′-((2-methoxy-naphthalene-1-yl)methylene) pyrimidine-5-carbohydrazide (H5) and

Table 11 Quantum calculation parameters: highest occupied molecular orbital energy (E_HOMO), lowest unoccupied molecular orbital energy (E_LUMO), the energy gap (ΔE), global hardness (η), softness (σ), electronegativity (χ), global electrophilicity (ω), ΔN, and Δw

Code	H4	H5	H6
Program	MS-Dmol6	MS-Dmol6	MS-Dmol6
Method	DFT	DFT	DFT
Basis set	DNP (4.4)	DNP (4.4)	DNP (4.4)
Function	GGA-RPBE	GGA-RPBE	GGA-RPBE
EHOMO (eV)	−5.24731	−5.13616	−4.56568
ELUMO (eV)	−3.29443	−2.96602	−3.16616
ΔE = ELUMO − EHOMO	1.95288	2.17013	1.39953
η = ΔE/2	0.97644	1.08507	0.69976
σ(S) = 1/η	1.02413	0.92160	1.42905
Pi = (EHOMO + ELUMO)/2	−4.27087	−4.05109	−3.86592
X = −Pi	4.27087	4.05109	3.86592
ΔNmax	2.18695	1.86674	2.76230
ΔN (FET)	0.28119	0.35432	0.68172
ω	9.34019	7.56235	10.67884
ε	0.10706	0.13223	0.09364
ΔE back-donation	−0.24411	−0.27127	−0.17494

	Fig. 15 Images of HOMO, LUMO electron density and optimized geometry configuration for inhibitor molecules resulted from computational chemical calculations.

Table 12 Orbital colors interpretation

Orbital type	Color	Refer to
Molecular orbital	Red/Blue	Filled orbitals
	Yellow	Unfilled orbitals
Electrostatic potential	Red	Negative sites
	Blue	Positive sites
Electrophilic (HOMO)	Blue	Site most susceptible to attack by a electrophile
Nucleophilic (LUMO)	Blue	Site most susceptible to attack by a nucleophile

3.8 Molecular simulation results

The computational Monte Carlo (MC) method was performed to study the adsorption behavior of (H4 & H5 & H6) on the carbon steel surface in the solution presence of H₃O⁺, Cl⁻ and H₂O molecules. Side and top views of the adsorbed (H4 & H5 & H6) molecules on Fe surface are shown in Fig. 16. Computer simulations were achieved to understand how the inhibitors react with the carbon steel surface and how the geometrical structures of inhibitors molecules are arranged in Fe surface, the figure shows the protonated form of the inhibitor in solution. All inhibitor molecules arranged in superficial orientation on the Fe surface, which aids to form ideal coverage of the carbon steel surface. Adsorption could be understood as sending and receiving electrons at the HOMO and LUMO locations, which create coordination and back donation bonds, and electrostatic interaction can be represented as the other adsorbed sites on the Fe surface. The findings in Tables 13 and 14 give the calculated energies in vacuum and acid solution, respectively, such as total energy, adsorption energy, rigid absorption energy, and deformation energies. The (H4 & H5 & H6) inhibitor molecules are adsorbed in flat mode on the clean surface of Fe, allowing perfect interactions of N & O atoms and π-electrons with the surface of carbon steel. In addition, the inhibitor molecules detach H₂O molecules from their adsorption sites and take their place, meaning that more inhibitors will displace H₂O molecules by increasing the inhibitor molecules in the solution.⁴⁹ The adsorption energy is, in fact, equal to the sum of the rigid energy of adsorption and the energy of deformation which is used to describe the adsorption of molecules on the metal surface. When the inhibitor molecules are adsorbed on the surface of Fe, the energy generated or adsorbed is called the rigid energy of adsorption. In all simulations, negative values of adsorption energy refer to a strong binding of (H4 & H5 & H6) on the Fe surface.⁵⁰ This is because of the atoms (N and O), where the bonds of coordination can take place by granting the unoccupied iron orbital their unpaired electrons and a conjugated π-electron. The findings show that the negative adsorption energy values of inhibitors on the surface of carbon steel are in the following order: (H5 > H4 > H6), which means that the strong adsorption arrangement may be described as H6 > H4 > H5. This order is in line with the three inhibitors' practical % IE. The wet simulation (HCl) adsorption energy is greater than the medium of H₃O and the medium of H₂O. This indicates that the adsorption is enhanced by the involvement of water with the species Cl⁻.

	Fig. 16 Top and side perspectives of the most stable structure of studied samples on the metal surface under acid solution conditions for MC simulations.

Table 13 Simulation results (total energy, adsorption energy, rigid absorption energy, and deformation energies) in vacuum