Communication: Linear-expansion shooting techniques for accelerating self-consistent field convergence

Abstract

Based on the corrected Hohenberg-Kohn-Sham total energy density functional [Y. A. Zhang and Y. A. Wang, J. Chem. Phys. 130, 144116 (2009)], we have developed two linear-expansion shooting techniques (LIST)- direct LIST (LISTd) and indirect LIST (LISTi), to accelerate the convergence of self-consistent field (SCF) calculations. Case studies show that overall LISTi is the most robust and efficient algorithm for accelerating SCF convergence, whereas LISTd is advantageous in the early stage of an SCF process. More importantly, LISTi outperforms Pulay's direct inversion in the iterative subspace (DIIS) [P. Pulay, J. Comput. Chem. 3, 556 (1982)] and its two recent improvements, energy-DIIS [K. N. Kudin, G. E. Scuseria, and E. Cancès, J. Chem. Phys. 116, 8255 (2002)] and augmented Roothaan-Hall energy-DIIS [X. Hu and W. Yang, J. Chem. Phys. 132, 054109 (2010)].

On the equivalence of LIST and DIIS methods for convergence acceleration

“…[11][12][13] This behavior arises because minimization of an approximate error function does not force convergence. A way to solve this issue is to use methods that ensure a decrease in the energy at every iteration, as this guarantees convergence to a local minimum.…”

“…However, this explanation is still unsatisfactory because the magnitudes of the coefficients would need to be extremely large-comparable to the inverse of machine precision-and LISTd performs poorly even when far from convergence. 11,12 The argument for the improved acceleration of LISTb over LISTd based on the minimization of approximate error functions seems, therefore, much more plausible than the aforementioned explanations and simultaneously clarifies why LISTi is also better than LISTd.…”

“…These results also clarify the reason for the poor convergence acceleration of LISTd. [11][12][13] In brief, the better convergence properties of LISTb and LISTi as compared to LISTd can be understood in terms of the properties of the error function being minimized; a SCF solution minimizes the error function in LISTb and LISTi, but not necessarily in LISTd.…”

“…There are three LIST methods: LISTi, LISTd, 11 and LISTb. 12 We first show how LISTi is equivalent to the general DIIS described above.…”

“…5 Since Pulay's seminal paper, 1 there has been an enormous amount of work to refine, improve, and complement DIIS [6][7][8][9][10] but the core of the technique remains the same, as originally formulated. More recently, there have been claims in the literature 11,12 that a family of recently proposed LIST (linear expansion shooting techniques) methods supersedes the performance of DIIS. We, here, prove that the LIST methods are mathematically equivalent to the general form of DIIS 2 (with specific error vectors) and are not superior to the commonly used commutator-DIIS (CDIIS), as shown numerically in our previous work.…”