Scalar Relativity Research Articles

Scalar Relativistic All-Electron and Pseudopotential Ab Initio Study of a Minimal Nitrogenase [Fe(SH)4H]- Model Employing Coupled-Cluster and Auxiliary-Field Quantum Monte Carlo Many-Body Methods.

Nitrogenase is the only enzyme that can cleave the triple bond in N2, making nitrogen available to organisms. The detailed mechanism of this enzyme is currently not known, and computational studies are complicated by the fact that different density functional theory (DFT) methods give very different energetic results for calculations involving nitrogenase models. Recently, we designed a [Fe(SH)4H]- model with the fifth proton binding either to Fe or S to mimic different possible protonation states of the nitrogenase active site. We showed that the energy difference between these two isomers (ΔE) is hard to estimate with quantum-mechanical methods. Based on nonrelativistic single-reference coupled-cluster (CC) calculations, we estimated that the ΔE is 101 kJ/mol. In this study, we demonstrate that scalar relativistic effects play an important role and significantly affect ΔE. Our best revised single-reference CC estimates for ΔE are 85-91 kJ/mol, including energy corrections to account for contributions beyond triples, core-valence correlation, and basis-set incompleteness error. Among coupled-cluster approaches with approximate triples, the canonical CCSD(T) exhibits the largest error for this problem. Complementary to CC, we also used phaseless auxiliary-field quantum Monte Carlo calculations (ph-AFQMC). We show that with a Hartree-Fock (HF) trial wave function, ph-AFQMC reproduces the CC results within 5 ± 1 kJ/mol. With multi-Slater-determinant (MSD) trials, the results are 82-84 ± 2 kJ/mol, indicating that multireference effects may be rather modest. Among the DFT methods tested, τ-HCTH, r2SCAN with 10-13% HF exchange with and without dispersion, and O3LYP/O3LYP-D4, and B3LYP*/B3LYP*-D4 generally perform the best. The r2SCAN12 (with 12% HF exchange) functional mimics both the best reference MSD ph-AFQMC and CC ΔE results within 2 kJ/mol.

Scalar Relativistic Effects with Multiwavelets: Implementation and Benchmark.

The importance of relativistic effects in quantum chemistry is widely recognized, not only for heavier elements but throughout the periodic table. At the same time, relativistic effects are strongest in the nuclear region, where the description of electrons through a linear combination of atomic orbitals becomes more challenging. Furthermore, the choice of basis sets for heavier elements is limited compared with lighter elements where precise basis sets are available. Thanks to the framework of multiresolution analysis, multiwavelets provide an appealing alternative to overcoming this challenge: they lead to robust error control and adaptive algorithms that automatically refine the basis set description until the desired precision is reached. This allows one to achieve a proper description of the nuclear region. In this work, we extended the multiwavelet-based code MRChem to the scalar zero-order regular approximation framework. We validated our implementation by comparing the total energies for a small set of elements and molecules. To confirm the validity of our implementation, we compared both against a radial numerical code for atoms and the plane-wave-based code EXCITING.

Relativistic effects on the energetic stability of $$\hbox {Pb}_5$$ clusters

In this work, we study isomers of small lead clusters with five atoms, $$\hbox {Pb}_5$$ , at different levels of approximation namely Scalar-Relativistic (SR), Scalar-Relativistic plus Spin–Orbit coupling interaction (SR + SO) and four-component Dirac–Hartree–Fock (4c-DHF), in order to analyze the effects of relativity in these heavy molecular systems. The exploration of potential energy surface (PES) with a genetic algorithm produces four possible equilibrium structures, and we find that when Relativity is included at a major level in calculations, the global minimum energy structure changes from S4 isomer with $$\hbox {D}_{3\mathrm{h}}$$ symmetry at SR level to S1 isomer with $$\hbox {C}_2$$ symmetry at 4c-DHF level; this change is related to modifications in the electronic structure and geometric parameters. We explain this significant result using two methodologies in order to analyze the electronic structure and strength of chemical bonds, like energy decomposition analysis (EDA) and Quantum Theory Atoms In Molecules (QTAIM). On the one hand, in the framework of EDA, results at SR + SO level show significant differences on the steric and orbital interactions compared with SR ones, with which the S1 isomer is more stable than S4; this means that SO effects stabilize the interactions on S1 isomer more than S4. The HOMO–LUMO gap also shows a drastic reduction due to the SO effects on S4 isomer, while for the other systems remains unchanged. This result can be associated with the lower stability of S4 isomer with respect to the others when Relativity is included at a major level. On the other hand, in the framework of QTAIM, calculations with SR + SO scheme show the formation of two new critical points compared with SR for S1 isomer, which is reflected in a greater stability of this system.

