arXiv subjects

Stella Stopkowicz

Publications and source records attributed to Stella Stopkowicz.

At least 19 recordsLinked to original sources

arXiv2026

Use of Cholesky decomposition in the CFOUR program package

An overview is given about our efforts to speed up high-level computations of molec- ular energies and properties using Cholesky decomposition (CD). We describe the cor- responding developments in the CFOUR quantum-chemical package with a focus on (a) the use of CD in complete-active space self-consistent-field (CASSCF) and coupled- cluster (CC) computations, (b) the implementation of analytic CC gradients using CD, (c) the efficient calculation of magnetic properties (NMR shieldings and magnetizabili- ties) using CD when carried out with explicit magnetic-field dependent basis functions, i.e., the so-called gauge-including atomic orbitals (GIAOs), (d) the efficient computa- tion of CASSCF response properties using CD, (e) the use of CD in relativistic two- and four-component computations, as well as (f) computations for molecules in finite magnetic fields using GIAOs together with CD. An outlook on future developments concerning the CFOUR package is given.

physics.chem-ph↗

arXiv2026

Exploitation of complex Abelian point groups in quantum-chemical calculations

Quantum-chemical calculations often make use of point-group theory to exploit molecular symmetry, resulting in a reduction of the computational cost and in insights into the electronic structure. This exploitation is often limited to subgroups of $D_{2h}$ which are Abelian with real characters. Here, we extend the symmetry exploitation to Abelian point groups with complex characters. Such point groups are often encountered in calculations that involve finite magnetic fields, though their occurrence is not limited to these cases alone. We present the evaluation of integrals over symmetry-adapted orbitals using the double-coset decomposition, as well as the use of these symmetries in the contractions needed within post Hartree Fock calculations in the context of block tensors. Efficiency gains are discussed for four simple hydrocarbons that exhibit a complex Abelian point group in the presence of a magnetic field.

physics.chem-ph↗

arXiv2025

Polaritonic Coupled Cluster Theory for Unpolarized Cavities Exploiting Point Group Symmetry

We introduce a generalization of the quantum electrodynamic coupled cluster (QED-CC)wave function ansatz, to describe the strongly coupled light-matter system in an unpolarized optical Fabry-P\'erot cavity. This is achieved by explicitly treating two cavity modes in our calculation with perpendicular polarizations and demonstrate that this ansatz preserves the symmetry of an unpolarized cavity. Furthermore, exploiting point-group symmetry enables the assignment of polaritonic excited states as well as their targeted calculation. Using our implementation, the aromatic species benzene, fluorobenzene and azulene are investigated. We demonstrate that molecules in unpolarized cavities have a complicated excited-state landscapes with a plethora of avoided-crossings. We compare the results for a cavity with a single polarization to those of an unpolarized cavity described by two perpendicular polarization vectors using the excited states of the H$_2$ molecule as an example.

quant-ph↗

arXiv2025

Unitary Coupled-Cluster theory for the treatment of molecules in strong magnetic fields

In Coupled-Cluster (CC) theory, unphysical complex energies may arise in the presence of strong magnetic fields, near conical intersections, or in systems exhibiting complex Abelian point group symmetries. This issue originates from the non-Hermitian nature of the CC energy expression. A promising solution is provided by unitary Coupled-Cluster (UCC) theory, which retains the advantages of an exponential parameterization while ensuring real-valued energy eigenvalues. In this work, we present an implementation of finite-field second-order (ff-UCC2) and third-order (ff-UCC3) UCC theory. We assess the performance of these truncation levels in comparison to conventional finite-field CC methods, using the methylidyne ion, water, and boric acid.

physics.chem-ph↗

arXiv2025

Equation-of-motion coupled-cluster variants in combination with perturbative triples corrections in strong magnetic fields

