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Singlet–triplet (ST) gaps are key descriptors of carbenes, because their properties and reactivity are strongly spin-dependent. However, the theoretical prediction of ST gaps is challenging and generally thought to require elaborate correlated wave function methods or double-hybrid density functionals. By evaluating two recent test sets of arylcarbenes (AC12 and AC18), we show that local hybrid functionals based on the “common t” local mixing function (LMF) model achieve mean absolute errors below 1 kcal/mol at a computational cost only slightly higher than that of global hybrid functionals. An analysis of correlation contributions to the ST gaps suggests that the accuracy of the common t-LMF model is mainly due to an improved description of nondynamical correlation which, unlike exchange, is not additive in each spin-channel. Although spin-nonadditivity can be achieved using the local spin polarization alone, using the “common”, i.e., spin-unresolved, iso-orbital indicator t for constructing the LMF is found to be critical for consistent accuracy in ST gaps of arylcarbenes. The results support the view of LHs as vehicles to improve the description of nondynamical correlation rather than sophisticated exchange mixing approaches. 

Treatment of the scandium(II) metallocene Cpttt2Sc (Cpttt = C5H2tBu3) with CO or the isocyanide CNXyl (Xyl = C6H3Me2-2,6) yields the carbonyl complex Cpttt2Sc(CO), 1, or the isocyanide complex Cpttt2Sc(CNXyl), 2, which were identified by X-ray crystallography. Isotopic labeling with 13CO shows the CO stretch of 1 at 1875 cm−1 shifts to 1838 cm−1 in 1-13CO. The CN stretch in 2 is shifted to 1939 cm−1 compared to 2118 cm−1 for the free isocyanide. The 80.1 MHz (28.7 G) 45Sc hyperfine coupling in 1 and 74.7 MHz (26.8 G) in 2 are similar to the 82.6 MHz (29.6 G) coupling constant in Cpttt2Sc and indicate that 1 and 2 are Sc(II) complexes. A comprehensive analysis of the electronic structures of 1 and 2 using DFT calculations is reported. 

To expand the range of donor atoms known to stabilize 4fⁿ5d¹Ln(ii) ions beyond C, N, and O first row main group donor atoms, the Ln(iii) terphenylthiolate iodides, Ln^III(SAr^iPr6)₂I (Ar^iPr6= C₆H₃-2,6-(C₆H₂-2,4,6-ⁱPr₃)₂, Ln = La, Nd) were reduced to Ln^II(SAr^iPr6)₂complexes. 

Computational studies of the coordination chemistry and bonding of lanthanides have grown in recent decades as the need for understanding the distinct physical, optical, and magnetic properties of these compounds increased. Density functional theory (DFT) methods offer a favorable balance of computational cost and accuracy in lanthanide chemistry and have helped to advance the discovery of novel oxidation states and electronic configurations. This Frontier article examines the scope and limitations of DFT in interpreting structural and spectroscopic data of low-valent lanthanide complexes, elucidating periodic trends, and predicting their properties and reactivity, presented through selected examples. 

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

The synthesis of previously unknown bis(cyclopentadienyl) complexes of the first transition metal, i.e., Sc(II) scandocene complexes, has been investigated using C5H2(tBu)3 (Cpttt), C5Me5 (Cp*), and C5H3(SiMe3)2 (Cp″) ligands. Cpttt 2ScI, 1, formed from ScI3 and KCpttt, can be reduced with potassium graphite (KC8) in hexanes to generate dark-red crystals of the first crystallographically characterizable bis(cyclopentadienyl) scandium(II) complex, Cpttt 2Sc, 2. Complex 2 has a 170.6° (ring centroid)-Sc-(ring centroid) angle and exhibits an eight-line EPR spectrum characteristic of Sc(II) with Aiso = 82.6 MHz (29.6 G). It sublimes at 200 °C at 10−4 Torr and has a melting point of 268−271 °C. Reductions of Cp*2ScI and Cp″2ScI under analogous conditions in hexanes did not provide new Sc(II) complexes, and reduction of Cp*2ScI in benzene formed the Sc(III) phenyl complex, Cp*2Sc(C6H5), 3, by C−H bond activation. However, in Et2O and toluene, reduction of Cp*2ScI at −78 °C gives a dark-red solution, 4, which displays an eight-line EPR pattern like that of 1, but it did not provide thermally stable crystals. Reduction of Cp″2ScI, in THF or Et2O at −35 °C in the presence of 2.2.2-cryptand, yields the green Sc(II) metallocene iodide complex, [K(crypt)][Cp″2ScI], 5, which was identified by X-ray crystallography and EPR spectroscopy and is thermally unstable. The analogous reaction of Cp*2ScI with KC8 and 18-crown-6 in Et2O gave the ligand redistribution product, [Cp*2Sc(18- crown-6-κ2O,O′)][Cp*2ScI2], 6, as the only crystalline product. Density functional theory 

