Calculation of lower bounds to energy eigenvalues by reduced density matrices and the representability problem

The expectation value of the Hamiltonian H for a many‐electron system may be expressed in terms of the reduced second‐order density matrix and the “reduced” Hamiltonian K for a two‐electron system. It may be shown that the antisymmetric component H AS of the original Hamiltonian is an “outer projection” of K with respect to the antisymmetrizer O , so that H AS = OKO . This result implies that the eigenvalues of K are, in order, lower bounds to the eigenvalues of H AS , i.e. to the eigenvalues of H associated with the antisymmetry requirement. The expectation values of K are hence often below the ground state energy of H , and the eigenvalues may be used for the calculation of lower bounds. Some numerical applications to atoms serve as an illustration, and it is shown that the lower bounds become worse as the number of electrons increases. The implications for the “representability problem” are discussed.