Infinite-order Two-component Research Articles

Advancing Our Understanding of the Excited States of the Tantalum Anion.

Our study provides a comprehensive theoretical examination of the energy levels associated with the neutral tantalum atom and its ions in various charge states (Ta, Ta+, and Ta-), employing the multiconfiguration Dirac-Hartree-Fock (MCDHF) method, and relativistic infinite order two-component (IOTC) method with multiconfiguration complete active space self-consistent field (CASSCF) followed by the second-order single-state multireference perturbation (CASPT2) methods. The effect of spin-orbit (SO) coupling is introduced via the restricted active space state interaction (RASSI) method, utilizing atomic mean field SO integrals (AMFI). Through IOTC CASSCF/CASPT2 RASSI calculations, we determined the electron affinity (EA) of the tantalum atom to be 0.321 eV, which stands among the most accurate theoretical values achieved to date. This result closely aligns with the experimental measurement of 0.329 eV. Our investigation highlights potential discrepancies between the predicted symmetry of the excited states of the tantalum anion and experimental observations. Additionally, we calculated the bonding energies for transitions from Ta- to Ta and identified four potential bound or quasi-bound states in the tantalum anion.

Matrix-decomposed two-electron integrals in the infinite-order two-component Hamiltonian

The Cholesky decomposition (CD) and lower–upper decomposition (LUD) of two-electron integral (TEI) matrices were applied to a spin-free infinite-order two-component (IOTC) relativistic Hamiltonian. In the Hamiltonian, the evaluation of three types of TEIs of primitive basis functions and the unitary transformation of TEIs are required. CD was used to calculate the symmetric TEI matrices: Coulomb-like and specific spin-free interaction terms. LUD was used for the asymmetric TEI matrix: Darwin-like term. Unitary transformation was performed on the decomposed matrices. Numerical assessments suggest that the proposed method can reduce the computational cost without loss of accuracy.

Douglas-Kroll and infinite order two-component transformations of Dirac-Fock operator.

We extended the conventional Douglas-Kroll (DK) and infinite order two-component (IOTC) methods to a technique applicable to Fock matrices, called extended DK (EDK) and extended IOTC (EIOTC), respectively. First, we defined a strategy to divide the Dirac-Fock operator into zero- and first-order terms. We then demonstrated that the first-order extended DK transformation, which is the Foldy-Wouthuysen transformation for the zero-order term, as well as the second- and third-order EDK and EIOTC, could be well defined. The EDK- and EIOTC-transformed Fock matrix, kinetic energy operator, nuclear attraction operator, and density matrix were derived. These equations were numerically evaluated, and it was found that these methods were accurate. In particular, EIOTC was consistent with the four-component approach. Four-component and extended two-component calculations are more expensive than non-relativistic calculations due to small-component-type two-electron integrals. We developed a new approximation formula, RIS-V, for small-component-type two-electron integrals, including the spin-orbit interaction between electrons. These results suggest that the RIS-V formula effectively accelerates the four-component and extended two-component methods.

AuSi molecule revisited: IOTC/CASSCF/CASPT2 calculations

AbstractThe results of theoretical calculations for the ground and low‐lying excited states of the AuSi molecule are presented. Calculations were carried out using the spin‐free relativistic infinite‐order two‐component (IOTC) Hamiltonian in conjunction with the multistate CASPT2 method. For comparison, the ground state was also calculated using the IOTC‐CCSD(T) method. The spin‐orbit coupling is introduced via the restricted active space state interaction method with the use of the atomic mean‐field SO integrals. The results are compared with the available experimental and theoretical results. The spectroscopic parameters derived from the potential energy curves are presented. The results obtained suggest that the experimental data should be reconsidered.

