CDS&E: Two-electron Reduced Density Matrices in Quantum Chemistry and Physics

芝加哥大学的David Mazziotti获得了化学理论、模型和计算方法项目的一项奖励，以开发量子化学的新方法。要发现最适合从太阳获取能量或与人体疾病作斗争的新型分子和材料，需要详细了解它们的电子分布。然而，电子是非常小的粒子；事实上，它们的质量比单个氢原子小1000倍。因此，电子遵循量子力学定律。就像薛定谔著名的猫一样，电子可以同时以两种或两种以上的物理状态被发现，这是自相矛盾的，它同时一半活着，一半死了。当分子中的电子占据的态比电子的数量多得多时，科学家将分子中的电子描述为强关联(或纠缠)。强关联电子存在于一些最吸引人的分子和材料中，从固氮中的过渡金属催化剂到高温超导体中的铜氧化物平面。然而，尽管强关联电子的重要性，传统上用量子力学定律描述它们的计算成本随着电子数量的增加而指数级增加，这极大地限制了可用理论化学和物理描述的强关联系统的数量。在这个项目中，Mazziotti和他的研究小组正在进一步开发一种新的方法来研究强关联分子，在这种方法中，能量和性质只作为剩余电子的电荷海中两个电子的函数来计算。这些所谓的双电子约化密度矩阵(2-RDM)方法能够高效和准确地处理强关联分子，并且计算成本不会随着系统变大而呈指数增长。最近，2-RDM方法已被应用于几个有趣且非常不同类型的化学体系，例如模拟萤火虫生物发光中能量的有效释放。Mazziotti和他的研究小组目前正在努力将2-RDM方法扩展到更大的分子系统以及依赖时间的现象，这对描述从材料到医学的研究中的分子具有重要意义。Mazziotti还为高中生制作了一本名为E=mc2的期刊，发表学生的研究成果。该杂志提供了一个论坛，鼓励高中生追求数学科学的职业生涯。多电子波函数包含的信息比计算分子和材料的重要能量和性质所需的信息多得多。然而，由于电子在两两相互作用下是不可区分的，作为量子力学基本变量的波函数可以用两电子约化密度矩阵(2-RDM)来代替。在NSF的支持下，Mazziotti和他的研究小组正在他们在2-RDM方法方面的最新进展的基础上，重点开发和改进处理基态和激发态以及依赖时间的量子系统的2-RDM方法。在直接计算强关联体系的2-RDM方面取得了几项重大进展，包括：(1)低标度2-RDM方法，它可以强关联数百个轨道，对应于用传统方法难以处理的波函数；(2)稳态2-RDM方法，即使在存在强电子关联的情况下，它也可以确定分子或材料的电导。这项研究在模拟和预测一系列大的强关联分子体系的电子能量和性质，包括电导率方面具有重要的应用。

David Mazziotti of the University of Chicago funded by an award from the Chemical Theory, Models and Computational Methods program for developing new approaches to quantum chemistry. Discovering novel molecules and materials that are optimal for capturing energy from the sun or fighting disease in the human body requires detailed knowledge of the distributions of their electrons. Electrons, however, are very small particles; in fact, they are 1000 times less massive than a single hydrogen atom. Consequently, electrons obey the laws of quantum mechanics. Like Schrödinger's famous cat, who is paradoxically both half alive and half dead simultaneously, electrons can be found in two or more physical states at the same time. When electrons in a molecule occupy many more states than the number of electrons, the electrons in the molecule are described by scientists as being strongly correlated (or entangled). Strongly correlated electrons are present in some of the most fascinating molecules and materials from the transition-metal catalysts in nitrogen fixation to the copper-oxide planes in high-temperature superconductors. Despite the importance of strongly correlated electrons, however, the computational cost of describing them by the laws of quantum mechanics traditionally increases exponentially with the number of electrons, dramatically limiting the number of strongly correlated systems describable by theoretical chemistry and physics. In this project, Mazziotti and his research group are further developing a novel approach to strongly correlated molecules in which the energies and properties are computed as functions of only two electrons in the "sea" of charge from the remaining electrons. These so-called two-electron reduced density matrix (2-RDM) methods are able to treat strongly correlated molecules efficiently and accurately with a computational cost that does not grow exponentially when the system becomes large. Recently, the 2-RDM method has been applied to several interesting and very different types of chemical systems, for example to model the efficient release of energy in firefly bioluminescence. Mazziotti and his research group are currently working to extend the 2-RDM methods to larger molecular systems as well as time-dependent phenomena with important implications for describing molecules in studies ranging from materials to medicine. Mazziotti also produces a journal for high school students named E = mc2 which publishes research by students. The journal provides a forum for encouraging high school students to pursue careers in the mathematical sciences. The many-electron wave function contains much more information than necessary to compute the important energies and properties of molecules and materials. Because electrons are indistinguishable with pairwise interactions, however, the wave function as the basic variable of quantum mechanics can be replaced by the two-electron reduced density matrix (2-RDM). With support from the NSF, Mazziotti and his research group are building upon their recent advances in 2-RDM methods with an emphasis on developing and improving 2-RDM methods for treating ground and excited states as well as time-dependent quantum systems. Several significant advances in the direct calculation of the 2-RDM for strongly correlated systems are being pursued including: (1) a low-scaling 2-RDM method, which can strongly correlate hundreds of orbitals, corresponding to wave functions that are intractable by conventional methods, and (2) a steady-state 2-RDM method that can determine the conductance of a molecule or material even in the presence of strong electron correlation. The research has important applications to modeling and predicting the electronic energies and properties including conductivity of a wide range of large strongly correlated molecular systems.