Dirac's equation and its implications for density functional theory based calculations of materials containing heavy elements

Electronic structure calculations based on density functional theory (DFT) give quantitatively accurate predictions of properties of most materials containing light elements. For heavy materials, and in particular for $f$-electron systems, DFT based methods can fail both qualitatively and quantitatively for two distinct reasons: their failure to describe confinement effects arising from localized $f$-electron behavior and their incomplete or approximate treatment of relativity. In addition, different methods for incorporating relativistic effects, which give identical results in most light materials, can give different predictions in heavy elements. In order to develop a quantitative capability for calculating the properties of these materials, it is essential to separate the predictions of the underlying equations from the uncertainty introduced in approximations used in computation. Working toward that goal, we have developed a code, called dirac-fp, which is based directly on solving the Dirac-Kohn-Sham equations and uses the full-potential linear muffin-tin orbital (FP-LMTO) approach to electronic structure. In order to assess the performance of dirac-fp, we perform calculations on three different face-centered cubic materials using different approximate treatments of relativity: the scalar relativistic (SR) approach commonly used in most solid-state DFT codes, the scalar relativistic plus spin-orbit coupling corrections (SR+SO) approach which includes spin-orbit coupling self-consistently using the SR states inside the muffin tins, and the Dirac-Kohn-Sham (Dirac) approach implemented in dirac-fp. Performing calculations on thorium, in which relativistic effects should be strong, aluminum, in which relativistic effects should be negligible, and gold, in which relativistic effects play an intermediate role, we find that the Dirac approach is able to provide theoretically consistent results in the electronic structure and ground-state properties across all three materials.

Investigating the influence of relativistic effects on absorption spectra for platinum complexes with light-activated activity against cancer cells.

We report the first systematic investigation of relativistic effects on the UV-vis spectra of two prototype complexes for so-called photo-activated chemotherapy (PACT), trans-trans-trans-[Pt(N3)2(OH)2(NH3)2] and cis-trans-cis-[Pt(N3)2(OH)2(NH3)2]. In PACT, design of new drugs requires in-depth understanding of the photo-activation mechanisms. A first step is usually to rationalize their UV-vis spectra for which time-dependent density functional theory (TD-DFT) is an indispensable tool. We carried out TD-DFT calculations with a systematic series of non-relativistic (NR), scalar-relativistic (SR), and four-component (4c) Hamiltonians. As expected, large differences are found between spectra calculated within 4c and NR frameworks, while the most intense features (found at higher energies below 300 nm) can be reasonably well reproduced within a SR framework. It is also shown that effective core potentials (ECPs) yield essentially similar results as all-electron SR calculations. Yet the underlying transitions can be strongly influenced by spin-orbit coupling, which is only present in the 4c framework: while this can affect both intense and less intense transitions in the spectra, the effect is most pronounced for weaker transitions at lower energies, above 300 nm. Since the investigated complexes are activated with light of wavelengths above 300 nm, employing a method with explicit inclusion of spin-orbit coupling may be crucial to rationalize the activation mechanism.

Single photon ionization of methyl isocyanide and the subsequent unimolecular decomposition of its cation: experiment and theory.

