Accelerating the Convergence of Self-Consistent Field Calculations Using the Many-Body Expansion

The balance between cost-effective and sufficiently accurate methods represents the proverbial “promised land” for quantum chemistry calculations. The burden thus falls upon theoretical and computational chemists to provide such alternatives to mitigate the issues that arise from the employ of finite computing resources. In this paper, we attempt to demonstrate the importance of the quality of the initial guess for the self-consistent field (SCF) calculation when considering cost reduction techniques. We broach this challenge by using the many body expansion (MBE) to yield high quality density matrices (DMs) which, in turn, are applied as an SCF initial guess. The MBE-DM approaches combined with purification schemes and distance-based cutoff schemes can serve as initial guesses to reduce the SCF cycles necessary for convergence or derive energy directly through one Fock build. To this end, four unique types of clusters including water clusters, fluoride anion water clusters, sodium cation water clusters, and ammonium-bisulfate salt clusters have been used to test the performance of MBE-DM where its truncation at three-body expansion, MBE(3)-DM, shows vast improvement for those four clusters with reductions in the number of SCF cycles up to 40% as compared with the traditional superposition of atomic densities (SAD) guess. Other types of typical initial guesses, superposition of atomic potentials (SAP) and basis set projection (BSP), perform much worse than MBE-DM and SAD. In addition, the MBE-DM shows consistency across an array of fragment types irrespective of charges, size, level of theory, and basis set selection. Through MBE­(3)-DM with the distance cutoff and the average purification scheme, the energy can be obtained directly with a mere 3.2 mH of the mean absolute deviation (MAD) for (H2O)N=6–55 which is at least 73 times better than the energy prediction using the typical initial guesses (SAD, SAP, and BSP). The corresponding MAD per monomer is only 0.14 mH which reaches the threshold of the “dynamical accuracy”. The promising results of the methods outlined in this paper not only indicate two direct routes for computational cost reduction but also lay the possible foundation for composite techniques (i.e., ab initio sampling) that make best use of near converged values as their starting point.对于量子化学计算而言，兼顾成本效益与足够精度的方法，正是学界公认的“理想境地”。因此，理论化学与计算化学家需要开发此类替代方案，以缓解有限计算资源使用中产生的各类问题。本文旨在探讨：在考量计算成本优化技术时，自洽场（self-consistent field, SCF）计算初始猜测的质量有多关键。为此，我们采用多体展开（many body expansion, MBE）方法生成高质量密度矩阵（density matrices, DMs），并将其作为SCF计算的初始猜测，以此应对这一挑战。将MBE-DM方法与纯化方案、基于距离的截断方案相结合，可作为初始猜测，要么减少收敛所需的SCF迭代次数，要么仅通过一次福克（Fock）构建即可直接得到体系能量。为此，我们选取四类典型团簇进行测试：水团簇、氟阴离子水团簇、钠阳离子水团簇以及硫酸氢铵盐团簇。结果表明，采用三体展开截断的MBE(3)-DM方法性能优异：与传统的原子密度叠加（superposition of atomic densities, SAD）初始猜测相比，该方法可将四类团簇的SCF迭代次数最多降低40%。其余两类典型初始猜测——原子势叠加（superposition of atomic potentials, SAP）与基组投影（basis set projection, BSP），其表现均远逊于MBE-DM与SAD。此外，MBE-DM方法的性能具有普适性：无论体系带电量、尺寸、理论方法或基组选择如何变化，其表现均保持稳定。结合距离截断与平均纯化方案的MBE(3)-DM方法，可直接计算(H₂O)ₙ（n=6~55）体系的能量，其平均绝对偏差（mean absolute deviation, MAD）仅为3.2毫哈特（mH），相较于传统初始猜测（SAD、SAP与BSP）的能量预测结果，精度提升至少73倍。对应的单体系平均绝对偏差仅为0.14毫哈特，达到了“动态精度”的阈值标准。本文所提出方法的优异结果，不仅提供了两条直接降低计算成本的路径，还为以近收敛值为起点的复合技术（如从头算采样（ab initio sampling））奠定了潜在的研究基础。