SparseMaps—A systematic infrastructure for reduced-scaling electronic structure methods. III. Linear-scaling multireference domain-based pair natural orbital N-electron valence perturbation theory

Spinless formulation of linearized adiabatic connection approximation and its comparison with the second order N-electron valence state perturbation theory

The adiabatic connection (AC) approximation, along with its linearized variant AC0, was introduced as a method of obtaining dynamic correlation energy. When using a complete active space self-consistent field (CASSCF) wave function as a reference, the AC0 approximation is considered one of the most efficient multi-reference perturbation theories. It only involves the use of 1st- and 2nd-order reduced density matrices. However, some numerical results have indicated that the excitation energies predicted by AC0 are not as reliable as those from the second-order N-electron valence state perturbation theory (NEVPT2). In this study, we develop a spinless formulation of AC0 based on the Dyall Hamiltonian and provide a detailed comparison between AC0 and NEVPT2 approaches. We demonstrate the components within the correlation energy expressions that are common to both methods and those unique to either AC0 or NEVPT2. We investigate the role of the terms exclusive to NEVPT2 and explore the possibility of enhancing AC0's performance in this regard.

State-of-the-art local correlation methods enable affordable gold standard quantum chemistry for up to hundreds of atoms

In this feature, we review the current capabilities of local electron correlation methods up to the coupled cluster model with single, double, and perturbative triple excitations [CCSD(T)], which is a gold standard in quantum chemistry. The main computational aspects of the local method types are assessed from the perspective of applications, but the focus is kept on how to achieve chemical accuracy (i.e., <1 kcal mol-1 uncertainty), as well as on the broad scope of chemical problems made accessible. The performance of state-of-the-art methods is also compared, including the most employed DLPNO and, in particular, our local natural orbital (LNO) CCSD(T) approach. The high accuracy and efficiency of the LNO method makes chemically accurate CCSD(T) computations accessible for molecules of hundreds of atoms with resources affordable to a broad computational community (days on a single CPU and 10-100 GB of memory). Recent developments in LNO-CCSD(T) enable systematic convergence and robust error estimates even for systems of complicated electronic structure or larger size (up to 1000 atoms). The predictive power of current local CCSD(T) methods, usually at about 1-2 order of magnitude higher cost than hybrid density functional theory (DFT), has become outstanding on the palette of computational chemistry applicable for molecules of practical interest. We also review more than 50 LNO-based and other advanced local-CCSD(T) applications for realistic, large systems across molecular interactions as well as main group, transition metal, bio-, and surface chemistry. The examples show that properly executed local-CCSD(T) can contribute to binding, reaction equilibrium, rate constants, etc. which are able to match measurements within the error estimates. These applications demonstrate that modern, open-access, and broadly affordable local methods, such as LNO-CCSD(T), already enable predictive computations and atomistic insight for complicated, real-life molecular processes in realistic environments.

Non-Kekulé meta-Quinodimethane Singlet Diradicals Based on Classical N-Heterocyclic Carbenes

Diradicals based on a meta-quinodimethane (m-QDM) scaffold generally have a triplet ground state and are rather scarce. Herein, m-QDM-based non-Kekulé diradicals [3,3'-(NHC)2BP] (3-NHC) (NHC = SIPr = C{N(Dipp)CH2}2; IPr = C{N(Dipp)CH}2, Me-IPr = C{N(Dipp)CMe}2; Dipp = 2,6-iPr2C6H3; BP = 1,1'-C6H4C6H4) featuring N-heterocyclic carbene (NHC) pendants are reported as crystalline solids. The EPR spectra of 3-NHC show both allowed (Δms = 1) and forbidden (Δms = 2; 'half-field') transitions characteristic for triplet diradicals. Variable temperature EPR studies however reveal a singlet ground state for 3-SIPr. Consistent with the EPR spectra, calculations predict a remarkably small singlet-triplet energy gap (ΔEST ≤ 0.26 kcal/mol) for the 3-NHC compounds. The calculated singlet diradical character for the ground states of the 3-NHC compounds amounts to ~99 %.

Lanthanide(III) SMMs with cationic and anionic complex fragments formed by a Schiff base: structure, luminescence, magnetic properties and ab initio calculations

The syntheses, structures, luminescence and magnetic properties of a new series of Ln(III) complexes of the formula [Ln(L)(H2O)2(DMF)2][Ln(L)2] (in which H2L is N,N'-ethylaminebis[1-phenyl-3-methyl-4-formylimino-2-pyrazoline-5-one]; Ln(III) - Gd (1), Tb (2), or Dy (3) ions). The crystal structures were determined by single-crystal X-ray diffraction measurements for all the above-mentioned complexes. The crystals of these compounds consist of cationic [Ln(L)(H2O)2(DMF)2]+ and anionic [Ln(L)2] moieties which form a 3D supramolecular architecture by the H-bonds and electrostatic forces. Luminescence emission in the visible range was observed for Tb(III) and Dy(III) compounds upon ligand sensitization, with moderate quantum yields of 3.2% for the Dy complex and 24.2% for the Tb analogue. Moreover the Tb(III) complex demonstrates triboluminescence activity. The dynamic magnetization studies revealed that 1 and 2 demonstrate field-induced magnetic relaxation with effective energy barriers, ΔE|kB = 24 K (for 1) and 85 K (for 2), while the Dy complex 3 exhibits slow relaxation of magnetization in zero field with an activation energy of 256 K.

Performant automatic differentiation of local coupled cluster theories: Response properties and ab initio molecular dynamics

In this work, we introduce a differentiable implementation of the local natural orbital coupled cluster (LNO-CC) method within the automatic differentiation framework of the PySCFAD package. The implementation is comprehensively tuned for enhanced performance, which enables the calculation of first-order static response properties on medium-sized molecular systems using coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)]. We evaluate the accuracy of our method by benchmarking it against the canonical CCSD(T) reference for nuclear gradients, dipole moments, and geometry optimizations. In addition, we demonstrate the possibility of property calculations for chemically interesting systems through the computation of bond orders and Mössbauer spectroscopy parameters for a [NiFe]-hydrogenase active site model, along with the simulation of infrared spectra via ab initio LNO-CC molecular dynamics for a protonated water hexamer.

Highly entangled polyradical nanographene with coexisting strong correlation and topological frustration

Open-shell nanographenes exhibit unconventional π-magnetism arising from topological frustration or strong electron-electron interaction. However, conventional design approaches are typically limited to a single magnetic origin, which can restrict the number of correlated spins or the type of magnetic ordering in open-shell nanographenes. Here we present a design strategy that combines topological frustration and electron-electron interactions to fabricate a large fully fused 'butterfly'-shaped tetraradical nanographene on Au(111). We employ bond-resolved scanning tunnelling microscopy and spin-excitation spectroscopy to resolve the molecular backbone and reveal the strongly correlated open-shell character, respectively. This nanographene contains four unpaired electrons with both ferromagnetic and anti-ferromagnetic interactions, harbouring a many-body singlet ground state and strong multi-spin entanglement, which is well described by many-body calculations. Furthermore, we study the magnetic properties and spin states in the nanographene using a nickelocene magnetic probe. The ability to imprint and characterize many-body strongly correlated spins in polyradical nanographenes paves the way for future advancements in quantum information technologies.

