Exact Two-component Hamiltonian Research Articles

Comparison of state-interaction and spinor-representation calculations of spin-orbit coupling within exact two-component coupled-cluster theories

A benchmark study of state-interaction and spinor-representation calculations of spin-orbit coupling using the exact two-component Hamiltonians with atomic mean-field integrals (the X2CAMF schemes) at the equation-of-motion coupled-cluster singles and doubles level is reported. We adopt a version of the X2CAMF scheme with spin-orbit integrals correct to first order for the state-interaction calculations and a version correct to infinite order for the spinor-representation calculations. The differences between the state-interaction calculations with minimum active spaces to account for (quasi-)degeneracy and the spinor-representation calculations thus correspond to higher-order spin-orbit contributions. This state-interaction approach with scalar-relativistic effects accurately included in the reference functions and in the spin-orbit integrals is shown to exhibit robust performance for elements across the periodic table. On the other hand, the more rigorous spinor representation shows more rapid convergence with respect to the number of correlated electrons and is the preferred choice for accurate calculations for heavy elements.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Relativistic calculations on the transition electric dipole moments and radiative lifetimes of the spin-forbidden transitions in the antimony hydride molecule

Calculations on the spectroscopic constants and transition properties of the first three states (a1Δ, b1Σ+, and X3Σ−) of the SbH molecule were performed under the relativistic framework using the exact two-component Hamiltonian (X2C). The potential energy curves in the Franck–Condon region were computed and compared with the previous values. Furthermore, the transition dipole moments for the weak spin-forbidden transitions (b0+–X10+, b0+–X21, X10+–X21, and X21–a2) were reported. The spontaneous radiative lifetime of the b1Σ+ (υ′ = 0) state was calculated as 163.5±7.5 μ s, which is in reasonable agreement with the latest experimental value of 173±3 μ s. The spontaneous radiative lifetimes of the X21 (υ′ = 0) state and the a2 (υ′ = 0) state were calculated to be 48.6 s and ∼ 8 ms, respectively. Our study is expected to be a benchmark transition property computation for comparison with other theoretical and experimental results. The datasets presented in this paper, including the transition dipole moments, are openly available at https://dx.doi.org/10.11922/sciencedb.j00113.00018.

Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element Compounds.

We present an efficient implementation of paramagnetic NMR shielding tensors and shifts in a nonrelativistic and scalar-relativistic density functional theory framework. For the latter, we make use of the scalar exact two-component Hamiltonian in its local approximation, and generally we apply the well established (multipole-accelerated) resolution of the identity approximation and the seminumerical exchange approximation. The perturbed density matrix of a paramagnetic NMR shielding calculation is further used to study the magnetically induced current density and ring currents of open-shell systems as illustrated for [U@Bi12]3-. [U@Bi12]3- features delocalized highest occupied molecular orbitals and sustains a net diatropic ring current of ca. 18 nA/T through the Bi12 torus similar to the all-metal aromatic heavy-element cluster [Th@Bi12]4-.

Electron-Nucleus Hyperfine Coupling Calculated from Restricted Active Space Wavefunctions and an Exact Two-Component Hamiltonian.

Exact two-component (X2C) relativistic nuclear hyperfine magnetic field operators were incorporated in X2C ab initio wavefunction calculations at the multireference restricted active space (RAS) level for calculations of nuclear hyperfine magnetic properties. Spin-orbit coupling was treated via RAS state interaction (SO-RASSI). The method was tested by calculations of electron-nucleus hyperfine coupling constants. The approach, implemented in the OpenMolcas program, overcomes a major limitation of a previous SO-RASSI implementation for hyperfine coupling that relied on nonrelativistic hyperfine operators [J. Chem. Theor. Comput. 2015, 11, 538-549] and therefore had limited applicability. Results from calculations on systems with light and heavy main group elements, transition metals, lanthanides, and one actinide complex demonstrate reasonably good agreement with experimental data, where available, as long as the active space can generate sufficient spin polarization.

A study of non-iterative triples contributions in relativistic equation-of-motion coupled-cluster calculations using an exact two-component Hamiltonian with atomic mean-field spin-orbit integrals: Application to uranyl and other heavy-element compounds.

