Gaussians for Electronic and Rovibrational Quantum Dynamics.

The assumptions underpinning the adiabatic Born-Oppenheimer (BO) approximation are broken for molecules interacting with attosecond laser pulses, which generate complicated coupled electronic-nuclear wave packets that generally will have components of electronic and dissociation continua as well as bound-state contributions. The conceptually most straightforward way to overcome this challenge is to treat the electronic and nuclear degrees of freedom on equal quantum-mechanical footing by not invoking the BO approximation at all. Explicitly correlated Gaussian (ECG) basis functions have proved successful for non-BO calculations of stationary molecular states and energies, reproducing rovibrational absorption spectra with very high accuracy. In this Article, we present a proof-of-principle study of the ability of fully flexible ECGs (FFECGs) to capture the intricate electronic and rovibrational dynamics generated by short, high-intensity laser pulses. By fitting linear combinations of FFECGs to accurate wave function histories obtained on a large real-space grid for a regularized 2D model of the hydrogen atom and for the 2D Morse potential, we demonstrate that FFECGs provide a very compact description of laser-driven electronic and rovibrational dynamics.

Magnetic optical rotation from real-time simulations in finite magnetic fields.

We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet's linearity remains valid up to roughly 10-20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao-Perdew-Staroverov-Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation.

S-Diagnostic─An a Posteriori Error Assessment for Single-Reference Coupled-Cluster Methods.

We propose a novel a posteriori error assessment for the single-reference coupled-cluster (SRCC) method called the S-diagnostic. We provide a derivation of the S-diagnostic that is rooted in the mathematical analysis of different SRCC variants. We numerically scrutinized the S-diagnostic, testing its performance for (1) geometry optimizations, (2) electronic correlation simulations of systems with varying numerical difficulty, and (3) the square-planar copper complexes [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+. Throughout the numerical investigations, the S-diagnostic is compared to other SRCC diagnostic procedures, that is, the T1, D1, max T2, and D2 diagnostics as well as different indices of multideterminantal and multireference character in coupled-cluster theory. Our numerical investigations show that the S-diagnostic outperforms the T1, D1, max T2 and D2 diagnostics and is comparable to the indices of multideterminantal and multireference character in coupled-cluster theory in their individual fields of applicability. The experiments investigating the performance of the S-diagnostic for geometry optimizations using SRCC reveal that the S-diagnostic correlates well with different error measures at a high level of statistical relevance. The experiments investigating the performance of the S-diagnostic for electronic correlation simulations show that the S-diagnostic correctly predicts strong multireference regimes. The S-diagnostic, moreover, correctly detects the successful SRCC computations for [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnostics in the past. This shows that the S-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations.

Cost-Efficient High-Resolution Linear Absorption Spectra through Extrapolating the Dipole Moment from Real-Time Time-Dependent Electronic-Structure Theory.

We present a novel function fitting method for approximating the propagation of the time-dependent electric dipole moment from real-time electronic structure calculations. Real-time calculations of the electronic absorption spectrum require discrete Fourier transforms of the electric dipole moment. The spectral resolution is determined by the total propagation time, i.e., the trajectory length of the dipole moment, causing a high computational cost. Our developed method uses function fitting on shorter trajectories of the dipole moment, achieving arbitrary spectral resolution through extrapolation. Numerical testing shows that the fitting method can reproduce high-resolution spectra by using short dipole trajectories. The method converges with as little as 100 a.u. dipole trajectories for some systems, though the difficulty converging increases with the spectral density. We also introduce an error estimate of the fit, reliably assessing its convergence and hence the quality of the approximated spectrum.

Adiabatic extraction of nonlinear optical properties from real-time time-dependent electronic-structure theory.

Real-time simulations of laser-driven electron dynamics contain information about molecular optical properties through all orders in response theory. These properties can be extracted by assuming convergence of the power series expansion of induced electric and magnetic multipole moments. However, the accuracy relative to analytical results from response theory quickly deteriorates for higher-order responses due to the presence of high-frequency oscillations in the induced multipole moment in the time domain. This problem has been ascribed to missing higher-order corrections. We here demonstrate that the deviations are caused by nonadiabatic effects arising from the finite-time ramping from zero to full strength of the external laser field. Three different approaches, two using a ramped wave and one using a pulsed wave, for extracting electrical properties from real-time time-dependent electronic-structure simulations are investigated. The standard linear ramp is compared to a quadratic ramp, which is found to yield highly accurate results for polarizabilities, and first and second hyperpolarizabilities, at roughly half the computational cost. Results for the third hyperpolarizability are presented along with a simple, computable measure of reliability.

Linear and Nonlinear Optical Properties from TDOMP2 Theory.