Methyl isocyanide, CH3NC, is a key compound in astrochemistry and astrobiology. A combined theoretical and experimental investigation of the single photon ionization of gas phase methyl isocyanide and its fragmentation pathways is presented. Vacuum ultraviolet (VUV) synchrotron radiation based experiments are used to measure the threshold photoelectron photoion coincidence (TPEPICO) spectra between 10.6 and 15.5 eV. This allowed us to experimentally determine the adiabatic ionization energy (AIE) and fragment ion appearance energies (AE) of gas-phase methyl isocyanide. Its AIE has been measured with a precision never achieved before. It is found to be AIEexp = 11.263 ± 0.005 eV. We observe a vibrational progression upon ionization corresponding to the population of vibrational levels of the ground state of the methyl isocyanide cation. In addition, four fragment ion appearance energies (AEs) were measured to be AE (m/z 40) = 12.80 ± 0.05 eV, AE (m/z 39) = 13.70 ± 0.05, AE (m/z 15) = 13.90 ± 0.05 eV, AE (m/z 14) 13.85 ± 0.05 eV, respectively. In order to interpret the experimental data, we performed state-of-the-art computations using the explicitly correlated coupled cluster approach. We also considered the zero-point vibrational energy (ZPVE), core-valence (CV) and scalar relativistic (SR) effects. The results of theoretical calculations of the AIE and AEs are in excellent agreement with the experimental findings allowing for assignment of the fragmentations to the loss of neutral H, H2, CN and HCN upon ionization of CH3NC. The computations show that in addition to the obvious bond breakings, some of the corresponding ionic fragments result from rearrangements - upon photon absorption - either before or after electron ejection.

Role of Ligand Straining in Complexation of Eu3+-Am3+ Ions by TPEN and PPDEN, Scalar Relativistic DFT Exploration in Conjunction with COSMO-RS.

To search for new ligands suitable for the separation of minor actinides (MA) from lanthanides (Ln) in nuclear waste reprocessing, theoretical (density functional theory) studies were carried out on the complexation (structures, bonding, and thermodynamics) of La3+, Sm3+, Eu3+, and Am3+ complexes with moderately soft donor ligands TPEN [N,N,N′,N′-tetrakis(2-pyridylmethyl)ethylenediamine] and PPDEN [N,N,N′,N″,N″-pentakis(2-pyridylmethyl) diethylenetriamine] in aqueous and nitrobenzene solutions. B3LYP level of theory was used in conjunction with the conductor-like screening model for real systems (COSMO-RS). The metal ions in [M(NO3)2(TPEN)]NO3 and [M(NO3)(PPDEN)](NO3)2 complexes were deca-coordinated with both TPEN and PPDEN. The enthalpy of the complexation with TPEN in an aqueous solution was found to be negative, indicating the exothermic nature of the reaction as observed in the experiments. The calculated values of free energy of complexation follow the experimental trend: Am3+ > Sm3+ > La3+. Furthermore, the calculated free energy with PPDEN is reduced compared to that with TPEN, which may be attributed to the ligand straining during complex formation, which is also reflected in greater residual charges on both the Eu3+ and Am3+ central ions in the complexes of octadentate PPDEN compared to hexadentate TPEN. The experimental complexation selectivity of Am3+ over Eu3+ with TPEN is established by employing COSMO-RS. Furthermore, TPEN is Am3+-selective, whereas PPDEN is Eu3+-selective, which could be exploited for the efficient separation of MA from Ln.

Rovibrational spectrum calculations of four electronic states in carbon monoxide molecule: Comparison of two effect correction methods

Accurate calculation of molecular energy is of great significance for studying molecular spectral properties. In this work, the potential energy curve and rovibrational spectrum (Gν) of the ground state X1∑+ and the excited states a3Π, a'3∑+ and A1Π of carbon monoxide molecule are calculated by the multi-reference configuration interaction method. In the calculation, the core-valence correlation correction (CV) effect and scalar relativistic (SR) effect are included.In order to obtain an accurate energy of molecule, two computational schemes are adopted. In the first scheme, i.e. (m MRCI+Q/CBS(TQ5)+CV+SR), the molecular orbital wavefunction is obtained from the Hartree-Fock self-consistent field method by using the basis set aug-cc-pVnZ. The wavefunction is first calculated by the state-averaged complete active space self-consistent field approach. Then the multi-reference configuration interaction method (MRCI) is adopted to calculate the dynamic correlation energy in the potential energy curve. Finally, we use the basis set cc-pCVQZ and aug-cc-pVQZ to calculate the CV effect and SR effect by the MRCI method. In the second scheme (aug-cc-pwCVnZ-DK (n=T, Q, 5)), the potential energy curves (PECs) of these four electronic states are calculated by the MRCI method whose basis set (aug-cc-pwCVnZ-DK) contains the CV effect and SR effect. Finally, in order to reduce the error caused by the basis set, we extrapolate the basis sets of the two computational schemes to the complete basis set. On the basis of the PECs plotted by the different methods, we obtain the spectroscopic parameters of the X1∑+, a3Π, a'3∑+ and A1Π states of the carbon monoxide by solving the internuclear Schrödinger equations through utilizing the numerical integration program “LEVEL”.In this paper, we calculate the SR effect and the CV effect by using different schemes, and the latter is indispensable for accurately calculating the molecular structure. For the lowest two electronic states, we consider the dependence of the two effects on the calculation of the Gaussian basis group (Method B), and find that the accuracy of the rovibrational spectrum is improved. It can be seen that these electronic states have higher requirements for electronic correlation calculation. For higher electronic states, the electron cloud distribution is relatively loose, and the electronic correlation obtained by a single Gaussian basis group can achieve the corresponding calculation accuracy. Of course, since the calculation of the rovibrational spectra is essentially only the relative energy, the offset effect of the electronic correlation effect of different electronic states is also included here in this paper.

