Infrared action spectroscopy as tool for probing gas-phase dynamics: Protonated Dimethyl Ether, (CH$_3$)$_2$OH$^+$, formed by the reaction of CH$_3$OH$_{2}^{+}$ with CH$_3$OH

Methanol is one of the most abundant interstellar Complex Organic Molecules (iCOMs) and it represents a major building block for the synthesis of increasingly complex oxygen-containing molecules. The reaction between protonated methanol and its neutral counterpart, giving protonated dimethyl ether, (CH$_3$)$_2$OH$^+$, along with the ejection of a water molecule, has been proposed as a key reaction in the synthesis of dimethyl ether in space. Here, gas phase vibrational spectra of the (CH$_3$)$_2$OH$^+$ reaction product and of the [C$_2$H$_9$O$_2$]$^+$ intermediate complex(es), formed under different pressure and temperature conditions, are presented. The widely tunable free electron laser for infrared experiments, FELIX, was employed to record their vibrational fingerprint spectra using different types of infrared action spectroscopy in the $600-1700$ cm$^{-1}$ frequency range, complemented with measurements using an OPO/OPA system to cover the O-H stretching region $3400-3700$ cm$^{-1}$. The formation of protonated dimethyl ether as a product of the reaction is spectroscopically confirmed, providing the first gas-phase vibrational spectrum of this potentially relevant astrochemical ion.