Semi-stochastic Diagonalization Approach to Quantum Chemistry

康奈尔大学的Cyrus Umrigar获得了化学理论，模型和计算方法计划的奖项，以扩展电子结构的高级算法，包括配置相互作用。 强相关系统，例如，具有拉伸键的分子，对高级量子化学方法提出了特别严峻的挑战，这是新方法可能产生相当大影响的领域。阿拉维小组已经在10^16（10的16次方）行列式空间中进行了全组态相互作用（FCI）计算，远远超出了传统迭代对角化技术可行的10^10行列式。问题的难度不仅取决于空间的大小，还取决于波函数权的分布和符号问题的严重性。工作围绕改进，以提高该方法的适用范围。目前正在开发一种确定性与随机性相结合的混合方法，该方法保留了两种方法的理想特征（减少了确定性成分造成的统计误差，以及通过随机方法探索非常大的空间的能力）。初步的工作表明，改进的试探波函数可以产生一个很大的减少计算的期望值的统计误差。由于行走是在行列式空间中进行的，费米子符号问题，这是严重的竞争方法，由于玻色子波函数相对于费米子波函数的指数增长，在本方法中是不存在的。该方法确实有一个不太严重的形式的标志问题和实际的方法进行了演示，以保持控制。一个程序，以改善收敛方面的基础规模也正在制定中。这些改进使得迄今为止用现有量子化学方法不可行的分子系统的高精度计算成为可能，例如过渡金属二聚体的解离曲线。所要解决的一般数学问题远远超出了量子化学的范畴，即找到一个矩阵的主本征值，这个矩阵太大了，以至于连矩阵的一列都无法存储在计算机内存中。尽管量子力学的基本方程薛定谔方程已经被知道了将近世纪，寻找该方程的精确解的有效方法仍然是一个活跃的研究领域。这些方法对于对技术感兴趣的分子和材料的各种性质进行有用的预测至关重要。这项工作的目的是改进最近由剑桥大学的Alavi小组开发的方法，该方法使用概率技术来有效地使用比迄今为止可能的多几个数量级的基函数（以获得更少的误差）。

Cyrus Umrigar of Cornell University is supported by an award from the Chemical Theory, Models and Computational Methods Program in extending a high-level algorithm for electronic structure including configuration interaction. Strongly correlated systems, for example, molecules with stretched bonds, pose a particularly severe challenge for high-level quantum chemistry methods, and this is an area where the new method may have considerable impact. The Alavi group has already performed full configuration interaction (FCI) calculations in a space of 10^16 (ten to the sixteenth) determinants, well beyond the 10^10 determinants feasible with traditional iterative diagonalization techniques. The diffculty posed by a problem depends not only on the size of the space but also upon distribution of wave function weights and the severity of the sign problem. The work centers around improvements to enhance the range of applicability of the method. A hybrid deterministic-cum-stochastic approach is being developed that preserves the desirable features of both approaches (reduced statistical error from having a deterministic component as well as the ability to explore very large spaces made possible by a stochastic approach). Preliminary work has demonstrated that improved trial wavefunctions can yield a large reduction in the statistical errors of the computed expectation values. Since the walk is done in determinant space, the Fermion sign problem, which is severe in competing methods due to the exponential growth of the Bosonic wavefunction relative to the Fermionic wavefunction, is absent in the present method. The method does have a less severe form of the sign problem and practical ways are demonstrated for keeping that under control. A procedure for improving the convergence with respect to basis size is also being developed. These improvements are to make possible highly accurate calculations on molecular systems that were hitherto not feasible with existing quantum chemistry methods, such as the dissociation curves for transition metal dimers. The general mathematical problem being addressed extends well beyond the realm of quantum chemistry, namely that of finding the dominant eigenvalue of a matrix that is so large that even a single column of the matrix cannot be stored in computer memory.Although the basic equation of quantum mechanics, the Schroedinger equation, has been known for almost a century, efficient methods for finding accurate solutions to this equation continue to be an active area of research. Such methods are essential for making useful predictions about a variety of properties of molecules and materials of technological interest. It is the purpose of this work to improve upon a method recently developed by the Alavi group at Cambridge University that uses probabilistic techniques to effectively employ orders of magnitude more basis functions (to get less error) than was hitherto possible.