The application of the quantum-chemical density-matrix renormalization group (DMRG) algorithm is cumbersome for complex electronic structures with many active orbitals. The high computational cost is mainly due to the poor convergence of standard DMRG calculations. A factor which affects the convergence behavior of the calculations is the choice of the start-up procedure. In this start-up step matrix representations of operators have to be calculated in a guessed many-electron basis of the DMRG environment block. Different possibilities for the construction of these basis states exist, and we first compare four procedures to approximate the environment states using Slater determinants explicitly. These start-up procedures are applied to DMRG calculations on a sophisticated test system: the chromium dimer. It is found that the converged energies and the rate of convergence depend significantly on the choice of the start-up procedure. However, since already the most simple start-up procedure, which uses only the Hartree-Fock determinant, is comparatively good, Slater determinants, in general, appear not to be a good choice as approximate environment basis states for convergence acceleration. Based on extensive test calculations it is demonstrated that the computational cost can be significantly reduced if the number of total states m is successively increased. This is done in such a way that the environment states are built up stepwise from system states of previous truncated DMRG sweeps for slowly increasing m values.
The application of the quantum-chemical density-matrix renormalization group (DMRG) algorithm is cumbersome for complex electronic structures with many active orbitals. The high computational cost is mainly due to the poor convergence of standard DMRG calculations. A factor which affects the convergence behavior of the calculations is the choice of the start-up procedure. In this start-up step matrix representations of operators have to be calculated in a guessed many-electron basis of the DMRG environment block. Different possibilities for the construction of these basis states exist, and we first compare four procedures to approximate the environment states using Slater determinants explicitly. These start-up procedures are applied to DMRG calculations on a sophisticated test system: the chromium dimer. It is found that the converged energies and the rate of convergence depend significantly on the choice of the start-up procedure. However, since already the most simple start-up procedure, which uses only the Hartree-Fock determinant, is comparatively good, Slater determinants, in general, appear not to be a good choice as approximate environment basis states for convergence acceleration. Based on extensive test calculations it is demonstrated that the computational cost can be significantly reduced if the number of total states m is successively increased. This is done in such a way that the environment states are built up stepwise from system states of previous truncated DMRG sweeps for slowly increasing m values.
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.
The density-matrix renormalization group algorithm has emerged as a promising new method in ab initio quantum chemistry. However, many problems still need to be solved before this method can be applied routinely. At the start of such a calculation, the orbitals originating from a preceding quantum chemical calculation must be placed in a specific order on a one-dimensional lattice. This ordering affects the convergence of the density-matrix renormalization group iterations significantly. In this paper, we present two approaches to obtain optimized orderings of the orbitals. First, we use a genetic algorithm to optimize the ordering with respect to a low total electronic energy obtained at a predefined stage of the density-matrix renormalization group algorithm with a given number of total states kept. In addition to that, we derive orderings from the one- and two-electron integrals of our test system. This test molecule is the chromium dimer, which is known to possess a complicated electronic structure. For this molecule, we have carried out calculations for the various orbital orderings obtained. The convergence behavior of the density-matrix renormalization group iterations is discussed in detail.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
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.
It is well-known that not only the orbital ordering but also the choice of the orbitals itself as the basis may significantly influence the computational efficiency of density-matrix renormalization group (DMRG) calculations. In this study, for assessing the efficiency of using various natural orbitals (NOs) as the DMRG basis, we performed benchmark DMRG calculations with different bases, which included the NOs obtained by various traditional electron correlation methods, as well as NOs acquired from preliminary moderate DMRG calculations (e.g., preserved states less than 500). The tested systems included N2, transition metal Cr2 systems, as well as 1D hydrogen polyradical chain systems under equilibrium and dissociation conditions and 2D hydrogen aggregates. The results indicate that a good compromise between the requirement for low computational costs of acquiring NOs and the demand for high efficiency of NOs as the basis of DMRG calculations may be very dependent on the studied systems' diverse electron correlation characteristics and the size of the active space. It is also shown that a DMRG-complete active space configuration interaction (DMRG-CASCI) calculation in a basis of carefully chosen NOs can provide a less expensive alternative to the standard DMRG-complete active space self-consistent field (DMRG-CASSCF) calculation and avoid the convergence difficulties of orbital optimization for large active spaces. The effect of different NO ordering schemes on DMRG-CASCI calculations is also discussed.