We present a derivation of real-time (RT) time-dependent orbital-optimized Møller-Plesset (TDOMP2) theory and its biorthogonal companion, time-dependent non-orthogonal OMP2 theory, starting from the time-dependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time-independent molecular electronic structure theory. We apply the TDOMP2 method to extract absorption spectra and frequency-dependent polarizabilities and first hyperpolarizabilities from RT simulations, comparing the results with those obtained from conventional time-dependent coupled-cluster singles and doubles (TDCCSD) simulations and from its second-order approximation, TDCC2. We also compare our results with those from CCSD and CC2 linear and quadratic response theories. Our results indicate that while TDOMP2 absorption spectra are of the same quality as TDCC2 spectra, including core excitations where optimized orbitals might be particularly important, frequency-dependent polarizabilities and hyperpolarizabilities from TDOMP2 simulations are significantly closer to TDCCSD results than those from TDCC2 simulations.

Interpretation of Coupled-Cluster Many-Electron Dynamics in Terms of Stationary States.

We demonstrate theoretically and numerically that laser-driven many-electron dynamics, as described by bivariational time-dependent coupled-cluster (CC) theory, may be analyzed in terms of stationary-state populations. Projectors heuristically defined from linear response theory and equation-of-motion CC theory are proposed for the calculation of stationary-state populations during interaction with laser pulses or other external forces, and conservation laws of the populations are discussed. Numerical tests of the proposed projectors, involving both linear and nonlinear optical processes for He and Be atoms and for LiH, CH+, and LiF molecules show that the laser-driven evolution of the stationary-state populations at the coupled-cluster singles-and-doubles (CCSD) level is very close to that obtained by full configuration interaction (FCI) theory, provided that all stationary states actively participating in the dynamics are sufficiently well approximated. When double-excited states are important for the dynamics, the quality of the CCSD results deteriorates. Observing that populations computed from the linear response projector may show spurious small-amplitude, high-frequency oscillations, the equation-of-motion projector emerges as the most promising approach to stationary-state populations.

Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems.

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

Numerical stability of time-dependent coupled-cluster methods for many-electron dynamics in intense laser pulses.

We investigate the numerical stability of time-dependent coupled-cluster theory for many-electron dynamics in intense laser pulses, comparing two coupled-cluster formulations with full configuration interaction theory. Our numerical experiments show that orbital-adaptive time-dependent coupled-cluster doubles (OATDCCD) theory offers significantly improved stability compared with the conventional Hartree-Fock-based time-dependent coupled-cluster singles-and-doubles (TDCCSD) formulation. The improved stability stems from greatly reduced oscillations in the doubles amplitudes, which, in turn, can be traced to the dynamic biorthonormal reference determinants of OATDCCD theory. As long as these are good approximations to the Brueckner determinant, OATDCCD theory is numerically stable. We propose the reference weight as a diagnostic quantity to identify situations where the TDCCSD and OATDCCD theories become unstable.

Symplectic integration and physical interpretation of time-dependent coupled-cluster theory.

The formulation of the time-dependent Schrödinger equation in terms of coupled-cluster theory is outlined, with emphasis on the bivariational framework and its classical Hamiltonian structure. An indefinite inner product is introduced, inducing physical interpretation of coupled-cluster states in the form of transition probabilities, autocorrelation functions, and explicitly real values for observables, solving interpretation issues which are present in time-dependent coupled-cluster theory and in ground-state calculations of molecular systems under the influence of external magnetic fields. The problem of the numerical integration of the equations of motion is considered, and a critical evaluation of the standard fourth-order Runge-Kutta scheme and the symplectic Gauss integrator of variable order are given, including several illustrative numerical experiments. While the Gauss integrator is stable even for laser pulses well above the perturbation limit, our experiments indicate that a system-dependent upper limit exists for the external field strengths. Above this limit, time-dependent coupled-cluster calculations become very challenging numerically, even in the full configuration interaction limit. The source of these numerical instabilities is shown to be rapid increases of the amplitudes as ultrashort high-intensity laser pulses pump the system out of the ground state into states that are virtually orthogonal to the static Hartree-Fock reference determinant.

Divide-Expand-Consolidate Second-Order Møller-Plesset Theory with Periodic Boundary Conditions.

We present a generalization of the divide-expand-consolidate (DEC) framework for local coupled-cluster calculations to periodic systems and test it at the second-order Møller-Plesset (MP2) level of theory. For simple model systems with periodicity in one, two, and three dimensions, comparisons with extrapolated molecular calculations and the local MP2 implementation in the Cryscor program show that the correlation energy errors of the extended DEC (X-DEC) algorithm can be controlled through a single parameter, the fragment optimization threshold. Two computational bottlenecks are identified: the size of the virtual orbital spaces and the number of pair fragments required to achieve a given accuracy of the correlation energy. For the latter, we propose an affordable algorithm based on cubic splines interpolation of a limited number of pair-fragment interaction energies to determine a pair cutoff distance in accordance with the specified fragment optimization threshold.

