A black-box, general purpose quadratic self-consistent field code with and without Cholesky decomposition of the two-electron integrals

Finite-field Cholesky decomposed coupled-cluster techniques (ff-CD-CC): theory and application to pressure broadening of Mg by a He atmosphere and a strong magnetic field

For the interpretation of spectra of magnetic stellar objects such as magnetic white dwarfs (WDs), highly accurate quantum chemical predictions for atoms and molecules in finite magnetic field are required. Especially the accurate description of electronically excited states and their properties requires established methods such as those from coupled-cluster (CC) theory. However, respective calculations are computationally challenging even for medium-sized systems. Cholesky decomposition (CD) techniques may be used to alleviate memory bottlenecks. In finite magnetic field computations, the latter are increased due to the reduction of permutational symmetry within the electron-repulsion-integrals (ERIs) as well as the need for complex-valued data types. CD enables a memory-efficient, approximate description of the ERIs with rigorous error control and thus the treatment of larger systems at the CC level becomes feasible. In order to treat molecules in a finite magnetic field, we present in this work the working equations of the left and right-hand side equations for finite field (ff)-EOM-CD-CCSD for various EOM flavours as well as for the approximate ff-EOM-CD-CC2 method. The methods are applied to the study of the modification of the spectral lines of a magnesium transition by a helium atmosphere that can be found on magnetic WD stars.

Toward Polaritonic Molecular Orbitals for Large Molecular Systems

A comprehensive understanding of electron-photon correlation is essential for describing the reshaping of molecular orbitals in quantum electrodynamics (QED) environments. The strong coupling QED Hartree-Fock (SC-QED-HF) theory tackles these aspects by providing consistent molecular orbitals in the strong coupling regime. The previous implementation, however, has significant convergence issues that limit the applicability. In this work, we introduce two second-order algorithms that significantly reduce the computational requirements, thereby enhancing the modeling of large molecular systems in QED environments. Furthermore, the implementation will enable the development of correlated methods based on a reliable molecular orbital framework as well as multi-level methodologies able to model the inclusion of solvent effects in this kind of complex systems.

Cholesky Decomposition in Spin-Free Dirac-Coulomb Coupled-Cluster Calculations

We present an implementation for the use of Cholesky decomposition (CD) of two-electron integrals within the spin-free Dirac-Coulomb (SFDC) scheme that enables to perform high-accuracy coupled-cluster (CC) calculations at costs almost comparable to those of their nonrelativistic counterparts. While for nonrelativistic CC calculations, atomic-orbital (AO)-based algorithms, due to their significantly reduced disk-space requirements, are the key to efficient large-scale computations, such algorithms are less advantageous in the SFDC case due to their increased computational cost in that case. Here, molecular-orbital (MO)-based algorithms exploiting the CD of the two-electron integrals allow us to reduce disk-space requirements and lead to computational cost in the CC step that is more or less the same as in the nonrelativistic case. The only remaining overhead in a CD-SFDC-CC calculation is due to the need to compute additional two-electron integrals, the somewhat higher cost of the Hartree-Fock calculation in the SFDC case, and additional cost in the transformation of the Cholesky vectors from the AO to the MO representation. However, these additional costs typically amount to less than 5-15% of the total wall time and are thus acceptable. We illustrate the efficiency of our CD scheme for SFDC-CC calculations on a series of illustrative calculations for the X(CO)4 molecules with X = Ni, Pd, Pt.

Geometric Optimization of Restricted-Open and Complete Active Space Self-Consistent Field Wave Functions

We explore Riemannian optimization methods for Restricted-Open-shell Hartree-Fock (ROHF) and Complete Active Space Self-Consistent Field (CASSCF) methods. After showing that ROHF and CASSCF can be reformulated as optimization problems on so-called "flag manifolds", we review Riemannian optimization basics and their application to these specific problems. We compare these methods to traditional ones and find robust convergence properties without fine-tuning of numerical parameters. Our study suggests that Riemannian optimization is a valuable addition to orbital optimization for ROHF and CASSCF, warranting further investigation.

