On the Efficiency of Algorithms for Solving Hartree–Fock and Kohn–Sham Response Equations

Extension of the D3 and D4 London dispersion corrections to the full actinides series

Efficient dispersion corrections are an indispensable component of modern density functional theory, semi-empirical quantum mechanical, and even force field methods. In this work, we extend the well established D3 and D4 London dispersion corrections to the full actinides series, francium, and radium. To keep consistency with the existing versions, the original parameterization strategy of the D4 model was only slightly modified. This includes improved reference Hirshfeld atomic partial charges at the ωB97M-V/ma-def-TZVP level to fit the required electronegativity equilibration charge (EEQ) model. In this context, we developed a new actinide data set called AcQM, which covers the most common molecular actinide compound space. Furthermore, the efficient calculation of dynamic polarizabilities that are needed to construct CAB6 dispersion coefficients was implemented into the ORCA program package. The extended models are assessed for the computation of dissociation curves of actinide atoms and ions, geometry optimizations of crystal structure cutouts, gas-phase structures of small uranium compounds, and an example extracted from a small actinide complex protein assembly. We found that the novel parameterizations perform on par with the computationally more demanding density-dependent VV10 dispersion correction. With the presented extension, the excellent cost-accuracy ratio of the D3 and D4 models can now be utilized in various fields of computational actinide chemistry and, e.g., in efficient composite DFT methods such as r2SCAN-3c. They are implemented in our freely available standalone codes (dftd4, s-dftd3) and the D4 version will be also available in the upcoming ORCA 6.0 program package.

Standard Response Calculation of Electric Dipole Polarizability and Specific Optical Rotation Power Densities

A time-dependent method has been developed to solve the standard response equation for the calculation of dynamic molecular property densities, endowed with the characteristic of being origin-invariant, entirely in the atomic orbital basis at both HF and DFT level of theory. The method has been tuned in particular for the calculation of origin-independent electric dipole polarizability density and specific rotation power density. Some demonstrations are given for the hexabenzocoronene molecule and the Tröger's base.

Approximate Exponential Integrators for Time-Dependent Equation-of-Motion Coupled Cluster Theory

With a growing demand for time-domain simulations of correlated many-body systems, the development of efficient and stable integration schemes for the time-dependent Schrödinger equation is of keen interest in modern electronic structure theory. In this work, we present two approaches for the formation of the quantum propagator for time-dependent equation-of-motion coupled cluster theory based on the Chebyshev and Arnoldi expansions of the complex, nonhermitian matrix exponential, respectively. The proposed algorithms are compared with the short-iterative Lanczos method of Cooper et al. [J. Phys. Chem. A 2021 125, 5438-5447], the fourth-order Runge-Kutta method, and exact dynamics for a set of small but challenging test problems. For each of the cases studied, both of the proposed integration schemes demonstrate superior accuracy and efficiency relative to the reference simulations.

responsefun: Fun with Response Functions in the Algebraic Diagrammatic Construction Framework

We present the open-source responsefun package, which implements a universally applicable procedure for computing molecular response properties within the algebraic diagrammatic construction (ADC) framework, exploiting the intermediate state representation (ISR) approach. With symbolic mathematics, the user can simply enter textbook sum-over-states (SOS) expressions from time-dependent perturbation theory, which are then automatically translated into the corresponding symbolic ADC/ISR formulations. Using the data structures provided by the hybrid Python/C++ module adcc for calculating excited states with ADC, the specified response property is directly evaluated, and the result is returned to the user. Employing the novel responsefun package, we present the first ADC/ISR calculations of second-order hyperpolarizability tensors and three-photon-absorption matrix elements.

