Decomposition of density matrix renormalization group states into a Slater determinant basis

Abstract

The quantum chemical density matrix renormalization group (DMRG) algorithm is difficult to analyze because of the many numerical transformation steps involved. In particular, a decomposition of the intermediate and the converged DMRG states in terms of Slater determinants has not been accomplished yet. This, however, would allow one to better understand the convergence of the algorithm in terms of a configuration interaction expansion of the states. In this work, the authors fill this gap and provide a determinantal analysis of DMRG states upon convergence to the final states. The authors show that upon convergence, DMRG provides the same complete-active-space expansion for a given set of active orbitals as obtained from a corresponding configuration interaction calculation. Additional insight into DMRG convergence is provided, which cannot be obtained from the inspection of the total electronic energy alone. Indeed, we will show that the total energy can be misleading as a decrease of this observable during DMRG microiteration steps may not necessarily be taken as an indication for the pickup of essential configurations in the configuration interaction expansion. One result of this work is that a fine balance can be shown to exist between the chosen orbital ordering, the guess for the environment operators, and the choice of the number of renormalized states. This balance can be well understood in terms of the decomposition of total and system states in terms of Slater determinants.

Density matrix renormalization group calculations on relative energies of transition metal complexes and clusters

“…15 A fresh look on the algorithm including detailed flow charts was presented in Ref. 16 very recently. The studies of Chan and co-workers comprise benchmark calculations on the dinitrogen ground state potential energy curve, 10 an extension of the DMRG algorithm for nonorthogonal orbitals, 17 a quadratic scaling algorithm, 18 and a harmonic Davidson algorithm.…”

The ab-initio density matrix renormalization group in practice

“…Since then, many groups have independently implemented and improved on the ab-initio DMRG algorithm. Some of these improvements include parallelization, 8,20 nonAbelian symmetry and spin-adaptation, 7,[21][22][23] orbital ordering 5,[24][25][26] and optimization, 9,27-29 more sophisticated initial guesses, 5,24,25,30,31 better noise algorithms, 5,32 extrapolation procedures, 5,33,34 response theories, 35,36 as well as the combination of the DMRG with various other quantum chemistry methods such as perturbation theory, 37 canonical transformations, 38 configuration interaction, 39 and relativistic Hamiltonians. 40 In the ecosystem of quantum chemistry, the DMRG occupies a unique spot.…”

Spin-adapted density matrix renormalization group algorithms for quantum chemistry

“…chemistry. 8,11,[17][18][19][20][21] Here, the ability to utilise spin symmetry is an important advantage. This is because the large number of unpaired electrons often leads to many low lying spin states in a very narrow energy window, which can only be efficiently resolved by targeting a specific spin sector.…”