Ab initio relativistic effective potentials with spin-orbit operators. VI. Fr through Pu

Ab initio averaged relativistic effective core potentials (AREP ), spin-orbit (SO ) operators, and valence basis sets are reported for the elements Fr through Pu in the form of expansions in Gaussian-type functions. Gaussian basis sets with expansion coefficients for the low-energy states of each atom are given. Atomic orbital energies calculated under the j–j coupling scheme within the self-consistent field approximation and employing the AREP 'S in their unaveraged form (REP 'S) agree to within 10% of orbital energies due to numerical all-electron Dirac–Fock calculations. The accuracy of the AREP 'S and so operators is also shown to be good through comparisons of calculated so splitting energies with all-electron Dirac–Fock results.     

Electronic structure of molecules using one-component wave functions and relativistic effective core potentials

A one-component approach to molecular electronic structure is discussed that includes the dominant relativistic effects on valence electrons and yet allows the use of the traditional quantum-chemistry techniques. The approach starts with one-component Cowan–Griffin relativistic orbitals that successfully incorporate the effects of the mass-velocity and Darwin terms present in more complicated wave functions such as the Dirac–Hartree–Fock. The approach then constructs “relativistic” effective core potentials (RECPS ) from these orbitals, and uses these to bring the relativistic effects into the molecular electronic calculations. The use of effective one-electron spin-orbit operators in conjunction with these one-component wave functions to include the effects of spin-orbit coupling is discussed. Applications to molecular systems involving heavy atoms and comparisons with available spectroscopic data on molecular geometries and excitation energies are presented. Finally, a new approach to the construction of RECPS encompassing the Hamiltonian and shapeconsistent approach is presented together with a novel analysis of the long-range behavior of the RECPS .     

Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.     

A two-step method of calculation of the electronic structure of molecules with heavy atoms: Theoretical aspect

An approach for a space-separated calculation of the wave function in the valence and core regions of a molecule is proposed. As the first step, the calculation of the orbitals (or two-component spinors in the relativistic case) in the valence region by the effective core potential (ECP ) method is performed. Then, it is followed by a restoration of orbitals (four-component spinors) expanded on spherical harmonics in the core regions of heavy atoms. Theoretical questions of the variational calculation of the molecular orbitals are considered in some core region limited by a sphere. Inclusion from the electronic cloud outside this region is reduced by the necessity of taking into account the orthonormality and boundary conditions together with an effective external field in respect to the selected core region. This method may be used for calculation of matrix elements of operators that are singular near nuclei (P, T-odd interactions, hyperfine structure, etc.). A substantial computational saving can be reached because the method enables, by the most optimal way, to combine the advantages of two well-developed approaches: molecular ECP calculations in the Gaussian basis set and one-center numerical atomic calculations with an external field. It is especially important when the relativistic effects are taken into account. © 1996 John Wiley & Sons, Inc.     

Comparative study of relativistic density functional methods applied to actinide species AcO(2)(2+) and AcF(6) for Ac = U,Np

A two-component relativistic density functional method based on the Douglas-Kroll-Hess transformation has been applied to the actinyls and hexafluorides of U and Np. All-electron scalar relativistic calculations as well as calculations including spin-orbit interaction have been compared to results obtained with a pseudopotential approach. In addition, several exchange-correlation potentials have been applied to examine their performance for the bond lengths and vibrational frequencies of the title compounds. The calculations confirm the well-known accuracy of the LDA approach for geometries and frequencies, which is corroborated for the hexafluorides where gas phase experimental data are available. Comparison with results of accurate wave function based methods provides further confirmation of this finding. Gradient-corrected functionals tend to overestimate bond lengths and underestimate frequencies also for actinide compounds. The results obtained with Stoll-Preuss (small core) effective core potentials agree very well with those of all-electron calculations, while calculations with Hay-Martin large core pseudopotentials are somewhat less accurate. For all molecules and properties considered, spin-orbit effects have been found negligible concomitant with the closed-shell electronic structure of the U(VI) compounds and the open-shell situation of the Np(VI) compounds with a single valence f electron.     

Darwin and mass-velocity relativistic corrections in non-Born-Oppenheimer variational calculations

The Pauli approach to account for the mass-velocity and Darwin relativistic corrections has been applied to the formalism for quantum mechanical molecular calculations that does not assume the Born-Oppenheimer (BO) approximation regarding separability of the electronic and nuclear motions in molecular systems. The corrections are determined using the first order perturbation theory and are derived for the non-BO wave function of a diatomic system expressed in terms of explicitly correlated Gaussian functions with premultipliers in the form of even powers of the internuclear distance. As a numerical example we used calculations of the transition energies for pure vibrational states of the HD(+) ion.     

