Hierarchy among the weak noncovalent interactions such as van der Waals, electrostatic, hydrogen bonding, etc. dictates the secondary and tertiary structures of proteins as well as their interactions with various ligands. In this work, we investigate the competition between conventional (N-H center dot center dot center dot O) hydrogen bonds, unconventional (C-H center dot center dot center dot O) hydrogen bonds, and the van der Waals interaction in the model compounds of the chromophores of the amino acids, tryptophan, and histidine. These include indole (IND), benzimidazole (BIM), and its N-methylated analog (N-methylbenzimidazole, MBIM), which present multiple docking sites. The binary complexes of these molecules with ethers (dimethyl ether, diethyl ether, and tetrahydrofuran), which possess high proton affinity but lack acidic protons (thereby only act as hydrogen bond acceptors), are investigated. The complexes are formed in a supersonic jet and jointly studied by electronic and vibrational spectroscopy as well as quantum chemical calculations. Only the N-H center dot center dot center dot O bound structures are observed for the complexes of IND and BIM with ethers, although computations predict reasonably competent C-H center dot center dot center dot O type of structures. Remarkably, IND and BIM produce three (N-H center dot center dot center dot O) conformers with Me2O but single conformers with Et2O and THF. In the case of MBIM, which lacks a conventional hydrogen bond donor, no evidence for C(2)-H center dot center dot center dot O hydrogen bonds is seen; instead, the complexes are found to be bound purely by van der Waals interactions. The results indicate that strong N-H center dot center dot center dot O and even weak van der Waals interactions are thermodynamically favored over C(2)-H center dot center dot center dot O bound structures in these binary gas-phase complexes.