A point-charge model for the nuclear quadrupole moment: Coupled-cluster, Dirac–Fock, Douglas–Kroll, and nonrelativistic Hartree–Fock calculations for the Cu and F electric field gradients in CuF

Calculation of electric field gradients with the exact two-component (X2C) quasi-relativistic method and its local approximations

When calculating electric field gradients (EFGs), relativistic and electron correlation effects are crucial for obtaining accurate results, and the commonly used density functional methods produce unsatisfactory results, especially for heavy elements and/or strongly correlated systems. In this work, a stand-alone program is presented, which enables calculation of EFGs from the molecular orbitals supplied by an external high accuracy quantum chemical calculation and includes relativistic effects through the exact two-component (X2C) formalism and efficient local approximations to it. Application to BiN and BiP molecules shows that a high precision can be achieved in the calculation of nuclear quadrupole coupling constants of 209Bi by combining advanced ab initio methods with the X2C approach. For seventeen iron compounds, the Mössbauer nuclear quadrupole splittings (NQS) of 57Fe calculated using a double-hybrid functional method are in very good agreement with the experimental values. It is shown that, for strongly correlated molecules, the double-hybrid functionals are much more accurate than the commonly used hybrid functionals. The computer program developed in this study furnishes a useful utility for obtaining EFGs and related nuclear properties with high accuracy.

Density-Corrected Density Functional Theory for Molecular Properties

Density-corrected (DC) density functional theory (DFT) has been proposed to overcome difficulties related to the self-interaction error. The procedure uses the Hartree-Fock electron density (matrix) non-self-consistently in conjunction with an approximate functional. DC-DFT has so far mainly been tested for total energy differences, whereas other types of molecular properties have not been evaluated systematically. This work focuses on the performance of DC-DFT for molecular properties, namely, dipole moments, static polarizabilities, and electric field gradients (EFGs) at atomic nuclei. Accurate reference data were generated from coupled-cluster theory to assess the performance of DC and self-consistent DFT calculations for twelve molecules, including diatomics with transition metals. DC-DFT does no harm in dipole moment calculations, but it negatively impacts the polarizability in at least one case. DC-DFT performs well for EFGs, even for the difficult case of CuCl.

Quadratic Unitary Coupled-Cluster Singles and Doubles Scheme: Efficient Implementation, Benchmark Study, and Formulation of an Extended Version

An efficient implementation of the quadratic unitary coupled-cluster singles and doubles (qUCCSD) scheme for calculations of electronic ground and excited states using an unrestricted molecular spin-orbital formulation and an efficient tensor contraction library is reported. The accuracy of the qUCCSD scheme and the efficiency of the present implementation are demonstrated using extensive benchmark calculations of excitation energies and an application to S₀ → S₁ vertical excitation energies for cis- and trans-4a,4b-dihydrotriphenylene. The qUCCSD scheme has been shown to provide improved excitation energies compared with the UCC3 scheme formulated based on perturbation theory. A UCC truncation scheme that can provide excitation energies correct through the fourth order is also presented to further improve the accuracy of the qUCCSD scheme.

Analytic evaluation of energy first derivatives for spin-orbit coupled-cluster singles and doubles augmented with noniterative triples method: General formulation and an implementation for first-order properties

A formulation of analytic energy first derivatives for the coupled-cluster singles and doubles augmented with noniterative triples [CCSD(T)] method with spin-orbit coupling included at the orbital level and an implementation for evaluation of first-order properties are reported. The standard density-matrix formulation for analytic CC gradient theory adapted to complex algebra has been used. The orbital-relaxation contributions from frozen core, occupied, virtual, and frozen virtual orbitals to analytic spin-orbit CCSD(T) gradients are fully taken into account and treated efficiently, which is of importance to calculations of heavy elements. Benchmark calculations of first-order properties including dipole moments and electric-field gradients using the corresponding exact two-component property integrals are presented for heavy-element containing molecules to demonstrate the applicability and usefulness of the present analytic scheme.

Picture-change correction in relativistic density functional theory

Relativistic quantum chemical calculations are performed based on one of two physical pictures, namely the Dirac picture and the Schrödinger picture. With regard to the latter, the so-called picture-change effect (PCE) and picture-change correction (PCC) have been studied. The PCE, which is the change in the expectation value associated with the transformation, is not commonly a minor effect. The electron density, which is given by the expectation value of the density operator, is a fundamental variable in relativistic density functional theory (RDFT). Thus, performing the PCC in RDFT calculations is essential not only in terms of numerical agreement with the Dirac picture, but also from the viewpoint of fundamental theory. This paper explains theories and numerical studies of PCE and PCC in RDFT after overviewing those in properties, which involves the authors' works on the development of RDFT in the Schrödinger picture and relativistic exchange-correlation functionals based on picture-change-corrected variables.

