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Jacob’s ladder is an analogy of a method to search for the divine functional: the exchange-correlation functional which would solve the Kohn-Sham equations equivalently to the many-body Shroedinger equation. The lowest rung of the ladder is Local-Density Approximation, which uses the electron density at every point. Higher rungs keep the approximations made by the lower rungs and refine them. At the top of the ladder is the divine functional.
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