Theoretical studies of the ground and excited states of the NaLi molecule by the IOTC CASPT2 method

Theoretical description of excited states of molecules is still not a routine issue, and searching for methods best suited for this purpose is still an open problem. The molecular calculations are performed using the CASSCF/CASPT2 method for the potential energy curves of the ground and excited states of NaLi molecule. The derived spectroscopic constants are compared with the results of the MR FSCC and PP methods and with the experimental values. The relativistic infinite-order two-component (IOTC) method is used to show its proper convergence to the nonrelativistic limit.

Relativistic calculations of AuSi+ and AuSi−

AbstractTheoretical calculations of the electronic structure of the ground state and a series of excited states of the AuSi+ and AuSi− molecules are presented. The calculations were carried out with the spin‐free relativistic infinite‐order two‐component (IOTC) method and high‐level complete active space self‐consistent field/complete active space perturbation theory correlated methods. The spin‐orbit (SO) coupling was introduced via the restricted active space state interaction method with the use of the atomic mean‐field SO integrals. The work presents the spectroscopic parameters of calculated states and full potential energy curves of the ionic AuSi+ and AuSi− structures for the first time. Electrostatic potential maps projected on the electron density surface illustrate the significant relativistic effects on going from nonrelativistic to scalar relativistic treatments.

Spin-Orbit Coupling Constants in Atoms and Ions of Transition Elements: Comparison of Effective Core Potentials, Model Core Potentials, and All-Electron Methods.

The spin-orbit coupling constants (SOCC) in atoms and ions of the first- through third-row transition elements were calculated for the low-lying atomic states whose main electron configuration is [ nd] q ( q = 1-4 and 6-9, n = the principal quantum number), using four different approaches: (1) a nonrelativistic Hamiltonian used to construct multiconfiguration self-consistent field (MCSCF) wave functions utilizing effective core potentials and their associated basis sets within the framework of second-order configuration interaction (SOCI) to calculate spin-orbit couplings (SOC) using one-electron Breit-Pauli Hamiltonian (BPH), (2) a nonrelativistic Hamiltonian used to construct MCSCF wave functions utilizing model core potentials and their associated basis sets within the framework of SOCI to calculate SOC using the full BPH, (3) nonrelativistic and spin-independent relativistic Hamiltonians used to construct MCSCF wave functions utilizing all-electron (AE) basis sets within the framework of SOCI to calculate SOC using the full BPH, and (4) a relativistic Hamiltonian given by the exact two-component (X2C) transformation for construction of Kramers-restricted relativistic configuration interaction wave functions. In this investigation, these four approaches are referred to as ECP, MCP, AE, and X2C methods, respectively. The ECP, MCP, and AE methods are so-called two-step approaches (TSA), while the X2C method is a one-step approach (OSA). In the AE method, three different calculations-relativistic elimination of small components (RESC), third-order Douglas-Kroll-Hess (DKH3), and infinite-order two-component (IOTC) relativistic correction-were performed for the estimation of the scalar relativistic components in addition to those of the nonscalar relativistic (NSR) contributions. The calculated SOCC are compared to the available experimental data via the Landé interval rule. Although there are several exceptions, including states whose main configuration is [ nd]5, the average differences between the ECP and AE (IOTC) SOCC and between the ECP and the X2C SOCC are mostly less than 20%. The differences between the ECP and the experimental SOCC are even smaller. No serious discrepancy was found between the TSA and OSA predictions of SOCC for the first- and second-row transition elements in comparison to experiment. For atoms and ions of the third-row transition elements, the SOCC calculated through the Landé interval rule are not reliable. The low-energy spin-mixed (SM) states originating from a [5d] q configuration ( q = 2-4) have a larger energy lowering due to the SOC effects, in comparison with those for atoms and ions of the first- and second-row transition elements. For the spin-mixed (SM) states originating from a [5d] q configuration ( q = 6-8), the energy lowering of all 4F7/2, 5D1, and 5D3 states due to the SOC effects is smaller than those of the other SM states. This difficulty, which also arises for the MCP, AE, and X2C (OSA) approaches, suggests that the LS-coupling scheme is inappropriate.

Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth.

This study investigates the possibility of detecting relativistic effects and electron correlation in single-crystal X-ray diffraction experiments using the examples of diphenyl mercury (HgPh2) and triphenyl bismuth (BiPh3). In detail, the importance of electron correlation (ECORR), relativistic effects (REL) [distinguishing between total, scalar and spin-orbit (SO) coupling relativistic effects] and picture change error (PCE) on the theoretical electron density, its topology and its Laplacian using infinite order two component (IOTC) wave functions is discussed. This is to develop an understanding of the order of magnitude and shape of these different effects as they manifest in the electron density. Subsequently, the same effects are considered for the theoretical structure factors. It becomes clear that SO and PCE are negligible, but ECORR and scalar REL are important in low- and medium-order reflections on absolute and relative scales-not in the high-order region. As a further step, Hirshfeld atom refinement (HAR) and subsequent X-ray constrained wavefunction (XCW) fitting have been performed for the compound HgPh2 with various relativistic and nonrelativistic wave functions against the experimental structure factors. IOTC calculations of theoretical structure factors and relativistic HAR as well as relativistic XCW fitting are presented for the first time, accounting for both scalar and spin-orbit relativistic effects.

Picture change error in quasirelativistic electron/spin density, Laplacian and bond critical points

The change of picture of the quasirelativistic Hartree–Fock wave functions is considered for electron/spin densities, the negative Laplacian of electron density and the appropriate bond critical point characteristics from the Quantum Theory of Atoms In Molecules (QTAIM). [OsCl5(Hpz)]− and [RuCl5(NO)]2− transition metal complexes are considered. Both, scalar relativistic and spin–orbit effects have been accounted for using the Infinite Order Two Component (IOTC) Hamiltonian. Picture change error (PCE) correction in the electron and spin densities and the Laplacian of electron density are treated analytically. Generally, PCE is found significant only in the core region of the atoms for the electron/spin density as well as Laplacian.

Relativistic calculations of X-ray photoelectron spectra and the accuracy of the IOTC method*

In this paper, the authors have applied the infinite-order two-component (IOTC) method to study the X-ray spectra of noble gases. The effect of spin–orbit coupling is introduced via the restricted active space interaction (RASSI) method with the use of the atomic mean-field AMFI integrals. For the deep core energies the discrepancies between the experimental and calculated values can be still quite large. To explain these differences the quantum electrodynamic corrections have been calculated such as Breit energy, self-energy and vacuum polarisation. The comparison between results of the IOTC method and the Douglas–Kroll–Hess (DKHn) approximation has been presented as well.

Study of the picture change error at the 2nd order Douglas Kroll Hess level of theory. Electron and spin density and structure factors of the Bis[bis(methoxycarbimido) aminato] copper (II) complex

The analytic correction and the extent of the picture change error (PCE) at the scalar 2nd order Douglas–Kroll–Hess level of theory is considered. The one-dimensional (1D), two-dimensional (2D) spin/electron densities and/or difference densities, structure factors and Mulliken populations of the Bis[bis-(methoxycarbimido)aminato]copper(II) model compound are presented. For further comparison the radial distributions of the electron and spin density of the copper atom (as well as of the copper di-cation) are presented. In addition, the infinite order two component (IOTC) radial distributions of electron and spin density of the copper atom and copper dication are presented as well. The PCE is almost hidden in the 2D densities of the studied model compound. The 1D electron/spin difference densities along the Cu–N bond show the extent of PCE in more detail than the 2D densities, nevertheless the small significance of PCE in the region further from the nucleus of the copper atom is confirmed. The DKH2 and IOTC electron/spin densities of Cu/Cu2+ do overlap with each other, unless in the vicinity of nucleus where the difference is visible, but still not as significant as PCE or the relativistic effects. The structure factors are found an order of magnitude less affected by PCE than by the relativistic effects. Furthermore, PCE in the Mulliken populations and in the expectation value of the 〈S^2〉 operator is briefly discussed.