Designing potentially singlet fission materials with an anti-Kasha behaviour

Singlet fission (SF) compounds offer a promising avenue for improving the performance of solar cells. Using TD-DFT methods, anti-Kasha azulene derivatives that could carry out SF have been designed. For this purpose, substituted azulenes with a donor (-OH) and/or an acceptor group (-CN) have been systematically studied using the S2 ≥ 2T1 formula. We have found that -CN (-OH) substituents on electrophilic (nucleophilic) carbons result in improved SF properties when compared to azulene.

Frozen Natural Orbitals for the State-Averaged Driven Similarity Renormalization Group

We present a reduced-cost implementation of the state-averaged driven similarity renormalization group (SA-DSRG) based on the frozen natural orbital (FNO) approach. The natural orbitals (NOs) are obtained by diagonalizing the one-body reduced density matrix from SA-DSRG second-order perturbation theory (SA-DSRG-PT2). We consider three criteria to truncate the virtual NOs for the subsequent electron correlation treatment beyond SA-DSRG-PT2. An additive second-order correction is applied to the SA-DSRG Hamiltonian to reintroduce correlation effects from the discarded orbitals. The FNO SA-DSRG method is benchmarked on 35 small organic molecules in the QUEST database. When keeping 98-99% of the cumulative occupation numbers, the mean absolute error in the vertical transition energies due to FNO is less than 0.01 eV. Using the same FNO threshold, we observe a speedup of 9 times compared to the conventional SA-DSRG implementation for nickel carbonyl with a quadruple-ζ basis set. The FNO approach enables nonperturbative SA-DSRG computations on chloroiron corrole [FeCl(C19H11N4)] with more than 1000 basis functions, surpassing the current limit of a conventional implementation.

Synthesis and Reactivity of a Dialane-Bridged Diradical

Radicals of the lightest group 13 element, boron, are well established and observed in numerous forms. In contrast to boron, radical chemistry involving the heavier group 13 elements (aluminum, gallium, indium, and thallium) remains largely underexplored, primarily attributed to the formidable synthetic challenges associated with these elements. Herein, we report the synthesis and isolation of planar and twisted conformers of a doubly CAAC (cyclic alkyl(amino)carbene)-radical-substituted dialane. Extensive characterization through spectroscopic analyses and X-ray crystallography confirms their identity, while quantum chemical calculations support their open-shell nature and provide further insights into their electronic structures. The dialane-connected diradicals exhibit high susceptibility to oxidation, as evidenced by electrochemical measurements and reactions with o-chloranil and a variety of organic azides. This study opens a previously uncharted class of dialuminum systems to study, broadening the scope of diradical chemistry and its potential applications.

Engineering Clock Transitions in Molecular Lanthanide Complexes

Molecular lanthanide (Ln) complexes are promising candidates for the development of next-generation quantum technologies. High-symmetry structures incorporating integer spin Ln ions can give rise to well-isolated crystal field quasi-doublet ground states, i.e., quantum two-level systems that may serve as the basis for magnetic qubits. Recent work has shown that symmetry lowering of the coordination environment around the Ln ion can produce an avoided crossing or clock transition within the ground doublet, leading to significantly enhanced coherence. Here, we employ single-crystal high-frequency electron paramagnetic resonance spectroscopy and high-level ab initio calculations to carry out a detailed investigation of the nine-coordinate complexes, [HoIIIL1L2], where L1 = 1,4,7,10-tetrakis(2-pyridylmethyl)-1,4,7,10-tetraaza-cyclododecane and L2 = F- (1) or [MeCN]0 (2). The pseudo-4-fold symmetry imposed by the neutral organic ligand scaffold (L1) and the apical anionic fluoride ion generates a strong axial anisotropy with an mJ = ±8 ground-state quasi-doublet in 1, where mJ denotes the projection of the J = 8 spin-orbital moment onto the ∼C4 axis. Meanwhile, off-diagonal crystal field interactions give rise to a giant 116.4 ± 1.0 GHz clock transition within this doublet. We then demonstrate targeted crystal field engineering of the clock transition by replacing F- with neutral MeCN (2), resulting in an increase in the clock transition frequency by a factor of 2.2. The experimental results are in broad agreement with quantum chemical calculations. This tunability is highly desirable because decoherence caused by second-order sensitivity to magnetic noise scales inversely with the clock transition frequency.

Improved CPS and CBS Extrapolation of PNO-CCSD(T) Energies: The MOBH35 and ISOL24 Data Sets

Computation of heats of reaction of large molecules is now feasible using the domain-based pair natural orbital (PNO)-coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] theory. However, to obtain agreement within 1 kcal/mol of experiment, it is necessary to eliminate basis set incompleteness error, which comprises both the AO basis set error and the PNO truncation error. Our investigation into the convergence to the canonical limit of PNO-CCSD(T) energies with the PNO truncation threshold T shows that errors follow the model E ( T ) = E + A T 1 / 2 . Therefore, PNO truncation errors can be eliminated using a simple two-point CPS extrapolation to the canonical limit so that subsequent CBS extrapolation is not limited by the residual PNO truncation error. Using the ISOL24 and MOBH35 data sets, we find that PNO truncation errors are larger for molecules with significant static correlation and that it is necessary to use very tight thresholds of T = 10 - 8 to ensure that errors do not exceed 1 kcal/mol. We present a lower-cost extrapolation scheme that uses information from small basis sets to estimate the PNO truncation errors for larger basis sets. In this way, the canonical limit of CCSD(T) calculations on sizable molecules with large basis sets can be reliably estimated in a practical way. Using this approach, we report near complete basis set (CBS)-CCSD(T) reaction energies for the full ISOL24 and MOBH35 data sets.

Toward Post-Hartree-Fock Accuracy for Protein-Ligand Affinities Using the Molecules-in-Molecules Fragmentation-Based Method

The complexity and size of large molecular systems, such as protein-ligand complexes, pose computational challenges for accurate post-Hartree-Fock calculations. This study delivers a thorough benchmarking of the Molecules-in-Molecules (MIM) method, presenting a clear and accessible strategy for layer/theory selections in post-Hartree-Fock computations on substantial molecular systems, notably protein-ligand complexes. An approach is articulated, enabling augmented computational efficiency by strategically canceling out common subsystem energy terms between complexes and proteins within the supermolecular equation. Employing DLPNO-based post-Hartree-Fock methods in conjunction with the three-layer MIM method (MIM3), this study demonstrates the achievement of protein-ligand binding energies with remarkable accuracy (errors <1 kcal mol-1), while significantly reducing computational costs. Furthermore, noteworthy correlations between theoretically computed interaction energies and their experimental equivalents were observed, with R2 values of approximately 0.90 and 0.78 for CDK2 and BZT-ITK sets, respectively, thus validating the efficacy of the MIM method in calculating binding energies. By highlighting the crucial role of diffuse or small Pople-style basis sets in the middle layer for reducing energy errors, this work provides valuable insights and practical methodologies for interaction energy computations in large molecular complexes and opens avenues for their application across a diverse range of molecular systems.

