Relativistic Effects on Electron–Nucleus Hyperfine Coupling Studied with an Exact 2‑Component (X2C) Hamiltonian

An exact 2-component (X2C) transformation of the one-electron Hamiltonian is used to transform nuclear hyperfine magnetic field operators from the 4-component Dirac picture to 2-component form. Numerical applications are concerned with hyperfine coupling constants of one-electron and many-electron atoms, as well as the HgH radical, using spin-unrestricted scalar X2C Hartree–Fock and Kohn–Sham theory. Reference data for 2-component generalized-collinear X2C calculations, including spin–orbit coupling, are also provided for selected cases. Calculations for one-electron atomic <i>n</i> s states with <i>n</i> = 1–3 show that the X2C transformed hyperfine operators give accurate hyperfine coupling constants. Kohn–Sham one-electron self-interaction errors for these states are small. The performance of the X2C transformed hyperfine operator for many-electron systems is also promising. The method is straightforward to implement in codes using spin-unrestricted (1-component) or 2-component spinor orbitals

Orbitals: Some Fiction and Some Facts

The use of electron orbitals in quantum theory and chemistry is discussed. Common misconceptions are highlighted. Suggestions are made how chemistry educators may describe orbitals in the first and second year college curriculum more accurately without introducing unwanted technicalities. A comparison is made of different ways of graphically representing orbitals. The connection of orbital delocalization and electron delocalization is explained by using graphical representations of canonical and localized molecular orbitals for water, benzene, and linear hexatriene

Longest-Wavelength Electronic Excitations of Linear Cyanines: The Role of Electron Delocalization and of Approximations in Time-Dependent Density Functional Theory

The lowest-energy/longest-wavelength electronic singlet excitation energies of linear cyanine dyes are examined, using time-dependent density functional theory (TDDFT) and selected wave function methods in comparison with literature data. Variations of the bond-length alternation obtained with different optimized structures produce small differences of the excitation energy in the limit of an infinite chain. Hybrid functionals with range-separated exchange are optimally ‘tuned’, which is shown to minimize the delocalization error (DE) in the cyanine π systems. Much unlike the case of charge-transfer excitations, small DEs are not strongly correlated with better performance. A representative cyanine is analyzed in detail. Compared with accurate benchmark data, TDDFT with ‘pure’ local functionals gives too high singlet excitation energies for all systems, but DFT-based ΔSCF calculations with a local functional severely underestimates the energies. TDDFT strongly overestimates the difference between singlet and triplet excitation energies. An analysis points to systematically much too small magnitudes of integrals from the DFT components of the exchange-correlation response kernel as the likely culprit. The findings support previous suggestions that the differential correlation energy between the ground and excited state is not correctly produced by TDDFT with most functionals

Puzzling Lack of Temperature Dependence of the PuO₂ Magnetic Susceptibility Explained According to Ab Initio Wave Function Calculations

The electronic structure and the magnetic properties of solid PuO₂ are investigated by wave function theory calculations, using a relativistic complete active space (CAS) approach including spin–orbit coupling. The experimental magnetic susceptibility is well reproduced by calculations for an embedded PuO₈^12– cluster model. The calculations indicate that the surprising lack of temperature dependence of the magnetic susceptibility χ of solid PuO₂ can be rationalized based on the properties of a single Pu⁴⁺ ion in the cubic ligand field of the surrounding oxygen ions. Below ∼300 K, the only populated state is the nonmagnetic ground state, leading to standard temperature-independent paramagnetism (TIP). Above 300 K, there is an almost perfect cancellation of temperature-dependent contributions to χ that depends delicately on the mixing of ion levels in the electronic states, their relative energies, and the magnetic coupling between them

Ligand NMR Chemical Shift Calculations for Paramagnetic Metal Complexes: 5f¹ vs 5f² Actinides