Analytic high-order Douglas-Kroll-Hess electric field gradients

In this work we present a comprehensive study of analytical electric field gradients in hydrogen halides calculated within the high-order Douglas-Kroll-Hess (DKH) scalar-relativistic approach taking picture-change effects analytically into account. We demonstrate the technical feasibility and reliability of a high-order DKH unitary transformation for the property integrals. The convergence behavior of the DKH property expansion is discussed close to the basis set limit and conditions ensuring picture-change-corrected results are determined. Numerical results are presented, which show that the DKH property expansion converges rapidly toward the reference values provided by four-component methods. This shows that in closed-shell cases, the scalar-relativistic DKH(2,2) approach which is of second order in the external potential for both orbitals and property operator yields a remarkable accuracy. As a parameter-dependence-free high-order DKH model, we recommend DKH(4,3). Moreover, the effect of a finite-nucleus model, different parametrization schemes for the unitary matrices, and the reliability of standard basis sets are investigated.

Nuclear electric quadrupole moment of gold

The nuclear quadrupole moment for (197)Au has been determined on the base of the state-of-art relativistic molecular calculations. The experimental shifts in the nuclear coupling constants in the series of molecules AuF, XeAuF, KrAuF, ArAuF, (OC)AuF, and AuH have been combined with highly accurate determinations of the electric field gradient (EFG) at the gold nucleus, obtained by molecular relativistic Dirac-Coulomb-Gaunt Hartree-Fock calculations. The electronic correlation contribution to the EFG is included with the CCSD(T) and CCSD-T approaches, also in the four-component framework, using a finite-difference method. In order to estimate the accuracy of their approach the authors have thoroughly investigated the convergence of the results with respect to the basis set employed and the size of the correlated orbital space. The effect of the full Breit electron-electron interaction on the nuclear quadrupole moment of gold has also been considered explicitly for the AuF molecule. They obtain for (197)Au a nuclear quadrupole moment of 510+/-15 mb, which deviates by about 7% from the currently accepted muonic value.

The Nuclear Quadrupole Moment of 14N from Accurate Electric Field Gradient Calculations and Microwave Spectra of NP Molecule

Extensive series of relativistic coupled cluster calculations of the electric field gradient at N in NP has been carried out. The accurate value of the calculated electric field gradient, combined with the highly accurate experimental value of the nuclear quadrupole coupling constant for the 14N nucleus, gives the 'molecular' value of the nuclear quadrupole moment Q(14N) = 20.46 mb. This result perfectly agrees with the value (20.44 ± 0.03 mb) determined from atomic calculations and atomic spectra. The present study involves also extensive investigations of basis sets which must be used in highly accurate calculations of electric field gradients.

The nuclear electric quadrupole moment of antimony from the molecular method

Relativistic Dirac-Coulomb (DC) Hartree-Fock calculations are employed to obtain the analytic electric field gradient (EFG) on the antimony nucleus in the SbN, SbP, SbF, and SbCl molecules. The electronic correlation contribution to the EFGs is included with the DC-CCSD(T) and DC-CCSD-T approaches, also in the four-component framework, using a finite-difference method. The total EFG results, along with the experimental nuclear quadrupole coupling constants from microwave spectroscopy, allow to derive the nuclear quadrupole moments of (121)Sb and (123)Sb, respectively, as -543(11) and -692(14) mb.

The quadrupole moment of the Sb nucleus from molecular microwave data and calculated relativistic electric-field gradients

The recently determined accurate values of the nuclear quadrupole coupling constant of the Sb nucleus in SbN, SbP, SbF, and SbCl and the calculated electric-field gradients at Sb in these molecules are used to obtain the nuclear quadrupole moment of 121Sb and 123Sb. The calculation of the electric-field gradient has been carried out by using the infinite-order two-component relativistic method in the scalar approximation. The accompanying change of picture of the electric-field gradient operator has been accounted for by employing the shifted nucleus model of nuclear quadrupoles. The electron correlation effects are calculated at the level of the coupled cluster approximation. The present calculations give the "molecular" value of the nuclear quadrupole moment of 121Sb equal to -556+/-24 mb which is considerably different from the old "recommended" value of -360+/-40 mb and also differs from the recent "solid-state" result (-669+/-15 mb). The validation of the present data is comprehensively discussed.

