Mazziotti David A
Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Dec;68(6 Pt 2):066701. doi: 10.1103/PhysRevE.68.066701. Epub 2003 Dec 3.
Generalizations of McWeeny's purification formula are developed within the formalism of the particle-hole duality from the theory of reduced density matrices. Each of the generalized purification formulas is expressed as a sum of the one-particle reduced density matrix (1-RDM) and a finite series in the product of the one-particle and the one-hole RDMs, a product which vanishes in the limit that the 1-RDM is idempotent. Two categories of purification formulas are explored: (i) formulas which treat the "occupied" and the "virtual" occupation numbers equivalently and (ii) formulas which treat these occupation numbers differently. The latter category includes and extends the purification formulas derived in the context of the 1,2-contracted Schrödinger equation [D. A. Mazziotti, J. Chem. Phys. 115, 8305 (2001)]. While the McWeeny purification minimizes the absolute error in the occupation numbers quadratically, the generalized purification formulas exhibit faster than quadratic convergence of the 1-RDM towards idempotency. Application of these purification formulas in existing algorithms for linear scaling will be explored and discussed including illustrative calculations on sodium wires of length 10, 20, 30, and 40 atoms.
在约化密度矩阵理论的粒子-空穴对偶形式体系内，对麦克韦尼纯化公式进行了推广。每个广义纯化公式都表示为单粒子约化密度矩阵（1-RDM）与单粒子和单空穴RDM乘积的有限级数之和，在1-RDM幂等的极限情况下，该乘积为零。探讨了两类纯化公式：（i）对“占据”和“虚”占据数同等对待的公式，以及（ii）对这些占据数区别对待的公式。后一类包括并扩展了在1,2-收缩薛定谔方程背景下推导的纯化公式[D. A. 马佐蒂，《化学物理杂志》115, 8305 (2001)]。虽然麦克韦尼纯化使占据数的绝对误差在二次方上最小化，但广义纯化公式显示出1-RDM向幂等性的收敛速度快于二次方收敛。将探讨并讨论这些纯化公式在现有线性标度算法中的应用，包括对长度为10、20、30和40个原子的钠线的示例计算。
