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There are a few issues with the derivation: (not necessarily a mistake, but) please note that if you mean the Coulomb Hamiltonian, the RHS of (1) should involve the reciprocals of $|r_i|$ etc., not $r_i^2$ etc. The Coulomb interaction enters the Hamiltonian in the form of potential energy, rather than in the form of force. Please ignore this comment if you ...

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