arXiv subjects

Markus Reiher

Publications and source records attributed to Markus Reiher.

At least 109 records · Page 6Linked to original sources

arXiv2018

Comprehensive Analysis of the Neglect of Diatomic Differential Overlap Approximation

Many modern semiempirical molecular orbital models are built on the neglect of diatomic differential overlap (NDDO) approximation. An in-depth understanding of this approximation is therefore indispensable to rationalize the success of these semiempirical molecular orbital models and to develop further improvements on them. The NDDO approximation provides a recipe to approximate electron-electron repulsion integrals (ERIs) in a symmetrically orthogonalized basis based on a far smaller number of ERIs in a locally orthogonalized basis. We first analyze the NDDO approximation by comparing ERIs in both bases for a selection of molecules and for a selection of basis sets. We find that the errors in Hartree-Fock and second-order Moller-Plesset perturbation theory energies grow roughly linearly with the number of basis functions. We then examine different approaches to correct for the errors caused by the NDDO approximation and propose a strategy to directly correct for them in the two-electron matrices that enter the Fock operator.

physics.chem-ph↗

arXiv2018

Quantum System Partitioning at the Single-Particle Level

We discuss the partitioning of a quantum system by subsystem separation through unitary block-diagonalization (SSUB) applied to a Fock operator. Our separation can be formulated in a very general way. It can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system-environment decoupling approach. Moreover, for SSUB there exists no restriction with respect to the number of subsystems that are generated - in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. The fact that a Fock operator depends on its eigenvectors poses challenges to all system-environment decoupling approaches and is discussed in terms of the SSUB framework. Apart from improved conceptual understanding, these relations bring about technical advances as developments in different fields can immediately cross-fertilize one another. As an important example we discuss the atomic decomposition of the unitary block-diagonalization matrix in X2C-type approaches that can inspire approaches for the efficient partitioning of large total systems based on SSUB.

physics.chem-ph↗

arXiv2018

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations

Numerical projection methods are elaborated for the calculation of eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with selected rotational and parity quantum numbers employing shifted explicitly correlated Gaussian functions, which are, in general, not eigenfunctions of the total angular momentum and parity operators. The increased computational cost of numerically projecting the basis functions onto the irreducible representations of the three dimensional rotation-inversion group is the price to pay for the increased flexibility of the basis functions. This increased flexibility allowed us to achieve a substantial improvement for the variational upper bound to the Pauli-allowed ground-state energy of the H$_3^+=\{$p$^+,$p$^+,$p$^+,$e$^-,$e$^-\}$ molecular ion treated as an explicit five-particle system. We compare our pre-Born-Oppenheimer result for this molecular ion with rovibrational results including non-adiabatic corrections.

physics.chem-ph↗

arXiv2018

Generalized Pauli constraints in small atoms

The natural occupation numbers of fermionic systems are subject to non-trivial constraints, which include and extend the original Pauli principle. A recent mathematical breakthrough has clarified their mathematical structure and has opened up the possibility of a systematic analysis. Early investigations have found evidence that these constraints are exactly saturated in several physically relevant systems, e.g., in a certain electronic state of the Beryllium atom. It has been suggested that in such cases, the constraints, rather than the details of the Hamiltonian, dictate the system's qualitative behaviour. Here, we revisit this question with state-of-the-art numerical methods for small atoms. We find that the constraints are, in fact, not exactly saturated, but that they lie much closer to the surface defined by the constraints than the geometry of the problem would suggest. While the results seem incompatible with the statement that the generalized Pauli constraints drive the behaviour of these systems, they suggest that the qualitatively correct wave-function expansions can in some systems already be obtained on the basis of a limited number of Slater determinants, which is in line with numerical evidence from quantum chemistry.

