Coupled Perturbed Approach to Dual Basis Sets for Molecules and Solids. II: Energy and Band Corrections for Periodic Systems

When trying to reach convergence of quantum chemical calculations toward the complete basis set limit, crystalline solids generally prove to be more challenging than molecules. This is due both to the closer packing of atoms─hence, to linear dependencies─and to the problematic behavior of Ewald techniques used for dealing with the infinite character of Coulomb sums. Thus, a dual basis set approach is even more desirable for periodic systems than for molecules. In such an approach, the self-consistent procedure is implemented in a small basis set, and the effect of the enlargement of the basis set is estimated <i>a posteriori</i>. In this paper, we extend to crystalline solids our previous coupled perturbed dual basis set approach [<i>J. Chem. Theory Comput.</i> <b>2020</b>, <i>16</i>, 1, 340-353] in which the basis set enlargement is treated as a perturbation. Among the notable features of this approach are (i) the possibility of obtaining not only a correction to the energy but also to energy bands and electron density; (ii) the absence of a diagonalization step for the full Fock matrix in the large basis set; and (iii) the possibility of extrapolating low order perturbation energy corrections to infinite order. We also present here the first periodic implementation of the dual basis set method of Liang and Head-Gordon [<i>J. Phys. Chem. A</i> <b>2004</b>, <i>108</i>, 3206-3210]. The effectiveness of both approaches is, then, compared on a small, but representative, set of solids.