Unknown Cave

Fly to Snowbelle City and exit to the southwest to reach Route 20. Head to the Pokemon Village and surf on the river to the west. (The TM22 Psychic is found on a cliff above the waterfall too). Disembark on the eastern side of the waterfall and go to the right a bit and you'll find the entrance to the Unknown Cave, where Mewtwo is waiting.


Before engaging it to battle, you must be prepared:

• SAVE YOUR GAME FIRST.
• Have a powerful, primary Pokemon that has high SP Def and not weak to Psychic or Fighting attacks.
• That Pokemon should have a move that can put a Pokemon to Sleep. Having the capability to use False Swipe is also recommended.
• Bring a lot of Pokeballs, preferably Ultra Balls and Timer Balls.
• Use Capture Power.
• Bring a lot of healing items, even the cheap and effective ones like Moomoo Milk.
• Have your primary Pokemon hold an item that will help boost its attack or accuracy. Wide Lens is recommended since it will boost the accuracy of your moves; very helpful in helping land more hits, especially Hypnosis.

Mewtwo can be a tricky Pokemon to capture and can pose problems if you're using weaker Pokemon. Its a LV70 Psychic Pokemon with the following movesets:

Move	Type	Category	Power	Accuracy	Effect
Aura Sphere	Fighting	Special	80	- -	Deals damage and ignores changes to accuracy and evasion.
Psychic	Psychic	Special	90	100	Deals damage and has a 10% chance of lowering the target's Sp Def stat
Recover	Normal	Status	- -	- -	Recovers 50% of the user's max HP
Barrier	Psychic	Status	- -	- -	Increases the user's defense by two stages. It can only be raised to a maximum of 6 stages (3 uses)

Mewtwo's got really high SP def and Def stats already and great overall stats all around. It will use Barrier three times to maximize its usage, and will probably use it again if ever the stats were reduced by an opponent's move. It will use Recover as well but not that effectively. Do not use moves that will poison or burn it, as Mewtwo may not use Recover when it is critically injured and those status effects faints it on the next turn. If possible, use moves that will increase your own Pokemon's stats, especially its SP Def and Attack. Your chances of catching it increases considerably once you got it down to 1 HP (via False Swipe) and while it is sleeping. As the battle takes longer, your Timer Balls will become even more effective.

As a reference, I'll share the Pokemon I used:

LV92 Gallade (Psychic/Fighting)
Held Item: Wide Lens
Moves:

• Power Up Punch
• False Swipe
• Hypnosis
• Mean Look

Since Gallade is a Psychic/Fighting Pokemon, Mewtwo's offensive attacks will just deal normal damage. Gallade also has high sp Def and even higher Atk stats – perfect for boosting physical offensive moves such as False Swipe. I opted using Power Up Punch. Even with this move's 40 base damage, the Atk stat boost it provides after every use is very useful. The high level ensured that I'll be able to deal more damage and take more hits. You can even reach LV100 by grinding in the Battle Chateau. Of course you can use other Pokemon of your choice. However, using False Swipe and any sleeping move is still the best way to go.

After defeating Mewtwo, you'll also get Mewtwonite X/Y, depending on the version of your game.