Effective one-particle energies from generalized Kohn-Sham random phase approximation: A direct approach for computing and analyzing core ionization energies

Taming Negative Ion Resonances Using Nonlocal Exchange-Correlation Functionals

The characterization of negative ion resonances poses a fundamental challenge to density functional methods due to the unbound nature of resonances. To overcome this challenge, we propose one-particle nonlocal exchange-correlation (xc) potentials combining the exact-exchange (EXX) and the random phase approximation (RPA) correlation potentials. The negative ion resonances are identified by perturbing the real Hermitian nonlocal xc potentials using complex absorbing local potentials. Our studies show that the nonlocal EXX+RPA potential significantly enhances the description of positions and widths of negative ion resonance states compared to potentials that exclude dynamic polarization in RPA or include only EXX. The use of low-scaling algorithms simplifies the computation of the RPA potential, thereby providing a practical solution for resonance-state characterization within the density functional framework. A theoretical framework and the underlying assumptions required for combining real Hermitian nonlocal xc potentials with complex local potentials are discussed.

Single-Pole Polarization Models: Rapid Evaluation of Electron Affinities of Solvated-Electron and Superatomic Molecular Anionic States

We propose a single-parameter effective one-particle potential, termed the single-pole exchange-correlation (1p-XC), to rapidly evaluate electron affinities (EAs) of nonvalence electronic states of molecular clusters and nanoassemblies. The model combines exact-exchange and the random phase approximation (RPA) correlation potential with a single-pole approximation to model the frequency-dependent polarization function. It captures long-range static and dynamic-frequency effects in the correlation potential, with mean absolute errors of 0.06 eV for EAs of hydrated- and ammoniated-electron clusters with EA values in the range 0.24-1.77 eV. The 1p-XC approximation enables EA estimation with a computational wall-time similar to that of hybrid functionals. The model also provides a compressed-basis, which significantly reduces the rank of higher-level parameter-free one-particle Hamiltonians and further simplifies the computation of EAs. The compressed-basis approach is used to model the hybridization of superatomic molecular states of (C60)2- and (C60)3-, thereby verifying previous model Hamiltonian studies.

Accelerating Core-Level GW Calculations by Combining the Contour Deformation Approach with the Analytic Continuation of W

In recent years, the GW method has emerged as a reliable tool for computing core-level binding energies. The contour deformation (CD) technique has been established as an efficient, scalable, and numerically stable approach to compute the GW self-energy for deep core excitations. However, core-level GW calculations with CD face the challenge of higher scaling with respect to system size N compared to the conventional quartic scaling in valence-state algorithms. In this work, we present the CD-WAC method [CD with W analytic continuation (AC)], which reduces the scaling of CD applied to the inner shells from O(N5) to O(N4) by employing an AC of the screened Coulomb interaction W. Our proposed method retains the numerical accuracy of CD for the computationally challenging deep core case, yielding mean absolute errors <5 meV for well-established benchmark sets, such as CORE65, for single-shot GW calculations. More extensive testing for different GW flavors proves the reliability of the method. We have confirmed the theoretical scaling by performing scaling experiments on large acene chains and amorphous carbon clusters, achieving speedups of up to 10× for structures of only 116 atoms. This improvement in computational efficiency paves the way for more accurate and efficient core-level GW calculations on larger and more complex systems.

XPS Core-Level Chemical Shift by Ab Initio Many-Body Theory

X-ray photoemission spectroscopy (XPS) provides direct information on atomic composition and stoichiometry by measuring core-electron binding energies. Moreover, from the shift of the binding energy, the so-called chemical shift, the precise chemical type of bonds can be inferred, which brings additional information on the local structure. In this work, we present a theoretical study of the chemical shift first by comparing different theories, from Hartree-Fock and density functional theory to many-body perturbation theory approaches like the GW approximation and its static version (COHSEX). The accuracy of each theory is assessed with respect to a carbon 1s chemical shift experimental benchmark measured on a set of gas-phase molecules. More importantly, by decomposing the chemical shift into different contributions according to terms in the total Hamiltonian, classical electrostatics is identified as the major contributor to the chemical shift, one order of magnitude larger than the correlation.

