量子蒙特卡罗波函数的能量最小化优化
Optimization of quantum Monte Carlo wave functions by energy minimization
摘要：
在变分蒙特卡罗框架下研究了三种基于能量最小化的波函数优化方法：牛顿法、线性法和微扰法。在牛顿法中，参数变化是从能量梯度和Hessian中计算出来的，对后者使用一个减少方差的统计估计。在线性方法中，通过对角化由波函数及其关于参数的导数所跨越的空间中的哈密顿矩阵的非对称估计来发现参数变化，利用强零方差原理。在计算量较小的摄动方法中，通过非正交摄动理论近似求解线性方法的广义特征值方程来计算参数变化。这些一般的方法在这里示出的波函数的优化组成的Jastrow因子乘以在配置态函数（CSF）的C $_2$分子，包括在计算中的价电子和芯电子的扩展。牛顿法和线性法是优化Jastrow、CSF和轨道参数的有效方法。微扰方法是一个很好的替代优化的CSF和轨道参数。虽然优化是在变分蒙特卡罗水平上进行的，但我们观察到，对于这里研究的C $_2$分子和我们研究的其他系统，随着试验波函数中更多参数的优化，扩散蒙特卡罗总能量单调提高，这意味着节点超曲面也单调提高。
英文摘要：
We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a non-symmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C$_2$ molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C$_2$ molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
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