Complex Linear Response Functions for a Multiconfigurational Self-Consistent Field Wave Function in a High Performance Computing Environment

We present novel developments for the highly efficient evaluation of complex linear response functions of a multiconfigurational self-consistent field (MCSCF) wave function as implemented in MultiPsi. Specifically, expressions for the direct evaluation of linear response properties at given frequencies using the complex polarization propagator (CPP) approach have been implemented, within both the Tamm-Dancoff approximation (TDA) and the random phase approximation (RPA). Purely real algebra with symmetric and antisymmetric trial vectors in a shared subspace is used wherein the linear response equations are solved. Two bottlenecks of large scale MC-CPP calculations, namely, the memory footprint and computational time, are addressed. The former is addressed by limiting the size of the subspace of trial vectors by using singular value decomposition (SVD) on either orbital or CI subspaces. The latter is addressed using an efficient parallel implementation as well as the strategy of dynamically adding linear response equations at near-convergence to neighboring roots. Furthermore, a novel methodology for decomposing MC-CPP spectra in terms of intuitive orbital excitations in an approximate fashion is presented. The performance of the code is illustrated with several numerical examples, including the X-ray spectrum of a molecule with nearly one hundred atoms. Additionally, for X-ray spectroscopy, the effect of including or excluding the core orbital in the active space on small covalent metal complexes is discussed.

Libkrylov: A modular open-source software library for extremely large on-the-fly matrix computations

We present the design and implementation of libkrylov, an open-source library for solving matrix-free eigenvalue, linear, and shifted linear equations using Krylov subspace methods. The primary objectives of libkrylov are flexible API design and modular structure, which enables integration with specialized matrix-vector evaluation "engines." Libkrylov features pluggable preconditioning, orthonormalization, and tunable convergence control. Diagonal (conjugate gradient, CG), Davidson, and Jacobi-Davidson preconditioners are available, along with orthonormal and nonorthonormal (nKs) schemes. All functionality of libkrylov is exposed via Fortran and C application programming interfaces (APIs). We illustrate the performance of libkrylov for eigenvalue calculations arising in time-dependent density functional theory (TDDFT) in the Tamm-Dancoff approximation (TDA) and discuss the convergence behavior as a function of preconditioning and orthonormalization methods.

Solving response expressions in the ADC/ISR framework

We present an implementation for the calculation of molecular response properties using the algebraic-diagrammatic construction (ADC)/intermediate state representation approach. For the second-order ADC model [ADC(2)], a memory-efficient ansatz avoiding the storage of double excitation amplitudes is investigated. We compare the performance of different numerical algorithms for the solution of the underlying response equations for ADC(2) and show that our approach also strongly improves the convergence behavior for the investigated algorithms compared with the standard implementation. All routines are implemented in an open-source Python library.

Nontrivial spectral band progressions in electronic circular dichroism spectra of carbohelicenes revealed by linear response calculations

We demonstrate that contemporary computational resources allow for accurate theoretical studies of systems matching recent advances in experimental helicene chemistry. Concerned with first-principles calculations of carbohelicenes, our work surpasses CH[12] as the largest system investigated to date and unravels trends in the electronic structure of the low-lying states of the homologous series. Utilizing a highly efficient implementation of linear response algorithms, we present electronic circular dichroism (CD) spectra of carbohelicenes ranging from CH[5] to CH[30] at the level of Kohn-Sham density-functional theory. Our results for a systematic increase in system size show the emergence of new CD bands that subsequently rise to intensities dominating the spectrum. The spectral band progressions exhibit a periodicity directly linked to the number of overlapping layers of conjugation. While our findings rectify the current understanding of the electronic structure of carbohelicenes, they also serve as a general call for caution regarding the extrapolation of trends from small system ranges.

