Legendary Pokémon can be caught after being defeated in Legendary Raids, but their Capture Chance is very slim. With a Base Capture Chance of 3%, Articuno, Zapdos and Moltres are a bit easier than Lugia (2% BCR).
This page goes into details on how each ball/berry/throw type influences your chances. Big thanks to Epimetheos, the original author of the capture chart.
For convenience, we’ve dissected the chart into numerical values. Of course, there are a lot more of possible combinations than we included. If you want to dive really, really deep into this, take a look at GamePress Catch Rate Calculator.
In short, always use a Golden Razz Berry and try to curve.
Articuno, Moltres and Zapdos Catch Chances
On July 25 2017, BCR was updated for the three Kanto Birds, up to 3% from the old 2% rate. Therefore, we have two separate tables for Kanto Birds and Lugia/Ho-Oh.
Lugia and Ho-Oh catch rate
The GAME_MASTER still shows 2% BCR for Lugia and Ho-Oh, resulting in a different table for these two Pokémon.