Poelmans Ward, Van Raemdonck Mario, Verstichel Brecht, De Baerdemacker Stijn, Torre Alicia, Lain Luis, Massaccesi Gustavo E, Alcoba Diego R, Bultinck Patrick, Van Neck Dimitri
Center for Molecular Modeling, Ghent University , Technologiepark 903, 9052 Zwijnaarde, Belgium.
Department of Inorganic and Physical Chemistry, Ghent University , Krijgslaan 281 (S3), 9000 Gent, Belgium.
J Chem Theory Comput. 2015 Sep 8;11(9):4064-76. doi: 10.1021/acs.jctc.5b00378. Epub 2015 Aug 14.
We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index N-representability P-, Q-, and G-conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly occupied many-electron wave function, i.e., a singlet wave function for which the Slater determinant decomposition only contains determinants in which spatial orbitals are doubly occupied. We rederive the two-index N-representability conditions first found by Weinhold and Wilson and apply them to various benchmark systems (linear hydrogen chains, He, N2, and CN(-)). This work is motivated by the fact that a doubly occupied many-electron wave function captures in many cases the bulk of the static correlation. Compared to the general case, the structure of doubly occupied two-particle density matrices causes the associate semidefinite program to have a very favorable scaling as L(3), where L is the number of spatial orbitals. Since the doubly occupied Hilbert space depends on the choice of the orbitals, variational calculation steps of the two-particle density matrix are interspersed with orbital-optimization steps (based on Jacobi rotations in the space of the spatial orbitals). We also point to the importance of symmetry breaking of the orbitals when performing calculations in a doubly occupied framework.
我们在二阶（双粒子）密度矩阵的一组受限的双指标N可表示性P、Q和G条件下，对多电子系统对应的二阶（双粒子）密度矩阵进行直接变分确定。此外，我们施加了一组必要的约束条件，即双粒子密度矩阵必须可从双占据多电子波函数导出，也就是从斯莱特行列式分解仅包含空间轨道被双占据的行列式的单重态波函数导出。我们重新推导了最初由温霍尔德和威尔逊发现的双指标N可表示性条件，并将其应用于各种基准系统（线性氢链、氦、氮气和氰根离子）。这项工作的动机在于，在许多情况下，双占据多电子波函数捕捉到了大部分的静态关联。与一般情况相比，双占据双粒子密度矩阵的结构使得相关的半定规划具有非常有利的L(3)缩放比例，其中L是空间轨道的数量。由于双占据希尔伯特空间取决于轨道的选择，双粒子密度矩阵的变分计算步骤穿插着轨道优化步骤（基于空间轨道空间中的雅可比旋转）。我们还指出了在双占据框架下进行计算时轨道对称性破缺的重要性。
