Exact decoupling of the Dirac Hamiltonian. I. General theory

Towards reliable and efficient modeling of [Cu2O2]2+-based compound electronic structures with the partially fixed reference space protocols

This work reports a computationally efficient approach for reliable modeling of complex electronic structures based on [Cu2O2]2+ moieties. Specifically, we explore the recently developed partially fixed reference space (PFRS) protocol to minimize the active space size, taking into account the double d-shell effects. We show that the ground-state electronic structure of the core [Cu2O2]2+ model system is dominated by the d9/d10 occupations. The PFRS-crafted active spaces are further used to generate the reference wave functions for the multi-reference coupled cluster, configuration interaction, and multi-reference perturbation theory calculations. Specifically, we demonstrate that the bare [Cu2O2]2+ core can be modeled qualitatively using active spaces as small as CAS(2,2)PFRS. To obtain quantitative agreement with the reference DMRG(32,62)CI calculations, the CAS(4,4) has to be used in conjunction with the MRCCSD correction on top of it. This reliable and computationally efficient protocol is further used to model the electronic structure and properties of ammonia coordinated [Cu2O2]2+ complexes. Finally, based on the large amount of available experimental data regarding the oxo-peroxo equilibrium of [Cu2O2]2+-based systems, it is possible to formulate educated guesses regarding the effect of each experimental variable over each d-occupancy-specific state. With a large sample size of d-occupancy-specific state dependence with ligands and solvents, it should be possible to propose new ligands with specific d-occupancy and, therefore, oxidative properties based on the d-occupancy energy gaps of relatively low-cost calculations.

An ab initio diabatic study of rovibronic spectra of CN

An ab initio study of the rovibronic spectra for the cyano radical (CN) based on a diabatic representation is presented. This work considers 17 electronic states, 59 dipole moment curves, 88 spin-orbit coupling curves, and 30 electronic angular momentum coupling curves, which are obtained using the internally contracted multireference configuration interaction method including the Davidson correction (icMRCI + Q) with the aug-cc-pwCV5Z-DK basis set. The diabatic transformations are performed based on a property-based diabatization method to remove the avoided crossings for the D2Π-H2Π and b4Π-24Π pairs. Ab initio potential energy curves of the X2Σ+, B2Σ+, E2Σ+, A2Π, D2Π, H2Π, F2Δ and J2Δ electronic states are shifted to match the experimental electronic excitation energies and the equilibrium internuclear distances. The coupled nuclear motion Schrödinger equations are then solved to obtain the rovibronic spectra of CN for wavenumbers from 0 to 80 000 cm-1. At wavenumbers of 0-30 000 cm-1, our absorption cross sections agree well with available theoretical data. For wavenumbers above 30 000 cm-1, our cross sections are larger than previous data in view of the fact that the transitions involving high-lying electronic states are considered. This work provides an overall prediction of the rovibronic spectrum of CN. Our results are suitable for temperatures below 8000 K and could be useful for the investigations of planetary exploration.

Relativistic Douglas-Kroll-Hess calculations of hyperfine interactions within first-principles multireference methods

A relativistic magnetic hyperfine interaction Hamiltonian based on the Douglas-Kroll-Hess (DKH) theory up to the second order is implemented within the ab initio multireference methods, including spin-orbit coupling in the Molcas/OpenMolcas package. This implementation is applied to calculate relativistic hyperfine coupling (HFC) parameters for atomic systems and diatomic radicals with valence s or d orbitals by systematically varying active space size in the restricted active space self-consistent field formalism with restricted active space state interaction for spin-orbit coupling. The DKH relativistic treatment of the hyperfine interaction reduces the Fermi contact contribution to the HFC due to the presence of kinetic factors that regularize the singularity of the Dirac delta function in the nonrelativistic Fermi contact operator. This effect is more prominent for heavier nuclei. As the active space size increases, the relativistic correction of the Fermi contact contribution converges well to the experimental data for light and moderately heavy nuclei. The relativistic correction, however, does not significantly affect the spin-dipole contribution to the hyperfine interaction. In addition to the atomic and molecular systems, the implementation is applied to calculate the relativistic HFC parameters for large trivalent and divalent Tb-based single-molecule magnets (SMMs), such as Tb(III)Pc2 and Tb(II)(CpiPr5)2 without ligand truncation using well-converged basis sets. In particular, for the divalent SMM, which has an unpaired valence 6s/5d hybrid orbital, the relativistic treatment of HFC is crucial for a proper description of the Fermi contact contribution. Even with the relativistic hyperfine Hamiltonian, the divalent SMM is shown to exhibit strong tunability of HFC via an external electric field (i.e., strong hyperfine Stark effect).

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

High-temperature spectra of the PNO molecule based on robust first-principles methods

The PNO molecule is an important species found in the interstellar medium, and its spectroscopic information is helpful for its detection. We present the first line list of PNO (X1Σ+) using robust first-principles methods. The analytical potential energy surface and the dipole moment surface were constructed based on 11 942 ab initio points. The variational nuclear motion calculation was implemented in TROVE to obtain the rovibrational energy levels, Einstein A coefficients and other parameters. The J-dependent Coriolis-decoupled Hamiltonian was adopted with k ≤ 15, and the l-type doubling was considered for the bending vibration of the linear molecule. The line list contained almost 5.87 billion transitions between 3.61 million levels with rotational excitation up to J = 200 and was used to generate the PNO spectrum below 3000 K in the wavenumber range from 0 to 6000 cm-1. The millimetre wave spectrum agrees well with available experimental benchmarks. The Fermi resonance effects in the PNO spectrum are universal and complex, resulting in significant intensity increment of the related weak transition. This line list may be helpful for the spectroscopic characterization and possible astronomical detection of PNO, especially in high-temperature environments.

