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In a classical system, we have the analytical approximation, only one type of approximation, and the exact solution. In a quantum system, we have two approximations, one based on a product of single electron wave functions which differs from the exact solution (ground state) by the exchange and correlation energy and one based on the sum of products of single electron wave functions, which differs from the true ground state by the correlation energy. Therefore the quantum and classical system differ in having exchange energy.
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