“２－矩阵”的本征值问题及其应用

从波函数约化的二阶密度矩阵，简称"2-矩阵"，具有计算能量的所有信息。 杨振宁曾研究"2-矩阵"非对角元的长程相关性，本征值的最大可能值，并把它同超导和波色-爱因斯坦凝聚体联系起来。这说明"2-矩阵"本征值问题是一个具有重要科学意义的研究课题。然而，同密度泛函理论方法比较，"2-矩阵"的本征值问题一直没有被广泛地研究和应用。近年来越来越多的人认识到密度泛函理论中描述电子关联的泛函模型存在诸多问题，很难突破。有些人开始寻找基于"1-矩阵"的泛函理论，但是"1-矩阵"泛函理论同样存在泛函模型的不确定性。为此，本项目建议开展"2-矩阵"本征值问题的研究，通过"2-矩阵"的本征值和本征函数来构造泛函理论。我们将通过具体原子与分子系统的波函数计算，分析"2-矩阵"本征值问题，获得有关的基本知识。同时，发展基于"2-矩阵"本征值和本征函数的泛函理论，计算方法和计算程序。

The reduced second order density matrix or "2-matrix" is associated with the calculation of energy of a many-electron system. According to C-N Yang, the off-diagonal long range order in the "2-matrix" implies a larger eigenvalue, which can characterize superconductivity and Bose-Einstein condensate. The eigenvalue problem of "2-matrix" is thus an important scientific topic. However, compared to the density funcitonal theory, the eigenvalue problem of "2-matrix" has not been explored extensively. In recent years, more and more people realize the difficulty to imporve the description of electron correlation in density functional, a few people even began to look for alternative "1-matrix" functional theory. But "1-matrix" functional theory also incurs the problem of uncertainty in its functional. So in this program we propose to study the eigenvalue problem of the "2-matrix", and develop a functional theory based on its eigenvalues and eigen functions. Starting from calculated wave functions of several atoms and molecules, the eigenvalue problem of "2-matrix" will be analyzed to get a basic knowledge of the eigenvalue problem. At the same time, computational algorithms and a computation program will be developed for the functional theory based on the eigenvalues and eigen functions of "2-matrix".