Polarizable Embedding Complex Polarization Propagator in Four- and Two-Component Frameworks

Explicit embedding methods combined with the complex polarization propagator (CPP) enable the modeling of spectroscopy for increasingly complex systems with a high density of states. We present the first derivation and implementation of the CPP in four- and exact-two-component (X2C) polarizable embedding (PE) frameworks. We denote the developed methods PE-4c-CPP and PE-X2C-CPP, respectively. We illustrate the methods by estimating the solvent effect on ultraviolet-visible (UV-vis) and X-ray atomic absorption (XAS) spectra of [Rh(H₂O)₆]³⁺ and [Ir(H₂O)₆]³⁺ immersed in aqueous solution. We moreover estimate solvent effects on UV-vis spectra of a platinum complex that can be photochemically activated (in water) to kill cancer cells. Our results clearly show that the inclusion of the environment is required: UV-vis and (to a lesser degree) XAS spectra can become qualitatively different from vacuum calculations. Comparison of PE-4c-CPP and PE-X2C-CPP methods shows that X2C essentially reproduces the solvent effect obtained with the 4c methods.

Self-assembling, structure and nonlinear optical properties of fluorescent organic nanoparticles in water

Owing to their intense emission, low toxicity and solubility in aqueous medium, fluorescent organic nanoparticles (FONs) have emerged as promising alternatives to inorganic ones for the realization of exogenous probes for bioimaging applications. However, the intimate structure of FONs in solution, as well as the role played by intermolecular interactions on their optical properties, remains challenging to study. Following a recent Second-Harmonic Scattering (SHS) investigation led by two of us [Daniel et al., ACS Photonics, 2015, 2, 1209], we report herein a computational study of the structural organization and second-order nonlinear optical (NLO) properties of FONs based on dipolar chromophores incorporating a hydrophobic triphenylamine electron-donating unit and a slightly hydrophilic aldehyde electron-withdrawing unit at their extremities. Molecular dynamics simulations of the FON formation in water are associated with quantum chemical calculations, to provide insight into the molecular aggregation process, the molecular orientation of the dipolar dyes within the nanoparticles, and the dynamical behavior of their NLO properties. Moreover, the impact of intermolecular interactions on the NLO responses of the FONs is investigated by employing the tight-binding version of the recently developed simplified time-dependent density functional theory (sTD-DFT) approach, allowing the all-atom quantum mechanics treatment of nanoparticles.

Size-dependent polarizabilities and van der Waals dispersion coefficients of fullerenes from large-scale complex polarization propagator calculations

While the anomalous non-additive size-dependencies of static dipole polarizabilities and van der Waals C₆ dispersion coefficients of carbon fullerenes are well established, the widespread reported scalings for the latter (ranging from N^2.2 to N^2.8) call for a comprehensive first-principles investigation. With a highly efficient implementation of the linear complex polarization propagator, we have performed Hartree-Fock and Kohn-Sham density functional theory calculations of the frequency-dependent polarizabilities for fullerenes consisting of up to 540 carbon atoms. Our results for the static polarizabilities and C₆ coefficients show scalings of N^1.2 and N^2.2, respectively, thereby deviating significantly from the previously reported values obtained with the use of semi-classical/empirical methods. Arguably, our reported values are the most accurate to date as they represent the first ab initio or first-principles treatment of fullerenes up to a convincing system size.

Efficient implementation of isotropic cubic response functions for two-photon absorption cross sections within the self-consistent field approximation

Within the self-consistent field approximation, computationally tractable expressions for the isotropic second-order hyperpolarizability have been derived and implemented for the calculation of two-photon absorption cross sections. The novel tensor average formulation presented in this work allows for the evaluation of isotropic damped cubic response functions using only ∼3.3% (one-photon off-resonance regions) and ∼10% (one-photon resonance regions) of the number of auxiliary Fock matrices required when explicitly calculating all the needed individual tensor components. Numerical examples of the two-photon absorption cross section in the one-photon off-resonance and resonance regions are provided for alanine-tryptophan and 2,5-dibromo-1,4-bis(2-(4-diphenylaminophenyl)vinyl)-benzene. Furthermore, a benchmark set of 22 additional small- and medium-sized organic molecules is considered. In all these calculations, a quantitative assessment is made of the reduced and approximate forms of the cubic response function in the one-photon off-resonance regions and results demonstrate a relative error of less than ∼5% when using the reduced expression as compared to the full form of the isotropic cubic response function.

