Size extensivity of the variational reduced-density-matrix method

Abstract

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 $M=32$; ${\text{Be}}_{M}$, ${({\text{CH}}_{4})}_{M}$, and ${({\text{N}}_{2})}_{M}$. 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 $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ and $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\text{ }\text{a}\text{.u}\text{.}$ for ${({\text{CH}}_{4})}_{M}$ and ${({\text{N}}_{2})}_{M}$ using the STO-6G basis set, respectively. In the examples studied, $E/M$ approaches a finite value as $M\ensuremath{\rightarrow}\ensuremath{\infty}$. The intermolecular elements of 2-RDMs do not satisfy the cumulant expansion.