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 $M=32$; ${\text{Be}}_M$, ${\left({\text{CH}}_4\right)}_M$, and ${\left({\text{N}}_2\right)}_M$. The energy per molecule $\left(E/M\right)$ monotonically decreases as $M$ increases, showing the lack of size extensivity in this method. The inextensive contributions to energies are $3\times {10}^{-4}$ and $3\times {10}^{-3}\text{a}\text{.u}\text{.}$ for ${\left({\text{CH}}_4\right)}_M$ and ${\left({\text{N}}_2\right)}_M$ using the STO-6G basis set, respectively. In the examples studied, $E/M$ approaches a finite value as $M\to \infty$. The intermolecular elements of 2-RDMs do not satisfy the cumulant expansion.

Figure 1

Energy per ${\text{N}}_2$ molecule of a noninteracting ${\left({\text{N}}_2\right)}_M$ cluster as a function of $M^{-2}$. Cross $(\times )$ and plus $(+)$ show the upper and lower bounds of the variational energy (per ${\text{N}}_2$), respectively. The energies are shifted by $-108.712489\text{a}\text{.u}\text{.}$, which is the upper bound of the isolated ${\text{N}}_2$. A size-extensive theory gives the $M$ independent energy (horizontal line). The inset is the enlarged figure near $M\to \infty$.

Figure 2

In the case of a ${\left({\text{CH}}_4\right)}_M$ cluster, the energies are shifted by $-40.2105267\text{a}\text{.u}\text{.}$, the upper bound of the isolated ${\text{CH}}_4$. The thin line is used to guide the reader’s eyes. Refer to Fig. 1 for details.