Powers
The Accelerator:
Accelerator is named after his ability. It creates a thin field of AIM around him, which allows him to modify the vector values of anything he touches. After the events of Volume 5, his brain has been damaged, and he has to rely on a choker-style electrode that connects him to the Misaka Network. The brainwave network allows him to use language, as well as making up for his lost ability to make calculations, although it is stated that it's less than half of what he once had. Furthermore the network enables Accelerator to reflect ultraviolet rays even when he isn't in esper mode. When using the network in esper mode, the battery of the electrode initially only lasts fifteen minutes. That changes in when he joins GROUP and its technical team cranks his time up to thirty minutes. He does not need the Misaka Network to use his abilities when in an awakened state.
He is shown to be far stronger than any other Level 5 in Academy City, being able to easily defeat the #2 Level 5 esper Kakine Teitoku without the use of his black wings. It should be noted that this fight took place after Accelerator's brain was damaged and he was forced to start relying on the Misaka Network to do his calculations, which reduced him to having less than half of his former calculation abilities. Accelerator has stated that as long as he has experienced it, he can analyze forces unknown to him as long as his body has been in contact with it, by following the vectors that have been applied (almost similar to a vaccine shot's case). He used this ability to understand Aiwass' "inexplicable laws."
Redirection:
He can change the vector values of anything by touch; for example, he kills a Sister clone by reversing the flow of blood inside her body after putting his finger inside her wound. Furthermore, this ability is automatic, so projectiles like bullets or explosives have no effect on him, as they will be redirected, even if he is unaware of their presence. He must consciously allow things like gravity and light to affect him. It is because of his ability that he has an albino appearance, as his "reflection" blocks all unnecessary ultraviolet radiation, thereby preventing the skin from manufacturing melanin. However, if one can find their way "around" his redirection ability (as demonstrated by Touma and Kihara Amata), it is shown to be rather useless, marking Accelerator as a Glass Cannon due to having a strong ability but a somewhat weak body (Touma/Kihara Amata managed to bring him down to his feet in only around 7-10 blows).
Wings:
He gains black wings towards the end of volume 13 when he had a epiphany right before he was nearly killed by Kihara Amata, and fully awakened in his fight with the other level 5, Kakine Teitoku in Volume 15, during which he displayed a massive difference in power between them. When he Awakens, Accelerator usually loses almost all conscious control of himself and begins to speak in the language of the Angels. He gains the ability to understand laws that are hitherto unknown and unexplainable to him, and no longer needs to be touching vectors to control them. When under tremendous emotional pressure like in extreme rage, he manifests massive black wings, stated to be made of the same "Dark Matter" that the #2 Level 5 Kakine Teitoku creates with his ability. These wings do all the vector-related calculations for him, implied to be a much faster rate than his original calculating abilities- perhaps infinitely so, and as a result he does not need the Misaka Network to manipulate vectors. In volume 22, due to his pillar of support being changed by an epiphany after being defeated by Touma again, he manifests white wings and a halo and is able to block a massive amount of Telesma that Fiamma of the Right launched from the Star of Bethlehem. The Angelic nature of his ability is described as "The One Who Wields the Power of God".
Magic:
Accelerator has demonstrated the ability to use magic. In volume 22, he used his vector ability to interpret the song used by Index to save Last Order and repeated the breathing method, casting the spell. As this was a pure magic spell, he suffered serious harm but saved Last Order. This is a possible factor that lead to Accelerator gaining White Wings, replacing his Black ones. After an explanation on the theory of magic provided by Leivinia Birdway, Accelerator displays his extreme delicate control over magic to having a side effect of using magic as unnaturally sweating rather than rupture of blood vessels and organ damage. An example of this is even when he has only a rough idea what feng shui was he is able to use a limited magic detection and then combine it with his knowledge of group psychology to make it become accurate. However, Magic still presents a great foil to Accelerator, as when he was tricked into fulfilling the conditions to perform a spell by magician disguised as a Trident member he suffered from the rejection reaction without having taken steps to lower the damage beforehand, which caused the blood to burst from his right arm. The extreme pain involved lowered his calculation ability to a point where he was unable to use his Reflection ability, leaving him helpless.
High-Level Intellect:
Accelerator has shown frequently to possess a very high intellect. For starters, his esper ability is shown to require complex calculations to function, with him being skilled enough in this field to come up with a new application on the fly during his battle against Touma. He also appears to have a very good memory, as he memorized the codes for Last Order's default mind state and was able to remotely return her to that condition, thus erasing the virus planted into her by Amai Ao. Later, he also starts to display good strategies in battle, like when he was able to take out the Hound Dogs using guns, which he utilized for the first time against them, along with exploiting his immediate environment and psychological tactics to his advantage, without using his esper ability. Even with less than half of his calculation ability, it is great enough to allow Accelerator to calculate and analyze all the 25,000 different kinds of energy of Kakine Teitoku's Dark Matter, while fighting at par with him and protecting the bystanders, to update his ability. Finally, he was even able to figure out a way to block the signal used to shut off his electrode and to figure out ways to diminish the harmful influence magic has on his body as an esper, which is a feat accomplished so far by only another individual. He is also fairly skilled at indentifying spies as well as being a skilled interrogator, as he was seen to easily break a Russian spy with Misaka Worst's help.
Machine Ingenuity:
Accelerator is also quite competent with machinery and technology, evident in when he modified his cane with various gravity sensors, and small motors to help better support himself. He also implanted in it a mechanism to overcome the Board of Directors' and their subordinates' means to shut off his choker-style electrode which works by memorizing the wavelength they used and blocking that wavelength afterwards.
Marksmanship:
After his fight with Kihara Amata, Accelerator realizes that he can no longer depend on his ability all the time. As such, he gets training with firearms and is shown to be extremely proficient in their use. Using his left, weaker hand, he was able to finish off a round of practice in 70 seconds. He has commented that reloading a pistol in two seconds with just one hand and his teeth is too slow, even though he was just testing the guns.
Driving Skill:
Accelerator displayed enough skill to drive a motorbike, seemingly without issues.
Equipment
Choker-style electrode:
After he was shot by a bullet inside his skull, Accelerator, despite having survived, had his frontal lobe was badly damaged and lost his ability to walk or better use his motor skills, as well as his verbal and calculation abilities, unabling him even to properly discern text and speech. This unique electrode, which is attached in his head, connects him with the Misaka Network and thanks to the limited connection to the brainwave network he has gained enough calculation ability to think normally, understand and speak human language, walk, etc. The choker have three modes: off, normal mode, esper mode, where the battery runs out in 15 minutes after he enters this mode. Later modified to 30 minutes, but it was also tampered with at the same time with a safety device that allows for someone to remotely to disable the device from switching to esper mode.
Modified Crutch:
It is the crutch Accelerator always uses to support his body when his esper ability is off. It contains a small motor and weight distribution sensors, allowing for it to determine the center of gravity. Instead of a single leg like a regular crutch, it ends with four legs attached to its end like a microphone stand, which according to the narration "holds the ground" helping Accelerator to get him up when he is down. The staff along with the four legs can be retracted in and out of the cuff. But its more important feature is a component inserted by Accelerator that blocks other interference devices. The crutch allows for Accelerator to block specific wavelengths after successfully analyzing it and using the equipment of the crutch, an interference wavelength can be reverse- engineered to specifically block that wavelength.