The present disclosure provides systems and methods that combine direct capture of one or more moieties from a gaseous mixture with one or both of calcium oxide production and power production. The systems and methods can utilize combinations of a capture unit, a regeneration unit, a calcination unit, a slaking unit, a heat exchange unit, a separation unit, and a power production unit. The present disclosure provides the ability to remove carbon dioxide and other moieties from air or other gaseous mixtures in a truly carbon negative manner by utilizing electricity from a power production unit that is operated in a carbon neutral or carbon negative manner and simultaneously provide useful products, such as calcium oxide and calcium hydroxide.