Organic Molecules Mimic Alkali Metals Enabling Spontaneous Harpoon Reactions with Halogens

The harpoon mechanism has been a milestone in molecular reaction dynamics. Until now, the entity from which electron harpooning occurs has been either alkali metal atoms or non-metallic analogs in their excited states. In this work, we demonstrate that a common organic molecule, octamethylcalix[4] pyrrole (omC4P), behaves just like alkali metal atoms, enabling the formation of charge-separated ionic bonding complexes with halogens omC4P+ ⋅ X- (X=F-I, SCN) via the harpoon mechanism. Their electronic structures and chemical bonding were determined by cryogenic photoelectron spectroscopy of the corresponding anions and confirmed by theoretical analyses. The omC4P+ ⋅ X- could be visualized to form from the reactants omC4P+X via electron harpooning from omC4P to X at a distance defined by the energy difference between the ionization potential of omC4P and electron affinity of X.

Unlocking Biradical Character in Diborepins

Systems that possess open- and closed-shell behavior attract significant attention from researchers due to their inherent redox and charge transport properties. Herein, we report the synthesis of the first diborepin biradicals. They display tunable biradical character based on the steric and electronic profile of the stabilizing ligand and the resulting geometric deviation of the diborepin core from planarity. While there are numerous all-carbon-based biradical systems, boron-based biradical compounds are comparatively rare, particularly ones in which the radical sites are disjointed. Calculations using density functional theory (DFT) and multireference methods demonstrate that the fused diborepin scaffold exhibits high biradical character, up to 95%. Use of a nonsterically demanding diaminocarbene promotes the planarization of the pentacyclic framework, resulting in the synthetic realization of a diborepin containing a dibora-quinoidal core, which possesses a closed-shell ground state and thermally accessible triplet state. The biradicals were structurally authenticated and characterized by both solution and solid-state electron paramagnetic resonance (EPR) spectroscopy. Half-field transitions were observed at low temperatures (about 170 K), confirming the presence of the triplet state. Initial reactivity studies of the biradicals led to the isolation and structural characterization of bis(borepin hydride) and bis(borepin dianion).

Generation, Spectroscopic Characterization, and Computational Analysis of a Six-Coordinate Cobalt(III)-Imidyl Complex with an Unusual S = 3/2 Ground State that Promotes N-Group and Hydrogen Atom-Transfer Reactions with Exogenous Substrates

We report the synthesis and characterization of a mononuclear nonheme cobalt(III)-imidyl complex, [Co(NTs)(TQA)(OTf)]+ (1), with an S = 3/2 spin state that is capable of facilitating exogenous substrate modifications. Complex 1 was generated from the reaction of CoII(TQA)(OTf)2 with PhINTs at -20 °C. A flow setup with ESI-MS detection was used to explore the kinetics of the formation, stability, and degradation pathway of 1 in solution by treating the Co(II) precursor with PhINTs. Co K-edge XAS data revealed a distinct shift in the Co K-edge compared to the Co(II) precursor, in agreement with the formation of a Co(III) intermediate. The unusual S = 3/2 spin state was proposed based on EPR, DFT, and CASSCF calculations and Co Kβ XES results. Co K-edge XAS and IR photodissociation (IRPD) spectroscopies demonstrate that 1 is a six-coordinate species, and IRPD and resonance Raman spectroscopies are consistent with 1 being exclusively the isomer with the NT ligand occupying the vacant site trans to the TQA aliphatic amine nitrogen atom. Electronic structure calculations (broken symmetry DFT and CASSCF/NEVPT2) demonstrate an S = 3/2 oxidation state resulting from the strong antiferromagnetic coupling of an •NTs spin to the high-spin S = 2 Co(III) center. Reactivity studies of 1 with PPh3 derivatives revealed its electrophilic characteristic in the nitrene-transfer reaction. While the activation of C-H bonds by 1 was proved to be kinetically challenging, 1 could oxidize weak O-H and N-H bonds. Complex 1 is, therefore, a rare example of a Co(III)-imidyl complex capable of exogenous substrate transformations.

Linear-Scaling Local Natural Orbital CCSD(T) Approach for Open-Shell Systems: Algorithms, Benchmarks, and Large-Scale Applications

The extension of the highly optimized local natural orbital (LNO) coupled cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented for high-spin open-shell molecules based on restricted open-shell references. The techniques enabling the outstanding efficiency of the closed-shell LNO-CCSD(T) variant are adopted, including the iteration- and redundancy-free second-order Møller-Plesset and (T) formulations as well as the integral-direct, memory- and disk use-economic, and OpenMP-parallel algorithms. For large molecules, the efficiency of our open-shell LNO-CCSD(T) method approaches that of its closed-shell parent method due to the application of restricted orbital sets for demanding integral transformations and a novel approximation for higher-order long-range spin-polarization effects. The accuracy of open-shell LNO-CCSD(T) is extensively tested for radicals and reactions thereof, ionization processes, as well as spin-state splittings, and transition-metal compounds. At the size range where the canonical CCSD(T) reference is accessible (up to 20-30 atoms), the average open-shell LNO-CCSD(T) correlation energies are found to be 99.9 to 99.95% accurate, which translates into average absolute deviations of a few tenths of kcal/mol in the investigated energy differences already with the default settings. For more extensive molecules, the local errors may grow, but they can be estimated and decreased via affordable systematic convergence studies. This enables the accurate modeling of large systems with complex electronic structures, as illustrated on open-shell organic radicals and transition-metal complexes of up to 179 atoms as well as on challenging biochemical systems, including up to 601 atoms and 11,000 basis functions. While the protein models involve difficulties for local approximations, such as the spin states of a bounded iron ion or an extremely delocalized singly occupied orbital, the corresponding single-node LNO-CCSD(T) computations were feasible in a matter of days with 10s to 100 GB of memory use. Therefore, the new LNO-CCSD(T) implementation enables highly accurate computations for open-shell systems of unprecedented size and complexity with widely accessible hardware.

Accurate Calculation of Isomerization and Conformational Energies of Larger Molecules Using Explicitly Correlated Local Coupled Cluster Methods in Molpro and ORCA

An overview of the approximations in the explicitly correlated local coupled cluster methods PNO-LCCSD(T)-F12 in Molpro and DLPNO-CCSD(T)F12 in ORCA is given. Options to select the domains of projected atomic orbitals (PAOs), pair natural orbitals (PNOs), and triples natural orbitals (TNOs) in both programs are described and compared in detail. The two programs are applied to compute isomerization and conformational energies of the ISOL24 and ACONFL test sets, where the former is part of the GMTKN55 benchmark suite. Thorough studies of basis set effects are presented for selected systems. These revealed large intramolecular basis set superposition effects that make it practically impossible to reliably determine the complete basis set (CBS) limits without including explicitly correlated terms. The latter strongly reduce the basis set dependence and at the same time also errors caused by the local domain approximations. On the basis of these studies, the PNO-LCCSD(T)-F12 method is applied to determine new reference energies for the above-mentioned benchmark sets. We are confident that our results should agree within a few tenths of a kcal mol-1 with the (unknown) CCSD(T)/CBS values, which therefore allowed us to define computational settings for accurate explicitly correlated local coupled cluster methods with moderate computational effort. With these protocols, especially PNO-LCCSD(T)-F12b/AVTZ', reliable reference values for comprehensive benchmark sets can be generated efficiently. This can significantly advance the development and evaluation of the performance of approximate electronic structure methods, especially improved density functional approximations or machine learning approaches.

