An accurate treatment of the long-range electron correlation energy, including van der Waals (vdW) or dispersion interactions, is essential for describing the structure, dynamics, and function of a wide variety of systems. Among the most accurate models for including dispersion into density functional theory (DFT) is the range-separated many-body dispersion (MBD) method [A. Ambrosetti et al., J. Chem. Phys., 2014, >140, 18A508], in which the correlation energy is modeled at short-range by a semi-local density functional and at long-range by a model system of coupled quantum harmonic oscillators. In this work, we develop analytical gradients of the MBD energy with respect to nuclear coordinates, including all implicit coordinate dependencies arising from the partitioning of the charge density into Hirshfeld effective volumes. To demonstrate the efficiency and accuracy of these MBD gradients for geometry optimizations of systems with intermolecular and intramolecular interactions, we optimized conformers of the benzene dimer and isolated small peptides with aromatic side-chains. We find excellent agreement with the wavefunction theory reference geometries of these systems (at a fraction of the computational cost) and find that MBD consistently outperforms the popular TS and D3(BJ) dispersion corrections. To demonstrate the performance of the MBD model on a larger system with supramolecular interactions, we optimized the C60@C60H28 buckyball catcher host–guest complex. In our analysis, we also find that neglecting the implicit nuclear coordinate dependence arising from the charge density partitioning, as has been done in prior numerical treatments, leads to an unacceptable error in the MBD forces, with relative errors of ∼20% (on average) that can extend well beyond 100%.
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机译：包括范德华（vdW）或色散相互作用在内的长距离电子相关能的准确处理对于描述各种系统的结构，动力学和功能至关重要。在将弥散纳入密度泛函理论（DFT）的最精确模型中，有一个范围分隔的多体弥散（MBD）方法[A. Ambrosetti et al。，J.Chem。 Phys。，2014，> 140 ，18A508]，其中，相关能量通过半局部密度泛函在短范围内建模，而在远距离范围内通过耦合量子谐波振荡器的模型系统建模。在这项工作中，我们开发了MBD能量相对于核坐标的解析梯度，包括由于将电荷密度划分为Hirshfeld有效体积而产生的所有隐式坐标依赖性。为了证明这些MBD梯度对于分子间和分子内相互作用的系统几何优化的效率和准确性，我们优化了苯二聚体的构象异构体和带有芳香族侧链的分离的小肽。我们发现这些系统的波函数理论参考几何形状具有极好的一致性（仅占计算成本的一小部分），并且发现MBD始终优于流行的TS和D3（BJ）色散校正。为了演示MBD模型在具有超分子相互作用的更大系统上的性能，我们优化了C60 @ C60H28布基球捕手捕手—来宾复合体。在我们的分析中，我们还发现忽略电荷密度分区所引起的隐式核坐标依赖性（如先前的数值处理所进行的那样）会导致MBD力出现不可接受的误差，相对误差约为20％（平均），可以扩展到100％以上。
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