Collaborative Research: CMG: Quantum Monte Carlo Calculations of Deep Earth Materials

量子蒙特卡罗（QMC）模拟中的新数学方法将被应用和发展，以获得比当前电子结构方法更精确的地球材料特性。QMC是目前已知最精确的研究物理和化学实际材料的技术之一，与传统的密度泛函理论（DFT）方法相比，它提供了显著的精度增益。这将使电子结构量子蒙特卡罗方法在质量上更适用于复杂材料，并提高精度。新的QMC技术将应用于地球材料性质的前沿问题，以获得地球物理学高度关注的固体材料的精确状态、相变和弹性方程。目前QMC方法的一个重要限制是高计算需求，这使得包括固体解决方案在内的大型系统的应用程序非常昂贵。QMC计算的很大一部分花费在形成和评估由单粒子轨道构造的Slater行列式上。该团队计划开发和应用不同的定位转换，以获得稀疏行列式。本文将发展两种线性代数方法，以有效地评估它们。首先，Krylov的迭代求行列式的方法将被纳入QMC。其次，用蒙特卡罗方法计算行列式的轨迹。这两种技术都将进一步实现order-N QMC技术的目标，这种技术更有效，适用于更广泛的材料，远远超出目前的可能性。计算矿物物理学是利用计算机模拟来预测和理解地球性质的巨大努力的一部分。新的数学技术将衍生，使先进的量子蒙特卡罗模拟更精确，更有效。重要的地球材料在高压下的性质将以前所未有的精度得到预测。这项工作还将导致对地球深部固体中一些基本相变的更精确描述。这个项目是数学家、数学物理学家和地球物理学家之间的密切合作。它将为地球科学带来新的应用数学方法，广泛适用于所有地球材料。更广泛的影响将包括开发适用于许多领域的新计算技术，以及对研究生和博士后的最新材料模拟教育，在研究生水平课程和将组织的两个讲习班中教授新的计算技术。

New mathematical methods in quantum Monte Carlo (QMC) simulation will be applied and developed to obtain more accurate properties of Earth materials than is possible using current electronic structure methods. QMC is one among the most precise known techniques to study realistic materials in physics and chemistry and provide a significant gain in precision compared traditional density functional theory (DFT) approaches. This will bring electronic structure quantum Monte Carlo methodology to a qualitatively higher level of applicability to complex materials and lead to increased accuracy. The new QMC techniques will be applied to forefront problems in the properties of Earth materials in order to obtain accurate equations of state, phase transitions, and elasticity of solid materials that are of high interest in geophysics. One significant limitation of today's QMC methods is the high computational demand, which currently makes applications to larger systems including solid solutions prohibitively expensive. A substantial portion of the QMC computation is spent on forming and evaluating a Slater determinant, which is constructed from one-particle orbitals. The team plans to develop and apply different localization transformations in order to obtain a sparse determinant. Two linear algebra methods will be developed for their efficient evaluation. First, Krylov's method for the iterative evaluation of a determinant will be incorporated into QMC. Secondly, the trace of the determinant will be calculated with Monte Carlo methods. Both techniques will further the goal of obtaining an order-N QMC techniques that are more efficient and applicable to a wider range of materials, well beyond the current possibilities.Computational mineral physics is part of the large effort to use computer simulations to predict and understand properties of the Earth. New mathematical techniques will be derived to make advance quantum Monte Carlo simulations more precise and significantly more efficient. The properties of important Earth materials at high pressure will be predicted with unprecedented accuracy. This work will also lead to a more precise description of a number of fundamental phase transitions in solid deep Earth. This project is a close collaboration between a mathematician, mathematical physicists, and geophysicists. It will bring new applied math methods into geoscience, with broad applicability to all Earth materials. The broader impact will include the development of new computational techniques applicable to many areas as well as the education of graduate students and post-docs in state-of-the-art materials simulations, teaching of new computational techniques in graduate level classes and during two workshops that will be organized.