Picture change error correction in the radial distributions of canonical orbital densities and total electron density of radon atom: the effect of the size of nucleus and the basis set limit

The 2nd order Douglas-Kroll-Hess (DKH2) and the Infinite Order Two Component (IOTC) radial distributions of electron density of canonical Hartree-Fock (HF) orbitals of radon atom are presented. Furthermore, the total electron density is revisited. The picture change error (PCE) correction is investigated by analytical means. The point charge model of nucleus and the Gaussian nucleus model are employed. The basis set is extrapolated by means of including tight s and also p Gaussians within the original triple zeta basis set. It is found that the DKH1 PCE corrected DKH2 total electron and s orbital contact densities are negative for the point charge model of nucleus if tight enough s Gaussians are included in the basis set. It is shown that this failure is caused due to the missing terms of the second order Douglas-Kroll transformation for the DKH2 electron density. PCE is found the most striking in the DKH2/IOTC electron density of s orbitals close to the nucleus. The radial distributions of the 2-component p 1/2 orbital densities are considerably affected by PCE at the nucleus as well. Furthermore, the PCE corrected DKH2/IOTC scalar p orbital densities have a non-zero value of electron density at nucleus and can be considered as an spin-orbit (SO) average of the p 1/2 and p 3/2 orbitals. The d and f orbitals are affected by PCE in the vicinity of the nucleus only little. The PCE corrected DKH2 and IOTC radial distributions of orbital densities are nodeless, which is completely in agreement with the radial distribution of the analytic or numeric DCH orbital densities.

Picture change error correction of radon atom electron density

The importance of the picture change error (PCE) correction in the quasirelativistic electron density of radon atom is presented. PCE correction is considered for the infinite order two-component (IOTC) and second order Douglas-Kroll-Hess (DKH2) wave functions. Implementation details of PCE correction of electron density are outlined. The result section presents the radial distributions of electron density ρ(r) and the 4πr(2)ρ(r) function of the radon atom in the nuclear region. The PCE corrected and contaminated DKH2, IOTC electron densities are presented and compared with the Dirac-Coulomb Hamiltonian and nonrelativistic electron densities. Besides, some additional effects in electron density and SCF energy are considered, such as spin-orbit coupling, the inclusion of the Gaunt term, and the finite nucleus model effects. The effects of p(2)Vp(2) and p(2)ρ(r)p(2) analytic integral classes within IOTC Hamiltonian and PCE correction of IOTC electron density are considered.

Recovering four-component solutions by the inverse transformation of the infinite-order two-component wave functions

The two-component Hamiltonian of the infinite-order two-component (IOTC) theory is obtained by a unitary block-diagonalizing transformation of the Dirac-Hamiltonian. Once the IOTC spin orbitals are calculated, they can be back transformed into four-component solutions. The transformed four component solutions are then used to evaluate different moments of the electron density distribution. This formally exact method may, however, suffer from certain approximations involved in its numerical implementation. As shown by the present study, with sufficiently large basis set of Gaussian functions, the Dirac values of these moments are fully recovered in spite of using the approximate identity resolution into eigenvectors of the p(2) operator.

Non-iterative approach to the infinite-order two-component (IOTC) relativistic theory and the non-symmetric algebraic Riccati equation

The formal framework of the recently developed infinite-order two-component (IOTC) method of relativistic quantum chemistry is analysed. The former iterative scheme of the IOTC approach is replaced by the non-iterative method which leads to the complete block-diagonalization of the algebraic Dirac Hamiltonian and permits to built the matrix representation of the equivalent two-component one-electron energy operator. The method is equivalent to solving the so-called non-symmetric algebraic Riccati equation. Its solution is illustrated numerically.

An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation

The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.

Two-component relativistic methods for the heaviest elements.

Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.