The implementation of an equation-of-motion coupled-cluster singles and doubles augmented with a noniterative triples correction [EOM-CCSD(T)(a)*] method [D. A. Matthews and J. F. Stanton, J. Chem. Phys. 145, 124102 (2016)] with an exact two-component Hamiltonian using atomic mean-field spin-orbit integrals (X2CAMF) is reported. Benchmark calculations show that both the correction from triple excitations in the cluster operator to the similarity-transformed Hamiltonian and the contribution from triple excitations in excited-state eigenvalue equations make important contributions to computed excitation energies of molecules containing heavy elements. X2CAMF-EOM-CCSD(T)(a)* excitation energies and excited-state properties for low-lying excited states of As2, Sb2, and Bi2 are in significantly better agreement with corresponding experimental values than EOM-CCSD ones. X2CAMF-EOM-CCSD(T)(a)* calculations have also been demonstrated to provide accurate results for excitation energies of low-lying excited states of bare uranyl ion UO2 2+, hereby resolving a long-standing debate on this topic.

Exact two-component equation-of-motion coupled-cluster singles and doubles method using atomic mean-field spin-orbit integrals.

A new semi-atomic-orbital- based algorithm for a two-component spin-orbit (SO) equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method using mean-field SO integrals is reported. The new algorithm removes the major computational bottlenecks of a SO-EOM-CCSD calculation associated with the evaluation, storage, and processing of the H¯ab,cd elements in the similarity-transformed Hamiltonian involving four virtual orbital labels. The partial recovery of spin symmetry in the present algorithm reduces the storage requirement by an order of magnitude and the floating point operation count for the evaluation of the ladder-like term by a factor of three to four. EOM-CCSD calculations of excited states in the triiodide ion (I3 -) using the exact two-component Hamiltonian in combination with atomic mean-field SO integrals (X2CAMF) are reported as a validation of the implementation and also as a demonstration of the capability of the new algorithm to correlate extended virtual spaces. X2CAMF-EOM-CCSD calculations of the ground and excited states in As2, Sb2, and Bi2 are also presented and compared with the available experimental studies. An analysis based on the computed spectroscopic constants as well as the compositions of the excited-state wavefunctions strongly supports a new assignment for the lowest 2u and 0u - levels in the photoelectron spectrum of Bi2.

Relativistic Polarizable Embedding.

Most chemistry, including chemistry where relativistic effects are important, occurs in an environment, and in many cases, this environment has a significant effect on the chemistry. In nonrelativistic quantum chemistry, a lot of progress has been achieved with respect to including environments such as a solvent or protein in the calculations, and now is the time to extend the possibilities for also doing this in relativistic quantum chemistry. The polarizable embedding (PE) model efficiently incorporates electrostatic effects of the environment by describing it as a collection of localized electric multipoles and polarizabilities obtained through quantum chemical calculations. In this article, we present the theory and implementation of four- and exact two-component Hamiltonians within a PE framework. We denote the methods the PE-4c-DFT and PE-X2C-DFT models. The models include a linear response formalism to calculate time-dependent (TD) properties: PE-TD-4c-DFT and PE-TD-X2C-DFT. With this first implementation, we calculate the PE-TD-4c-PBE0 excitation energies of the TcO4- and ReO4- ions in an explicit water solvent. This initial investigation focuses on the relative size of relativistic and solvent contributions to the excitation energies. The solvent effect is divided into an indirect solvent effect due to the structural perturbation of the XO4- ion and a direct electrostatic effect. The relativistic effects as well as both types of solvent effects are found to contribute to a shift in the excitation energies, but they do so to different extents depending on the ion and the electronic transition in question.

Real time propagation of the exact two component time-dependent density functional theory.

We report the development of a real time propagation method for solving the time-dependent relativistic exact two-component density functional theory equations (RT-X2C-TDDFT). The method is fundamentally non-perturbative and may be employed to study nonlinear responses for heavy elements which require a relativistic Hamiltonian. We apply the method to several group 12 atoms as well as heavy-element hydrides, comparing with the extensive theoretical and experimental studies on this system, which demonstrates the correctness of our approach. Because the exact two-component Hamiltonian contains spin-orbit operators, the method is able to describe the non-zero transition moment of otherwise spin-forbidden processes in non-relativistic theory. Furthermore, the two-component approach is more cost effective than the full four-component approach, with similar accuracy. The RT-X2C-TDDFT will be useful in future studies of systems containing heavy elements interacting with strong external fields.

Exact two-component Hamiltonians revisited

Two routes for deriving the exact two-component Hamiltonians are compared. In the first case, as already known, we start directly from the matrix representation of the Dirac operator in a restricted kinetically balanced basis and make a single block diagonalization. In the second case, not considered before, we start instead from the Foldy-Wouthuysen operator and make proper use of resolutions of the identity. The expressions are surprisingly different. It turns out that a mistake was made in the former formulation when going from the Dirac to the Schrodinger picture. The two formulations become equivalent after the mistake is corrected.