In this paper, we report on the implementation of the EOM spin-flip (SF), ionization-potential (IP) and electron-affinity (EA) coupled cluster singles doubles (CCSD) methods for atoms and molecules in strong magnetic fields for energies as well as one-electron properties. Moreover, non-perturbative triples corrections using the EOM-CCSD(T)(a)* scheme were implemented in the finite-field framework for the EE, SF, IP, and EA variants. These developments allow the access to a large variety of electronic states as well as the investigation of IPs and EAs in a strong magnetic field. The latter two indicate the relative stability of the different oxidation states of elements. The increased flexibility to target challenging electronic states and the access to the electronic states of the anion and cation are important for the assignment of spectra from strongly magnetic White Dwarf (WD) stars. Here, we investigate the development of the IPs and EAs in the presence of a magnetic field for the elements of the first and second row of the periodic table. In addition, we study the development of the electronic structure of Na, Mg, and Ca that aided in the assignment of metal lines in a magnetic WD. Lastly, we investigate the electronic excitations of CH in different magnetic-field orientations and strengths, a molecule that has been found in the atmospheres of WD stars.

physics.chem-ph↗

arXiv2024

Cholesky Decomposition and the Second-Derivative Two-Electron Integrals Required for the Computation of Magnetizabilities using Gauge-Including Atomic Orbitals

The computation of magnetizability tensors using gauge-including atomic orbitals is discussed in the context of Cholesky decomposition for the two-electron repulsion integrals with a focus on the involved doubly differentiated integrals. Three schemes for their handling are suggested: the first exploits the DF aspect of Cholesky decomposition, the second uses expressions obtained by differentiating the CD expression for the unperturbed two electron integrals, while the third addresses the issue that the first two schemes are not able to represent the doubly differentiated integrals with arbitrary accuracy. This scheme uses a separate Cholesky decomposition for the cross terms in the doubly differentiated two-electron integrals. Test calculations reveal that all three schemes are able to represent the integrals with similar accuracy and yield indistinguishable results for the values of the computed magnetizability tensor elements. Thus, we recommend our first scheme which has the lowest computational cost for routine computations. The applicability of our CD schemes is further shown in large-scale Hartree-Fock calculations of the magnetizability tensor of coronene (C24H12) with a doubly polarized triple-zeta basis consisting of 684 basis functions.

physics.chem-ph↗

arXiv2024

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-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 (ff), we present in this work the working equations of the left and right-hand side equations for 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 in magnetic WD stars.

physics.chem-ph↗

arXiv2024

Diagrams and symmetry in polaritonic coupled cluster theory

We present a diagrammatic notation to derive the quantum-electrodynamic coupled cluster (QED-CC) equations needed for the description for polaritonic ground and excited states. Our presented notation is a generalization of the existing diagrammatic notation of standard electronic coupled-cluster theory. Further, we discuss point-group symmetry inside a cavity and its exploitation within a QED-CC code for efficiency and the targeted treatment of excited states.

physics.chem-ph↗

arXiv2023

The approximate Coupled-Cluster methods CC2 and CC3 in a finite magnetic field

In this paper, we report on the implementation of CC2 and CC3 in the context of molecules in finite magnetic fields. The methods are applied to the investigation of atoms and molecules through spectroscopic predictions and geometry optimizations for the study of the atmospheres of highly-magnetized White Dwarfs (WDs). We show that ground-state finite-field (ff) CC2 is a reasonable alternative to CCSD for energies and, in particular for geometrical properties. For excited states ff-CC2 is shown to perform well for states with predominant single-excitation character. Yet, for cases in which the excited-state wavefunction has double-excitation character with respect to the reference, ff-CC2 can easily Ff-CC3, however, is shown to reproduce the CCSDT behaviour very well and enables the treatment of larger systems at a high accuracy.

physics.chem-ph↗

arXiv2023

A Cholesky decomposition-based implementation of relativistic two-component coupled-cluster methods for medium-sized molecules

A Cholesky decomposition (CD)-based implementation of relativistic two-component coupled-cluster (CC) and equation-of-motion CC (EOM-CC) methods using an exact two-component Hamiltonian augmented with atomic-mean-field spin-orbit integrals (the X2CAMF scheme) is reported. The present CD-based implementation of X2CAMF-CC and EOM-CC methods employs atomic-orbital-based algorithms to avoid the construction of two-electron integrals and intermediates involving three and four virtual indices. Our CD-based implementation extends the applicability of X2CAMF-CC and EOM-CC methods to medium-sized molecules with the possibility to correlate around 1000 spinors. Benchmark calculations for uranium-containing small molecules have been performed to assess the dependence of the CC results on the Cholesky threshold. A Cholesky threshold of $10^{-4}$ is shown to be sufficient to maintain chemical accuracy. Example calculations to illustrate the capability of the CD-based relativistic CC methods are reported for the bond-dissociation energy of the uranium hexafluoride molecule, UF$_6$, with up to quadruple-zeta basis sets, and the lowest excitation energy in solvated uranyl ion [UO$_2^{2+}$(H$_2$O)$_{12}$].