The error due to neglecting the core spin–orbit splitting in valence ZORA calculations with frozen core approximation and its elimination

In valence zeroth-order regular approximation (ZORA) calculations with frozen core approximation, when the basis set optimized to the related scalar relativistic ZORA calculations is used, neglecting the core spin–orbit splitting may result in additional basis set truncation errors. It is found that the error is negligible for most elements except the 6p-block elements. When the basis set is extended by a p-type STO function put on the 6p element atoms with the ζ value proper to 5p_1/2 orbitals, the error can be reduced to be negligible. The calculated atomic properties related to valence orbitals can be improved greatly by use of this extended basis set. The frozen core approximation calculations of some molecules containing Tl, Pb and Bi with closed shells show that neglecting the core spin–orbit splitting only slightly affects the calculated bond lengths and bond energies, and the calculated molecular property can also be improved slightly by use of the extended basis sets.     

Electronic structure of UH,UF, and their ions

A relativistic effective core potential (REP) has been generated for the uranium atom and used in self-consistent-field calculations of the A states of UH, UF, and their ions. Energy curves were calculated at the base configuration level which ensures the dissociating atoms are described by Hartree–Fock wavefunctions. The electronic bonding of these molecules is found to be similar to that of comparable alkaline–earth hydrides and fluorides. The uranium 6p, 6d, and 5f orbitals retain their atomic character but the orbitals extend into the bonding region and are distorted by overlap repulsion and electrostatic effects. Nonetheless, the atomic energetic coupling determines that low energy states will have the maximum spin multiplicity and maximum orbital angular momentum projection consonant with the charge-transfer bonding.     

All-electron scalar relativistic basis sets for the elements Rb–Xe

Segmented all-electron relativistically contracted (SARC) basis sets are presented for the elements ₃₇Rb–₅₄Xe, for use with the second-order Douglas–Kroll–Hess approach and the zeroth-order regular approximation. The basis sets have a common set of exponents produced with established heuristic procedures, but have contractions optimized individually for each scalar relativistic Hamiltonian. Their compact size and loose segmented contraction, which is in line with the construction of SARC basis sets for heavier elements, makes them suitable for routine calculations on large systems and when core spectroscopic properties are of interest. The basis sets are of triple-zeta quality and come in singly or doubly polarized versions, which are appropriate for both density functional theory and correlated wave function theory calculations. The quality of the basis sets is assessed against large decontracted reference basis sets for a number of atomic and ionic properties, while their general applicability is demonstrated with selected molecular examples.     

Relativistic effective potential SCF calculations of AgH and AuH

Relativistic effective potentials (REP) are now widely used in molecular electronic structure calculations. Tests of these REP are needed to assess their accuracy. This can now be done for AgH and AuH since Lee and McLean have published Dirac-Fock calculations for these molecules. Comparative SCF calculations have been performed for two types of effective potential. Satisfactory agreement between the effective potential results and Dirac-Fock values is found for spectroscopic constants and dipole moments, which supports the use of these potentials for heavy atom containing molecules.     

Generalized relativistic effective core potentials for superheavy elements

The generalized relativistic effective core potentials (GRECPs) for calculations of electronic structure and physical-chemical properties of compounds containing superheavy elements (Z ≥ 104) are presented. Features of accounting for the finite nuclear size effects which are unusually large for superheavy elements are discussed in details. Accuracy of the GRECPs is analyzed in atomic calculations compared to all-electron studies with the Dirac-Coulomb-Breit Hamiltonian. Applications of the GRECP method in molecular and cluster calculations are surveyed.     

The Jahn-Teller effect: a case of incomplete theory for d4 complexes?

We present relativistic calculations at the four-component Dirac-DFT level for the geometries of the series of group 9 monoanionic hexafluorides MF(6)(-), M = Co, Rh, Ir. Highly correlated four-component relativistic CCSD(T) energies were also calculated for the optimized geometries. Spin-orbit coupling effects influence the geometrical preferences for molecular structures: relativistic calculations predict ground states with octahedral symmetries O(h)* for all hexafluorides in this study, while at the nonrelativistic limit, a structural deviation toward D(4h) ground state symmetries is predicted. Our findings suggest that relativistic effects have an important role in molecular structure preferences for the title hexafluorides.     

Full spin-orbit configuration interaction calculations on electronic states of LiBe

Ab initio calculations are reported for the simplest heteronuclear metal cluster, LiBe. Full spin-orbit configuration interaction calculations in the context of relativistic effective core potentials lead to accurate potential energy curves for low-lying states. Results are compared with recent experimental observations and with all electron multi-reference configuration interaction calculations.     

