Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: Exact formula and its approximate treatment

Abstract

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.

Analytic gradient and molecular dynamics simulations using the fragment molecular orbital method combined with effective potentials

“…The reduction of scaling is a major problem in computational chemistry. For this purpose, a variety of quantum-mechanical methods has been developed based on fragmentation of large systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], whose relation has been introduced by Nagata et al [26] and discussed in more detail in the recent review [27].…”

\mathcal{O}(N) methods in electronic structure calculations

“…Forces have been derived for the method [250], though the choice of applying the divide and conquer approach to the total system force (rather than differentiating the divide and conquer energy) may lead to slight discrepancies between the force and energy gradients. A more recent study of forces [251] showed that it is possible, though time-consuming, to derive an exact gradient, and proposed another approximate solution.…”

Graph-based linear scaling electronic structure theory

“…5. Such problems may occur because of difficulties controlling the error in the force evaluations 6,74 as atoms move across the local zone boundaries and as the electronic FIG. 6.…”