Ligand paramagnetic NMR (pNMR) chemical shifts of the 5f¹ complexes UO₂(CO₃)₃^5– and NpO₂(CO₃)₃^4–, and of the 5f² complexes PuO₂(CO₃)₃^4– and (C₅H₅)₃UCH₃ are investigated by wave function theory calculations, using a recently developed sum-over-states approach within complete active space and restricted active space paradigm including spin–orbit (SO) coupling [<i>J. Phys. Chem. Lett.</i> <b>2015</b>, 20, 2183-2188]. The experimental ¹³C pNMR shifts of the actinyl tris-carbonate complexes are well reproduced by the calculations. The results are rationalized by visualizing natural spin orbitals (NSOs) and spin-magnetizations generated from the SO wave functions, in comparison with scalar relativistic spin densities. The analysis reveals a complex balance between spin-polarization, spin and orbital magnetization delocalization, and spin-compensation effects due to SO coupling. This balance creates the magnetization due to the electron paramagnetism around the nucleus of interest, and therefore the pNMR effects. The calculated proton pNMR shifts of the (C₅H₅)₃UCH₃ complex are also in good agreement with experimental data. Because of the nonmagnetic ground state of (C₅H₅)₃UCH₃, the ¹H pNMR shifts arise mainly from the magnetic coupling contributions between the ground state and low-energy excited states belonging to the 5f manifold, along with the thermal population of degenerate excited states at ambient temperatures

Tuned Range-Separated Time-Dependent Density Functional Theory Applied to Optical Rotation

For range-separated hybrid density functionals, the consequences of using system-specific range-separation parameters (γ) in calculations of optical rotations (ORs) are investigated. Computed ORs at three wavelengths are reported for methyloxirane, norbornenone, β-pinene, [6]helicene, [7]helicene, and two derivatives of [6]helicene. The γ parameters are adjusted such that Kohn–Sham density functional calculations satisfy the condition −ε^HOMO(<i>N</i>) = IP. For β-pinene, the behavior of the energy as a function of fractional total charge is also tested. For the test set of molecules, comparisons of ORs with available coupled-cluster and experimental data indicate that the γ “tuning” leads to improved results for β-pinene and the helicenes and does not do too much harm in other cases

Computational Study and Molecular Orbital Analysis of NMR Shielding, Spin–Spin Coupling, and Electric Field Gradients of Azido Platinum Complexes

¹⁹⁵Pt, ¹⁴N, and ¹⁵N NMR data for five azido (N₃^–) complexes are studied using relativistic density functional theory (DFT). Good agreement with experiment is obtained for Pt and N chemical shifts as well as Pt–N <i>J</i>-coupling constants. Calculated ¹⁴N electric field gradients (EFGs) reflect experimentally observed trends for the line broadening of azido ¹⁴N NMR signals. A localized molecular orbital analysis of the nitrogen EFGs and chemical shifts is performed to explain some interesting trends seen experimentally and in the first-principles calculations: (i) ¹⁴N NMR signals for the Pt-coordinating (N_α) nuclei in the azido ligands are much broader than for the central (N_β) or terminal (N_γ) atoms. The N_β signals are particularly narrow; (ii) compared to N_γ, the N_α nuclei are particularly strongly shielded; (iii) N_β nuclei have much larger chemical shifts than N_α and N_γ ; and (iv) The Pt–N_α <i>J</i>-coupling constants are small in magnitude when considering the formal sp hybridization of N_α . It is found that for N_α a significant shielding reduction due to formation of the dative N_α–Pt bond is counterbalanced by an increased shielding from spin–orbit (SO) coupling originating at Pt. Upon coordination, the strongly delocalized π system of free azide localizes somewhat on N_β and N_γ . This effect, along with rehybridization at N_α upon bond formation with Pt, is shown to cause a deshielding of N_γ relative to N_α and a strong increase of the EFG at N_α . The large 2p character of the azide σ bonds is responsible for the particularly high N_β chemical shifts. The nitrogen s-character of the Pt–N_α bond is low, which is the reason for the small <i>J</i>-coupling. Similar bonding situations are likely to be found in azide complexes with other transition metals