Regime Switch in the Dual-Catalyzed Coupling of Alkyl Silicates with Aryl Bromides

Metallaphotoredox catalyzed cross-coupling of an arylbromide (Ar-Br) with an alkyl bis(catecholato)silicate (R-Si⊖ ) has been analyzed in depth using a continuum of analytical techniques (EPR, fluorine NMR, electrochemistry, photophysics) and modeling (micro-kinetics and DFT calculations). These studies converged on the impact of four control parameters consisting in the initial concentrations of the iridium photocatalyst ([Ir]0 ), nickel precatalyst ([Ni]0 ) and silicate ([R-Si⊖ ]0 ) as well as light intensity I0 for an efficient reaction between Ar-Br and R-Si⊖ . More precisely, two regimes were found to be possibly at play. The first one relies on an equimolar consumption of Ar-Br with R-Si⊖ smoothly leading to Ar-R, with no side-product from R-Si⊖ and a second one in which R-Si⊖ is simultaneously coupled to Ar-Br and degraded to R-H. This integrative approach could serve as a case study for the investigation of other metallaphotoredox catalysis manifolds of synthetic significance.

Bis(9-Boraphenanthrene) and Its Stable Biradical

Selective and site-specific boron-doping of polycyclic aromatic hydrocarbon frameworks often give rise to redox and/or photophysical properties that are not easily accessible with the analogous all-carbon systems. Herein, we report ligand-mediated control of boraphenanthrene closed- and open-shell electronic states, which has led to the first structurally characterized examples of neutral bis(9-boraphenanthrene) (2-3) and its corresponding biradical (4). Notably, compounds 2 and 3 show intramolecular charge transfer absorption from the 9-boraphenanthrene units to p-quinodimethane, exhibiting dual (red-shifted) emission in solution due to excited state conjugation enhancement (ESCE). Moreover, while boron-centered monoradicals are ubiquitous, biradical 4 represents a rare type of open-shell singlet compound with 95% biradical character, among the highest of any reported boron-based polycyclic species with two radical sites.

Polyhedral Dicobaltadithiaboranes and Dicobaltdiselenaboranes as Examples of Bimetallic Nido Structures without Bridging Hydrogens

The geometries and energetics of the n-vertex polyhedral dicobaltadithiaboranes and dicobaltadiselenaboranes Cp2Co2E2Bn−4Hn−4 (E = S, Se; n = 8 to 12) have been investigated via the density functional theory. Most of the lowest-energy structures in these systems are generated from the (n + 1)-vertex most spherical closo deltahedra by removal of a single vertex, leading to a tetragonal, pentagonal, or hexagonal face depending on the degree of the vertex removed. In all of these low-energy structures, the chalcogen atoms are located at the vertices of the non-triangular face. Alternatively, the central polyhedron in most of the 12-vertex structures can be derived from a Co2E2B8 icosahedron with adjacent chalcogen (E) vertices by breaking the E–E edge and 1 or more E–B edges to create a hexagonal face. Examples of the arachno polyhedra with two tetragonal and/or pentagonal faces derived from the removal of two vertices from isocloso deltahedra were found among the set of lowest-energy Cp2Co2E2Bn−4Hn−4 (E = S, Se; n = 8 and 12) structures.

Aromaticity in P8 allotropes and (CH)8 analogues: significance of their 40 valence electrons?

The currently unknown phosphorus allotrope P₈ is of interest since its 40 total valence electrons is a "magic number" corresponding to a filled 1S²1P⁶1D¹⁰1S²1F¹⁴2P⁶ shell such as found in the relatively stable main group element clusters Al₁₃^- and Ge₉^4-. However, P₈ still remains as an elusive structure not realized experimentally. The lowest energy P₈ structure by a margin of ∼9 kcal mol^-1 is shown by density functional theory to be a cuneane analogue with no PP double bonds and two each of P₅, P₄, and P₃ rings. Higher energy P₈ structures are polycyclic systems having at most a single PP double bond. These P₈ systems are not "carbon copies" of the corresponding (CH)₈ hydrocarbons with exactly one hydrogen atom bonded to each carbon atom. Thus the lowest energy (CH)₈ structure is cyclooctatetraene with four CC bonds followed by benzocyclobutene with three CC bonds. The cuneane (CH)₈ structure is a relatively high energy isomer lying ∼36 kcal mol^-1 above cyclooctatetraene. The cubane P₈ and (CH)₈ structures are even higher energy structures, lying ∼37 and ∼74 kcal mol^-1 in energy above the corresponding global minima. Our results demonstrate differences in medium sized aggregates of elemental phosphorus and isolobal hydrocarbon species.

The magnetic properties of MAl4(OH)12SO4·3H2O with M = Co2+, Ni2+, and Cu2+ determined by a combined experimental and computational approach

The magnetic properties of the nickelalumite-type layered double hydroxides (LDH), MAl₄(OH)₁₂(SO₄)·3H₂O (MAl₄-LDH) with M = Co²⁺ (S = 3/2), Ni²⁺ (S = 1), or Cu²⁺ (S = 1/2) were determined by a combined experimental and computational approach. They represent three new inorganic, low-dimensional magnetic systems with a defect-free, structurally ordered magnetic lattice. They exhibit no sign of magnetic ordering down to 2 K in contrast to conventional hydrotalcite LDH. Detailed insight into the complex interplay between the choice of magnetic ion (M²⁺) and magnetic properties was obtained by a combination of magnetic susceptibility, heat capacity, neutron scattering, solid-state NMR spectroscopy, and first-principles calculations. The NiAl₄- and especially CoAl₄-LDH have pronounced zero-field splitting (ZFS, easy-axis and easy-plane, respectively) and weak ferromagnetic nearest-neighbour interactions. Thus, they are rare examples of predominantly zero-dimensional spin systems in dense, inorganic matrices. In contrast, CuAl₄-LDH (S = 1/2) consists of weakly ferromagnetic S = 1/2 spin chains. For all three MAl₄-LDH, good agreement is found between the experimental magnetic parameters (J, D, g) and first-principles quantum chemical calculations, which also predict that the interchain couplings are extremely weak (< 0.1 cm^-1). Thus, our approach will be valuable for evaluation and prediction of magnetic properties in other inorganic materials.

Tailored Charge Transfer Kinetics in Precursors for Organic Radical Batteries: A Joint Synthetic-Theoretical Approach

The development of sustainable energy storage devices is crucial for the transformation of our energy management. In this scope, organic batteries attracted considerable attention. To overcome the shortcomings of typically applied materials from the classes of redox-active conjugated polymers (i. e., unstable cell voltages) and soft matter-embedded stable organic radicals (i. e., low conductivity), a novel design concept was introduced, integrating such stable radicals within a conductive polymer backbone. In the present theory-driven design approach, redox-active (2,2,6,6-tetramethylpiperidin-1-yl)oxyls (TEMPOs) were incorporated in thiophene-based polymer model systems, while structure-property relationships governing the thermodynamic properties as well as the charge transfer kinetics underlying the charging and discharging processes were investigated in a systematical approach. Thereby, the impact of the substitution pattern, the length as well as the nature of the chemical linker, and the ratio of TEMPO and thiophene units was studied using state-of-the-art quantum chemical and quantum dynamical simulations for a set of six molecular model systems. Finally, two promising candidates were synthesized and electrochemically characterized, paving the way to applications in the frame of novel organic radical batteries.