On Resolution-of-the-Identity Electron Repulsion Integral Approximations and Variational Stability.

The definiteness of the Mulliken and Dirac electron repulsion integral (ERI) matrices is examined for different classes of resolution-of-the-identity (RI) ERI approximations with particular focus on local fitting techniques. For global RI, robust local RI, and nonrobust local RI we discuss the definiteness of the approximated ERI matrices as well as the resulting bounds of Hartree, exchange, and total energies. Lower bounds of Hartree and exchange energy contributions are crucial as their absence may lead to variational instabilities, causing severe convergence problems or even convergence to a spurious state in self-consistent-field optimizations. While the global RI approximation guarantees lower bounds of Hartree and exchange energies, local RI approximations are generally unbounded. The robust local RI approximation guarantees a lower bound of the exchange energy but not of the Hartree energy. The nonrobust local RI approximation guarantees a lower bound of the Hartree energy but not of the exchange energy. These issues are demonstrated by sample calculations on carbon dioxide and benzene using the pair atomic RI approximation.

Redox Activity of Oxo-Bridged Iridium Dimers in an N,O-Donor Environment: Characterization of Remarkably Stable Ir(IV,V) Complexes.

Chemical and electrochemical oxidation or reduction of our recently reported Ir(IV,IV) mono-µ-oxo dimers results in the formation of fully characterized Ir(IV,V) and Ir(III,III) complexes. The Ir(IV,V) dimers are unprecedented and exhibit remarkable stability under ambient conditions. This stability and modest reduction potential of 0.99 V vs NHE is in part attributed to complete charge delocalization across both Ir centers. Trends in crystallographic bond lengths and angles shed light on the structural changes accompanying oxidation and reduction. The similarity of these mono-µ-oxo dimers to our Ir "blue solution" water-oxidation catalyst gives insight into potential reactive intermediates of this structurally elusive catalyst. Additionally, a highly reactive material, proposed to be a Ir(V,V) µ-oxo species, is formed on electrochemical oxidation of the Ir(IV,V) complex in organic solvents at 1.9 V vs NHE. Spectroelectrochemistry shows reversible conversion between the Ir(IV,V) and proposed Ir(V,V) species without any degradation, highlighting the exceptional oxidation resistance of the 2-(2-pyridinyl)-2-propanolate (pyalk) ligand and robustness of these dimers. The Ir(III,III), Ir(IV,IV) and Ir(IV,V) redox states have been computationally studied both with DFT and multiconfigurational calculations. The calculations support the stability of these complexes and provide further insight into their electronic structures.

Analytical State-Average Complete-Active-Space Self-Consistent Field Nonadiabatic Coupling Vectors: Implementation with Density-Fitted Two-Electron Integrals and Application to Conical Intersections.

Analytical state-average complete-active-space self-consistent field derivative (nonadiabatic) coupling vectors are implemented. Existing formulations are modified such that the implementation is compatible with Cholesky-based density fitting of two-electron integrals, which results in efficient calculations especially with large basis sets. Using analytical nonadiabatic coupling vectors, the optimization of conical intersections is implemented within the projected constrained optimization method. The standard description and characterization of conical intersections is reviewed and clarified, and a practical and unambiguous system for their classification and interpretation is put forward. These new tools are subsequently tested and benchmarked for 19 different conical intersections. The accuracy of the derivative coupling vectors is validated, and the information that can be drawn from the proposed characterization is discussed, demonstrating its usefulness.

Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method.

We compare the performance of three approximate methods for speeding up evaluation of the exchange contribution in Hartree-Fock and hybrid Kohn-Sham calculations: the chain-of-spheres algorithm (COSX; Neese , F. Chem. Phys. 2008 , 356 , 98 - 109 ), the pair-atomic resolution-of-identity method (PARI-K; Merlot , P. J. Comput. Chem. 2013 , 34 , 1486 - 1496 ), and the auxiliary density matrix method (ADMM; Guidon , M. J. Chem. Theory Comput. 2010 , 6 , 2348 - 2364 ). Both the efficiency relative to that of a conventional linear-scaling algorithm and the accuracy of total, atomization, and orbital energies are compared for a subset containing 25 of the 200 molecules in the Rx200 set using double-, triple-, and quadruple-ζ basis sets. The accuracy of relative energies is further compared for small alkane conformers (ACONF test set) and Diels-Alder reactions (DARC test set). Overall, we find that the COSX method provides good accuracy for orbital energies as well as total and relative energies, and the method delivers a satisfactory speedup. The PARI-K and in particular ADMM algorithms require further development and optimization to fully exploit their indisputable potential.

Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table.

In this report, we summarize and describe the recent unique updates and additions to the Molcas quantum chemistry program suite as contained in release version 8. These updates include natural and spin orbitals for studies of magnetic properties, local and linear scaling methods for the Douglas-Kroll-Hess transformation, the generalized active space concept in MCSCF methods, a combination of multiconfigurational wave functions with density functional theory in the MC-PDFT method, additional methods for computation of magnetic properties, methods for diabatization, analytical gradients of state average complete active space SCF in association with density fitting, methods for constrained fragment optimization, large-scale parallel multireference configuration interaction including analytic gradients via the interface to the Columbus package, and approximations of the CASPT2 method to be used for computations of large systems. In addition, the report includes the description of a computational machinery for nonlinear optical spectroscopy through an interface to the QM/MM package Cobramm. Further, a module to run molecular dynamics simulations is added, two surface hopping algorithms are included to enable nonadiabatic calculations, and the DQ method for diabatization is added. Finally, we report on the subject of improvements with respects to alternative file options and parallelization.

Analytical gradients of the state-average complete active space self-consistent field method with density fitting.

An efficient implementation of the state-averaged complete active space self-consistent field (SA-CASSCF) gradients employing density fitting (DF) is presented. The DF allows a reduction both in scaling and prefactors of the different steps involved. The performance of the algorithm is demonstrated on a set of molecules ranging up to an iron-Heme b complex which with its 79 atoms and 811 basis functions is to our knowledge the largest SA-CASSCF gradient computed. For smaller systems where the conventional code could still be used as a reference, both the linear response calculation and the gradient formation showed a clear timing reduction and the overall cost of a geometry optimization is typically reduced by more than one order of magnitude while the accuracy loss is negligible.

Orbital entanglement and CASSCF analysis of the Ru-NO bond in a Ruthenium nitrosyl complex.

Complete active space self-consistent field (CASSCF) wavefunctions and an orbital entanglement analysis obtained from a density-matrix renormalisation group (DMRG) calculation are used to understand the electronic structure, and, in particular, the Ru-NO bond of a Ru nitrosyl complex. Based on the configurations and orbital occupation numbers obtained for the CASSCF wavefunction and on the orbital entropy measurements evaluated for the DMRG wavefunction, we unravel electron correlation effects in the Ru coordination sphere of the complex. It is shown that Ru-NO π bonds show static and dynamic correlation, while other Ru-ligand bonds feature predominantly dynamic correlation. The presence of static correlation requires the use of multiconfigurational methods to describe the Ru-NO bond. Subsequently, the CASSCF wavefunction is analysed in terms of configuration state functions based on localised orbitals. The analysis of the wavefunctions in the electronic singlet ground state and the first triplet state provides a picture of the Ru-NO moiety beyond the standard representation based on formal oxidation states. A distinct description of the Ru and NO fragments is advocated. The electron configuration of Ru is an equally weighted superposition of Ru(II) and Ru(III) configurations, with the Ru(III) configuration originating from charge donation mostly from Cl ligands. However, and contrary to what is typically assumed, the electronic configuration of the NO ligand is best described as electroneutral.

Analytical gradients of complete active space self-consistent field energies using Cholesky decomposition: geometry optimization and spin-state energetics of a ruthenium nitrosyl complex.

We present a formulation of analytical energy gradients at the complete active space self-consistent field (CASSCF) level of theory employing density fitting (DF) techniques to enable efficient geometry optimizations of large systems. As an example, the ground and lowest triplet state geometries of a ruthenium nitrosyl complex are computed at the DF-CASSCF level of theory and compared with structures obtained from density functional theory (DFT) using the B3LYP, BP86, and M06L functionals. The average deviation of all bond lengths compared to the crystal structure is 0.042 Å at the DF-CASSCF level of theory, which is slightly larger but still comparable with the deviations obtained by the tested DFT functionals, e.g., 0.032 Å with M06L. Specifically, the root-mean-square deviation between the DF-CASSCF and best DFT coordinates, delivered by BP86, is only 0.08 Å for S0 and 0.11 Å for T1, indicating that the geometries are very similar. While keeping the mean energy gradient errors below 0.25%, the DF technique results in a 13-fold speedup compared to the conventional CASSCF geometry optimization algorithm. Additionally, we assess the singlet-triplet energy vertical and adiabatic differences with multiconfigurational second-order perturbation theory (CASPT2) using the DF-CASSCF and DFT optimized geometries. It is found that the vertical CASPT2 energies are relatively similar regardless of the geometry employed whereas the adiabatic singlet-triplet gaps are more sensitive to the chosen triplet geometry.