Super heavy element Copernicium: Cohesive and electronic properties revisited

First principles scalar relativistic (SR) calculations with and without including the spin orbit (SO) interactions have been performed for solid Copernicium (Cn) to determine its ground state equilibrium structure, volume, bulk modulus, pressure derivative of the bulk modulus, density of states and band structure. Both SR and SR+SO calculations have been performed with 6p levels treated as part of core electrons and also as part of valence electrons. These calculations have been performed for the rhombohedral, BCT, FCC, HCP, BCC and SC structures. Results have been compared with the results for Hg which is lighter homologue of Cn in the periodic table. We find hcp to be the stable crystal structure at SR level of theory and also at SR+SO level of theory when the 6p electrons are treated as part of core electrons. With 6p as part of valence electrons, SR+SO level of computations, however, yield bcc structure to be the most stable structure. Equilibrium volume (V0) of the most stable crystal structure at SR level of theory viz. hcp structure is 188.66 a.u.3whereas its value for the bcc structure, the equilibrium ground state structure at SR+SO level of theory is 165.71 a.u.3 i.e a large change due to relativistic effects is seen. The density of states at Fermi level is much smaller in Cn than in Hg, making it a poorer metal than mercury. In addition the cohesive energy of Cn is computed to be almost two times that of Hg for SR+SO case.

Effects of All-Electron Basis Sets and the Scalar Relativistic Corrections in the Structure and Electronic Properties of Niobium Clusters.

In this paper, an augmented all-electron double-ζ basis set is used in calculations of the structure and electronic properties of small niobium clusters. The B3PW91 and M06 DFT functionals with and without second order Douglas-Kroll-Hess (DKH) scalar relativistic corrections are also utilized. Furthermore, an additional d Gaussian type function is introduced in the standard basis sets in order to improve the description of the clusters orbitals in the valence band. Our findings show that the extra d function is important to yield accurate results of electronic properties and, in addition, the DKH corrections can be relevant when the all-electron basis sets are used in the calculations. Our best results are obtained with the M06 functional together with the DKH second order corrections and with the extra d function added to the all-electron basis set.

Relativistic Approximations to Paramagnetic NMR Chemical Shift and Shielding Anisotropy in Transition Metal Systems.