quant-ph↗

arXiv2018

Calculation of Ligand Dissociation Energies in Large Transition-Metal Complexes

Despite the importance of ligand dissociation energies for computational chemistry, obtaining accurate ab initio reference data is difficult and density-functional methods of uncertain reliability are chosen for feasibility reasons. Here, we consider advanced coupled-cluster and multi-configurational approaches to reinvestigate our WCCR10 set of ten gas-phase ligand dissociation energies. We assess the potential multi-configurational character of all molecules involved in these reactions in order to determine where single-reference coupled-cluster approaches can be applied. For some reactions of the WCCR10 set, large deviations from density-functional results including semiclassical dispersion corrections from experimental reference data had been observed. We tackle the issue (i) by comparing to ab initio data that comprise dispersion effects on a rigorous first-principles footing and (ii) by a comparison of density-functional approaches that model dispersion interactions in various ways. Only for two reactions, species exhibiting nonnegligible static electron correlation were identified, and hence, we may choose standard single-reference coupled-cluster approaches to compare with density-functional methods for the other eight reactions. For WCCR10, the Minnesota M06-L functional yielded the smallest mean absolute deviation without additional dispersion corrections in comparison to the coupled-cluster results and the PBE0-D3 functional produced the overall smallest mean absolute deviation. The agreement of density-functional results with coupled-cluster data increases significantly upon inclusion of any type of dispersion correction. It is important to emphasize that different density-functional schemes available for this purpose perform equally well. The coupled-cluster dissociation energies, however, deviate significantly from experimental results.

physics.chem-ph↗

arXiv2018

Minimum Energy Paths and Transition States by Curve Optimization

Transition states and minimum energy paths are essential to understand and predict chemical reactivity. Double-ended methods represent a standard approach for their determination. We introduce a new double-ended method that optimizes reaction paths described by curves. Unlike other methods, our approach optimizes the curve parameters rather than distinct structures along the path. With molecular paths represented as continuous curves, the optimization can benefit from the advantages of an integral-based formulation. We call this approach ReaDuct and demonstrate its applicability for molecular paths parametrized by B-spline curves.

physics.chem-ph↗

arXiv2018

Statistical Analysis of Semiclassical Dispersion Corrections

Semiclassical dispersion corrections developed by Grimme and coworkers have become indispensable in applications of Kohn-Sham density functional theory. We present an in-depth assessment of the fit parameters present in semiclassical (D3-type) dispersion corrections by means of a statistically rigorous analysis. We find that the choice of the cost function generally has a small effect on the empirical parameters of D3-type dispersion corrections with respect to the reference set under consideration. Only in a few cases, the choice of cost function has a surprisingly large effect on the total dispersion energies. In particular, the weighting scheme in the cost function can significantly affect the reliability of predictions. In order to obtain unbiased (data-independent) uncertainty estimates for both the empirical fit parameters and the corresponding predictions, we carried out a nonparametric bootstrap analysis. The mean prediction uncertainty obtained by bootstrapping is not much larger than previously reported error measures. On the basis of a jackknife analysis, we find that the original reference set is slightly skewed, but our results also suggest that this feature hardly affects the prediction of dispersion energies. However, a rigorous analysis of error accumulation arising from different parameterizations reveals that error cancellation does not necessarily occur, leading to a monotonically increasing deviation in the dispersion energy with increasing molecule size. We discuss this issue in detail at the prominent example of the C60 buckycatcher. We find deviations between individual parameterizations of several tens of kcal/mol in some cases. Hence, in combination with any calculation of dispersion energies, we recommend to always determine the associated uncertainties for which we will provide a software tool.

physics.chem-ph↗

arXiv2018

Generalized elimination of the global translation from explicitly correlated Gaussian functions

This paper presents the multi-channel generalization of the center-of-mass kinetic energy elimination approach [Mol. Phys., 111 2086 (2013)] when the Schrödinger equation is solved variationally with explicitly correlated Gaussian functions. The approach has immediate relevance in many-particle systems which are handled without the Born--Oppenheimer approximation and can be employed also for Dirac-type Hamiltonians. The practical realization and numerical properties of solving the Schrödinger equation in laboratory-frame Cartesian coordinates are demonstrated for the ground rovibronic state of the H$_2^+=\lbrace\text{p}^+,\text{p}^+,\text{e}^+\rbrace$ ion and the H$_2=\lbrace\text{p}^+,\text{p}^+,\text{e}^+,\text{e}^+\rbrace$ molecule.

physics.comp-ph↗

arXiv2018

Integrated Reaction Path Processing from Sampled Structure Sequences

Sampled structure sequences obtained, for instance, from real-time reactivity explorations or first-principles molecular dynamics simulations contain valuable information about chemical reactivity. Eventually, such sequences allow for the construction of reaction networks that are required for the kinetic analysis of chemical systems. For this purpose, however, the sampled information must be processed to obtain stable chemical structures and associated transition states. The manual extraction of valuable information from such reaction paths is straightforward but unfeasible for large and complex reaction networks. For real-time quantum chemistry, this implies automatization of the extraction and relaxation process while maintaining immersion in the virtual chemical environment. Here, we describe an efficient path processing scheme for the on-the-fly construction of an exploration network by approximating the explored paths as continuous basis-spline curves.