Approximate versus Exact Embedding for Chiroptical Properties: Reconsidering Failures in Potential and Response

We investigate the suitability of subsystem time-dependent density-functional theory (sTDDFT) for describing chiroptical properties with a focus on optical rotation parameters. Our starting point is a new implementation of the recently proposed projection-based, coupled frozen-density embedding (FDEc) framework. We adapt the generalized, non-Hermitian formulation of TDDFT and derive corresponding expressions for regular and damped response properties from subsystem TDDFT. We verify that our implementation of this "exact" formulation allows to reproduce supermolecular results of electronic circular dichroism (ECD) spectra, of optical rotatory dispersion, and of polarizabilities. We present a systematic test of the main approximations typically introduced in practical frozen-density embedding (FDE) calculations of response properties: (i) the use of approximate nonadditive kinetic-energy (NAKE) functionals, which can be avoided through projection techniques, (ii) the use of monomer (subsystem) basis sets rather than supersystem basis sets, and (iii) the neglect of intersubsystem response coupling within the so-called uncoupled FDE (or FDEu) approximation. While approximation (i) is known to generally lead to large errors for covalently bound subsystems, we present cases in which either the basis set or the coupling step are similarly or even (much) more important. In particular, we explicitly demonstrate by comparison to a fully coupled calculation that missing intersubsystem response couplings are responsible for the failure of FDE reported in a previous study [ J. Chem. Theory Comput. 2015, 11, 5305-5315]. We show that good agreement with reference results can be obtained in this case even with standard NAKE approximations for the FDE potentials and efficient monomer basis sets, making calculations for larger systems well accessible.

CPPE: An Open-Source C++ and Python Library for Polarizable Embedding

We present a modular open-source library for polarizable embedding (PE) named CPPE. The library is implemented in C++, and it additionally provides a Python interface for rapid prototyping and experimentation in a high-level scripting language. Our library integrates seamlessly with existing quantum chemical program packages through an intuitive and minimal interface. Until now, CPPE has been interfaced to three packages, Q-Chem, Psi4, and PySCF. Furthermore, we show CPPE in action using all three program packages for a computational spectroscopy application. With CPPE, host program interfaces only require minor programming effort, paving the way for new combined methodologies and broader availability of the PE model.

Open-ended formulation of self-consistent field response theory with the polarizable continuum model for solvation

The study of high-order absorption properties of molecules is a field of growing importance. Quantum-chemical studies can help design chromophores with desirable characteristics. Given that most experiments are performed in solution, it is important to devise a cost-effective strategy to include solvation effects in quantum-chemical studies of these properties. We here present an open-ended formulation of self-consistent field (SCF) response theory for a molecular solute coupled to a polarizable continuum model (PCM) description of the solvent. Our formulation relies on the open-ended, density matrix-based quasienergy formulation of SCF response theory of Thorvaldsen, et al., [J. Chem. Phys., 2008, 129, 214108] and the variational formulation of the PCM, as presented by Lipparini et al., [J. Chem. Phys., 2010, 133, 014106]. Within the PCM approach to solvation, the mutual solute-solvent polarization is represented by means of an apparent surface charge (ASC) spread over the molecular cavity defining the solute-solvent boundary. In the variational formulation, the ASC is an independent, variational degree of freedom. This allows us to formulate response theory for molecular solutes in the fixed-cavity approximation up to arbitrary order and with arbitrary perturbation operators. For electric dipole perturbations, pole and residue analyses of the response functions naturally lead to the identification of excitation energies and transition moments. We document the implementation of this approach in the Dalton program package using a recently developed open-ended response code and the PCMSolver libraries and present results for one-, two-, three-, four- and five-photon absorption processes of three small molecules in solution.

