-For the first time you encounter and fight against Accelerator, defeating him is impossible due to his vector powers, which make him invulnerable and reflects back your attack. To defeat him, you will need a unique Dengeki Card that makes him vulnerable. First, lose once to him. Afterward, Accelerator will run away. Go back to the library in Dengeki Academy and go to Toaru Majutsu no Index 1 (Book #3 where you could revisit the worlds you have cleared). Go all the way at the top right part of the world (go up from the place where you had fought Styl Magnus in Chapter 3). You should see a girl. Speak to her and she will give you a unique Dengeki Card with a unique red symbol at the Dengeki Card if its equipped. Note: She will give you something else if you talk to her without losing to Accelerator at least once. After you receive that card, return to where you found Accelerator. He will be waiting for you. While fighting him, keep yourself alive while the unique Dengeki Card is loading; it will take longer than the normal Boss summon cards. After the unique card loads, use it and Accelerator will be vulnerable. He will not be able to do anything, except for the special move when you reduce his HP to about 50% to 60%. While he is disabled, do your chain burst (try to end the chain burst with only a normal helper attack Dengeki card or with the quickest Boss summon card) as fast as possible before his vector powers return. Be careful if you attempt to use attack or Boss summon cards (except for support Boss summon card or the fastest Boss summon cards) after the chain burst. It might take you long enough to complete your chain burst, depending on what characters you are using. His vector powers might be up at the same time you use your attack or boss summon cards, which will end up damaging or killing you, depending how powerful it is. Bring recovery cards as well. Repeat this process until you defeat him.
-After making Accelerator vulnerable, use Misaka Mikoto's Railgun two or three times before his vector powers return. Using it a fourth time might end up killing you since his vector powers might be back at the same time you are using it. You could use Boss summon cards along with Railgun spamming for about two times as well. This way, it is slower to finish him unlike the chain bursting tactic, but you could continue hitting him without giving him a chance to use his only special move which he will use when his HP reaches about 50% to 60% while he is vulnerable. Bring recovery cards as well. Repeat this process until you defeat him.
After completing the game, face the following hidden bosses by performing the corresponding tasks:
Hecate(Shakugan no Shana) Speak with Enomoto (person from Iriya no Sora; the real version of the fake version you fought in Chapter 1) in Dengeki Academy (found somewhere in the hallways where Dokuro-chan is found). Go to Iriya no Sora 2 (Book #9 in your bookshelves where you can revisit the worlds you completed before). Then, go inside the building where the colored elevators are found. Take the yellow elevator and she'll be there.
Accelerator(Toaru Majutsu no Index) Speak with Shizu (person from Kino's Journey; the man with a dog next to him, the real version of the fake version you fought in Chapter 2) in Dengeki Academy (located outside of the Dengeki Academy. Note: You may need to go out one more time in order to trigger him if you did not see him somewhere outside Dengeki Academy). Then, go to Kino's Journey 1 (Book #2 in your bookshelves where you can revisit the worlds you completed before). Then, go to the place where you fought Shizu and you will see the closed gate behind Shizu from before is open. Enter that gate and he will be there.
Aleister Crowley(Toaru Majutsu no Index) After you defeat Accelerator, in Dengeki Academy, go to the place where you purchase items and Dengeki Cards. Once you enter that room, a cinematic will be triggered. Aleister will be there and he will challenge you.