Context. The 3.3 $mu$m aromatic C-H stretching band of polycyclic aromatic hydrocarbon (PAH) molecules seen in a wide variety of astrophysical regions is often accompanied by a series of weak satellite bands at ~3.4-3.6 $mu$m. One of these sources, IRAS 21282+5050, a planetary nebula, also exhibits a weak band at ~1.68 $mu$m. While the satellite features at ~3.4-3.6 $mu$m are often attributed to the anharmonicities of PAHs, it is not clear whether overtones or combination bands dominate the 1.68 $mu$m feature. Aims. In this work, we examine the anharmonic spectra of eight PAH molecules, including anthracene, tetracene, pentacene, phenanthrene, chrysene, benz[a]anthracene, pyrene, and perylene, to explore the origin of the infrared bands in the 1.6-1.7 $mu$m waveelngth region. Methods. Density Functional Theory (DFT) in combination with the vibrational second-order perturbation theory (VPT2) is utilized for computing the anharmonic spectra of PAHs. To simulate the vibrational excitation process of PAHs, the Wang-Landau random walk technique is employed. Results. All the dominant bands in the 1.6-1.7 $mu$m wavelength range and in the 3.1-3.5 $mu$m C-H stretching region are calculated and tabulated. It is demonstrated that combination bands dominate the 1.6-1.7 $mu$m region, while overtones are rare and weak in this region. We also calculate the intensity ratios of the 3.1-3.5 $mu$m C-H stretching features to the bands in the 1.6-1.7 $mu$m region, $I_{3.1-3.5}/I_{1.6-1.7}$, for both ground and vibrationally excited states. On average, we obtain $langle I_{3.1-3.5}/I_{1.6-1.7} rangle$ $approx$ 12.6 and $langle I_{3.1-3.5}/I_{1.6-1.7} rangle$ $approx$ 17.6 for PAHs at ground states and at vibrationally excited states, respectively.
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