Approaching the Complete Basis Set Limit for Spin-State Energetics of Mononuclear First-Row Transition Metal Complexes

Convergence to the complete basis set (CBS) limit is analyzed for the problem of spin-state energetics in mononuclear first-row transition metal (TM) complexes by taking under scrutiny a benchmark set of 18 energy differences between spin states for 13 chemically diverse TM complexes. The performance of conventional CCSD(T) and explicitly correlated CCSD(T)-F12a/b calculations in approaching the CCSD(T)/CBS limits is systematically studied. An economic computational protocol is developed based on the CCSD-F12a approximation and (here proposed) modified scaling of the perturbative triples term (T#). This computational protocol recovers the relative spin-state energetics of the benchmark set in excellent agreement with the reference CCSD(T)/CBS limits (mean absolute deviation of 0.4, mean signed deviation of 0.2, and maximum deviation of 0.8 kcal/mol) and enables performing canonical CCSD(T) calculations for mononuclear TM complexes sized up to ca. 50 atoms, which is illustrated by application to heme-related metalloporphyrins. Furthermore, a good transferability of the basis set incompleteness error (BSIE) is demonstrated for spin-state energetics computed using CCSD(T) and other wave function methods (MP2, CASPT2, CASPT2/CC, NEVPT2, and MRCI + Q), which justifies efficient focal-point approximations and simplifies the construction of multimethod benchmark studies.

Generalized Foldy-Wouthuysen transformation for relativistic two-component methods: Systematic analysis of two-component Hamiltonians

The generalized Foldy-Wouthuysen (GFW) transformation was proposed as a generic form that unifies four types of transformations in relativistic two-component methods: unnormalized GFW(UN), and normalized form 1, form 2, and form 3 (GFW(N1), GFW(N2), and GFW(N3)). The GFW transformation covers a wide range of transformations beyond the simple unitary transformation of the Dirac Hamiltonian, allowing for the systematic classification of all existing two-component methods. New two-component methods were also systematically derived based on the GFW transformation. These various two-component methods were applied to hydrogen-like and helium-like ions. Numerical errors in energy were evaluated and classified into four types: the one-electron Hamiltonian approximation, the two-electron operator approximation, the newly defined "picture difference error (PDE)," and the error in determining the transformation, and errors in multi-electron systems were discussed based on this classification.

Renormalized-Residue-Based Multireference Configuration Interaction Method for Strongly Correlated Systems

The implementation of multireference configuration interaction (MRCI) methods in quantum systems with large active spaces is hindered by the expansion of configuration bases or the intricate handling of reduced density matrices (RDMs). In this work, we present a spin-adapted renormalized-residue-based MRCI (RR-MRCI) approach that leverages renormalized residues to effectively capture the entanglement between active and inactive orbitals. This approach is reinforced by a novel efficient algorithm, which also facilitates an efficient deployment of spin-adapted matrix product state MRCI (MPS-MRCI). The RR-MRCI framework possesses several advantages: (1) It considers the orbital entanglement and utilizes highly compressed MPS structure, improving computational accuracy and efficiency compared with internally contracted (ic) MRCI. (2) Utilizing small-sized buffer environments of a few external orbitals as probes based on quantum information theory, it enhances computational efficiency over MPS-MRCI and offers potential application to large molecular systems. (3) The RR framework can be implemented in conjunction with ic-MRCI, eliminating the need for high-rank RDMs, by using distinct renormalized residues. We evaluated this method across nine diverse molecular systems, including Cu2O22+ with an active space of (24e,24o) and two complexes of lanthanide and actinide with active space (38e,36o), demonstrating the method's versatility and efficacy.

Unified construction of relativistic Hamiltonians

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

Spin-Adapted Externally Contracted Multireference Configuration Interaction Method Based on Selected Reference Configurations

As one kind of approximation of the full configuration interaction solution, the selected configuration interaction (sCI) methods have been shown to be valuable for large active spaces. However, the inclusion of dynamic correlation beyond large active spaces is necessary for more quantitative results. Since the sCI wave function can provide a compact reference for multireference methods, previously, we proposed an externally contracted multireference configuration interaction method using the sCI reference reconstructed from the density matrix renormalization group wave function [J. Chem. Theory Comput. 2018, 14, 4747-4755]. The DMRG2sCI-EC-MRCI method is promising for dealing with more than 30 active orbitals and large basis sets. However, it suffers from two drawbacks: spin contamination and low efficiency when using Slater determinant bases. To solve these problems, in this work, we adopt configuration state function bases and introduce a new algorithm based on the hybrid of tree structure for convenient configuration space management and the graphical unitary group approach for efficient matrix element calculation. The test calculation of naphthalene shows that the spin-adapted version could achieve a speed-up of 6.0 compared with the previous version based on the Slater determinant. Examples of dinuclear copper(II) compound as well as Ln(III) and An(III) complexes show that the sCI-EC-MRCI can give quantitatively accurate results by including dynamic correlation over sCI for systems with large active spaces and basis sets.

Range Separation Method for Density Functional Theory Based on Two-Electron Infinite-Order Two-Component Hamiltonian

The range separation method for density functional theory (DFT) was extended to a two-component relativistic theory based on the unitary transformation of one- and two-electron operators and a density operator. In the framework of the spin-free infinite-order two-component Hamiltonian, we implemented several types of two-electron integrals of range-separated two-electron interactions arising from the unitary transformation. Numerical assessments were performed using long-range-corrected (LC)-DFT, which utilizes the range separation of an exchange functional. The present method successfully reproduced the reference values obtained by the four-component LC-DFT calculations when the whole unitary transformations of one-electron, full-range, and range-separated two-electron operators and a density operator were considered. An efficient scheme for the unitary transformation, which is termed the local unitary transformation (LUT), was also applied to the range-separated two-electron term and other operators. The LUT method reduced the computational costs of the LC-DFT calculations significantly without any loss of accuracy.

Magnetic properties of a europium(III) complex - possible multiplet crossover

A dinuclear complex [(H2O)Zn(LH)Eu(NO3)3] containing a hexadentate Schiff-base {N2O4}-donor ligand LH2- was prepared and characterized by X-ray structural analysis and IR, electronic and fluorescence spectroscopy. DC magnetic data show that upon heating the diamagnetic complex with the ground state Eu(III)-7F0 and Zn(II)-1S switches to paramagnetic species due to the population of 7FJ (J = 1 to 6) magnetic multiplets. The magnetic susceptibility increases from zero, passes through a maximum, and then decreases upon heating. This behaviour can be explained using a spin-orbit Hamiltonian with an axial distortion term. There is an alternative interpretation of the susceptibility data based on a two-level model similar to that used in the spin crossover theory.

Toward a correct treatment of core properties with local hybrid functionals

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.

