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The reduced-density-matrix method is an promising candidate for the nextgeneration electronic structure calculation method; it is equivalent to solvethe Schr\"odinger equation for the ground state. The number of variables is thesame as a four electron system and constant regardless of the electrons in thesystem. Thus many researchers have been dreaming of a much simpler method forquantum mechanics. In this chapter, we give a overview of the reduced-densitymatrix method; details of the theories, methods, history, and some newcomputational results. Typically, the results are comparable to the CCSD(T)which is a sophisticated traditional approach in quantum chemistry.

Variational approach for the electronic structure calculation on the second-order reduced density matrices and the $N$-representability problem