Natural determinant reference functional theory.

The natural determinant reference (NDR) or principal natural determinant is the Slater determinant comprised of the N most strongly occupied natural orbitals of an N-electron state of interest. Unlike the Kohn-Sham (KS) determinant, which yields the exact ground-state density, the NDR only yields the best idempotent approximation to the interacting one-particle reduced density matrix, but it is well-defined in common atom-centered basis sets and is representation-invariant. We show that the under-determination problem of prior attempts to define a ground-state energy functional of the NDR is overcome in a grand-canonical ensemble framework at the zero-temperature limit. The resulting grand potential functional of the NDR ensemble affords the variational determination of the ground state energy, its NDR (ensemble), and select ionization potentials and electron affinities. The NDR functional theory can be viewed as an "exactification" of orbital optimization and empirical generalized KS methods. NDR functionals depending on the noninteracting Hamiltonian do not require troublesome KS-inversion or optimized effective potentials.

TURBOMOLE: Today and Tomorrow.

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Room-Temperature Stable Ln(II) Complexes Supported by 2,6-Diadamantyl Aryloxide Ligands.

The sterically bulky aryloxide ligand OAr* (OAr* = -OC6H2-Ad2-2,6tBu-4; Ad = 1-adamantyl) has been used to generate Ln(II) complexes across the lanthanide series that are more thermally stable than complexes of any other ligand system reported to date for 4fnd1 Ln(II) ions. The Ln(III) precursors Ln(OAr*)3 (1-Ln) were synthesized by reacting 1.2 equiv of Ln(NR2)3 (R = SiMe3) with 3 equiv of HOAr* for Ln = La, Ce, Nd, Gd, Dy, Yb, and Lu. 1-Ce, 1-Nd, 1-Gd, 1-Dy, and 1-Lu were identified by single-crystal X-ray diffraction studies. Reductions of 1-Ln with potassium graphite (KC8) in tetrahydrofuran in the presence of 2.2.2-cryptand (crypt) yielded the Ln(II) complexes [K(crypt)][Ln(OAr*)3] (2-Ln). The 2-Ln complexes for Ln = Nd, Gd, Dy, and Lu were characterized by X-ray crystallography and found to have Ln-O bond distances 0.038-0.087 Å longer than those of their 1-Ln analogues; this is consistent with 4fn5d1 electron configurations. The structure of 2-Yb has Yb-O distances 0.167 Å longer than those predicted for 1-Yb, which is consistent with a 4f14 electron configuration. Although 2-La and 2-Ce proved to be challenging to isolate, with 18-crown-6 (18-c-6) as the potassium chelator, La(II) and Ce(II) complexes with OAr* could be isolated and crystallographically characterized: [K(18-c-6)][Ln(OAr*)3] (3-Ln). The Ln(II) complexes decompose at room temperature more slowly than other previously reported 4fn5d1 Ln(II) complexes. For example, only 30% decomposition of 2-Dy was observed after 30 h at room temperature compared to complete decomposition of [Dy(OAr')3]- and [DyCp'3]- under similar conditions (OAr' = OC6H2-2,6-tBu2-4-Me; Cp' = C5H4SiMe3).

Periodic Coupled-Cluster Green's Function for Photoemission Spectra of Realistic Solids.

We present an efficient implementation of the coupled-cluster Green's function (CCGF) method for simulating photoemission spectra of periodic systems. We formulate the periodic CCGF approach with Brillouin zone sampling in the Gaussian basis at the coupled-cluster singles and doubles (CCSD) level. To enable CCGF calculations of realistic solids, we propose an active-space self-energy correction scheme by combining CCGF with the cheaper many-body perturbation theory (GW) and implement the model order reduction (MOR) frequency interpolation technique. We find that the active-space self-energy correction and MOR techniques significantly reduce the computational cost of CCGF while maintaining the high accuracy. We apply the developed CCGF approaches to compute spectral properties and band structure of silicon (Si) and zinc oxide (ZnO) crystals using triple-ζ Gaussian basis sets and medium-size k-point sampling and find good agreement with experimental measurements.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance.

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C5Me5)2Y(µ-S)2Mo(µ-S)2Y(C5Me5)2]-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C6Cl5)4]- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

A 9.2-GHz clock transition in a Lu(II) molecular spin qubit arising from a 3,467-MHz hyperfine interaction.