Local Oxidation States in {FeNO}6-8 Porphyrins: Insights from DMRG/CASSCF-CASPT2 Calculations

A first DMRG/CASSCF-CASPT2 study of a series of paradigmatic {FeNO}6, {FeNO}7, and {FeNO}8 heme-nitrosyl complexes has led to substantial new insight as well as uncovered key shortcomings of the DFT approach. By virtue of its balanced treatment of static and dynamic correlation, the calculations have provided some of the most authoritative information available to date on the energetics of low- versus high-spin states of different classes of heme-nitrosyl complexes. Thus, the calculations indicate low doublet-quartet gaps of 1-4 kcal/mol for {FeNO}7 complexes and high singlet-triplet gaps of ≳20 kcal/mol for both {FeNO}6 and {FeNO}8 complexes. In contrast, DFT calculations yield widely divergent spin state gaps as a function of the exchange-correlation functional. DMRG-CASSCF calculations also help calibrate DFT spin densities for {FeNO}7 complexes, pointing to those obtained from classic pure functionals as the most accurate. The general picture appears to be that nearly all the spin density of Fe[P](NO) is localized on the Fe, while the axial ligand imidazole (ImH) in Fe[P](NO)(ImH) pushes a part of the spin density onto the NO moiety. An analysis of the DMRG-CASSCF wave function in terms of localized orbitals and of the resulting configuration state functions in terms of resonance forms with varying NO(π*) occupancies has allowed us to address the longstanding question of local oxidation states in heme-nitrosyl complexes. The analysis indicates NO(neutral) resonance forms [i.e., Fe(II)-NO0 and Fe(III)-NO0] as the major contributors to both {FeNO}6 and {FeNO}7 complexes. This finding is at variance with the common formulation of {FeNO}6 hemes as Fe(II)-NO+ species but is consonant with an Fe L-edge XAS analysis by Solomon and co-workers. For the {FeNO}8 complex {Fe[P](NO)}-, our analysis suggests a resonance hybrid description: Fe(I)-NO0 ↔ Fe(II)-NO-, in agreement with earlier DFT studies. Vibrational analyses of the compounds studied indicate an imperfect but fair correlation between the NO stretching frequency and NO(π*) occupancy, highlighting the usefulness of vibrational data as a preliminary indicator of the NO oxidation state.

Unraveling the Spin-State Energetics of FeN4 Complexes with Ab Initio Methods

A systematic analysis was conducted to explore the spin-state energetics of a series of 19 FeN4 complexes. The performance of a large number of multireference methods was assessed, highlighting the significant challenges associated with accurately describing the spin-state energetics of FeN4 complexes. Most multireference methods were found to be susceptible to errors originating from the reference CASSCF wavefunction, leading to an overstabilization of high-spin states. Nonetheless, a few multireference methods, namely, CASPT2/CC, DSRG-MRPT3, and LDSRG(2), demonstrated promising performance compared to the benchmark CCSD(T) method. Furthermore, our study revealed that FeN4 complexes having a quintet ground state are exceedingly rare. Accordingly, only one specific model (Fe(L2)) and one synthesized complex (Fe(OTBP)) have the quintet ground state among the studied complexes. This scarcity of quintet FeN4 complexes highlights the unique nature of these systems and raises intriguing questions regarding the factors influencing spin states, such as the size of the macrocycle cavity, the introduction of substituents, or the induction of out-of-plane deformation.

Spin-state energetics and magnetic anisotropy in penta-coordinated Fe(III) complexes with different axial and equatorial ligand environments

The penta-coordinated trigonal-bi-pyramidal (TBP) Fe(III) complex (PMe₂Ph)₂FeCl₃ shows a reduced magnetic anisotropy in its intermediate-spin (IS) state as compared to its methyl-analog (PMe₃)₂Fe(III)Cl₃. In this work, the ligand environment in (PMe₂Ph)₂FeCl₃ is systematically altered by replacing the axial -P with -N and -As, the equatorial -Cl with other halides, and the axial methyl group with an acetyl group. This has resulted in a series of Fe(III) TBP complexes modelled in their IS and high-spin (HS) states. Lighter ligands -N and -F stabilize the complex in the HS state, while the magnetically anisotropic IS state is stabilized by -P and -As at the axial site, and -Cl, -Br, and -I at the equatorial site. Larger magnetic anisotropies appear for complexes with nearly degenerate ground electronic states that are well separated from the higher excited states. This requirement, largely controlled by the d-orbital splitting pattern due to the changing ligand field, is achieved with a certain combination of axial and equatorial ligands, such as -P and -Br, -As and -Br, and -As and -I. In most cases, the acetyl group at the axial site enhances the magnetic anisotropy compared to its methyl counterpart. In contrast, the presence of -I at the equatorial site compromises the uniaxial type of anisotropy of the Fe(III) complex leading to an enhanced rate of quantum tunneling of magnetization.

Electron Detachments of NbSin-/0 (n = 1-3) Clusters from Density Matrix Renormalization Group-CASPT2 Calculations

The electronic states of NbSi_n^-/0/+ (n = 1-3) clusters have been explored using the state-of-the-art DMRG-CASPT2 method with relatively large active spaces. The leading configurations, bond distances, vibrational frequencies, and relative energies of the low-lying states were identified. Electron detachment energies of the anionic cluster and ionization energies of the neutral clusters were reported at the DMRG-CASPT2 level. The ground states of the NbSi_n^-/0/+ (n = 1-3) clusters were predicted as the ³Δ, ⁴Π, and ⁵Π states of the linear NbSi^-/0/+, the ³A₂, ⁴B₁, and ³B₁ states of cyclic NbSi₂^-/0/+, and the ¹A', ²A', and ³A″ states of tetrahedral NbSi₃^-/0/+ isomers. The first feature in the photoelectron spectrum of NbSi^- was attributed to the transitions from the anionic ³Δ ground state to the neutral ⁴Π, ⁴Δ, and ⁴Φ states, whereas the second feature was assigned to the transitions to the neutral ²Δ, ²Σ⁺, and ²Φ states. The first band in the photoelectron spectrum of NbSi₃^- was ascribed to the transition from the anionic ¹A' ground state to the neutral 1²A' and 1²A″ states; the second band was attributed to the transitions to 2²A', 2²A″, and 3²A' states; and the third band was assigned to the transition to 3²A' states.

