Robust mixing for ab initio quantum mechanical calculations

We study the general problem of mixing for ab initio quantum mechanical problems. Guided by general mathematical principles and the underlying physics, we propose a multisecant form of Broyden’s second method for solving the self-consistent field equations of Kohn-Sham density-functional theory. The algorithm is robust, requires relatively little fine tuning, and appears to outperform the current state of the art, converging for cases that defeat many other methods. We compare our technique to the conventional methods for problems ranging from simple to nearly pathological.

Figure 1

(Color online) Iterates for an ${\text{O}}_2$ molecule with atomic densities having a spin of $+2$ the other with 0. The figures show the difference between the spins (vertical axis) for the two atoms and the difference between the total charges (horizontal axis) within the muffin tins for iterates generated by the Pratt step [Eq. (3.2)] with different fixed step-length parameters $\lambda$. In frame (a) the Pratt step parameter is $\lambda =0.30$, in frame (b) the Pratt step parameter is $\lambda =0.40$.

Figure 2

(Color online) Plot of the convergence for models 1–5 [frames (a)–(e), respectively] using the multisecant update based on Broyden’s first method and second method, compared to Broyden’s second method and Broyden’s first method. In model 5 the only algorithm to converge is MSB2.