Benchmarking Post-Hartree-Fock Methods To Describe the Nonlinear Optical Properties of Polymethines: An Investigation of the Accuracy of Algebraic Diagrammatic Construction (ADC) Approaches

Third-order nonlinear optical (NLO) properties of polymethine dyes have been widely studied for applications such as all-optical switching. However, the limited accuracy of the current computational methodologies has prevented a comprehensive understanding of the nature of the lowest excited states and their influence on the molecular optical and NLO properties. Here, attention is paid to the lowest excited-state energies and their energetic ratio, as these characteristics impact the figure-of-merit for all-optical switching. For a series of model polymethines, we compare several algebraic diagrammatic construction (ADC) schemes for the polarization propagator with approximate second-order coupled cluster (CC2) theory, the widely used INDO/MRDCI approach and the symmetry-adapted cluster configuration interaction (SAC-CI) algorithm incorporating singles and doubles linked excitation operators (SAC-CI SD-R). We focus in particular on the ground-to-excited state transition dipole moments and the corresponding state dipole moments, since these quantities are found to be of utmost importance for an effective description of the third-order polarizability γ and two-photon absorption spectra. A sum-overstates expression has been used, which is found to quickly converge. While ADC(3/2) has been found to be the most appropriate method to calculate these properties, CC2 performs poorly.

K- and L-edge X-ray absorption spectrum calculations of closed-shell carbon, silicon, germanium, and sulfur compounds using damped four-component density functional response theory

X-ray absorption spectra of carbon, silicon, germanium, and sulfur compounds have been investigated by means of damped four-component density functional response theory. It is demonstrated that a reliable description of relativistic effects is obtained at both K- and L-edges. Notably, an excellent agreement with experimental results is obtained for L2,3-spectra-with spin-orbit effects well accounted for-also in cases when the experimental intensity ratio deviates from the statistical one of 2 : 1. The theoretical results are consistent with calculations using standard response theory as well as recently reported real-time propagation methods in time-dependent density functional theory, and the virtues of different approaches are discussed. As compared to silane and silicon tetrachloride, an anomalous error in the absolute energy is reported for the L2,3-spectrum of silicon tetrafluoride, amounting to an additional spectral shift of ∼1 eV. This anomaly is also observed for other exchange-correlation functionals, but it is seen neither at other silicon edges nor at the carbon K-edge of fluorine derivatives of ethene. Considering the series of molecules SiH4-XFX with X = 1, 2, 3, 4, a gradual divergence from interpolated experimental ionization potentials is observed at the level of Kohn-Sham density functional theory (DFT), and to a smaller extent with the use of Hartree-Fock. This anomalous error is thus attributed partly to difficulties in correctly emulating the electronic structure effects imposed by the very electronegative fluorines, and partly due to inconsistencies in the spurious electron self-repulsion in DFT. Substitution with one, or possibly two, fluorine atoms is estimated to yield small enough errors to allow for reliable interpretations and predictions of L2,3-spectra of more complex and extended silicon-based systems.

Beyond the electric-dipole approximation: A formulation and implementation of molecular response theory for the description of absorption of electromagnetic field radiation

We present a formulation of molecular response theory for the description of a quantum mechanical molecular system in the presence of a weak, monochromatic, linearly polarized electromagnetic field without introducing truncated multipolar expansions. The presentation focuses on a description of linear absorption by adopting the energy-loss approach in combination with the complex polarization propagator formulation of response theory. Going beyond the electric-dipole approximation is essential whenever studying electric-dipole-forbidden transitions, and in general, non-dipolar effects become increasingly important when addressing spectroscopies involving higher-energy photons. These two aspects are examined by our study of the near K-edge X-ray absorption fine structure of the alkaline earth metals (Mg, Ca, Sr, Ba, and Ra) as well as the trans-polyenes. In following the series of alkaline earth metals, the sizes of non-dipolar effects are probed with respect to increasing photon energies and a detailed assessment of results is made in terms of studying the pertinent transition electron densities and in particular their spatial extension in comparison with the photon wavelength. Along the series of trans-polyenes, the sizes of non-dipolar effects are probed for X-ray spectroscopies on organic molecules with respect to the spatial extension of the chromophore.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.