Hirshfeld Atom Refinement of Metal-Organic Complexes: Treatment of Hydrogen Atoms Bonded to Transition Metals

Hydrogen positions in hydrides play a key role in hydrogen storage materials and high-temperature superconductors. Our recently published study of five crystal structures of transition-metal-bound hydride complexes showed that using aspherical atomic scattering factors for Hirshfeld atom refinement (HAR) resulted in a systematic elongation of metal-hydrogen bonds compared to using spherical scattering factors with the Independent Atom Model (IAM). Even though only standard-resolution X-ray data was used, for the highest-quality data, we obtained excellent agreement between the X-ray and the neutron-derived bond lengths. We present an extended version of this study including 10 crystal structures of metal-organic complexes containing hydrogen atoms bonded to transition-metal atoms for which both X-ray and neutron data are available. The neutron structures were used as a benchmark, and the X-ray structures were refined by applying Hirshfeld atom refinement using various basis sets and DFT functionals in order to investigate the influence of the technical aspects on the length of metal-hydrogen bonds. The result of including relativistic effects in the Hamiltonian and using a cluster of multipoles simulating interactions with a crystal environment during wave function calculations was examined. The effect of the data quality on the final result was also evaluated. The study confirms that a high quality of experimental data is the key factor allowing us to obtain significant improvement in transition metal (TM)-hydrogen bond lengths from HAR in comparison with the IAM. Individual adjustments and better choices of the basis set can improve hydrogen positions. Average differences between TM-H bond lengths obtained with various DFT functionals upon including relativistic effects or between double-ζ and triple-ζ basis sets were not statistically significant. However, if all bonds formed by H atoms were considered, significant differences caused by different refinement strategies were observed. Finally, we examined the refinement of atomic thermal motions. Anisotropic refinement of hydrogen thermal motions with HAR was feasible only in some cases, and isotropically refined hydrogen thermal motions were in similar agreement with neutron values whether obtained with HAR or with the IAM.

A configuration interaction correction on top of pair coupled cluster doubles

Numerous numerical studies have shown that geminal-based methods are a promising direction to model strongly correlated systems with low computational costs. Several strategies have been introduced to capture the missing dynamical correlation effects, which typically exploit a posteriori corrections to account for correlation effects associated with broken-pair states or inter-geminal correlations. In this article, we scrutinize the accuracy of the pair coupled cluster doubles (pCCD) method extended by configuration interaction (CI) theory. Specifically, we benchmark various CI models, including, at most double excitations against selected CC corrections as well as conventional single-reference CC methods. A simple Davidson correction is also tested. The accuracy of the proposed pCCD-CI approaches is assessed for challenging small model systems such as the N₂ and F₂ dimers and various di- and triatomic actinide-containing compounds. In general, the proposed CI methods considerably improve spectroscopic constants compared to the conventional CCSD approach, provided a Davidson correction is included in the theoretical model. At the same time, their accuracy lies between those of the linearized frozen pCCD and frozen pCCD variants.

High-precision cavity-enhanced spectroscopy for studying the H2-Ar collisions and interactions

Information about molecular collisions is encoded in the shapes of collision-perturbed molecular resonances. This connection between molecular interactions and line shapes is most clearly seen in simple systems, such as the molecular hydrogen perturbed by a noble gas atom. We study the H₂-Ar system by means of highly accurate absorption spectroscopy and ab initio calculations. On the one hand, we use the cavity-ring-down-spectroscopy technique to record the shapes of the S(1) 3-0 line of molecular hydrogen perturbed by argon. On the other hand, we simulate the shapes of this line using ab initio quantum-scattering calculations performed on our accurate H₂-Ar potential energy surface (PES). In order to validate the PES and the methodology of quantum-scattering calculations separately from the model of velocity-changing collisions, we measured the spectra in experimental conditions in which the influence of the latter is relatively minor. In these conditions, our theoretical collision-perturbed line shapes reproduce the raw experimental spectra at the percent level. However, the collisional shift, δ₀, differs from the experimental value by 20%. Compared to other line-shape parameters, collisional shift displays much higher sensitivity to various technical aspects of the computational methodology. We identify the contributors to this large error and find the inaccuracies of the PES to be the dominant factor. With regard to the quantum scattering methodology, we demonstrate that treating the centrifugal distortion in a simple, approximate manner is sufficient to obtain the percent-level accuracy of collisional spectra.

Active Thermochemical Tables: Enthalpies of Formation of Bromo- and Iodo-Methanes, Ethenes and Ethynes

The thermochemistry of halocarbon species containing iodine and bromine is examined through an extensive interplay between new Feller-Peterson-Dixon (FPD) style composite methods and a detailed analysis of all available experimental and theoretical determinations using the thermochemical network that underlies the Active Thermochemical Tables (ATcT). From the computational viewpoint, a slower convergence of the components of composite thermochemistry methods is observed relative to species that solely contain first row elements, leading to a higher computational expense for achieving comparable levels of accuracy. Potential systematic sources of computational uncertainty are investigated, and, not surprisingly, spin-orbit coupling is found to be a critical component, particularly for iodine containing molecular species. The ATcT analysis of available experimental and theoretical determinations indicates that prior theoretical determinations have significantly larger uncertainties than originally reported, particularly in cases where molecular spin-orbit effects were ignored. Accurate and reliable heats of formation are reported for 38 halogen containing systems, based on combining the current computations with previous experimental and theoretical work via the ATcT approach.

Simulating Spin-Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

Multireference Wavefunction-Based Investigation of the Ground and Excited States of LrF and LrO

Complete active space self-consistent field (CASSCF) and multireference configuration interaction with Davidson correction (MRCI+Q) calculations have been carried out for lawrencium fluoride (LrF) and lawrencium oxide (LrO) molecules, detailing 19 and 20 electronic states for LrF and LrO, respectively. For LrF, two dissociation channels were considered, Lr(2P)+F(2P) and Lr(2D)+F(2P). However, due to the more complex electronic manifold of LrO, three dissociation channels were computed: Lr(2P)+O(3P), Lr(2D)+O(3P), and Lr(2P)+O(1D). In addition, equilibrium bond lengths, harmonic vibrational frequencies ωe, anharmonicity constants ωeχe, ΔG1/2 values, and excitation energies Te for the ground and several excited electronic states were calculated for both molecules, for the first time. Bond dissociation energies (BDEs) were calculated for LrF and LrO using several different levels of theory: unrestricted coupled-cluster with single, double, and perturbative triple excitations (UCCSD(T)), density functional theory (B3LYP, TPSS, M06-L, and PBE), and the correlation-consistent composite approach developed for f-elements (f-ccCA).