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.

Thermochemical evaluation of adaptive and fixed density functional theory quadrature schemes

A systematic study is made of the accuracy and efficiency of a number of existing quadrature schemes for molecular Kohn-Sham Density-Functional Theory (DFT) using 408 molecules and 254 chemical reactions. Included are the fixed SG-x (x = 0-3) grids of Gill et al., Dasgupta, and Herbert, the 3-zone grids of Treutler and Ahlrichs, a fixed five-zone grid implemented in Molpro, and a new adaptive grid scheme. While all methods provide a systematic reduction of errors upon extension of the grid sizes, significant differences are observed in the accuracies for similar grid sizes with various approaches. For the tests in this work, the SG-x fixed grids are less suitable to achieve high accuracies in the DFT integration, while our new adaptive grid performed best among the schemes studied in this work. The extra computational time to generate the adaptive grid scales linearly with molecular size and is negligible compared with the time needed for the self-consistent field iterations for large molecules. A comparison of the grid accuracies using various density functionals shows that meta-GGA functionals need larger integration grids than GGA functionals to reach the same degree of accuracy, confirming previous investigations of the numerical stability of meta-GGA functionals. On the other hand, the grid integration errors are almost independent of the basis set, and the basis set errors are mostly much larger than the errors caused by the numerical integrations, even when using the smallest grids tested in this work.

A local pair-natural orbital-based complete-active space perturbation theory using orthogonal localized virtual molecular orbitals

The multireference second-order perturbation theory (CASPT2) is known to deliver a quantitative description of various complex electronic states. Despite its near-size-consistent nature, the applicability of the CASPT2 method to large, real-life systems is mostly hindered by large computational and storage costs for the two-external tensors, such as two-electron integrals, amplitudes, and residuum. To this end, Menezes and co-workers developed a reduced-scaling CASPT2 scheme by incorporating the local pair-natural orbital (PNO) representation of the many-body wave functions using non-orthonormal projected atomic orbitals (PAOs) into the CASPT theory [F. Menezes et al., J. Chem. Phys. 145, 124115 (2016)]. Alternatively, in this paper, we develop a new PNO-based CASPT2 scheme using the orthonormal localized virtual molecular orbitals (LVMOs) and assess its performance and accuracy in comparison with the conventional PAO-based counterpart. Albeit the compactness, the LVMOs were considered to perform somewhat poorly compared to PAOs in the local correlation framework because they caused enormously large orbital domains. In this work, we show that the size of LVMO domains can be rendered comparable to or even smaller than that of PAOs by the use of the differential overlap integrals for domain construction. Optimality of the MOs from the CASSCF treatment is a key to reducing the LVMO domain size for the multireference case. Due to the augmented Hessian-based localization algorithm, an additional computational cost for obtaining the LVMOs is relatively minor. We demonstrate that the LVMO-based PNO-CASPT2 method is routinely applicable to large, real-life molecules such as Menshutkin SN2 reaction in a single-walled carbon nanotube reaction field.

Assessing the Experimental Hydrogen Bonding Energy of the Cyclic Water Dimer Transition State with a Cyclooctatetraene-Based Molecular Balance

We have conducted an experimental and computational study of cyclooctatetraene-1,4/1,6-dimethanol (1,4 and 1,6) as a molecular balance with the goal in mind to determine the otherwise inaccessible hydrogen bonding energy (HBE) of the cyclic water dimer, which constitutes a transition state. The 1,4/1,6 folding equilibrium is governed by an intramolecular hydrogen bond in the folded 1,6-isomer, in which the OH groups adopt a cyclic planar geometry, akin to the structure of the cyclic water dimer transition state. We characterized hydrogen bonding in 1,6 and reference complexes utilizing SAPT2 + (3)δMP2/aug-cc-pVTZ and selected quantum theory of atoms in molecule descriptors at M06-2XD3(0)/ma-def2-TZVPP. Additionally, we computed HBEs at the DLPNO-CCSD(T)/aug-cc-pVQZ level of theory. We find that hydrogen bonding in 1,6 is very similar to the interaction in the C_i symmetric cyclic water dimer TS, both in magnitude and character. We experimentally determined the Gibbs free energy of the folding process (ΔG_eq) in a variety of organic solvents via nuclear magnetic resonance spectroscopy measurements at room temperature. By combining experimentally obtained ΔG_eq values with corrections derived from accurate computational methods, we provide estimates for the HBE of cyclic water dimers and the cyclic water dimer TS, as the most stable cyclic water dimer.

Pair Natural Orbital Equation-of-Motion Coupled-Cluster Method for Core Binding Energies: Theory, Implementation, and Benchmark

We present the theory and implementation of a lower scaling core-valence separated equation-of-motion coupled-cluster approach based on domain-based local pair natural orbitals for core binding energies. The accuracy of the new method has been compared with that of the standard equation-of-motion coupled-cluster method and experimentally measured results. The use of pair natural orbitals significantly reduces the computation cost and can be applied to large molecules.

The nature of metal-metal bonding in Re-, Ru- and Os-corrole dimers

Studies of multiple bonding between transition metal complexes offer fundamental insight into the nature of bonding between metal ions and facilitate predictions of the physical properties and the reactivities of metal complexes containing metal-metal multiple bonds. Here we report a computational interrogation on the nature of the metal-metal bonding for neutral, oxidized, and reduced forms of dinuclear rhenium and osmium corrole complexes, [{Re[TpXPC]}2]0/1+/1- and [{Os[TpXPC]}2]0/1+/1-, using a complete active space self-consistent (CASSCF) methodology and density functional theory (DFT) calculations. For [{Re[TpXPC]}2]0, [{Ru[TpXPC]}2]0, and [{Os[TpXPC]}2]0, CASSCF calculations shows that the effective bond order is 3.29, 2.63, and 2.73, respectively. On their oxidized forms, [{Re[TpXPC]}2]1+, [{Ru[TpXPC]}2]1+, and [{Os[TpXPC]}2]1+ molecules, the results indicate an electron removal from a ligand-based orbital, where [{Re[TpXPC]}2]1+ gives slightly different geometry from its neutral form due to populating the δ* orbital. In this regard, the CASSCF calculations give an effective bond order of 3.25 which is slightly lower than in the [{Re[TpXPC]}2]0. On their reduced forms, the electron addition appears to be in the metal-based orbital for [{Re[TpXPC]}2]1- and [{Ru[TpXPC]}2]1- whereas in the ligand-based orbital for the Os-analogue which has no effect on the Os-Os bonding, an effective bond order of 3.18 and 2.17 is presented for the [{Re[TpXPC]}2]1- and [{Ru[TpXPC]}2]1-, respectively, within the CASSCF simulations. These results will further encourage theoreticians and experimentalists to design metalloporphyrin dimers with distinct metal-metal bonding.