physics.chem-ph↗

arXiv2017

Context-Driven Exploration of Complex Chemical Reaction Networks

The construction of a reaction network containing all relevant intermediates and elementary reactions is necessary for the accurate description of chemical processes. In the case of a complex chemical reaction (involving, for instance, many reactants or highly reactive species), the size of such network may grow rapidly. Here, we present a computational protocol that constructs such reaction networks in a fully automated fashion steered in an intuitive, graph-based fashion through a single graphical user interface. Starting from a set of initial reagents new intermediates are explored through intra- and intermolecular reactions of already explored intermediates or new reactants presented to the network. This is done by assembling reactive complexes based on heuristic rules derived from conceptual electronic-structure theory and exploring the corresponding approximate reaction path. A subsequent path refinement leads to a minimum-energy path which connects the new intermediate to the existing ones to form a connected reaction network. Tree traversal algorithms are then employed to detect reaction channels and catalytic cycles. We apply our protocol to the formose reaction to study different pathways of sugar formation and to rationalize its autocatalytic nature.

physics.chem-ph↗

arXiv2017

Self-Adaptive Tensor Network States with Multi-Site Correlators

We introduce the concept of self-adaptive tensor network states (SATNS) based on multi-site correlators. The SATNS ansatz gradually extends its variational space incorporating the most important next-order correlators into the ansatz for the wave function. The selection of these correlators is guided by entanglement-entropy measures from quantum information theory. By sequentially introducing variational parameters and adjusting them to the system under study, the SATNS ansatz achieves to keep their number significantly smaller than the total number of full-configuration interaction parameters. The SATNS ansatz is studied for manganocene in its lowest-energy sextet and doublet states, the latter of which is known to be difficult to describe. It is shown that the SATNS parametrization solves the convergence issues found for previous correlator-based tensor network states.

physics.chem-ph↗

arXiv2017

Vibrational Density Matrix Renormalization Group

Variational approaches for the calculation of vibrational wave functions and energies are a natural route to obtain highly accurate results with controllable errors. However, the unfavorable scaling and the resulting high computational cost of standard variational approaches limit their application to small molecules with only few vibrational modes. Here, we demonstrate how the density matrix renormalization group (DMRG) can be exploited to optimize vibrational wave functions (vDMRG) expressed as matrix product states. We study the convergence of these calculations with respect to the size of the local basis of each mode, the number of renormalized block states, and the number of DMRG sweeps required. We demonstrate the high accuracy achieved by vDMRG for small molecules that were intensively studied in the literature. We then proceed to show that the complete fingerprint region of the sarcosyn-glycin dipeptide can be calculated with vDMRG.

physics.chem-ph↗

arXiv2017

Reliable estimation of prediction uncertainty for physico-chemical property models

The predictions of parameteric property models and their uncertainties are sensitive to systematic errors such as inconsistent reference data, parametric model assumptions, or inadequate computational methods. Here, we discuss the calibration of property models in the light of bootstrapping, a sampling method akin to Bayesian inference that can be employed for identifying systematic errors and for reliable estimation of the prediction uncertainty. We apply bootstrapping to assess a linear property model linking the 57Fe Moessbauer isomer shift to the contact electron density at the iron nucleus for a diverse set of 44 molecular iron compounds. The contact electron density is calculated with twelve density functionals across Jacob's ladder (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS, B3LYP, B3PW91, PBE0, M06, TPSSh). We provide systematic-error diagnostics and reliable, locally resolved uncertainties for isomer-shift predictions. Pure and hybrid density functionals yield average prediction uncertainties of 0.06-0.08 mm/s and 0.04-0.05 mm/s, respectively, the latter being close to the average experimental uncertainty of 0.02 mm/s. Furthermore, we show that both model parameters and prediction uncertainty depend significantly on the composition and number of reference data points. Accordingly, we suggest that rankings of density functionals based on performance measures (e.g., the coefficient of correlation, r2, or the root-mean-square error, RMSE) should not be inferred from a single data set. This study presents the first statistically rigorous calibration analysis for theoretical Moessbauer spectroscopy, which is of general applicability for physico-chemical property models and not restricted to isomer-shift predictions. We provide the statistically meaningful reference data set MIS39 and a new calibration of the isomer shift based on the PBE0 functional.

