Several deep eutectic solvents (DESs) have been recently developed for extraction of hydrogen sulfide from natural gas. Among these newly designed DESs, a combination of caprolactam and tetrabutylammonium halides at a molar ratio of 1:1, as the DESs, has the highest desulfurization efficiencies. In this work, we explored the formation of caprolactam-based DESs using molecular dynamics (MD) simulations and ab initio computations. The results, based on the time average of the equilibrated production run of MD simulations, revealed similar to 15% decrease in the ionic interactions of tetrabutylammonium halides and, more interestingly, similar to 92% decrease in the hydrogen bonds between caprolactam, thereby explaining the strong depression in the melting point observed in experiments during the formation of the DES. Next, simulations of the DES with mixtures of methane and hydrogen sulfide were performed to mimic the natural gas desulfurization process. Efficient absorption of hydrogen sulfide from natural gas under different operating conditions (5000 and 10,000 ppm H2S, at 25 and 60 degrees C, and at 1 and 10 bar) can be observed from the simulations. The results further unveiled strong interactions between the anions of the DESs and hydrogen sulfide, with interaction energies 10-fold higher than those of methane/hydrogen sulfide, explaining the mechanism of desulfurization by these DESs. Additionally, two different DESs composed of monoethanolamines/methyltriphenylphosphonium bromide and urea/choline chloride were used to evaluate and compare their capacity to absorb hydrogen sulfide; however, we observed that the caprolactam-based DESs are highly efficient, particularly at low temperatures, low pressures, and low fuel/DES mole ratios. As the existing literature covers only experiments on the removal of hydrogen sulfide via the DESs, the simulations presented herein provide insights into the mechanism of absorption, thereby paving a way forward for better design of the DESs.