A combined first- and second-order optimization method for improving convergence of Hartree–Fock and Kohn–Sham calculations

In this work, we investigate the optimization of Hartree–Fock (HF) orbitals with our recently proposed combined first- and second-order (SO-SCI) method, which was originally developed for multi-configuration self-consistent field (MCSCF) and complete active space SCF (CASSCF) calculations. In MCSCF/CASSCF, it unites a second-order optimization of the active orbitals with a Fock-based first-order treatment of the remaining closed-virtual orbital rotations. In the case of the single-determinant wavefunctions, the active space is replaced by a preselected “second-order domain,” and all rotations involving orbitals in this subspace are treated at second-order. The method has been implemented for spin-restricted and spin-unrestricted Hartree–Fock (RHF, UHF), configuration-averaged Hartree–Fock (CAHF), as well as Kohn–Sham (KS) density functional theory (RKS, UKS). For each of these cases, various choices of the second-order domain have been tested, and appropriate defaults are proposed. The performance of the method is demonstrated for several transition metal complexes. It is shown that the SO-SCI optimization provides faster and more robust convergence than the standard SCF procedure but requires, in many cases, even less computation time. In difficult cases, the SO-SCI method not only speeds up convergence but also avoids convergence to saddle-points. Furthermore, it helps to find spin-symmetry broken solutions in the cases of UHF or UKS. In the case of CAHF, convergence can also be significantly improved as compared to a previous SCF implementation. This is particularly important for multi-center cases with two or more equal heavy atoms. The performance is demonstrated for various two-center complexes with different lanthanide atoms.

Effect of Ligand Chain Length for Tuning of Molecular Dimensionality and Magnetic Relaxation in Redox Active Cobalt(II) EDOT Complexes (EDOT = 3,4-Ethylenedioxythiophene)

Four cobalt(II) complexes, [Co(L1)2(NCX)2(MeOH)2] (X = S (1), Se (2)) and {[Co(L2)2(NCX)2]}n (X = S (3), Se (4)) (L1 = 2,5dipyridyl-3,4,-ethylenedioxylthiophene and L2 = 2,5diethynylpyridinyl-3,4-ethylenedioxythiophene), were synthesized by incorporating ethylenedioxythiophene based redox-active luminescence ligands. All these complexes have been well characterized using single-crystal X-ray diffraction analyses, spectroscopic and magnetic investigations. Magneto-structural studies showed that 1 and 2 adopt a mononuclear structure with CoN4O2 octahedral coordination geometry while 3 and 4 have a 2D [4 x 4] rhombic grid coordination networks (CNs) where each cobalt(II) center is in a CoN6 octahedral coordination environment. Static magnetic measurements reveal that all four complexes displayed a high spin (HS) (S = 3/2) state between 2 and 280 K which was further confirmed by X-band and Q-band EPR studies. Remarkably, along with the molecular dimensionality (0D and 2D) the modification in the axial coligands lead to a significant difference in the dynamic magnetic properties of the monomers and CNs at low temperatures. All complexes display slow magnetic relaxation behavior under an external dc magnetic field. For the complexes with NCS- as coligand observed higher energy barrier for spin reversal in comparison to the complexes with NCSe- as coligand, while mononuclear complex 1 exhibited a higher energy barrier than that of CN 3. Theoretical calculations at the DFT and CASSCF level of theory have been performed to get more insight into the electronic structure and magnetic properties of all four complexes.

Post-spin crossing dynamics determine the regioselectivity in open-shell singlet biradical recombination

Comprehensive computational studies reveal unique dynamic effects in a multi-spin-state reaction that determine the regioselectivity of a biradical recombination process.

High Magnetic Anisotropy of a Square-Planar Iron-Carbene Complex

Among Earth-abundant catalyst systems, iron-carbene intermediates that perform C-C bond forming reactions such as cyclopropanation of olefins and C-H functionalization via carbene insertion are rare. Detailed descriptions of the possible electronic structures for iron-carbene bonds are imperative to obtain better mechanistic insights and enable rational catalyst design. Here, we report the first square-planar iron-carbene complex (MesPDPPh)Fe(CPh2), where [MesPDPPh]2- is the doubly deprotonated form of [2,6-bis(5-(2,4,6-trimethylphenyl)-3-phenyl-1H-pyrrol-2-yl)pyridine]. The compound was prepared via reaction of the disubstituted diazoalkane N2CPh2 with (MesPDPPh)Fe(thf) and represents a rare example of a structurally characterized, paramagnetic iron-carbene complex. Temperature-dependent magnetic susceptibility measurements and applied-field Mössbauer spectroscopic studies revealed an orbitally near-degenerate S = 1 ground state with large unquenched orbital angular momentum resulting in high magnetic anisotropy. Spin-Hamiltonian analysis indicated that this S = 1 spin system has uniaxial magnetic properties arising from a ground MS = ±1 non-Kramers doublet that is well-separated from the MS = 0 sublevel due to very large axial zero-field splitting (D = -195 cm-1, E/D = 0.02 estimated from magnetic susceptibility data). This remarkable electronic structure gives rise to a very large, positive magnetic hyperfine field of more than +60 T for the 57Fe nucleus along the easy magnetization axis observed by Mössbauer spectroscopy. Computational analysis with complete active space self-consistent field (CASSCF) calculations provides a detailed electronic structure analysis and confirms that (MesPDPPh)Fe(CPh2) exhibits a multiconfigurational ground state. The majority contribution originates from a configuration best described as a singlet carbene coordinated to an intermediate-spin FeII center with a (dxy)2{(dxz),(dz2)}3(dyz)1(dx2-y2)0 configuration featuring near-degenerate dxz and dz2 orbitals.

First-Principles Study of Optical Absorption Energies, Ligand Field and Spin-Hamiltonian Parameters of Cr3+ Ions in Emeralds

Herein, we study the electronic structure, energies, and vibronic structure of optical d-d transitions of Cr³⁺ ions doped in beryl (Be₃Si₆Al₂O₁₈:Cr³⁺, emerald). A computational protocol is developed that combines periodic density functional theory (for modeling of the bulk crystalline lattice of emerald) and the multireference configuration interaction complete active space self-consistent field method supplemented with n-electron valence second-order perturbation theory (for the calculation of the energy levels, wave functions, and spin-Hamiltonian and ligand-field parameters of the trigonal Cr³⁺ centers in the [CrO₆]^9- clusters embedded in an extended point charge field). Ligand-field parameters were extracted from mapping the effective ligand-field Hamiltonian onto the full many-particle Hamiltonian from one side and from a direct fit to energies of computed d-d transitions on the other side. These have been analyzed using ab initio ligand-field theory. The quality of the theoretical predictions is critically assessed through a detailed comparison with the available experimental data.

