  We develop convergence acceleration procedures that enable a gradient
descent-type iteration method to efficiently simulate Hartree--Fock equations
for atoms interacting both with each other and with an external potential. Our
development focuses on three aspects: (i) optimization of a parameter in the
preconditioning operator; (ii) adoption of a technique that eliminates the
slowest-decaying mode to the case of many equations (describing many atoms);
and (iii) a novel extension of the above technique that allows one to eliminate
multiple modes simultaneously. We illustrate performance of the numerical
method for the 2D model of the first layer of helium atoms above a graphene
sheet. We demonstrate that incorporation of aspects (i) and (ii) above into the
``plain" gradient descent method accelerates it by at least two orders of
magnitude, and often by much more. Aspect (iii) -- a multiple-mode elimination
-- may bring further improvement to the convergence rate compared to aspect
(ii), the single-mode elimination. Both single- and multiple-mode elimination
techniques are shown to significantly outperform the well-known Anderson
Acceleration. We believe that our acceleration techniques can also be gainfully
employed by other numerical methods, especially those handling hard-core-type
interaction potentials.
