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

The accurate first-principles calculation of relative energies of transition metal complexes and clusters is still one of the great challenges for quantum chemistry. Dense lying electronic states and near degeneracies make accurate predictions difficult, and multireference methods with large active spaces are required. Often density functional theory calculations are employed for feasibility reasons, but their actual accuracy for a given system is usually difficult to assess (also because accurate ab initio reference data are lacking). In this work we study the performance of the density matrix renormalization group algorithm for the prediction of relative energies of transition metal complexes and clusters of different spin and molecular structure. In particular, the focus is on the relative energetical order of electronic states of different spin for mononuclear complexes and on the relative energy of different isomers of dinuclear oxo-bridged copper clusters.

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

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.

Construction of environment states in quantum-chemical density-matrix renormalization group calculations.

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.

Relativistic DMRG calculations on the curve crossing of cesium hydride.

Over the past few years, it has been shown in various studies on small molecules with only a few electrons that the density-matrix renormalization group (DMRG) method converges to results close to the full configuration-interaction limit for the total electronic energy. In order to test the capabilities of the method for molecules with complex electronic structures, we performed a study on the potential-energy curves of the ground state and the first excited state of 1sigma+ symmetry of the cesium hydride molecule. For cesium relativistic effects cannot be neglected, therefore we have used the generalized arbitrary-order Douglas-Kroll-Hess protocol up to tenth order, which allows for a complete decoupling of the Dirac Hamiltonian. Scalar-relativistic effects are thus fully incorporated in the calculations. The potential curves of the cesium hydride molecule feature an avoided crossing between the ground state and the first excited state, which is shown to be very well described by the DMRG method. Compared to multireference configuration-interaction results, the potential curves hardly differ in shape, for both the ground state and the excited state, but the total energies from the DMRG calculations are in general consistently lower. However, the DMRG energies are as accurate as corresponding coupled cluster energies at the equilibrium distance, but convergence to the full configuration-interaction limit is not achieved.

Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings.

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.