Population analysis and the effects of Gaussian basis set quality and quantum mechanical approach: main group through heavy element species

Atomic charge and its distribution across molecules provide important insight into chemical behavior. Though there are many studies on various routes for the determination of atomic charge, there are few studies that examine the broader impact of basis set and quantum method used over many types of population analysis methods across the periodic table. Largely, such a study of population analysis has focused on main-group species. In this work, atomic charges were calculated using several population analysis methods including orbital-based methods (Mulliken, Löwdin, and Natural Population Analysis), volume-based methods (Atoms-in-Molecules (AIM) and Hirshfeld), and potential derived charges (CHELP, CHELPG, and Merz-Kollman). The impact of basis set and quantum mechanical method choices upon population analysis has been considered. The basis sets utilized include Pople (6-21G**, 6-31G**, 6-311G**) and Dunning (cc-pVnZ, aug-cc-pVnZ; n = D, T, Q, 5) basis sets for main group molecules. For the transition metal and heavy element species examined, relativistic forms of the correlation consistent basis sets were used. This is the first time the cc-pVnZ-DK3 and cc-pwCVnZ-DK3 basis sets have been examined with respect to their behavior across all levels of basis sets for atomic charges for an actinide. The quantum methods chosen include two density functional (PBE0 and B3LYP), Hartree-Fock, and second-order Møller-Plesset perturbation theory (MP2) approaches.

Multiconfiguration Pair-Density Functional Theory for Chromium(IV) Molecular Qubits

Pseudotetrahedral organometallic complexes containing chromium(IV) and aryl ligands have been experimentally identified as promising molecular qubit candidates. Here we present a computational protocol based on multiconfiguration pair-density functional theory for computing singlet-triplet gaps and zero-field splitting (ZFS) parameters in Cr(IV) aryl complexes. We find that two multireference methods, multistate complete active space second-order perturbation theory (MS-CASPT2) and hybrid multistate pair-density functional theory (HMS-PDFT), perform better than Kohn-Sham density functional theory for singlet-triplet gaps. Despite the very small values of the ZFS parameters, both multireference methods performed qualitatively well. MS-CASPT2 and HMS-PDFT performed particularly well for predicting the trend in the ratio of the rhombic and axial ZFS parameters, |E/D|. We have also investigated the dependence and sensitivity of the calculated ZFS parameters on the active space and the molecular geometry. The methodologies outlined here can guide future prediction of ZFS parameters in molecular qubit candidates.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10^-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnF_n, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Accurate computed singlet-triplet energy differences for cobalt systems: implication for two-state reactivity

Accurate singlet-triplet energy differences for cobalt and rhodium complexes were calculated by using several wave function methods, such as MRCISD, CASPT2, CCSD(T) and BCCD(T). Relaxed energy differences were obtained by considering the singlet and triplet complexes, each at the minimum of their potential energy surfaces. Active spaces for multireference calculations were carefully checked to provide accurate results. The considered systems are built by increasing progressively the first coordination sphere around the metal. We included in our set two CpCoX complexes (Cp = cyclopentadienyl, X = alkenyl ligand), which have been suggested as intermediates in cycloaddition reactions. Indeed, cobalt systems have been used for more than a decade as active species in this kind of transformations, for which a two-state reactivity has been proposed. Most of the considered systems display a triplet ground state. However, in the case of a reaction intermediate, while a triplet ground state was predicted on the basis of Density Functional Theory results, our calculations suggest a singlet ground state. This stems from the competition between the exchange term (stabilising the triplet) and the accessibility of an intramolecular coordination (stabilising the singlet). This finding has an impact on the general mechanism of the cycloaddition reaction. Analogous rhodium systems were also studied and, as expected, they have a larger tendency to electron pairing than cobalt species.

A new generation of effective core potentials from correlated and spin-orbit calculations: selected heavy elements

We introduce new correlation consistent effective core potentials (ccECPs) for the elements I, Te, Bi, Ag, Au, Pd, Ir, Mo, and W with $4d$, $5d$, $6s$ and $6p$ valence spaces. These ccECPs are given as a sum of spin-orbit averaged relativistic effective potential (AREP) and effective spin-orbit (SO) terms. The construction involves several steps with increasing refinements from more simple to fully correlated methods. The optimizations are carried out with objective functions that include weighted many-body atomic spectra, norm-conservation criteria, and spin-orbit splittings. Transferability tests involve molecular binding curves of corresponding hydride and oxide dimers. The constructed ccECPs are systematically better and in a few cases on par with previous effective core potential (ECP) tables on all tested criteria and provide a significant increase in accuracy for valence-only calculations with these elements. Our study confirms the importance of the AREP part in determining the overall quality of the ECP even in the presence of sizable spin-orbit effects. The subsequent quantum Monte Carlo (QMC) calculations point out the importance of accurate trial wave functions which in some cases (mid series transition elements) require treatment well beyond single-reference.

The Chemistry of Mercury in the Stratosphere

Mercury, a global contaminant, enters the stratosphere through convective uplift, but its chemical cycling in the stratosphere is unknown. We report the first model of stratospheric mercury chemistry based on a novel photosensitized oxidation mechanism. We find two very distinct Hg chemical regimes in the stratosphere: in the upper stratosphere, above the ozone maximum concentration, Hg⁰ oxidation is initiated by photosensitized reactions, followed by second-step chlorine chemistry. In the lower stratosphere, ground-state Hg⁰ is oxidized by thermal reactions at much slower rates. This dichotomy arises due to the coincidence of the mercury absorption at 253.7 nm with the ozone Hartley band maximum at 254 nm. We also find that stratospheric Hg oxidation, controlled by chlorine and hydroxyl radicals, is much faster than previously assumed, but moderated by efficient photo-reduction of mercury compounds. Mercury lifetime shows a steep increase from hours in the upper-middle stratosphere to years in the lower stratosphere.

Ab Initio Composite Approaches for Heavy Element Energetics: Ionization Potentials for the Actinide Series of Elements

The first, second, and third gas-phase ionization potentials have been determined for the actinide series of elements using an ab initio composite scalar and fully relativistic approach, employing the coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) and Dirac Hartree-Fock (DHF) methods, extrapolated to the complete basis set (CBS) limit. The impact of electron correlation and basis set choice within this framework are examined. Additionally, the first three ionization potentials were obtained using an ab initio heavy element correlation-consistent Composite Approach (here referred to as α-ccCA). This is the first utilization of a ccCA for actinide species.

