Sliced Basis Density Matrix Renormalization Group for Electronic Structure

We introduce a hybrid approach to applying the density matrix renormalization group to continuous systems, combining a grid approximation along one direction with a finite Gaussian basis set for the remaining two directions. This approach is especially useful for chainlike molecules, where the grid is used in the long direction. For hydrogen chain systems, the computational time scales approximately linearly with the number of atoms, as we show with near-exact minimal basis set calculations with up to 1000 atoms. The linear scaling comes from both the localization of the basis and a compression method for the long-ranged two-electron interaction. For shorter hydrogen chains, we show results with up to triple-$\zeta$ bases.

Figure 1

The sliced basis set approach can be viewed as finely slicing the continuum into parallel two-dimensional planes, each spanned by a small set of transverse functions.

Figure 2

Electronic density in the $y-z$ plane of a linear chain of ten hydrogen atoms, equally spaced at a near-neighbor distance $R=2.4\mathrm{a}.\mathrm{u}.$, calculated in a sliced cc-pVDZ basis (with $N_o=4$). Dimerization induced by the open boundaries is visible, representing the strong tendency to form ${\mathrm{H}}_2$ molecules.

Figure 3

Energy of linear chains of ten hydrogen atoms, equally spaced by a distance $R$. Dashed lines show results using QCDMRG in standard basis sets. Solid lines with symbols are SBDMRG results in a sliced version of each basis set.

Figure 4

Energy per atom of $N$ hydrogen atom chains, equally spaced by a distance $R=3.6\mathrm{a}.\mathrm{u}.$ using standard versus sliced STO-6G basis sets. Inset: Average time per DMRG sweep with $m=100$, demonstrating linear scaling up to $N=1000$ atoms.