physics.chem-ph↗

arXiv2023

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.

physics.chem-ph↗

arXiv2023

A DZ white dwarf with a 30 MG magnetic field

Magnetic white dwarfs with field strengths below 10 MG are easy to recognise since the Zeeman splitting of spectral lines appears proportional to the magnetic field strength. For fields $\geq 100$ MG, however, transition wavelengths become chaotic, requiring quantum-chemical predictions of wavelengths and oscillator strengths with a non-perturbative treatment of the magnetic field. While highly accurate calculations have previously been performed for hydrogen and helium, the variational techniques employed become computationally intractable for systems with more than three to four electrons. Modern computational techniques, such as finite-field coupled-cluster theory, allow the calculation of many-electron systems in arbitrarily strong magnetic fields. Because around 25 percent of white dwarfs have metal lines in their spectra, and some of those are also magnetic, the possibility arises for some metals to be observed in very strong magnetic fields, resulting in unrecognisable spectra. We have identified SDSSJ114333.48+661531.83 as a magnetic DZ white dwarf, with a spectrum exhibiting many unusually shaped lines at unknown wavelengths. Using atomic data calculated from computational finite-field coupled-cluster methods, we have identified some of these lines arising from Na, Mg, and Ca. Surprisingly, we find a relatively low field strength of 30 MG, where the large number of overlapping lines from different elements make the spectrum challenging to interpret at a much lower field strength than for DAs and DBs. Finally we model the field structure of SDSSJ1143+6615 finding the data are consistent with an offset dipole.

astro-ph.SR↗

arXiv2022

Computation of NMR shieldings at the CASSCF level using gauge-including atomic orbitals and Cholesky decomposition

We present an implementation of coupled-perturbed complete active space self-consistent field (CP-CASSCF) theory for the computation of nuclear magnetic resonance chemical shifts using gauge-including atomic orbitals and Cholesky decomposed two-electron integrals. The CP-CASSCF equations are solved using a direct algorithm where the magnetic Hessian matrix-vector product is expressed in terms of one-index transformed quantities. Numerical tests on systems with up to about 1300 basis functions provide information regarding both the computational efficiency and limitations of our implementation.

physics.chem-ph↗

arXiv2022

Cholesky decomposition of two-electron integrals in quantum-chemical calculations with perturbative or finite magnetic fields using gauge-including atomic orbitals

A rigorous analysis is carried out concerning the use of Cholesky decomposition (CD) of two-electron integrals in the case of quantum-chemical calculations with finite or perturbative magnetic fields and gauge-including atomic orbitals. We investigate in particular how permutational symmetry can be accounted for in such calculations and how this symmetry can be exploited to reduce the computational requirements. A modified CD procedure is suggested for the finite-field case that roughly halves the memory demands for the storage of the Cholesky vectors. The resulting symmetry of the Cholesky vectors also enables savings in the computational costs. For the derivative two-electron integrals in case of a perturbative magnetic field we derive CD expressions by means of a first-order Taylor expansion of the corresponding finite magnetic-field formulas with the field-free case as reference point. The perturbed Cholesky vectors are shown to be antisymmetric (as already proposed by Burger et al. (J. Chem. Phys., 155, 074105 (2021))) and the corresponding expressions enable significant savings in the required integral evaluations (by a factor of about four) as well as in the actual construction of the Cholesky vectors (by means of a two-step procedure similar to the one presented by Folkestad et al. (J. Chem. Phys., 150, 194112 (2019)) and Zhang et al. (J. Phys. Chem. A, 125, 4258-4265 (2021))). Numerical examples with cases involving several hundred basis functions verify our suggestions concerning CD in case of finite and perturbative magnetic fields.