Intersystem Crossings in Late-Row Elements: A Perspective

Intersystem crossing (ISC), a vital component of the electronic and nuclear transitions that compose photophysics, has been successfully simulated in light elements and transition metal complexes. Derived from the Z-dependent spin-orbit coupling (SOC), ISC is expected to be of greater importance in heavier elements, but few attempts have been made at the simulation of ISC in lanthanides or actinides. In this work, we explore several of the challenges that will need to be overcome in order to treat ISC in late-row elements, including the loss of spin as a good quantum number, the need to include SOC variationally via two- or four-component electronic structure, and the high density of states present in late-row complexes. Density functional theory (DFT) calculations are used to illustrate several of these effects, while a model Hamiltonian is used to illustrate the importance of momentum rescaling in surface hopping simulations of strongly coupled states.

Exact-Two-Component Multiconfiguration Pair-Density Functional Theory

Molecules containing late-row elements exhibit large relativistic effects. To account for both relativistic effects and electron correlation in a computationally inexpensive way, we derived a formulation of multiconfiguration pair-density functional theory with the relativistic exact-two-component Hamiltonian (X2C-MC-PDFT). In this new method, relativistic effects are included during variational optimization of a reference wave function by exact-two-component complete active-space self-consistent-field (X2C-CASSCF) theory, followed by an energy evaluation using pair-density functional theory. Benchmark studies of excited-state and ground-state fine-structure splitting of atomic species show that X2C-MC-PDFT can significantly improve the X2C-CASSCF results by introducing additional state-specific electron correlation.

Simulation of Auger decay dynamics in the hard X-ray regime: HCl as a showcase

Auger decay after photoexcitation or photoemission of an electron from a deep inner shell in the hard X-ray regime can be rather complex, implying a multitude of phenomena such as multiple-step cascades, post-collision interaction (PCI), and electronic state-lifetime interference. Furthermore, in a molecule nuclear motion can also be triggered. Here we discuss a comprehensive theoretical method which allows us to analyze in great detail Auger spectra measured around an inner-shell ionization threshold. HCl photoexcited or photoionized around the deep Cl 1s threshold is chosen as a showcase. Our method allows calculating Auger cross sections considering the nature of the ground, intermediate and final states (bound or dissociative), and the evolution of the relaxation process, including both electron and nuclear dynamics. In particular, we show that we can understand and reproduce a so-called experimental 2D-map, consisting of a series of resonant Auger spectra measured at different photon energies, therefore obtaining a detailed picture of all above-mentioned dynamical phenomena at once.

Ionization Energies and Redox Potentials of Hydrated Transition Metal Ions: Evaluation of Domain-Based Local Pair Natural Orbital Coupled Cluster Approaches

Hydrated transition metal ions are prototypical systems that can be used to model properties of transition metals in complex chemical environments. These seemingly simple systems present challenges for computational chemistry and are thus crucial in evaluations of quantum chemical methods for spin-state and redox energetics. In this work, we explore the applicability of the domain-based pair natural orbital implementation of coupled cluster (DLPNO-CC) theory to the calculation of ionization energies and redox potentials for hydrated ions of all first transition row (3d) metals in the 2+/3+ oxidation states, in connection with various solvation approaches. In terms of model definition, we investigate the construction of a minimally explicitly hydrated quantum cluster with a first and second hydration layer. We report on the convergence with respect to the coupled cluster expansion and the PNO space, as well as on the role of perturbative triple excitations. A recent implementation of the conductor-like polarizable continuum model (CPCM) for the DLPNO-CC approach is employed to determine self-consistent redox potentials at the coupled cluster level. Our results establish conditions for the convergence of DLPNO-CCSD(T) energetics and stress the absolute necessity to explicitly consider the second solvation sphere even when CPCM is used. The achievable accuracy for redox potentials of a practical DLPNO-based approach is, on average, 0.13 V. Furthermore, multilayer approaches that combine a higher-level DLPNO-CCSD(T) description of the first solvation sphere with a lower-level description of the second solvation layer are investigated. The present work establishes optimal and transferable methodological choices for employing DLPNO-based coupled cluster theory, the associated CPCM implementation, and cost-efficient multilayer derivatives of the approach for open-shell transition metal systems in complex environments.

High Accuracy Transition Metal Effective Cores for the Many-Body Diffusion Monte Carlo Method

Practical applications of the real-space diffusion Monte Carlo (DMC) method require the removal of core electrons, where currently localization approximations of semilocal potentials are generally used in the projector. Accurate calculations of complex solids and large molecules demand minimizing the impact of approximated atomic cores. Prior works have shown that the errors from such approximations can be sizable in both finite and periodic systems. In this work, we show that a class of differential pseudopotentials, known as pseudo-Hamiltonians, can be constructed for the 3d transition metal atoms, entirely removing the need for any localization scheme in the DMC projector. As a proof of principle, we demonstrate the approach for the case of Co. In order to minimize errors in the pseudo-Hamiltonian at the many-body level, we generalize the recently proposed correlation-consistent pseudopotential generation scheme to successively close semilocal representations of the differential potentials. Our generation scheme successfully produces potentials tailored specifically for real space projector quantum Monte Carlo methods with low error at the many-body level, i.e., with many-body scattering properties very close to relativistic all-electron results. In particular, we show that the agreement with respect to atomic and molecular quantities reach chemical accuracy in many cases─on par with the most accurate semilocal pseudopotentials available. Further, our pseudo-Hamiltonian generation scheme utilizes standard quantum chemistry codes designed only to work with semilocal pseudopotentials, enabling straightforward generation of pseudo-Hamiltonians for additional elements in future works.