Although there are few reduced dinitrogen complexes of scandium, this metal has revealed a new structural type in reductive dinitrogen chemistry by reduction of bis(pentamethylcyclopentadienyl) scandium halides under N2. 

The importance of the specific trialkylsilyl substituent in the cyclopentadienyl chemistry of C5H4SiR3 ligands has been demonstrated by the synthesis of low oxidation-state thorium complexes. Although the structure of the disilyl-substituted cyclopentadienyl Th(III) complex, [C5H3(SiMe3)2]3ThIII (Cp″3ThIII), was reported in 1986, no monosilyl-substituted analogues, (C5H4SiR3)3ThIII (R = alkyl, aryl), have been isolated to date, even though analogues are well known in U(III) chemistry. We now report that crystalline tris(monosilyl-substituted cyclopentadienyl) Th(III) and Th(II) complexes can be isolated when R = isopropyl, i.e., using the (triisopropylsilyl)cyclopentadienyl ligand, C5H4SiiPr3 (CpTIPS). The salt metathesis reaction between three equiv of KCpTIPS and ThIVBr4(DME)2 (DME = 1,2-dimethoxyethane) afforded the colorless Th(IV) complex, CpTIPS3ThIVBr, 1, which was identified spectroscopically and crystallographically. KC8 reduction of 1 in THF produced dark blue CpTIPS3ThIII, 2, in crystalline form. The complex was identified by X-ray crystallography, EPR, and UV–visible spectroscopy in contrast to ″(C5H4SiMe3)3ThIII,″ which has never been isolated due to its instability. This Th(III) complex can be reduced further with KC8 in the presence of 2.2.2-cryptand (crypt) to make [K(crypt)][CpTIPS3ThII], 3, which is only the second crystallographically characterized Th(II) complex isolated since (Cp″3ThII)1– was discovered in 2014. Spectroscopic, crystallographic, and density functional theory (DFT) analyses are consistent with 6d1 and 6d2 electron configurations for the Th(III) and Th(II) complexes, respectively. The importance of the triisopropylsilyl substituent and the role that steric factors play in the successful isolation of Th(III) and Th(II) complexes were evaluated by Guzei solid angle calculations and electrochemical studies. The results suggest that both electronic and steric effects should be considered in the isolation of Th(III) and Th(II) complexes. 

An analytical implementation of static dipole polarizabilities within the generalized Kohn–Sham semicanonical projected random phase approximation (GKS-spRPA) method for spin-restricted closed-shell and spin-unrestricted open-shell references is presented. General second-order analytical derivatives of the GKS-spRPA energy functional are derived using a Lagrangian approach. By resolution-of-the-identity and complex frequency integration methods, an asymptotic [Formula: see text] scaling of operation count and [Formula: see text] scaling of storage is realized, i.e., the computational requirements are comparable to those for GKS-spRPA ground state energies. GKS-spRPA polarizabilities are assessed for small molecules, conjugated long-chain hydrocarbons, metallocenes, and metal clusters, by comparison against Hartree–Fock (HF), semilocal density functional approximations (DFAs), second-order Møller–Plesset perturbation theory, range-separated hybrids, and experimental data. For conjugated polydiacetylene and polybutatriene oligomers, GKS-spRPA effectively addresses the “overpolarization” problem of semilocal DFAs and the somewhat erratic behavior of post-PBE RPA polarizabilities without empirical adjustments. The ensemble averaged GKS-spRPA polarizabilities of sodium clusters (Na n for n = 2, 3, …, 10) exhibit a mean absolute deviation comparable to PBE with significantly fewer outliers than HF. In conclusion, analytical second-order derivatives of GKS-spRPA energies provide a computationally viable and consistent approach to molecular polarizabilities, including systems prohibitive for other methods due to their size and/or electronic structure.