722

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Values of the correlation coefficient (R2) between the octane number and the variables calculated in the quadratic statistic using the

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Methods	Model	Coefficients	Unstandardized coefficients	—	Standardized coefficients	t	Sig.
			B	Std. error	Beta
DFT	Model 5	(Constant)	460.239	341.223	-	1.349	0.249
		∆E	10.195	5.935	10.634	1.718	0.161
		HOMO	2899.223	1811.121	3.115	1.601	0.185
		∆G	-7497.620	4200.794	-11.097	-1.785	0.149
		µ	-2035.772	1407.053	-2.817	-1.447	0.221

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Quadruple statistics of octane number using DFT method

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Model summary	—	—	—	—	—
Methods	Model	R	R square	Adjusted R square	Std. error of the estimate
DFT	Model 5	0.890	0.793	0.585	14.877

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Characteristics of the quadratic statistical model used to calculate the octane number using the DFT method

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Normal P-P plot of regression standardized residual

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Simple scatter plot with fit line for model 5

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