We apply approximate relativistic methods to calculate the magnetic property tensors, i.e., the g-tensor, zero-field splitting (ZFS) tensor (D), and hyperfine coupling (HFC) tensors, for the purpose of constructing paramagnetic nuclear magnetic resonance (pNMR) shielding tensors. The chemical shift and shielding anisotropy are calculated by applying a modern implementation of the classic Kurland-McGarvey theory ( J. Magn. Reson. 1970 , 2 , 286 ), which formulates the shielding tensor in terms of the g- and HFC tensors obtained for the ground multiplet, in the case of higher than doublet multiplicity defined by the ZFS interaction. The g- and ZFS tensors are calculated by ab initio complete active space self-consistent field and N-electron valence-state perturbation theory methods with spin-orbit (SO) effects treated via quasidegenerate perturbation theory. Results obtained with the scalar relativistic (SR) Douglas-Kroll-Hess Hamiltonian used for the g- and ZFS tensor calculations are compared with nonrelativistically based computations. The HFC tensors computed using the fully relativistic four-component matrix Dirac-Kohn-Sham approach are contrasted against perturbationally SO-corrected nonrelativistic results in the density functional theory framework. These approximations are applied on paramagnetic metallocenes (MCp2) (M = Ni, Cr, V, Mn, Co, Rh, Ir), a Co(II) pyrazolylborate complex, and a Cr(III) complex. SR effects are found to be small for g and D in these systems. The HFCs are found to be more influenced by relativistic effects for the 3d systems. However, for some of the 3d complexes, nonrelativistic calculations give a reasonable agreement with the experimental chemical shift and shielding anisotropy. The influence of scalar relativity is strong for the 5d IrCp2 system. This mixed ab initio/DFT technique, with a fully relativistic method used for the critical HFC tensor, should be useful for the treatment of both electron correlation and relativistic effects at a reasonable computational cost to compute the pNMR shielding tensors in transition metal systems.

Theoretical studies on the structural, electronic and optical properties of orthorhombic perovskites CH3NH3PbX3(X = I, Br, Cl)

In this study, we report advanced density functional theory (DFT) calculations on the structural, electronic, and optical properties of orthorhombic perovskite CH3NH3PbX3(X=I,Br,Cl). We present basic parameters such as lattice constants, electronic structures, effective charge masses, optical absorption spectra, and excitonic binding energies to understand the fundamental properties of CH3NH3PbX3(X=I,Br,Cl). We show that scalar relativistic (SR) DFT calculations yield good band gaps for three systems, and fully relativistic spin-orbital coupling (SOC) DFT calculations give excellent reduced excitonic effective charge masses, as compared to experiments. Moreover, our calculations show that the absorption onset of CH3NH3PbX3 moves to short wavelength region with the increase of band gap. Finally, we discussed the relationship between excitonic binding energies and dielectric constants.

High-Level ab Initio Predictions for the Ionization Energies, Bond Dissociation Energies, and Heats of Formation of Titanium Oxides and Their Cations (TiOn/TiOn+, n = 1 and 2).

The ionization energies (IEs) of TiO and TiO2 and the 0 K bond dissociation energies (D0) and the heats of formation at 0 K (ΔH°f0) and 298 K (ΔH°f298) for TiO/TiO+ and TiO2/TiO2+ are predicted by the wave-function-based CCSDTQ/CBS approach. The CCSDTQ/CBS calculations involve the approximation to the complete basis set (CBS) limit at the coupled cluster level up to full quadruple excitations along with the zero-point vibrational energy (ZPVE), high-order correlation (HOC), core-valence (CV) electronic, spin-orbit (SO) coupling, and scalar relativistic (SR) effect corrections. The present calculations yield IE(TiO) = 6.815 eV and are in good agreement with the experimental IE value of 6.819 80 ± 0.000 10 eV determined in a two-color laser-pulsed field ionization-photoelectron (PFI-PE) study. The CCSDT and MRCI+Q methods give the best predictions to the harmonic frequencies: ωe (ωe+) = 1013 (1069) and 1027 (1059) cm-1 and the bond lengths re (re+) = 1.625 (1.587) and 1.621 (1.588) Å, for TiO (TiO+) compared with the experimental values. Two nearly degenerate, stable structures are found for TiO2 cation: TiO2+(C2v) structure has two equivalent TiO bonds, while the TiO2+(Cs) structure features a long and a short TiO bond. The IEs for the TiO2+(C2v)←TiO2 and TiO2+(Cs)←TiO2 ionization transitions are calculated to be 9.515 and 9.525 eV, respectively, giving the theoretical adiabatic IE value in good agreement with the experiment IE(TiO2) = 9.573 55 ± 0.000 15 eV obtained in the previous vacuum ultraviolet (VUV)-PFI-PE study of TiO2. The potential energy surface of TiO2+ along the normal vibrational coordinates of asymmetric stretching mode (ω3+) is nearly flat and exhibits a double-well potential with the well of TiO2+ (Cs) situated around the central well of TiO2+(C2v). This makes the theoretical calculation of ω3+ infeasible. For the symmetric stretching (ω1+), the current theoretical predictions overestimate the experimental value of 829.1 ± 2.0 cm-1 by more than 100 cm-1. This work together with the previous experimental and theoretical investigations supports the conclusion that the CCSDTQ/CBS approach is capable of providing reliable IE and D0 predictions for TiO/TiO+ and TiO2/TiO2+ with error limits less than or equal to 60 meV. The CCSDTQ/CBS calculations give the predictions of D0(Ti+-O) - D0(Ti-O) = 0.004 eV and D0(O-TiO) - D0(O-TiO+) = 2.699 eV, which are also consistent with the respective experimental determination of 0.008 32 ± 0.000 10 and 2.753 75 ± 0.000 18 eV.