The accuracy of hyperfine integrals in relativistic NMR computations based on the zeroth-order regular approximation

The accuracy of the hyperfine integrals obtained in relativistic NMR computations based on the zeroth–order regular approximation (ZORA) is investigated. The matrix elements of the Fermi contact operator and its relativistic analogs for s orbitals obtained from numerical nonrelativistic, ZORA, and four–component Hartree–Fock–Slater calculations on atoms are compared. It is found that the ZORA yields very accurate hyperfine integrals for the valence shells of heavy atoms, but performs rather poorly for the innermost core shells. Because the important observables of the NMR experiment—chemical shifts and spin–spin coupling constants—can be understood as valence properties it is concluded that ZORA computations represent a reliable tool for the investigations of these properties. On the other hand, absolute shieldings calculated with the ZORA might be substantially in error. Because applications to molecules have so far exclusively been based on basis set expansions of the molecular orbitals, ZORA hyperfine integrals obtained from atomic Slater-type basis set computations for mercury are compared with the accurate numerical values. It is demonstrated that the core part of the basis set requires functions with Slater exponents only up to 10⁴ in the case where errors in the hyperfine integrals of a few percent are acceptable.     

Relativistic effects on inversion barriers of pyramidal group 15 hydrides

Quantum chemistry is an important tool for determining general molecular properties, although relativistic corrections are usually required for systems containing heavy and super heavy elements. Non‐relativistic along with relativistic two‐ and four‐component electronic structure calculations done with the CCSD‐T method and the new RPF‐4Z basis set have therefore been applied for determining inversion barriers, corresponding to the change from a pyramidal (C_3v) ground‐state structure to the trigonal planar (D_3h) transition state, TS, of group 15 hydrides, XH₃ (X= N, P, As, Sb, and Bi). The ground‐state structure of the McH₃ molecule, which contains the super heavy element Moscovium, is also predicted as pyramidal (C_3v), with an atomization energy of 90.8 kcal mol⁻¹. However, although non‐relativistic calculations still provided a D_3h planar TS for McH₃, four‐component relativistic calculations based on single‐reference wave functions are unable to elucidate the definitive TS geometry in this case. Hence, the results show that relativistic effects are crucial for this barrier determination in those hydrides containing Bi and Mc. Moreover, while the scalar relativistic effects predominate, increasing barrier heights by as much as 17.6 kcal mol⁻¹ (32%) in BiH₃, the spin‐orbit coupling cannot be disregarded in those hydrides containing the heaviest group 15 elements, decreasing the barrier by 2.5 kcal mol⁻¹ (4.5%) in this same molecule.     

A theoretical study of the excited states of AmO2n+, n=1,2,3

The ground and excited states of the AmO(2) (+), AmO(2) (2+), and AmO(2) (3+) ions have been studied using the four-component configuration interaction singles doubles, spin-orbit complete active space self-consistent field, and spin-orbit complete active space-order perturbation theory methods. The roles of scalar relativistic effects and spin-orbit coupling are analyzed; results with different methods are carefully compared by a precise analysis of the wave functions. A molecular spinor diagram is used in relation to the four-component calculations while a ligand field model is used for the two-step method. States with the same number of electrons in the four nonbonding orbitals are in very good agreement with the two methods while ligand field and charge transfer states do not have the same excitation energies.     

An analysis of core effects on shape-consistent pseudopotentials

Large core (seven-valence electrons) shape-consistent averaged relativistic pseudopotentials (AREP) including core effects have been derived for the halogen series (Cl,Br,I,At). The influence of core effects on the spin-orbit splitting of the halogen and alkali atoms is clearly demonstrated within an all-electron four-component atomic reference calculation by means of a perturbation analysis. In particular, it is shown that AREPs extracted at the Dirac-Coulomb-Fock level, which already include spin-orbit polarization effects, give excellent results for atomic spectroscopy and equilibrium distances of halogen dimers. We also show that in our approach the core effects, included by configuration interaction using the numerical GRASP code, are transferred to the averaged orbital one-electron energy, defined in a perturbational way. This leads to a modification of the extracted AREPs by core effects, which is illustrated by calculations of the first atomic excited states using these AREPs. These results support the validity of including core effects directly in the AREPs extracted in a shape-consistent scheme. The transferability to the atomic excited states as well as to the molecular case is also verified.     

Formation of giant solvation shells around fulleride anions in liquid ammonia

Here, we measure the solvation structure of fulleride C605- anions in potassium ammonia solution using neutron diffraction. We find a very strong solvation structure consisting of two shells of ammonia densely packed around the anion. The system's structure is driven by the propensity of ammonia molecules to direct one of their hydrogen bonds to the center of the anion while retaining axial hydrogen bonding within the shells. This permits high concentrations of solvent separated fulleride anions.     

Relativistic Density Functional NMR Tensors Analyzed with Spin-free Localized Molecular Orbitals

The implementation of fast relativistic methods based on density functional theory, in conjunction with localized molecular orbital (LMO) based analysis, allows straightforward interpretations of NMR parameters in terms of contributions from core shells, lone pairs, and bonds, for compounds containing elements from across the periodic table. We present a conceptual review of a frequently used LMO analysis of NMR parameters calculated in the presence of spin-orbit interactions and other relativistic effects. An accompanying example focuses on the ¹⁵N shielding in a heavy metal complex.