The density matrix renormalisation group (DMRG) algorithm solves the many-particle electronic Schrödinger equation as accurately as possible in a full-configuration-interaction (FCI) sense for a given one-electron basis. The FCI limit is approached by increasing the number of many-particle DMRG states which span the Hilbert space of interest. In this work we investigate an automated DMRG error protocol which extrapolates the electronic energy using Richardson's deferred approach to the limit. The key idea is to consider the result of a DMRG algorithm as an analytical function of an adjustable parameter like the number of DMRG states. We can then probe this analytical function by performing calculations for different set sizes of renormalised DMRG basis states. None of these calculations has to actually provide the desired accuracy but after we have collected enough information about the function's behaviour, we can represent it by an analytic rational function that may then be used to extrapolate to the converged energy. The advantage of this approach is that it delivers an error estimate for the electronic energy which can be used for accuracy control and as a convergence criterion. In addition, one may detect convergence to local energy minima by automatically increasing the analytic parameter (i.e. the number of renormalised DMRG states). The error estimates also allows us to aim at relative energies from extrapolated total energies of similar accuracy that is then independent of the number of renormalised DMRG states or of the truncation error of the reduced density matrix.
The density matrix renormalisation group (DMRG) algorithm solves the many-particle electronic Schrödinger equation as accurately as possible in a full-configuration-interaction (FCI) sense for a given one-electron basis. The FCI limit is approached by increasing the number of many-particle DMRG states which span the Hilbert space of interest. In this work we investigate an automated DMRG error protocol which extrapolates the electronic energy using Richardson's deferred approach to the limit. The key idea is to consider the result of a DMRG algorithm as an analytical function of an adjustable parameter like the number of DMRG states. We can then probe this analytical function by performing calculations for different set sizes of renormalised DMRG basis states. None of these calculations has to actually provide the desired accuracy but after we have collected enough information about the function's behaviour, we can represent it by an analytic rational function that may then be used to extrapolate to the converged energy. The advantage of this approach is that it delivers an error estimate for the electronic energy which can be used for accuracy control and as a convergence criterion. In addition, one may detect convergence to local energy minima by automatically increasing the analytic parameter (i.e. the number of renormalised DMRG states). The error estimates also allows us to aim at relative energies from extrapolated total energies of similar accuracy that is then independent of the number of renormalised DMRG states or of the truncation error of the reduced density matrix.
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 CH2,H2O, and F2 as well as the first excited state of CH2. 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.
High Resolution Image Download MS PowerPoint Slide The density matrix renormalization group (DMRG) is a powerful numerical technique to solve strongly correlated quantum systems: it deals well with systems which are not dominated by a single configuration (unlike Coupled Cluster) and it converges rapidly to the Full Configuration Interaction (FCI) limit (unlike truncated Configuration Interaction (CI) expansions). In this work, we develop an algorithm integrating DMRG within the multiwavelet-based multiresolution analysis (MRA). Unlike fixed basis sets, multiwavelets offer an adaptive and hierarchical representation of functions, approaching the complete basis set limit to a specified precision. As a result, this combined technique leverages the multireference capability of DMRG and the complete basis set limit of MRA and multiwavelets. More specifically, we adopt a pre-existing Lagrangian optimization algorithm for orbitals represented in the MRA domain and improve its computational efficiency by replacing the original CI calculations with DMRG. Additionally, we substitute the reduced density matrices computation with the direct extraction of energy gradients from the DMRG tensors. We apply our method to small systems such H 2, He, HeH 2, BeH 2 and N 2 . The results demonstrate that our approach reduces the final energy while keeping the number of orbitals low compared to FCI calculations on an atomic orbital basis set.
This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.
We introduce the Nuclear-Electronic All-Particle Density Matrix Renormalization Group (NEAP-DMRG) method for solving the time-independent Schrödinger equation simultaneously for electrons and other quantum species. In contrast to the already existing multicomponent approaches, in this work, we construct from the outset a multi-reference trial wave function with stochastically optimized non-orthogonal Gaussian orbitals. By iterative refining of the Gaussians’ positions and widths, we obtain a compact multi-reference expansion for the multicomponent wave function. We extend the DMRG algorithm to multicomponent wave functions to take into account inter- and intra-species correlation effects. The efficient parameterization of the total wave function as a matrix product state allows NEAP-DMRG to accurately approximate the full configuration interaction energies of molecular systems with more than three nuclei and 12 particles in total, which is currently a major challenge for other multicomponent approaches. We present the NEAP-DMRG results for two few-body systems, i.e., H2 and H3+, and one larger system, namely, BH3.
In this paper, we investigate the convergence properties of density matrix renormalisation group (DMRG) calculations for vertical electronic excitations performed on three bimolecular complexes motivated by Szalay's recent benchmark results [JCTC. 2020;16:7]. Besides the high-level coupled-cluster reference spectrum, the extrapolated truncation-free DMRG results are compared against several alternative solutions based on multireference configuration interaction and coupled-cluster theories. By taking advantage of the error cancellation effects, which are inherent in the state-averaged computational scheme, we demonstrate that the extrapolated solution based on only low bond-dimensional DMRG data could already provide quantitative predictions even for large active spaces. We also show that orbital optimisation improves further the accuracy of the DMRG results which systematically approach the excitation energies of coupled cluster single-double-triple at similar computational costs.
We present a procedure to construct a configuration-interaction expansion containing arbitrary excitations from an underlying full-configuration-interaction-type wave function defined for a very large active space. Our procedure is based on the density-matrix renormalization group (DMRG) algorithm that provides the necessary information in terms of the eigenstates of the reduced density matrices to calculate the coefficient of any basis state in the many-particle Hilbert space. Since the dimension of the Hilbert space scales binomially with the size of the active space, a sophisticated Monte Carlo sampling routine is employed. This sampling algorithm can also construct such configuration-interaction-type wave functions from any other type of tensor network states. The configuration-interaction information obtained serves several purposes. It yields a qualitatively correct description of the molecule's electronic structure, it allows us to analyze DMRG wave functions converged for the same molecular system but with different parameter sets (e.g., different numbers of active-system (block) states), and it can be considered a balanced reference for the application of a subsequent standard multi-reference configuration-interaction method.