A DMRG/CASPT2 Investigation of Metallocorroles: Quantifying Ligand Noninnocence in Archetypal 3d and 4d Element Derivatives

Hybrid density functional theory (B3LYP) and density matrix renormalization group (DMRG) theory have been used to quantitatively compare the degree of ligand noninnocence (corrole radical character) in seven archetypal metallocorroles. The seven complexes, in decreasing order of corrole noninnocent character, are Mn[Cor]Cl > Fe[Cor]Cl > Fe[Cor](NO) > Mo[Cor]Cl2 > Ru[Cor](NO) ≈ Mn[Cor]Ph ≈ Fe[Cor]Ph ≈ 0, where [Cor] refers to the unsubstituted corrolato ligand. DMRG-based second-order perturbation theory calculations have also yielded detailed excited-state energetics data on the compounds, shedding light on periodic trends involving middle transition elements. Thus, whereas the ground state of Fe[Cor](NO) (S = 0) is best described as a locally S = 1/2 {FeNO}7 unit antiferromagnetically coupled to a corrole A' radical, the calculations confirm that Ru[Cor](NO) may be described as simply {RuNO}6-Cor3-, that is, having an innocent corrole macrocycle. Furthermore, whereas the ferromagnetically coupled S = 1{FeNO}7-Cor•2- state of Fe[Cor](NO) is only ∼17.5 kcal/mol higher than the S = 0 ground state, the analogous triplet state of Ru[Cor](NO) is higher by a far larger margin (37.4 kcal/mol) relative to the ground state. In the same vein, Mo[Cor]Cl2 exhibits an adiabatic doublet-quartet gap of 36.1 kcal/mol. The large energy gaps associated with metal-ligand spin coupling in Ru[Cor](NO) and Mo[Cor]Cl2 reflect the much greater covalent character of 4d-π interactions relative to analogous interactions involving 3d orbitals. As far as excited-state energetics is concerned, DMRG-CASPT2 calculations provide moderate validation for hybrid density functional theory (B3LYP) for qualitative purposes, but underscore the possibility of large errors (>10 kcal/mol) in interstate energy differences.

Calculation of the Zeeman Effect for Transition-Metal Complexes by Multiconfiguration Pair-Density Functional Theory

Spin-orbit coupling is especially critical for the description of magnetic anisotropy, electron paramagnetic resonance spectroscopy of inorganic radicals and transition-metal complexes, and intersystem crossing. Here, we show how spin-orbit coupling may be included in multiconfiguration pair-density functional theory (MC-PDFT), and we apply the resulting formulation to the calculation of magnetic g tensors (which govern the Zeeman effect) of molecules containing transition metals. MC-PDFT is an efficient method for including static and dynamic electronic correlation in the quantum mechanical treatment of molecules; here, we apply it with spin-orbit coupling by using complete active space self-consistent field (CASSCF) and complete active space configuration interaction (CASCI) wave functions and on-top density functionals. We propose a systematic CASCI scheme for the g tensor calculation of the ground state of the systems under consideration, and we show its superiority over the conventional CASSCF scheme. State interaction, which is important for degenerate and nearly degenerate states, is included by extended multi-state PDFT (XMS-PDFT). Applications are reported for the ground doublet states of 25 transition-metal complexes with d¹, d⁵, d⁷, and d⁹ configurations. The MC-PDFT methods are shown to be both efficient and accurate as compared with complete active space second-order perturbation theory.

Electronic States of CoSin-/0/+ (n = 1-3) Clusters from Density Matrix Renormalization Group-CASPT2 Calculations

Density matrix renormalization group-CASPT2 (DMRG-CASPT2), CASPT2, and density functional theory are employed to describe the complicated geometrical and electronic structures of CoSi_n^-/0/+ (n = 1-3) clusters. The active spaces of DMRG-CASPT2 are extended to 23 orbitals. The DMRG-CASPT2 method with such large active spaces is reasonable to provide highly accurate relative energies of the electronic states. The pure BP86, PBE, and TPSS functionals appear to be suitable to compute the relative energies of the electronic states of cobalt-doped silicon clusters. The leading configurations, bond distances, vibrational frequencies, normal modes, and relative energies of the electronic states are reported. The electron detachment energies of the removals of one electron from the anionic and neutral clusters are estimated. All six bands in the photoelectron spectrum of CoSi₃^- are interpreted based on the computed electron detachment energies and Franck-Condon factor simulations.

Interpretation of photoelectron spectra of CoGen- (n = 4, 5) clusters by multiconfigurational RASPT2 calculations

The low-lying electronic states CoGe_n^-/0 (n = 4, 5) have been investigated with density functional theory and the state-of-the-art RASSCF/RASPT2 method to give assignments for the anion photoelectron spectra. The BP86 functional was employed to optimize the geometrical structures of the electronic states, while the RASSCF/RASPT2 was applied to calculate the single-point energies. With the RASSCF/RASPT2 approach, the active spaces are extended to a size of 21 orbitals for CoGe₄^-/0 and 24 orbitals for CoGe₅^-/0. The ground states of CoGe₄^-/0 are determined to be ³A″ and ²A″ of a trigonal bipyramidal structure in which the Co atom is situated at the equatorial corner of the bipyramid. The vertical detachment energies of the transitions from the anionic ground state to the neutral ²A″, 1⁴A″, ²A', 2⁴A″, 3⁴A″, 1⁴A', 2⁴A', and 6⁴A″ states are evaluated to be 2.29, 2.39, 2.60, 2.83, 3.17, 3.24, 3.47, and 4.00 eV. For the CoGe₅^-/0 clusters, the ground states are computed to be ¹A₁ and 1²A₂ of an octahedral structure. The vertical detachment energies of the removal of one electron from the anionic ground state to result in the 1²A₂, 1²A₁, 2²A₁, 1²B₁, 1²B₂, 4²B₁, 4²B₂, and 6²A₂ states are estimated to be 2.16, 2.79, 2.84, 3.06, 3.06, 3.59, 3.59, and 4.22 eV. All features in the photoelectron spectra of CoGe₄^- and CoGe₅^- are interpreted based on the computed electron detachment energies of the anionic ground states.

Quantum chemical calculations of 77 Se and 125 Te nuclear magnetic resonance spectral parameters and their structural applications

An accurate quantum chemical (QC) modeling of ⁷⁷ Se and ¹²⁵ Te nuclear magnetic resonance (NMR) spectra is deeply involved in the NMR structural assignment for selenium and tellurium compounds that are of utmost importance both in organic and inorganic chemistry nowadays due to their huge application potential in many fields, like biology, medicine, and metallurgy. The main interest of this review is focused on the progress in QC computations of ⁷⁷ Se and ¹²⁵ Te NMR chemical shifts and indirect spin-spin coupling constants involving these nuclei. Different computational methodologies that have been used to simulate the NMR spectra of selenium and tellurium compounds since the middle of the 1990s are discussed with a strong emphasis on their accuracy. A special accent is placed on the calculations resorting to the relativistic methodologies, because taking into account the relativistic effects appreciably influences the precision of NMR calculations of selenium and, especially, tellurium compounds. Stereochemical applications of quantum chemical calculations of ⁷⁷ Se and ¹²⁵ Te NMR parameters are discussed so as to exemplify the importance of integrated approach of experimental and computational NMR techniques.