Assessment of the Accuracy of Density Functionals for Calculating Oxygen Reduction Reaction on Nitrogen-Doped Graphene

Experimental studies of the oxygen reduction reaction (ORR) at nitrogen-doped graphene electrodes have reported a remarkably low overpotential, on the order of 0.5 V, similar to Pt-based electrodes. Theoretical calculations using density functional theory have lent support to this claim. However, other measurements have indicated that transition metal impurities are actually responsible for the ORR activity, thereby raising questions about the reliability of both the experiments and the calculations. To assess the accuracy of the theoretical calculations, various generalized gradient approximation (GGA), meta-GGA, and hybrid functionals are employed here and calibrated against high-level wave-function-based coupled-cluster calculations (CCSD(T)) of the overpotential as well as self-interaction corrected density functional calculations and published quantum Monte Carlo calculations of O adatom binding to graphene. The PBE0 and HSE06 hybrid functionals are found to give more accurate results than the GGA and meta-GGA functionals, as would be expected, and for a low dopant concentration, 3.1%, the overpotential is calculated to be 1.0 V. The GGA and meta-GGA functionals give a lower estimate by as much as 0.4 V. When the dopant concentration is doubled, the overpotential calculated with hybrid functionals decreases, while it increases in GGA functional calculations. The opposite trends result from different potential-determining steps, the *OOH species being of central importance in the hybrid functional calculations, while the reduction of *O determines the overpotential obtained in GGA and meta-GGA calculations. The results presented here are mainly based on calculations of periodic representations of the system, but a comparison is also made with molecular flake models that are found to give erratic results due to finite size effects and geometric distortions during energy minimization. The presence of the electrolyte has not been taken into account explicitly in the calculations presented here but is estimated to be important for definitive calculations of the overpotential.

A Joint Venture of Ab Initio Molecular Dynamics, Coupled Cluster Electronic Structure Methods, and Liquid-State Theory to Compute Accurate Isotropic Hyperfine Constants of Nitroxide Probes in Water

The isotropic hyperfine coupling constant (HFCC, Aiso) of a pH-sensitive spin probe in a solution, HMI (2,2,3,4,5,5-hexamethylimidazolidin-1-oxyl, C9H19N2O) in water, is computed using an ensemble of state-of-the-art computational techniques and is gauged against X-band continuous wave electron paramagnetic resonance (EPR) measurement spectra at room temperature. Fundamentally, the investigation aims to delineate the cutting edge of current first-principles-based calculations of EPR parameters in aqueous solutions based on using rigorous statistical mechanics combined with correlated electronic structure techniques. In particular, the impact of solvation is described by exploiting fully atomistic, RISM integral equation, and implicit solvation approaches as offered by ab initio molecular dynamics (AIMD) of the periodic bulk solution (using the spin-polarized revPBE0-D3 hybrid functional), embedded cluster reference interaction site model integral equation theory (EC-RISM), and polarizable continuum embedding (using CPCM) of microsolvated complexes, respectively. HFCCs are obtained from efficient coupled cluster calculations (using open-shell DLPNO-CCSD theory) as well as from hybrid density functional theory (using revPBE0-D3). Re-solvation of "vertically desolvated" spin probe configuration snapshots by EC-RISM embedding is shown to provide significantly improved results compared to CPCM since only the former captures the inherent structural heterogeneity of the solvent close to the spin probe. The average values of the Aiso parameter obtained based on configurational statistics using explicit water within AIMD and from EC-RISM solvation are found to be satisfactorily close. Using either such explicit or RISM solvation in conjunction with DLPNO-CCSD calculations of the HFCCs provides an average Aiso parameter for HMI in aqueous solution at 300 K and 1 bar that is in good agreement with the experimentally determined one. The developed computational strategy is general in the sense that it can be readily applied to other spin probes of similar molecular complexity, to aqueous solutions beyond ambient conditions, as well as to other solvents in the longer run.

Modern multireference methods and their application in transition metal chemistry

Transition metal chemistry is a challenging playground for quantum chemical methods owing to the simultaneous presence of static and dynamic electron correlation effects in many systems. Wavefunction based multireference (MR) methods constitute a physically sound and systematically improvable Ansatz to deal with this complexity but suffer from some conceptual difficulties and high computational costs. The latter problem partially arises from the unfavorable scaling of the Full Configuration Interaction (Full-CI) problem which in the majority of MR methods is solved for a subset of the molecular orbital space, the so-called active space. In the last years multiple methods such as modern variants of selected CI, Full-CI Quantum Monte Carlo (FCIQMC) and the density matrix renormalization group (DMRG) have been developed that solve the Full-CI problem approximately for a fraction of the computational cost required by conventional techniques thereby significantly extending the range of applicability of modern MR methods. This perspective review outlines recent advancements in the field of MR electronic structure methods together with the resulting chances and challenges for theoretical studies in the field of transition metal chemistry. In light of its emerging importance a special focus is put on the selection of adequate active spaces and the concomitant development of numerous selection aides in recent years.

Model protein excited states: MRCI calculations with large active spaces vs CC2 method

Benchmarking calculations on excited states of models of phenylalanine protein chains are presented to assess the ability of alternative methods to the standard and most commonly used multiconfigurational wave function-based method, the complete active space self-consistent field (CASSCF), in recovering the non-dynamical correlation for systems that become not affordable by the CASSCF. The exploration of larger active spaces beyond the CASSCF limit is benchmarked through three strategies based on the reduction in the number of determinants: the restricted active space self-consistent field, the generalized active space self-consistent field (GASSCF), and the occupation-restricted multiple active space (ORMAS) schemes. The remaining dynamic correlation effects are then added by the complete active space second-order perturbation theory and by the multireference difference dedicated configuration interaction methods. In parallel, the approximate second-order coupled cluster (CC2), already proven to be successful for small building blocks of model proteins in one of our previous works [Ben Amor et al., J. Chem. Phys. 148, 184105 (2018)], is investigated to assess its performances for larger systems. Among the different alternative strategies to CASSCF, our results highlight the greatest efficiency of the GASSCF and ORMAS schemes in the systematic reduction of the configuration interaction expansion without loss of accuracy in both nature and excitation energies of both singlet ππ^* and nπ^* _CO excited states with respect to the equivalent CASSCF calculations. Guidelines for an optimum applicability of this scheme to systems requiring active spaces beyond the complete active space limit are then proposed. Finally, the extension of the CC2 method to such large systems without loss of accuracy is demonstrated, highlighting the great potential of this method to treat accurately excited states, mainly single reference, of very large systems.

iCAS: Imposed Automatic Selection and Localization of Complete Active Spaces

It is shown that in the spirit of "from fragments to molecule" for localizing molecular orbitals [J. Chem. Theory Comput. 2011, 7, 3643], a prechosen set of occupied/virtual valence/core atomic/fragmental orbitals can be transformed to an equivalent set of localized occupied/virtual pre-localized molecular orbitals (pre-LMO), which can then be taken as probes to select the same number of maximally matching localized occupied/virtual Hartree-Fock (HF) or restricted open-shell HF (ROHF) molecular orbitals as the initial local orbitals spanning the desired complete active space (CAS). In each cycle of the self-consistent field (SCF) calculation, the CASSCF orbitals can be localized by means of the noniterative "top-down least-change" algorithm for localizing ROHF orbitals [J. Chem. Phys. 2017, 146, 104104] such that the maximum matching between the orbitals of two adjacent iterations can readily be monitored, leading finally to converged localized CASSCF orbitals that overlap most the guess orbitals. Such an approach is to be dubbed as "imposed CASSCF" (iCASSCF or simply iCAS in short) for good reasons: (1) it has been assumed that only those electronic states that have largest projections onto the active space defined by the prechosen atomic/fragmental orbitals are to be targeted. This is certainly an imposed constraint but has wide applications in organic and transition metal chemistry where valence (or core) atomic/fragmental orbitals can readily be identified. (2) The selection of both initial and optimized local active orbitals is imposed from the very beginning by the pre-LMOs (which span the same space as the prechosen atomic/fragmental orbitals). Apart from the (imposed) automation and localization, iCAS has two additional merits: (a) the guess orbitals are guaranteed to be the same for all geometries, for the pre-LMOs do not change in character with geometry and (b) the use of localized orbitals facilitates the SCF convergence, particularly for large active spaces. Both organic molecules and transition-metal complexes are taken as showcases to reveal the efficacy of iCAS.

