Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
J Chem Phys. 2023 Apr 21;158(15). doi: 10.1063/5.0146431.
In Born-Oppenheimer molecular dynamics (BOMD) simulations based on the density functional theory (DFT), the potential energy and the interatomic forces are calculated from an electronic ground state density that is determined by an iterative self-consistent field optimization procedure, which, in practice, never is fully converged. The calculated energies and forces are, therefore, only approximate, which may lead to an unphysical energy drift and instabilities. Here, we discuss an alternative shadow BOMD approach that is based on backward error analysis. Instead of calculating approximate solutions for an underlying exact regular Born-Oppenheimer potential, we do the opposite. Instead, we calculate the exact electron density, energies, and forces, but for an underlying approximate shadow Born-Oppenheimer potential energy surface. In this way, the calculated forces are conservative with respect to the approximate shadow potential and generate accurate molecular trajectories with long-term energy stabilities. We show how such shadow Born-Oppenheimer potentials can be constructed at different levels of accuracy as a function of the integration time step, δt, from the constrained minimization of a sequence of systematically improvable, but approximate, shadow energy density functionals. For each energy functional, there is a corresponding ground state Born-Oppenheimer potential. These pairs of shadow energy functionals and potentials are higher-level generalizations of the original "zeroth-level" shadow energy functionals and potentials used in extended Lagrangian BOMD [Niklasson, Eur. Phys. J. B 94, 164 (2021)]. The proposed shadow energy functionals and potentials are useful only within this extended dynamical framework, where also the electronic degrees of freedom are propagated as dynamical field variables together with the atomic positions and velocities. The theory is quite general and can be applied to MD simulations using approximate DFT, Hartree-Fock, or semi-empirical methods, as well as to coarse-grained flexible charge models.
在基于密度泛函理论（DFT）的 Born-Oppenheimer 分子动力学（BOMD）模拟中，势能和原子间力是从电子基态密度计算得出的，而电子基态密度是通过迭代自洽场优化程序确定的，在实际应用中，该程序永远不会完全收敛。因此，计算出的能量和力只是近似的，这可能导致非物理的能量漂移和不稳定性。在这里，我们讨论了一种基于反向误差分析的替代阴影 BOMD 方法。我们不是为基础的精确正则 Born-Oppenheimer 势能计算近似解，而是做相反的事情。相反，我们计算电子密度、能量和力，但对于基础的近似阴影 Born-Oppenheimer 位能表面。通过这种方式，计算出的力相对于近似阴影位能是保守的，并生成具有长期能量稳定性的准确分子轨迹。我们展示了如何根据积分时间步长 δt 构建不同精度的阴影 Born-Oppenheimer 势，从一系列系统改进但近似的阴影能量密度泛函的约束最小化中构建。对于每个能量泛函，都有一个对应的基态 Born-Oppenheimer 势。这些阴影能量泛函和势能对是原始“零级”阴影能量泛函和势能的高级泛化，它们用于扩展 Lagrangian BOMD[Niklasson, Eur. Phys. J. B 94, 164 (2021)]。所提出的阴影能量泛函和势能仅在这个扩展的动力学框架内有用，在这个框架中，电子自由度也与原子位置和速度一起作为动力学场变量传播。该理论相当通用，可以应用于使用近似 DFT、Hartree-Fock 或半经验方法的 MD 模拟，以及粗粒化的柔性电荷模型。