Benchmark of GW Methods for Core-Level Binding Energies

The GW approximation has recently gained increasing attention as a viable method for the computation of deep core-level binding energies as measured by X-ray photoelectron spectroscopy. We present a comprehensive benchmark study of different GW methodologies (starting point optimized, partial and full eigenvalue-self-consistent, Hedin shift, and renormalized singles) for molecular inner-shell excitations. We demonstrate that all methods yield a unique solution and apply them to the CORE65 benchmark set and ethyl trifluoroacetate. Three GW schemes clearly outperform the other methods for absolute core-level energies with a mean absolute error of 0.3 eV with respect to experiment. These are partial eigenvalue self-consistency, in which the eigenvalues are only updated in the Green's function, single-shot GW calculations based on an optimized hybrid functional starting point, and a Hedin shift in the Green's function. While all methods reproduce the experimental relative binding energies well, the eigenvalue self-consistent schemes and the Hedin shift yield with mean absolute errors <0.2 eV the best results.

Modeling Nonresonant X-ray Emission of Second- and Third-Period Elements without Core-Hole Reference States and Empirical Parameters

Nonresonant X-ray emission (XE) energies and oscillator strengths are obtained using the effective potential of the generalized Kohn-Sham semi-canonical projected random phase approximation (GKS-spRPA) method. XE energies are estimated as a difference between the valence and core ionization eigenvalues, while the oscillator strengths are obtained within a frozen orbital approximation. This straightforward approach provides accurate XE energies without any need for core-hole reference states, empirical shifting parameters, or tuning of density functionals. To account for relativistic corrections to the core orbitals, we have formulated a scalar relativistic (sr) GKS-spRPA approach based on the spin-free X2C one-electron Hamiltonian. The sr-GKS-spRPA method provides highly reliable XE energies using uncontracted basis-sets on atoms where the core-hole is created prior to emission. For the largest basis-sets used in our study, using completely uncontracted polarized core-valence Dunning basis-sets, the mean absolute errors (MAEs) are within 0.7 eV compared to experimental reference values for a test-set consisting of 27 valence-to-core XE energies of molecules with second- and third-period elements. Considering a balance of accuracy and computational effort, we recommend the use of s-uncontracted def2-TZVP for second-period and all-uncontracted def2-TZVP for third-period elements. For this recommended basis-set, the MAE is 0.2 eV. The analytically continued sr-GKS-spRPA approach, with an O(N4) computational cost, enables efficient computation of XE spectra of molecules such as S₈ and C₆₀ with several core-hole states.

High-Resolution X-ray Photoelectron Spectroscopy of Organometallic (C5H4SiMe3)3LnIII and [(C5H4SiMe3)3LnII]1- Complexes (Ln = Sm, Eu, Gd, Tb)

The capacity of X-ray photoelectron spectroscopy (XPS) to provide information on the electronic structure of molecular organometallic complexes of Ln(II) ions (Ln = lanthanide) has been examined for the first time. XPS spectra were obtained on the air-sensitive molecular trivalent 4fⁿ Cp'₃Ln^III complexes (Ln = Sm, Eu, Gd, Tb; Cp' = C₅H₄SiMe₃) and compared to those of the highly reactive divalent complexes, [K(crypt)][Cp'₃Ln^II] (crypt = 2.2.2-cryptand), which have either 4fⁿ⁺¹ (Sm, Eu) or 4fⁿ5d¹ electron configurations (Gd, Tb). The Ln 4d, Si 2p, and C 1s regions of the Ln(III) and Ln(II) complexes were identified and compared. The metal 4d peaks of these molecular lanthanide complexes were used diagnostically to compare oxidation states. The valence region of the Gd(III) and Gd(II) complexes was also examined with XPS and density function theory/random phase approximation (DFT/RPA) calculations, and this led to the tentative assignment of a signal from the 5d¹ electron consistent with a 4f⁷5d¹ electron configuration for Gd(II).

