The computation of strongly correlated quantum systems is challenging because of its potentially exponential scaling in the number of electron configurations. Variational calculation of the two-electron reduced density matrix (2-RDM) without the many-electron wave function exploits the pairwise nature of the electronic Coulomb interaction to compute a lower bound on the ground-state energy with polynomial computational scaling. Recently, a dual-cone formulation of the variational 2-RDM calculation was shown to generate the ground-state energy, albeit not the 2-RDM, at a substantially reduced computational cost, especially for higher $N$-representability conditions such as the T2 constraint. Here we generalize the dual-cone variational 2-RDM method to compute not only the ground-state energy but also the 2-RDM. The central result is that we can compute the 2-RDM from a generalization of the Hellmann-Feynman theorem. Specifically, we prove that in the Lagrangian formulation of the dual-cone optimization the 2-RDM is the Lagrange multiplier. We apply the method to computing the energies and properties of strongly correlated electrons—including atomic charges, electron densities, dipole moments, and orbital occupations—in an illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual variational computation of the 2-RDM with T2 or higher $N$-representability conditions provides a polynomially scaling approach to strongly correlated molecules and materials with significant applications in atomic and molecular and condensed-matter chemistry and physics.

中文翻译：

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双电子降密度矩阵的双锥变分计算

强相关量子系统的计算具有挑战性，因为它在电子构型数量上可能呈指数比例缩放。不带多电子波函数的双电子降密度矩阵（2-RDM）的变分计算利用电子库仑相互作用的成对性质，通过多项式计算比例来计算基态能量的下限。最近，变分2-RDM计算的双锥公式显示出生成基态能量（尽管不是2-RDM），但是却大大降低了计算成本，尤其是对于$ñ$-可表示性条件，例如T2约束。在这里，我们概括了双锥变分2-RDM方法，不仅可以计算基态能量，还可以计算2-RDM。中心结果是，我们可以根据Hellmann-Feynman定理的推广来计算2-RDM。具体来说，我们证明在双锥优化的拉格朗日公式中，2-RDM是拉格朗日乘数。我们将该方法应用于示例性氢链和固氮催化剂FeMoco中的强相关电子的能量和性质计算，包括原子电荷，电子密度，偶极矩和轨道占据。T2或更高的2-RDM的对偶变分计算$ñ$可表示性条件为强关联的分子和材料提供了多项式缩放方法，在原子和分子以及凝聚态化学和物理学中具有重要应用。