The High Luminosity upgrade of the Large Hadron Collider (HL-LHC) aims to achieve an integrated luminosity of 200-300 fb(-1) per year, including the contribution from the upgrade of the injector chain. For the HL-LHC the larger crossing angle together with a smaller beta function at the collision point would result in more than 70% luminosity loss due to the incomplete geometric overlap of colliding bunches. To recover head-on collisions at the high-luminosity particle-physics detectors ATLAS and CMS and benefit from the very low beta* provided by the Achromatic Telescopic Squeezing (ATS) optics, a local crab cavity scheme provides transverse kicks to the proton bunches. The tight space constraints at the location of these cavities leads to designs which are axially non-symmetric, giving rise to high order multipoles components of the main deflecting mode and, since these kicks are harmonic in time, we expand them in a series of multipoles in a similar fashion as is done for static field magnets. In this work we calculate, for the first time, the higher order multipoles and their impact on beam dynamics for three different crab cavity prototypes. Different approaches to calculate the multipoles are presented. Furthermore, we perform the first calculation of their impact on the long term stability of the machine using the concept of dynamic aperture.