节约型路径积分
Economised path integrals
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中文总结 AI 辅助
研究如何优化Trotter路径积分，通过求解最小二乘问题得到节约型内部模式频率，计算量仅为二阶Trotter。以六角冰、MOF - 5为例，其收敛性与四阶Suzuki - Chin相当，且无需复杂计算和新估计器，实现简单。
中文摘要 AI 辅助
标准Trotter路径积分中环形聚合物弹簧势的海森矩阵是一个\(P\times P\)对称循环矩阵，其质心特征值为零。所有此类矩阵可交换，并通过相同的珠子到正常模式变换矩阵进行对角化，其特征值因对称性包含\(\lceil P/2\rceil - 1\)个简并对。仍可优化Trotter近似，即优化剩余\(\lfloor P/2\rfloor\)个独立非零正常模式频率，以拟合频率在\(0\le\omega\le\omega_{\rm max}\)范围内的谐波环形聚合物的精确量子力学回转半径。优化涉及求解一个简单的最小二乘问题以得到最优（节约型或“Eco”）内部模式频率。计算的其余部分与Trotter路径积分计算相同。以六角冰为例，Eco路径积分的收敛性与四阶Suzuki - Chin路径积分相当，但仅需二阶Trotter计算量。无需通过有限差分计算Suzuki - Chin方法中出现的投影海森矩阵，无需为可观测量开发新的估计器，一旦计算出Eco频率，Eco路径积分的实现只需更改Trotter路径积分代码的几行。在GPUMD中实现Eco方法并用于收敛具有机器学习神经进化势的MOF - 5的（负）热膨胀系数和恒压热容。
英文摘要
The Hessian of the ring polymer spring potential in the standard Trotter path integral is a $P\times P$ symmetric circulant matrix with a centroid eigenvalue of zero. All such matrices commute and are diagonalised by the same bead to normal mode transformation matrix, and their eigenvalues contain $\lceil P/2\rceil-1$ degenerate pairs by symmetry. However, this still leaves some freedom to improve on the Trotter approximation: one can optimise the remaining $\lfloor P/2\rfloor$ independent non-zero normal mode frequencies to fit the exact quantum mechanical radii of gyration of harmonic ring polymers with frequencies in the range $0\leω\leω_{\rm max}$, where $ω_{\rm max}$ is the maximum physical frequency in the problem of interest. The optimisation involves solving a simple least squares problem for the optimum (economised or "Eco") internal mode frequencies. The remainder of the calculation then proceeds in the same way as a Trotter path integral calculation. An example application to hexagonal ice shows that the convergence of the Eco path integral is comparable to that of the 4th order Suzuki-Chin path integral, but with purely 2nd order Trotter effort. There is no need to calculate the projected Hessians that arise in the Suzuki-Chin method by finite differences, there is no need to develop any new estimators for observables, and once the Eco frequencies have been calculated the implementation of the Eco path integral involves changing just a few lines of a Trotter path integral code. To provide a more impressive example we have implemented the Eco method in GPUMD and used it to converge the (negative) thermal expansion coefficient and the constant pressure heat capacity of MOF-5 with a machine-learned neuroevolution potential.
发表机构
- University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。