Relativistic two-component calculations of electronic g-tensors that include spin polarization

The first two-component relativistic density-functional approach for the calculation of electronic g-tensors is reported that includes spin polarization using noncollinear spin-density functionals. The method is based on the relativistic Douglas-Kroll-Hess Hamiltonian and has been implemented into the ReSpect program package. Using three self-consistent-field calculations with orthogonal orientations of total magnetization J, the full g-matrix may be obtained. In contrast to previous spin-restricted two-component treatments, results with the new approach agree excellently with spin-polarized one-component calculations for light-atom radicals. Additionally, unlike one-component approaches, the method also reproduces successfully the negative deltag(parallel)-values of heavy-atom 2sigma radicals and the negative deltag(perpendicular) components in cysteinyl. The new method removes effectively the dilemma existing up to now regarding the simultaneous inclusion of spin polarization and higher-order spin-orbit effects in g-tensor calculations. It is straightforwardly applicable to higher than doublet spin multiplicities and has been implemented with hybrid functionals.

Calculation of electric-field gradients based on higher-order generalized Douglas–Kroll transformations

In this paper, the calculation of electric-field-like properties based on higher-order Douglas-Kroll-Hess (DKH) transformations is discussed. The electric-field gradient calculated within the Hartree-Fock self-consistent field framework is used as a representative property. The properties are expressed as an analytic first derivative of the four-component Dirac energy and the nth-order DKH energy, respectively. The differences between a "forward" transformation of the relativistic energy or the "back transformation" of the wave function is discussed in some detail. Detailed test calculations were carried out on the electric-field gradient at the halogen nucleus in the series HX (X=F,Cl,Br,I,At) for which extensive reference data are available. The DKH method is shown to reproduce (spin-free) four-component Dirac-Fock results to an accuracy of better than 99% which is significantly closer than previous DKH studies. The calculations of both the Hamiltonian and the property operator are shown to be essentially converged after the second-order transformation, even for elements as heavy as At. In addition, we have obtained results within the density-functional framework using the DKHZ and zeroth-order regular approximation (ZORA) methods. The latter results included picture-change effects at the scalar relativistic variant of the ZORA-4 level and were shown to be in quantitative agreement with earlier results obtained by van Lenthe and Baerends. The picture-change effects are somewhat smaller for the ZORA method compared to DKH. For heavier elements significant differences in the field gradients predicted by the two methods were found. Based on comparison with four-component Dirac-Kohn-Sham calculations, the DKH results are more accurate. Compared to the spin-free Dirac-Kohn-Sham reference values, the ZORA-4 formalism did not improve the results of the ZORA calculations.

The quadrupole moment of the 3∕2+ nuclear ground state of Au197 from electric field gradient relativistic coupled cluster and density-functional theory of small molecules and the solid state

An attempt is made to improve the currently accepted muonic value for the 197Au nuclear quadrupole moment [+0.547(16)x10(-28) m2] for the 3/2+ nuclear ground state obtained by Powers et al. [Nucl. Phys. A230, 413 (1974)]. From both measured Mossbauer electric quadrupole splittings and solid-state density-functional calculations for a large number of gold compounds a nuclear quadrupole moment of +0.60x10(-28) m2 is obtained. Recent Fourier transform microwave measurements for gas-phase AuF, AuCl, AuBr, and AuI give accurate bond distances and nuclear quadrupole coupling constants for the 197Au isotope. However, four-component relativistic density-functional calculations for these molecules yield unreliable results for the 197Au nuclear quadrupole moment. Relativistic singles-doubles coupled cluster calculations including perturbative triples [CCSD(T) level of theory] for these diatomic systems are also inaccurate because of large cancellation effects between different field gradient contributions subsequently leading to very small field gradients. Here one needs very large basis sets and has to go beyond the standard CCSD(T) procedure to obtain any reliable field gradients for gold. From recent microwave experiments by Gerry and co-workers [Inorg. Chem. 40, 6123 (2001)] a significantly enhanced (197)Au nuclear quadrupole coupling constant in (CO)AuF compared to free AuF is observed. Here, these cancellation effects are less important, and relativistic CCSD(T) calculations finally give a nuclear quadrupole moment of +0.64x10(-28) m2 for 197Au. It is argued that it is currently very difficult to improve on the already published muonic value for the 197Au nuclear quadrupole moment.

Multireference CI calculation of nuclear quadrupole coupling constants of CN+ and CN-: rovibrational dependence

The 14N quadrupole coupling constants of rovibrational levels of the X1sigma+ and c1sigma+ states of CN+, and the ground electronic state of CN- are calculated from molecular wavefunctions which explicitly describe nuclear displacement. From the electronic states considered, the excited 1sigma+ state of CN is predicted to exhibit the strongest N coupling, at least in the ground vibrational state. Compared to the vibrational dependence of the 14N QCC's, which is found to be significant in all cases, the rotational dependence is predicted to be unimportant. Special attention is paid to the assessment of adequacy of the expectation value approach to the evaluation of the electric field gradient tensor within the applied multireference configuration interaction formalism. Spectroscopic constants are derived from corresponding potential energy curves to testify to the quality of the correlated wave functions used.