Spins in molecules are particularly attractive targets for next-generation quantum technologies, enabling chemically programmable qubits and potential for scale-up via self-assembly. Here we report the observation of one of the largest hyperfine interactions for a molecular system, Aiso = 3,467 ± 50 MHz, as well as a very large associated clock transition. This is achieved through chemical control of the degree of s-orbital mixing into the spin-bearing d orbital associated with a series of spin-½ La(II) and Lu(II) complexes. Increased s-orbital character reduces spin-orbit coupling and enhances the electron-nuclear Fermi contact interaction. Both outcomes are advantageous for quantum applications. The former reduces spin-lattice relaxation, and the latter maximizes the hyperfine interaction, which, in turn, generates a 9-GHz clock transition, leading to an increase in phase memory time from 1.0 ± 0.4 to 12 ± 1 µs for one of the Lu(II) complexes. These findings suggest strategies for the development of molecular quantum technologies, akin to trapped ion systems.

Hyperfine Coupling Constants in Local Exact Two-Component Theory.

We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete derivative of the X2C Hamiltonian is formed, that is, the derivatives of the unitary decoupling transformation are considered. This requires the solution of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the density functional approximation is carried out for a set of 12-17 transition-metal compounds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amount of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure density functional r2SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C6Cl5)4]- with more than 4700 primitive basis functions and four rare-earth single-molecule magnets: [La(OAr*)3]-, [Lu(NR2)3]-, [Lu(OAr*)3]-, and [TbPc2]-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the experimental findings of all Pt, La, Lu, and Tb molecules.

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.

Recent developments in the PySCF program package.

PySCF is a Python-based general-purpose electronic structure platform that supports first-principles simulations of molecules and solids as well as accelerates the development of new methodology and complex computational workflows. This paper explains the design and philosophy behind PySCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PySCF as a development environment. We then summarize the capabilities of PySCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PySCF across the domains of quantum chemistry, materials science, machine learning, and quantum information science.

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.

Theoretical Study of Divalent Bis(Pentaisopropylcyclopentadienyl) Actinocenes.

The existence of divalent bis(pentaisopropylcyclopentadienyl) actinocene compounds, An(CpiPr5)2 (An = Th, U, Pu, Am, Bk, No, and Lr), is assessed by density functional theory (DFT) calculations with scalar-relativistic small-core pseudopotentials. The calculations predict ground states with significant 6d occupation for Th, U, and Lr, whereas Am, Bk, and No exhibit 5f ground states. A mixed ground state with predominant 5f character is found for Pu. The complexes exhibit a linear coordination geometry and high S10 symmetry except for Pu(CpiPr5)2 and Am(CpiPr5)2, which are found to be bent by 11 and 12°, respectively. Absorption spectra are simulated with time-dependent density functional theory (TD-DFT) and compared to experimental spectra of known tris(C4H4SiMe3) and tris(C5H3(SiMe3)2) compounds [ J. Am. Chem. Soc. 2015 , 137 , 369 - 382 . DOI: 10.1021/ja510831n ] as well as recently synthesized divalent lanthanide analogs Dy(CpiPr5)2 and Tb(CpiPr5)2 [ J. Am. Chem. Soc. , 2019 , 141 , 12967 - 12973 . DOI: 10.1021/jacs.9b05816 ]. Thermodynamic stability is assessed by calculation of adiabatic reduction potentials of the trivalent precursors [An(CpiPr5)2]+, and the feasibility of further reduction to obtain as yet unknown monovalent molecular actinide complexes is discussed.

Synthesis and Magnetism of Neutral, Linear Metallocene Complexes of Terbium(II) and Dysprosium(II).

The divalent metallocene complexes Ln(CpiPr5)2 (Ln = Tb, Dy) were synthesized through the KC8 reduction of Ln(CpiPr5)2I intermediates and represent the first examples of neutral, linear metallocenes for these elements. X-ray diffraction analysis, density functional theory calculations, and magnetic susceptibility measurements indicate a 4fn5d1 electron configuration with strong s/d mixing that supports the linear coordination geometry. A comparison of the magnetic relaxation behavior of the two divalent metallocenes relative to salts of their trivalent counterparts, [Ln(CpiPr5)2][B(C6F5)4], reveals that lanthanide reduction has opposing effects for dysprosium and terbium, with magnetic relaxation times increasing from TbIII to TbII and decreasing from DyIII to DyII. The impact of this effect is most notably evident for Tb(CpiPr5)2, which displays an effective thermal barrier to magnetic relaxation of 1205 cm-1 and a 100-s blocking temperature of 52 K, the highest values yet observed for any nondysprosium single-molecule magnet.

Brominated Luciferins Are Versatile Bioluminescent Probes.

We report a set of brominated luciferins for bioluminescence imaging. These regioisomeric scaffolds were accessed by using a common synthetic route. All analogues produced light with firefly luciferase, although varying levels of emission were observed. Differences in photon output were analyzed by computation and photophysical measurements. The brightest brominated luciferin was further evaluated in cell and animal models. At low doses, the analogue outperformed the native substrate in cells. The remaining luciferins, although weak emitters with firefly luciferase, were inherently capable of light production and thus potential substrates for orthogonal mutant enzymes.