Cost-Effective Treatment of Scalar Relativistic Effects for Multireference Systems: A CASSCF Implementation Based on the Spin-free Dirac-Coulomb Hamiltonian.

We present an implementation of the complete active space-self-consistent field (CASSCF) method specifically designed to be used in four-component scalar relativistic calculations based on the spin-free Dirac-Coulomb (SFDC) Hamiltonian. Our implementation takes full advantage of the properties of the SFDC Hamiltonian that allow us to use real algebra and to exploit point-group and spin symmetry to their full extent while including in a rigorous way scalar relativistic effects in the treatment. The SFDC-CASSCF treatment is more expensive than its non-relativistic counterpart only in the orbital optimization step, while exhibiting the same computational cost for the rate-determining full configuration interaction part. The numerical aspects are discussed, and the capabilities of the SFDC-CASSCF methodology are demonstrated through a pilot application.

Implementation and Validation of Fully Relativistic GW Calculations: Spin-Orbit Coupling in Molecules, Nanocrystals, and Solids.

We present an implementation of G0W0 calculations including spin-orbit coupling (SOC) enabling investigations of large systems, with thousands of electrons, and we discuss results for molecules, solids, and nanocrystals. Using a newly developed set of molecules with heavy elements (called GW-SOC81), we find that, when based upon hybrid density functional calculations, fully relativistic (FR) and scalar-relativistic (SR) G0W0 calculations of vertical ionization potentials both yield excellent performance compared to experiment, with errors below 1.9%. We demonstrate that while SR calculations have higher random errors, FR calculations systematically underestimate the VIP by 0.1 to 0.2 eV. We further verify that SOC effects may be well approximated at the FR density functional level and then added to SR G0W0 results for a broad class of systems. We also address the use of different root-finding algorithms for the G0W0 quasiparticle equation and the significant influence of including d electrons in the valence partition of the pseudopotential for G0W0 calculations. Finally, we present statistical analyses of our data, highlighting the importance of separating definitive improvements from those that may occur by chance due to a limited number of samples. We suggest the statistical analyses used here will be useful in the assessment of the accuracy of a large variety of electronic structure methods.

Spin-Orbit Coupling Effects in AumPtn Clusters (m + n = 4).

A study of AumPtn(m + n = 4) clusters with and without spin-orbit (SO) coupling using scalar relativistic (SR) and two component methods with the ZORA Hamiltonian was carried out. We employed the PW91 functional in conjunction with the all-electron TZ2P basis set. This paper offers a detailed analysis of the SO effects on the cluster geometries, on the LUMO-HOMO gap, on the charge distribution, and on the relative energies for each relativistic method. In general, SO coupling led to an energetic rearrangement of the species, to changes in geometries and structural preferences, to changes in the structural identity of the global minimum for the Au3Pt, AuPt3 and Pt4 cases, and to a reduction of relative energies among the clusters, an effect that appears stronger as the amount of Pt increases.

Estimating the Impact of an All-Electron Basis Set and Scalar Relativistic Effects on the Structure, Stability, and Reactivity of Small Copper Clusters

Basis sets of valence double and quadruple zeta qualities and the Douglas-Kroll-Hess (DKH) approximation are used to estimate the impact of an all-electron basis set and scalar relativistic effects on the structure, stability, and electronic properties of small neutral copper clusters (Cun, n ≤ 8). At the Becke three-parameter for exchange and Perdew-Wang 91 for correlation (B3PW91) non-relativistic and relativistic levels of theory, the bond length, binding energy, ionization potential, electron affinity, chemical potential, chemical hardness, and electrophilicity index are calculated. The results show that the agreement with experiment improves significantly when the DKH Hamiltonian combined with an all-electron relativistic basis set is used. Polarizabilities and hyperpolarizability are also reported. At the B3PW91 level, all-electron basis sets are shown to be more reliable than effective core potential valence basis sets in the determination of the second hyperpolarizability of copper clusters.