Electronic Energy Gaps for π‑Conjugated Oligomers and Polymers Calculated with Density Functional Theory

In varying contexts, the terms “energy gap” (energy difference) or “band gap” may refer to different experimentally observable quantities or to calculated values that may or may not represent observable quantities. This work discusses various issues related to calculations of electronic energy gaps for organic π-conjugated oligomers and linear polymers by density functional theory (DFT). Numerical examples are provided, juxtaposing systematic versus fortuitous agreement of orbital energy gaps with observable fundamental (ionization vs electron attachment) or optical (electronic excitation) energy gaps. Successful applications of DFT using nonempirically tuned hybrid density functionals with range-separated exchange (RSE) for calculations of optical gaps, fundamental gaps, and electron attachment/detachment energies are demonstrated. The extent of “charge-transfer like” character in the longest-wavelength singlet electronic excitations is investigated

Theoretical Investigation of Paramagnetic NMR Shifts in Transition Metal Acetylacetonato Complexes: Analysis of Signs, Magnitudes, and the Role of the Covalency of Ligand–Metal Bonding

Ligand chemical shifts are calculated and analyzed for three paramagnetic transition metal tris-acetylacetonato (acac) complexes, namely high-spin Fe­(III) and Cr­(III), and low-spin Ru­(III), using scalar relativistic density functional theory (DFT). The signs and magnitudes of the paramagnetic NMR ligand chemical shifts are directly related to the extent of covalent acac oxygen-to-metal σ donation involving unoccupied metal valence d_σ acceptor orbitals. The role of delocalization of metal-centered spin density over the ligand atoms plays a minor secondary role. Of particular interest is the origin of the sign and magnitude of the methyl carbon chemical shift in the acac ligands, and the role played by the DFT delocalization error when calculating such shifts. It is found that the α versus β spin balance of oxygen σ donation to metal valence d acceptor orbitals is responsible for the sign and the magnitude of the ligand methyl carbon chemical shift. A problematic case is the methyl carbon shift of Fe­(acac)₃. Most functionals produce shifts in excess of 1400 ppm, whereas the experimental shift is approximately 279 ppm. Range-separated hybrid functionals that are optimally tuned for Fe­(acac)₃ based on DFT energetic criteria predict a lower limit of about 2000 ppm for the methyl carbon shift of the high-spin electronic configuration. Since the experimental value is based on a very strongly broadened signal it is possibly unreliable

Relativistic Density Functional Calculations of Hyperfine Coupling with Variational versus Perturbational Treatment of Spin–Orbit Coupling

Different approaches are compared for relativistic density functional theory (DFT) and Hartree–Fock (HF) calculations of electron–nucleus hyperfine coupling (HFC) in molecules with light atoms, in transition metal complexes, and in selected actinide halide complexes with a formal metal 5<i>f</i>¹ configuration. The comparison includes hybrid density functionals with range-separated exchange. Within the variationally stable zeroth-order regular approximation (ZORA) relativistic framework, the HFC is obtained (i) with a linear response (LR) method where spin–orbit (SO) coupling is treated as a linear perturbation, (ii) with a spin-polarized approach closely related to a DFT method for calculating magnetic anisotropy (MA) previously devised by van Wüllen et al. where SO coupling is included variationally, (iii) with a quasi-restricted variational SO method previously devised by van Lenthe, van der Avoird, and Wormer (LWA). The MA and LWA approaches for HFC calculations were implemented in the open-source NWChem quantum chemistry package as part of this study. The methodology extends recent implementations for calculations of electronic <i>g</i>-factors (<i>J. Chem. Theor. Comput.</i> <b>2013</b>, <i>9</i>, 1052). The impact of electron correlation (DFT vs HF) and DFT delocalization errors, the effects of spin-polarization, the importance of treating spin–orbit coupling beyond first-order, and the magnitude of finite-nucleus effects, are investigated. Similar to calculations of <i>g</i>-factors, the MA approach in conjunction with hybrid functionals performs reasonably well for theoretical predictions of HFC in a wide range of scenarios