Hi, SagSag:
Again, aperture coupled antenna is too complicated for any good analytical formula. If the patch is not far from the ground, half a wavelength is a good start point. However, for aperture coupled antennas, the patch to ground distance is too big. The patch size is normally significantly smaller than half a wavelength. Also, the slot may also be used to couple different resonances together to form a wide resonance for wide bandwidth. It is not easy to get a start point. I think using EM tools is the only good way. IE3D V12 has the FastEM Design Kit for real-time tuning and synthesis. If you are designing many such antennas with different frequency parameteres, you can make use of it for fast design. If you are designing individual one, the FastEM will not help. It is more for many designs with slightly different parameters such as center frequency. Anyway, modern computers are fast enough. You are able to optimize a reasonably sophisticated antenna in not-too-long time even using full the full-EM optimization on IE3D. Regards.