physics.chem-ph↗

arXiv2017

Second-Order Self-Consistent-Field Density-Matrix Renormalization Group

We present a matrix-product state (MPS)-based quadratically convergent density-matrix renormalization group self-consistent-field (DMRG-SCF) approach. Following a proposal by Werner and Knowles (JCP 82, 5053, (1985)), our DMRG-SCF algorithm is based on a direct minimization of an energy expression which is correct to second-order with respect to changes in the molecular orbital basis. We exploit a simultaneous optimization of the MPS wave function and molecular orbitals in order to achieve quadratic convergence. In contrast to previously reported (augmented Hessian) Newton-Raphson and super-configuration-interaction algorithms for DMRG-SCF, energy convergence beyond a quadratic scaling is possible in our ansatz. Discarding the set of redundant active-active orbital rotations, the DMRG-SCF energy converges typically within two to four cycles of the self-consistent procedure

physics.chem-ph↗

arXiv2017

Error Assessment of Computational Models in Chemistry

Computational models in chemistry rely on a number of approximations. The effect of such approximations on observables derived from them is often unpredictable. Therefore, it is challenging to quantify the uncertainty of a computational result, which, however, is necessary to assess the suitability of a computational model. Common performance statistics such as the mean absolute error are prone to failure as they do not distinguish the explainable (systematic) part of the errors from their unexplainable (random) part. In this paper, we discuss problems and solutions for performance assessment of computational models based on several examples from the quantum chemistry literature. For this purpose, we elucidate the different sources of uncertainty, the elimination of systematic errors, and the combination of individual uncertainty components to the uncertainty of a prediction.

physics.chem-ph↗

arXiv2017

Automated Identification of Relevant Frontier Orbitals for Chemical Compounds and Processes

Quantum-chemical multi-configurational methods are required for a proper description of static electron correlation, a phenomenon inherent to the electronic structure of molecules with multiple (near-)degenerate frontier orbitals. Here, we review how a property of these frontier orbitals, namely the entanglement entropy is related to static electron correlation. A subset of orbitals, the so-called active orbital space, is an essential ingredient for all multi-configurational methods. We proposed an automated selection of this active orbital space, that would otherwise be a tedious and error prone manual procedure, based on entanglement measures. Here, we extend this scheme to demonstrate its capability for the selection of consistent active spaces for several excited states and along reaction coordinates.

physics.chem-ph↗

arXiv2016

Steering Orbital Optimization out of Local Minima and Saddle Points Toward Lower Energy

The general procedure underlying Hartree-Fock and Kohn-Sham density functional theory calculations consists in optimizing orbitals for a self-consistent solution of the Roothaan-Hall equations in an iterative process. It is often ignored that multiple self-consistent solutions can exist, several of which may correspond to minima of the energy functional. In addition to the difficulty sometimes encountered to converge the calculation to a self-consistent solution, one must ensure that the correct self-consistent solution was found, typically the one with the lowest electronic energy. Convergence to an unwanted solution is in general not trivial to detect and will deliver incorrect energy and molecular properties, and accordingly a misleading description of chemical reactivity. Wrong conclusions based on incorrect self-consistent field convergence are particularly cumbersome in automated calculations met in high-throughput virtual screening, structure optimizations, ab initio molecular dynamics, and in real-time explorations of chemical reactivity, where the vast amount of data can hardly be manually inspected. Here, we introduce a fast and automated approach to detect and cure incorrect orbital convergence, which is especially suited for electronic structure calculations on sequences of molecular structures. Our approach consists of a randomized perturbation of the converged electron density (matrix) intended to push orbital convergence to solutions that correspond to another stationary point (of potentially lower electronic energy) in the variational parameter space of an electronic wave function approximation.

physics.chem-ph↗

arXiv2016

Measuring Multi-Configurational Character by Orbital Entanglement

One of the most critical tasks at the very beginning of a quantum chemical investigation is the choice of either a multi- or single-configurational method. Naturally, many proposals exist to define a suitable diagnostic of the multi-configurational character for various types of wave functions in order to assist this crucial decision. Here, we present a new orbital-entanglement based multi-configurational diagnostic termed $Z_{s(1)}$. The correspondence of orbital entanglement and static (or nondynamic) electron correlation permits the definition of such a diagnostic. We chose our diagnostic to meet important requirements such as well-defined limits for pure single-configurational and multi-configurational wave functions. The $Z_{s(1)}$ diagnostic can be evaluated from a partially converged, but qualitatively correct, and therefore inexpensive density matrix renormalization group wave function as in our recently presented automated active orbital selection protocol. Its robustness and the fact that it can be evaluated at low cost make this diagnostic a practical tool for routine applications.

physics.chem-ph↗