Exploring the Solvation of Acetic Acid in Water Using Liquid Jet X-ray Photoelectron Spectroscopy and Core Level Electron Binding Energy Calculations

Liquid jet X-ray photoelectron spectroscopy was used to investigate changes in the local electronic structure of acetic acid in the bulk of aqueous solutions induced by solvation effects. These effects manifest themselves as shifts in the difference in the carbon 1s binding energy (ΔBE) between the methyl and carboxyl carbons of acetic acid. Furthermore, molecular dynamics simulations, coupled with correlated electronic structure calculations of the first solvation sphere, provide insight into the number of water molecules directly interacting with the carboxyl group that are required to match the ΔBE from the photoelectron spectroscopy experiments. This comparison shows that a single water molecule in the first solvation shell describes the photoelectron ΔBE of acetic acid while at least 20 water molecules are required for the conjugate base, acetate, in aqueous solutions.

Selfconsistent random phase approximation methods

This Perspective reviews recent efforts toward selfconsistent calculations of ground-state energies within the random phase approximation (RPA) in the (generalized) Kohn-Sham (KS) density functional theory context. Since the RPA correlation energy explicitly depends on the non-interacting KS potential, an additional condition to determine the energy as a functional of the density is necessary. This observation leads to the concept of functional selfconsistency (FSC), which requires that the KS density equals the interacting density defined as the functional derivative of the ground-state energy with respect to the external potential. While all existing selfconsistent RPA schemes violate FSC, the recent generalized KS semicanonical projected RPA (GKS-spRPA) method takes a step toward satisfying it. This leads to systematic improvements in densities, binding energy curves, reference state stability, and molecular properties compared to non-selfconsistent RPA as well as optimized effective potential RPA. GKS-spRPA orbital energies accurately approximate valence and core ionization potentials, and even electron affinities of non-valence bound anions. The computational cost and performance of GKS-spRPA are compared to those of related selfconsistent schemes, including GW and orbital optimization methods, and limitations are discussed. Large differences between KS and interacting densities observed in the absence of FSC and the well-rounded performance of GKS-spRPA suggest that the KS potential as a density functional should be defined via the FSC condition for explicitly potential-dependent density functionals.

Molecular Electron Affinities Using the Generalized Kohn-Sham Semicanonical Projected Random Phase Approximation

The suitability of one-particle energies from the generalized Kohn-Sham semicanonical projected random phase approximation (GKS-spRPA) method for electron affinities of molecules is investigated. It is shown that the GKS-spPRA effective potential includes exact exchange and polarizability-dependent correlation terms that are necessary for the correct description of anionic systems. An O(N4) implementation that enables fast computation of electron affinities is presented. For model systems, I show that the GKS-spRPA approach is applicable for valence and nonvalence type anions with a maximum error of 0.13 eV for valence anionic states and 0.03 eV for nonvalence anionic states compared to equation of motion coupled cluster methods. For a series of perhalobenzene molecules, C₆X₆ (X= F, Cl, Br, and I), GKS-spRPA predicts that the ground-state character changes from a nonvalence-σ* type in C₆F₆^- to valence-π* in C₆Cl₆^- and valence-σ* in C₆Br₆^- and C₆I₆^-. Experimental implications of these findings are discussed.

TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations

TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chemical simulations of molecules, clusters, periodic systems, and solutions. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy-cost ratio, such as density functional theory including local hybrids and the random phase approximation (RPA), GW-Bethe-Salpeter methods, second-order Møller-Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resolution-of-the-identity approximation, imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent additions to TURBOMOLE's functionality, including excited-state methods, RPA and Green's function methods, relativistic approaches, high-order molecular properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE's current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE's development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted.

Accurate Absolute and Relative Core-Level Binding Energies from GW

We present an accurate approach to compute X-ray photoelectron spectra based on the GW Green's function method that overcomes the shortcomings of common density functional theory approaches. GW has become a popular tool to compute valence excitations for a wide range of materials. However, core-level spectroscopy is thus far almost uncharted in GW. We show that single-shot perturbation calculations in the G0W0 approximation, which are routinely used for valence states, cannot be applied for core levels and suffer from an extreme, erroneous transfer of spectral weight to the satellite spectrum. The correct behavior can be restored by partial self-consistent GW schemes or by using hybrid functionals with almost 50% of exact exchange as a starting point for G0W0. We also include relativistic corrections and present a benchmark study for 65 molecular 1s excitations. Our absolute and relative GW core-level binding energies agree within 0.3 and 0.2 eV with experiment, respectively.