Theoretical prediction of closed-shell paramagnetism for scandium and yttrium hydride

Following chemical intuition, one would expect that all closed-shell molecules are diamagnetic. However, it is known that this is not the case for some second-row hydrides with low-lying unoccupied π orbitals due to an unquenching of the total angular momentum in the presence of an external magnetic field. In this article, the transition-metal hydrides ScH and YH are investigated, assuming a similar unquenching effect involving low-lying unoccupied π and δ orbitals formed from the metal d orbitals rather than the p orbitals. We are comparing results obtained with various quantum-chemical methods (HF, CCSD, CCSD(T), CCSDT) and basis sets. The obtained positive values for the magnetizabilities clearly indicate paramagnetic behavior. Vibrational effects on the magnetizability tensor are also considered, but these effects are small and do not change the overall conclusion that both ScH and YH are further examples for closed-shell paramagnetism.

Economical quasi-Newton unitary optimization of electronic orbitals

We present an efficient quasi-Newton orbital solver optimized to reduce the number of gradient evaluations and other computational steps of comparable cost. The solver optimizes orthogonal orbitals by sequences of unitary rotations generated by the (preconditioned) limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm equipped with trust-region step restriction. The low-rank structure of the L-BFGS inverse Hessian is exploited when solving the trust-region problem. The efficiency of the proposed "Quasi-Newton Unitary Optimization with Trust-Region" (QUOTR) solver is compared to that of the standard Roothaan-Hall approach accelerated by the Direct Inversion of Iterative Subspace (DIIS), and other exact and approximate Newton solvers for mean-field (Hartree-Fock and Kohn-Sham) problems.

Heptazine, Cyclazine, and Related Compounds: Chemically-Accurate Estimates of the Inverted Singlet-Triplet Gap

Molecules that violate Hund's rule and exhibit an inverted gap between the lowest singlet S1 and triplet T1 excited states have attracted considerable attention due to their potential applications in optoelectronics. Among these molecules, the triangular-shaped heptazine, and its derivatives, have been in the limelight. However, conflicting reports have arisen regarding the relative energies of S1 and T1. Here, we employ highly accurate levels of theory, such as CC3, to not only resolve the debate concerning the sign but also quantify the magnitude of the S1-T1 gap. We also determined the 0-0 energies to evaluate the significance of the vertical approximation. In addition, we compute reference S1-T1 gaps for a series of 10 related molecules. This enables us to benchmark lower-order methods for future applications in larger systems within the same family of compounds. This contribution can serve as a foundation for the design of triangular-shaped molecules with enhanced photophysical properties.

A trust-region augmented Hessian implementation for state-specific and state-averaged CASSCF wave functions

In this work, we present a one-step second-order converger for state-specific (SS) and state-averaged (SA) complete active space self-consistent field (CASSCF) wave functions. Robust convergence is achieved through step restrictions using a trust-region augmented Hessian (TRAH) algorithm. To avoid numerical instabilities, an exponential parameterization of variational configuration parameters is employed, which works with a nonredundant orthogonal complement basis. This is a common approach for SS-CASSCF and is extended to SA-CASSCF wave functions in this work. Our implementation is integral direct and based on intermediates that are formulated in either the sparse atomic-orbital or small active molecular-orbital basis. Thus, it benefits from a combination with efficient integral decomposition techniques, such as the resolution-of-the-identity or the chain-of-spheres for exchange approximations. This facilitates calculations on large molecules, such as a Ni(II) complex with 231 atoms and 5154 basis functions. The runtime performance of TRAH-CASSCF is competitive with the other state-of-the-art implementations of approximate and full second-order algorithms. In comparison with a sophisticated first-order converger, TRAH-CASSCF calculations usually take more iterations to reach convergence and, thus, have longer runtimes. However, TRAH-CASSCF calculations still converge reliably to a true minimum even if the first-order algorithm fails.