physics.chem-ph↗

arXiv2021

Cholesky decomposition of complex two-electron integrals over GIAOs: Efficient MP2 computations for large molecules in strong magnetic fields

In large-scale quantum-chemical calculations the electron-repulsion integral (ERI) tensor rapidly becomes the bottleneck in terms of memory and disk space. When an external finite magnetic field is employed, this problem becomes even more pronounced because of the reduced permutational symmetry and the need to work with complex integrals and wave-function parameters. One way to alleviate the problem is to employ a Cholesky decomposition (CD) to the complex ERIs over gauge-including atomic orbitals. The CD scheme establishes favourable compression rates by selectively discarding linearly dependent product densities from the chosen basis set while maintaining a rigorous and robust error control. This error control constitutes the main advantage over conceptually similar methods such as density fitting which rely on employing pre-defined auxiliary basis sets. We implemented the use of the CD in the framework of finite-field (ff) Hartree-Fock and ff second-order Møller Plesset perturbation theory. Our work demonstrates that the CD compression rates are particularly beneficial in calculations in the presence of a finite magnetic field. The ff-CD-MP2 scheme enables the correlated treatment of systems with more than 2000 basis functions in strong magnetic fields within a reasonable time span.

physics.chem-ph↗

arXiv2021

NMR Chemical Shift Computations at Second-Order M{\o}ller-Plesset Perturbation Theory Using Gauge-Including Atomic Orbitals and Cholesky-Decomposed Two-Electron Integrals

We report on a formulation and implementation of a scheme to compute NMR shieldings at second-order Moller-Plesset (MP2) perturbation theory using gauge-including atomic orbitals (GIAOs) to ensure gauge-origin independence and Cholesky decomposition (CD) to handle unperturbed as well as perturbed two-electron integrals. We investigate the accuracy of the CD for the derivatives of the two-electron integrals with respect to an external magnetic field as well as for the computed NMR shieldings, before we illustrate the applicability of our CD based GIAO-MP2 scheme in calculations involving up to about one hundred atoms and more than one thousand basis functions.

physics.chem-ph↗

arXiv2021

Complex Ground-State and Excitation Energies in Coupled-Cluster Theory

Since in coupled-cluster (CC) theory ground-state and excitation energies are eigenvalues of a non-Hermitian matrix, these energies can in principle take on complex values. In this paper we discuss the appearance of complex energy values in CC calculations from a mathematical perspective. We analyze the behaviour of the eigenvalues of Hermitian matrices that are perturbed (in a non-Hermitian manner) by a real parameter. Based on these results we show that for CC calculations with real-valued Hamiltonian matrices the ground-state energy generally takes a real value. Furthermore, we show that in the case of real-valued Hamiltonian matrices complex excitation energies only occur in the context of conical intersections. In such a case, unphysical consequences are encountered such as a wrong dimension of the intersection seam, large numerical deviations from full configuration-interaction (FCI) results, and the square-root-like behaviour of the potential surfaces near the conical intersection. In the case of CC calculations with complex-valued Hamiltonian matrix elements, it turns out that complex energy values are to be expected for ground and excited states when no symmetry is present. We confirm the occurrence of complex energies by sample calculations using a six-state model and by CC calculations for the H2O molecule in a strong magnetic field. We furthermore show that symmetry can prevent the occurrence of complex energy values. Lastly, we demonstrate that in most cases the real part of the complex energy values provides a very good approximation to the FCI energy.

physics.chem-ph↗

arXiv2015

Coupled-cluster theory for atoms and molecules in strong magnetic fields

An implementation of coupled-cluster (CC) theory to treat atoms and molecules in finite magnetic fields is presented. The main challenges stem from the magnetic-field dependence in the Hamiltonian, or, more precisely, the appearance of the angular momentum operator, due to which the wave function becomes complex and which introduces a gauge-origin dependence. For this reason, an implementation of a complex CC code is required together with the use of gauge-including atomic orbitals to ensure gauge-origin independence. Results of coupled-cluster singles--doubles--perturbative-triples (CCSD(T)) calculations are presented for atoms and molecules with a focus on the dependence of correlation and binding energies on the magnetic field.

physics.chem-ph↗