Reinvestigation of high pressure polymorphism in hafnium metal

There has been a recent controversy about the high pressure polymorphism of Hafnium (Hf). Unlike, the earlier known α→ω structural transition at 38 ± 8 GPa, at ambient temperature, Hrubiak et al. [J. Appl. Phys. 111, 112612 (2012)] did not observe it till 51 GPa. They observed this transition only at elevated temperatures. We have reinvestigated the room temperature phase diagram of Hf, employing x-ray diffraction (XRD) and DFT based first principles calculations. Experimental investigations have been carried out on several pure and impure Hf samples and also with different pressure transmitting media. Besides demonstrating the significant role of impurity levels on the high pressure phase diagram of Hf, our studies re-establish room temperature α→ω transition at high pressures, even in quasi-hydrostatic environment. We observed this transition in pure Hf with equilibrium transition pressure Po = 44.5 GPa; however, with large hysteresis. The structural sequence, transition pressures, the lattice parameters, the c/a ratio and its variation with compression for the α and ω phases as predicted by our ab-initio scalar relativistic (SR) calculations are found to be in good agreement with our experimental results of pure Hf.

Accurate theoretical spectroscopy of the lowest electronic states of CP radical

The present theoretical calculations on CP X2Σ+, A2Π, 14Σ+, 14Δ, 14Π, 16Π and 16Σ+ electronic states use standard and explicitly correlated coupled cluster approaches, multi-reference configuration interaction (MRCI) techniques in connection with a large panel of basis sets. We also examined core–valence (CV) and scalar relativistic (SR) effects. For the ground electronic state, our calculations reveal that the MRCI+CV+SR method and the coupled cluster technique with perturbative treatment of triple excitations including scalar relativistic effects computed with the Douglas-Kroll approach (RCCSD[T]-DK) in connection with a large basis set lead to an accurate description of CP(X2Σ+). We computed the evolution along the internuclear distance of the diagonal and the off-diagonal spin–orbit integrals between the X2Σ+ and A2Π states. These integrals are then incorporated together with the corresponding potentials into variational treatment of the nuclear motion. We deduced hence the energy positions of the low and high vibronic levels of X2Σ+ and A2Π states. Finally, it is shown the necessity of considering the spin–orbit coupling for accurate prediction of the energy position of these levels.

Scalar Relativistic Computations and Localized Orbital Analyses of Nuclear Hyperfine Coupling and Paramagnetic NMR Chemical Shifts.

A method is reported by which calculated hyperfine coupling constants (HFCCs) and paramagnetic NMR (pNMR) chemical shifts can be analyzed in a chemically intuitive way by decomposition into contributions from localized molecular orbitals (LMOs). A new module for density functional calculations with nonhybrid functionals, global hybrids, and range-separated hybrids, utilizing the two-component relativistic zeroth-order regular approximation (ZORA), has been implemented in the parallel open-source NWChem quantum chemistry package. Benchmark results are reported for a test set of few-atom molecules with light and heavy elements. Finite nucleus effects on (199)Hg HFCCs are shown to be on the order of -11 to -15%. A proof of concept for the LMO analysis is provided for the metal and fluorine HFCCs of TiF3 and NpF6. Calculated pNMR chemical shifts are reported for the 2-methylphenyl-t-butylnitroxide radical and for five cyclopentadienyl (Cp) sandwich complexes with 3d metals. Nickelocene and vanadocene carbon pNMR shifts are analyzed in detail, demonstrating that the large carbon pNMR shifts calculated as +1540 for Ni (exptl.: +1514) and -443 for V (exptl.: -510) are caused by different spin-polarization mechanisms. For Ni, Cp to Ni π back-donation dominates the result, whereas for vanadocene, V to Cp σ donation with relaxation of the carbon 1s shells can be identified as the dominant mechanism.