A complex-polarization-propagator protocol for magneto-chiral axial dichroism and birefringence dispersion

A schematic representation of magneto-chiral effects.

Requirements of first-principles calculations of X-ray absorption spectra of liquid water

A computational benchmark study on X-ray absorption spectra of water has been performed by means of transition-potential density functional theory (TP-DFT), damped time-dependent density functional theory (TDDFT), and damped coupled cluster (CC) linear response theory. For liquid water, using TDDFT with a tailored CAM-B3LYP functional and a polarizable embedding, we find that an embedding with over 2000 water molecules is required to fully converge spectral features for individual molecules, but a substantially smaller embedding can be used within averaging schemes. TP-DFT and TDDFT calculations on 100 MD structures demonstrate that TDDFT produces a spectrum with spectral features in good agreement with experiment, while it is more difficult to fully resolve the spectral features in the TP-DFT spectrum. Similar trends were also observed for calculations of bulk ice. In order to further establish the performance of these methods, small water clusters have been considered also at the CC2 and CCSD levels of theory. Issues regarding the basis set requirements for spectrum simulations of liquid water and the determination of gas-phase ionization potentials are also discussed.

Wavelet formulation of the polarizable continuum model. II. Use of piecewise bilinear boundary elements

The sparsity pattern obtained in the wavelet Galerkin boundary element representation of the PCM boundary integral operators. By employing a wavelet basis on a smooth solvent-excluded molecular surface the method is able to guarantee solutions with high accuracy at a linear cost in memory and computational time.

Relative Stability of the La and Lb Excited States in Adenine and Guanine: Direct Evidence from TD-DFT Calculations of MCD Spectra

The relative position of La and Lb ππ* electronic states in purine nucleobases is a much debated topic, since it can strongly affect our understanding of their photoexcited dynamics. To assess this point, we calculated the absorption and magnetic circular dichroism (MCD) spectra of adenine, guanine, and their nucleosides in gas-phase and aqueous solution, exploiting recent developments in MCD computational technology within time-dependent density functional theory. MCD spectroscopy allows us to resolve the intense S0→ La transition from the weak S0→ Lb transition. The spectra obtained in water solution, by using B3LYP and CAM-B3LYP functionals and describing solvent effect by cluster models and by the polarizable continuum model (PCM), are in very good agreement with the experimental counterparts, thus providing direct and unambiguous evidence that the energy ordering predicted by TD-DFT, La < Lb, is the correct one.

Nuclear spin circular dichroism in fullerenes: a computational study

Chemically different carbons in C₇₀ give distinct signals in nuclear spin circular dichroism spectroscopy, a novel candidate for high-resolution identification of chemical compounds.

The Dalton quantum chemistry program system

Dalton is a powerful general-purpose program system for the study of molecular electronic structure at the Hartree-Fock, Kohn-Sham, multiconfigurational self-consistent-field, Møller-Plesset, configuration-interaction, and coupled-cluster levels of theory. Apart from the total energy, a wide variety of molecular properties may be calculated using these electronic-structure models. Molecular gradients and Hessians are available for geometry optimizations, molecular dynamics, and vibrational studies, whereas magnetic resonance and optical activity can be studied in a gauge-origin-invariant manner. Frequency-dependent molecular properties can be calculated using linear, quadratic, and cubic response theory. A large number of singlet and triplet perturbation operators are available for the study of one-, two-, and three-photon processes. Environmental effects may be included using various dielectric-medium and quantum-mechanics/molecular-mechanics models. Large molecules may be studied using linear-scaling and massively parallel algorithms. Dalton is distributed at no cost from http://www.daltonprogram.org for a number of UNIX platforms.