Size extensivity of the variational reduced-density-matrix method

With the density-matrix variational method, the ground-state energy of a many-electron system is calculated by minimizing the energy expectation value subject to the selected representability conditions for the second-order reduced-density matrix (2-RDM). There is a lack of size extensivity under the P, Q, and G conditions of this method by applying to M noninteracting atoms and molecules up to M=32; Be_M, (CH₄)_M, and (N₂)_M. The energy per molecule (E/M) monotonically decreases as M increases, showing the lack of size extensivity in this method. The inextensive contributions to energies are 3×10⁻⁴ and 3×10⁻³ a.u. for (CH₄)_M and (N₂)_M using the STO-6G basis set, respectively. In the examples studied, E/M approaches a finite value as M→∞. The intermolecular elements of 2-RDMs do not satisfy the cumulant expansion.