Direct coherent switching with decay of mixing for intersystem crossing dynamics of thioformaldehyde: The effect of decoherence

We evaluate the effect of electronic decoherence on intersystem crossing in the photodynamics of thioformaldehyde. First, we show that the state-averaged complete-active-space self-consistent field electronic structure calculations with a properly chosen active space of 12 active electrons in 10 active orbitals can predict the potential energy surfaces and the singlet-triplet spin-orbit couplings quite well for CH₂S, and we use this method for direct dynamics by coherent switching with decay of mixing (CSDM). We obtain similar dynamical results with CSDM or by adding energy-based decoherence to trajectory surface hopping, with the population of triplet states tending to a small steady-state value over 500 fs. Without decoherence, the state populations calculated by the conventional trajectory surface hopping method or the semiclassical Ehrenfest method gradually increase. This difference shows that decoherence changes the nature of the results not just quantitatively but qualitatively.

Spin-state energetics of metallocenes: How do best wave function and density functional theory results compare with the experimental data?

We benchmark the accuracy of quantum-chemical methods, including wave function theory methods [coupled cluster theory at the CCSD(T) level, multiconfigurational perturbation-theory (CASPT2, NEVPT2) and internally contracted multireference configuration interaction (MRCI)] and 30 density functional theory (DFT) approximations, in reproducing the spin-state splittings of metallocenes. The reference values of the electronic energy differences are derived from the experimental spin-crossover enthalpy for manganocene and the spectral data of singlet-triplet transitions for ruthenocene, ferrocene, and cobaltocenium. For ferrocene and cobaltocenium we revise the previous experimental interpretations regarding the lowest triplet energy; our argument is based on the comparison with the lowest singlet excitation energy and herein reported, carefully determined absorption spectrum of ferrocene. When deriving vertical energies from the experimental band maxima, we go beyond the routine vertical energy approximation by introducing vibronic corrections based on simulated vibrational envelopes. The benchmarking result confirms the high accuracy of the CCSD(T) method (in particular, for UCCSD(T) based on Hartree-Fock orbitals we find for our dataset: maximum error 0.12 eV, weighted mean absolute error 0.07 eV, weighted mean signed error 0.01 eV). The high accuracy of the single-reference method is corroborated by the analysis of a multiconfigurational character of the complete active space wave function for the triplet state of ferrocene. On the DFT side, our results confirm the non-universality problem with approximate functionals. The present study is an important step toward establishing an extensive and representative benchmark set of experiment-derived spin-state energetics for transition metal complexes.

Picture-change correction in relativistic density functional theory

Relativistic quantum chemical calculations are performed based on one of two physical pictures, namely the Dirac picture and the Schrödinger picture. With regard to the latter, the so-called picture-change effect (PCE) and picture-change correction (PCC) have been studied. The PCE, which is the change in the expectation value associated with the transformation, is not commonly a minor effect. The electron density, which is given by the expectation value of the density operator, is a fundamental variable in relativistic density functional theory (RDFT). Thus, performing the PCC in RDFT calculations is essential not only in terms of numerical agreement with the Dirac picture, but also from the viewpoint of fundamental theory. This paper explains theories and numerical studies of PCE and PCC in RDFT after overviewing those in properties, which involves the authors' works on the development of RDFT in the Schrödinger picture and relativistic exchange-correlation functionals based on picture-change-corrected variables.

Portably parallel construction of a configuration-interaction wave function from a matrix-product state using the Charm++ framework

The construction of configuration-interaction (CI) expansions from a matrix product state (MPS) involves numerous matrix operations and the skillful sampling of important configurations in a large Hilbert space. In this work, we present an efficient procedure for constructing CI expansions from MPS employing the parallel object-oriented Charm++ programming framework, upon which automatic load-balancing and object migrating facilities can be employed. This procedure was employed in the MPS-to-CI utility (Moritz et al., J. Chem. Phys. 2007, 126, 224109), the sampling-reconstructed complete active-space algorithm (SR-CAS, Boguslawski et al., J. Chem. Phys. 2011, 134, 224101), and the entanglement-driven genetic algorithm (EDGA, Luo et al., J. Chem. Theory Comput. 2017, 13, 4699). It enhances productivity and allows the sampling programs to evolve to their population-expansion versions, for example, EDGA with population expansion (PE-EDGA). Further, examples of 1,2-dioxetanone and firefly dioxetanone anion (FDO^- ) molecules demonstrated the following: (a) parallel efficiencies can be persistently improved by simply by increasing the proportions of the asynchronous executions and (b) a sampled CAS-type CI wave function of a bi-radical-state FDO^- molecule utilizing the full valence (30e,26o) active space can be constructed within a few hours with using thousands of cores.

Valence and Structure Isomerism of Al2FeO4+: Synergy of Spectroscopy and Quantum Chemistry

We provide spectroscopic and computational evidence for a substantial change in structure and gas phase reactivity of Al₃O₄⁺ upon Fe-substitution, which is correctly predicted by multireference (MR) wave function calculations. Al₃O₄⁺ exhibits a cone-like structure with a central trivalent O atom (C_3v symmetry). The replacement of the Al- by an Fe atom leads to a planar bicyclic frame with a terminal Al-O^•- radical site, accompanied by a change from the Fe^+III/O^-II to the Fe^+II/O^-I valence state. The gas phase vibrational spectrum of Al₂FeO₄⁺ is exclusively reproduced by the latter structure, which MR wave function calculations correctly identify as the most stable isomer. This isomer of Al₂FeO₄⁺ is predicted to be highly reactive with respect to C-H bond activation, very similar to Al₈O₁₂⁺ which also features the terminal Al-O^•- radical site. Density functional theory, in contrast, predicts a less reactive Al₃O₄⁺-like "isomorphous substitution" structure of Al₂FeO₄⁺ to be the most stable one, except for functionals with very high admixture of Fock exchange (50%, BHLYP).