An efficient Fock space multi-reference coupled cluster method based on natural orbitals: Theory, implementation, and benchmark

We present a natural orbital-based implementation of the intermediate Hamiltonian Fock space coupled-cluster method for the (1, 1) sector of Fock space. The use of natural orbitals significantly reduces the computational cost and can automatically choose an appropriate set of active orbitals. The new method retains the charge transfer separability of the original intermediate Hamiltonian Fock space coupled-cluster method and gives excellent performance for valence, Rydberg, and charge-transfer excited states. It offers significant computational advantages over the popular equation of motion coupled cluster method for excited states dominated by single excitations.

Tetrairon(II) extended metal atom chains as single-molecule magnets

Iron-based extended metal atom chains (EMACs) are potentially high-spin molecules with axial magnetic anisotropy and thus candidate single-molecule magnets (SMMs). We herein compare the tetrairon(ii), halide-capped complexes [Fe4(tpda)3Cl2] (1Cl) and [Fe4(tpda)3Br2] (1Br), obtained by reacting iron(ii) dihalides with [Fe2(Mes)4] and N2,N6-di(pyridin-2-yl)pyridine-2,6-diamine (H2tpda) in toluene, under strictly anhydrous and anaerobic conditions (HMes = mesitylene). Detailed structural, electrochemical and Mössbauer data are presented along with direct-current (DC) and alternating-current (AC) magnetic characterizations. DC measurements revealed similar static magnetic properties for the two derivatives, with χMT at room temperature above that for independent spin carriers, but much lower at low temperature. The electronic structure of the iron(ii) ions in each derivative was explored by ab initio (CASSCF-NEVPT2-SO) calculations, which showed that the main magnetic axis of all metals is directed close to the axis of the chain. The outer metals, Fe1 and Fe4, have an easy-axis magnetic anisotropy (D = -11 to -19 cm-1, |E/D| = 0.05-0.18), while the internal metals, Fe2 and Fe3, possess weaker hard-axis anisotropy (D = 8-10 cm-1, |E/D| = 0.06-0.21). These single-ion parameters were held constant in the fitting of DC magnetic data, which revealed ferromagnetic Fe1-Fe2 and Fe3-Fe4 interactions and antiferromagnetic Fe2-Fe3 coupling. The competition between super-exchange interactions and the large, noncollinear anisotropies at metal sites results in a weakly magnetic non-Kramers doublet ground state. This explains the SMM behavior displayed by both derivatives in the AC susceptibility data, with slow magnetic relaxation in 1Br being observable even in zero static field.

Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction

Over the last decade, the second-order N-electron valence state perturbation theory (NEVPT2) has developed into a widely used multireference perturbation method. To apply NEVPT2 to systems with large active spaces, the computational bottleneck is the construction of the fourth-order reduced density matrix. Both its generation and storage become quickly problematic beyond the usual maximum active space of about 15 active orbitals. To reduce the computational cost of handling fourth-order density matrices, the cumulant approximation (CU) has been proposed in several studies. A more conventional strategy to address the higher-order density matrices is the pre-screening approximation (PS), which is the default one in the ORCA program package since 2010. In the present work, the performance of the CU, PS, and extended PS (EPS) approximations for the fourth-order density matrices is compared. Following a pedagogical introduction to NEVPT2, contraction schemes, as well as the approximations to density matrices, and the intruder state problem are discussed. The CU approximation, while potentially leading to large computational savings, virtually always leads to intruder states. With the PS approximation, the computational savings are more modest. However, in conjunction with conservative cutoffs, it produces stable results. The EPS approximation to the fourth-order density matrices can reproduce very accurate NEVPT2 results without any intruder states. However, its computational cost is not much lower than that of the canonical algorithm. Moreover, we found that a good indicator of intrude states problems in any approximation to high order density matrices is the eigenspectra of the Koopmans matrices.

Electron Attachment to Cytosine: The Role of Water

We present an EOM-CCSD-based quantum mechanical/molecular mechanical (QM/MM) study on the electron attachment process to solvated cytosine. The electron attachment in the bulk solvated cytosine occurs through a doorway mechanism, where the initial electron is localized on water. The electron is subsequently transferred to cytosine by the mixing of electronic and nuclear degrees of freedom, which occurs on an ultrafast time scale. The bulk water environment stabilizes the cytosine-bound anion by an extensive hydrogen-bond network and drastically enhances the electron transfer rate from that observed in the gas phase. Microhydration studies cannot reproduce the effect of the bulk water environment on the electron attachment process, and one needs to include a large number of water molecules in the calculation to obtain converged results. The predicted adiabatic electron affinity and electron transfer rate obtained from our QM/MM calculations are consistent with the available experimental results.

Lerf-Klinowski-type models of graphene oxide and reduced graphene oxide are robust in analyzing non-covalent functionalization with porphyrins

Graphene-based nanohybrids are good candidates for various applications. However, graphene exhibits some unwanted features such as low solubility in an aqueous solution or tendency to aggregate, limiting its potential applications. On the contrary, its derivatives, such as graphene oxide (GO) and reduced graphene oxide (RGO), have excellent properties and can be easily produced in large quantities. GO/RGO nanohybrids with porphyrins were shown to possess great potential in the field of photocatalytic hydrogen production, pollutant photodegradation, optical sensing, or drug delivery. Despite the rapid progress in experimental research on the porphyrin-graphene hybrids some fundamental questions about the structures and the interaction between components in these systems still remain open. In this work, we combine detailed experimental and theoretical studies to investigate the nature of the interaction between the GO/RGO and two metal-free porphyrins 5,10,15,20-tetrakis(4-aminophenyl) porphyrin (TAPP) and 5,10,15,20-tetrakis(4-hydroxyphenyl) porphyrin (TPPH)]. The two porphyrins form stable nanohybrids with GO/RGO support, although both porphyrins exhibited a slightly higher affinity to RGO. We validated finite, Lerf-Klinowski-type (Lerf et al. in J Phys Chem B 102:4477, 1998) structural models of GO ([Formula: see text]) and RGO ([Formula: see text]) and successfully used them in ab initio absorption spectra simulations to track back the origin of experimentally observed spectral features. We also investigated the nature of low-lying excited states with high-level wavefunction-based methods and shown that states' density becomes denser upon nanohybrid formation. The studied nanohybrids are non-emissive, and our study suggests that this is due to excited states that gain significant charge-transfer character. The presented efficient simulation protocol may ease the properties screening of new GO/RGO-nanohybrids.

Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications

A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11 000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity.