Institute of Physics, University of Rostock, Albert Einstein Straße 23-24, 18059 Rostock, Germany.
J Chem Phys. 2019 Feb 28;150(8):084109. doi: 10.1063/1.5045293.
The Caldeira-Leggett model provides a compact characterization of a thermal environment in terms of a spectral density function, which has led to a variety of numerically exact quantum methods for reduced density matrix propagation. Since spectral densities are often computed from classical molecular dynamics simulations, we investigate in this paper whether quantum effects should be accounted for in the calculations. Therefore, we reformulate the recently developed Fourier method for spectral density calculations from semiclassical simulations which approximately allow for quantum effects. We propose two possible protocols based on either correlation functions or expectation values. These protocols are tested on a generic Calderra-Leggett model for the linearized semiclassical initial-value representation (LSC-IVR), the thawed Gaussian wave packet dynamics (TGWD), and hybrid schemes combining the two with the more accurate Herman-Kluk formula. Surprisingly, spectral densities from the LSC-IVR method, which treats the dynamics completely classically, are found to be extremely accurate, even in the quantum regime, where this method does not give a correct description of the correlation functions and expectation values. In contrast, the TGWD method turns out as too inaccurate for spectral density calculations, and the hybrid schemes perform well only if the system is close to the classical regime. This implies that, if the bath has a Caldeira-Leggett form, spectral densities are insensitive to quantum effects and any effort to approximately account for them rather leads to errors. Hence, in this case, spectral densities can be computed from classical simulations and used in a reduced quantum simulation as well.
卡尔德雷拉-莱格特模型（Caldeira-Leggett model）以谱密度函数（spectral density function）为特征，简洁地描述了热环境，这为简化密度矩阵传播的各种数值精确量子方法奠定了基础。由于谱密度通常是从经典分子动力学模拟中计算出来的，我们在本文中研究了在计算中是否应该考虑量子效应。因此，我们重新制定了最近开发的从半经典模拟中计算谱密度的傅里叶方法（Fourier method），该方法近似允许量子效应。我们提出了两种基于相关函数（correlation functions）或期望值（expectation values）的可能方案。这些方案在通用的卡尔德雷拉-莱格特线性化半经典初始值表示（linearized semiclassical initial-value representation，LSC-IVR）模型、解冻高斯波包动力学（thawed Gaussian wave packet dynamics，TGWD）和结合这两种方法的混合方案中进行了测试，其中混合方案采用更准确的赫尔曼-克鲁克公式（Herman-Kluk formula）。令人惊讶的是，完全从经典动力学处理的 LSC-IVR 方法的谱密度被发现是非常准确的，即使在量子区域也是如此，而这种方法不能正确描述相关函数和期望值。相比之下，TGWD 方法对于谱密度计算来说过于不准确，而混合方案只有在系统接近经典区域时才表现良好。这意味着，如果浴槽具有卡尔德雷拉-莱格特形式，那么谱密度对量子效应不敏感，任何近似考虑它们的努力反而会导致错误。因此，在这种情况下，可以从经典模拟中计算谱密度，并在简化的量子模拟中使用。
