Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach

We have applied the momentum space version of the density-matrix renormalization-group method (k-DMRG) in quantum chemistry in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to determine the desired accuracy of the method in advance of the calculations by dynamically controlling the truncation error and the number of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the number of orbitals and the fraction of filled orbitals. We have calculated the ground state of the molecules ${\mathrm{CH}}_2,$ ${\mathrm{H}}_2\mathrm{O},$ and ${\mathrm{F}}_2$ as well as the first excited state of ${\mathrm{CH}}_2.$ Our largest calculations were carried out with 57 orbitals, the largest number of block states was 1500–2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000–1 200 000.