Exact-Two-Component Multiconfiguration Pair-Density Functional Theory

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Dirac-Coulomb-Breit Molecular Mean-Field Exact-Two-Component Relativistic Equation-of-Motion Coupled-Cluster Theory

We present a relativistic equation-of-motion coupled-cluster with single and double excitation formalism within the exact two-component framework (X2C-EOM-CCSD), where both scalar relativistic effects and spin-orbit coupling are variationally included at the reference level. Three different molecular mean-field treatments of relativistic corrections, including the one-electron, Dirac-Coulomb, and Dirac-Coulomb-Breit Hamiltonian, are considered in this work. Benchmark calculations include atomic excitations and fine-structure splittings arising from spin-orbit coupling. Comparison with experimental values and relativistic time-dependent density functional theory is also carried out. The computation of the oscillator strength using the relativistic X2C-EOM-CCSD approach allows for studies of spin-orbit-driven processes, such as the spontaneous phosphorescence lifetime.

Overdestabilization vs Overstabilization in the Theoretical Analysis of f-Orbital Covalency

The complex nature of the f-orbital electronic structures and their interaction with the chemical environment pose significant computational challenges. Advanced computational techniques that variationally include scalar relativities and spin-orbit coupling directly at the molecular orbital level have been developed to address this complexity. Among these, variational relativistic multiconfigurational multireference methods stand out for their high accuracy and systematic improvement in studies of f-block complexes. Additionally, these advanced methods offer the potential for calibrating low-scaling electronic structure methods such as density functional theory. However, studies on the Cl K-edge X-ray absorption spectra of the [Ce(III)Cl6]3- and [Ce(IV)Cl6]2- complexes show that time-dependent density functional theory with approximate exchange-correlation kernels can lead to inaccuracies, resulting in an overstabilization of 4f orbitals and incorrect assessments of covalency. In contrast, approaches utilizing small active space wave function methods may understate the stability of these orbitals. The results herein demonstrate the need for large active space, multireference, and variational relativistic methods in studying f-block complexes.

Unveiling Hidden Shake-Up Features in the Uranyl M4-Edge Spectrum

The M4,5-edge high energy resolution X-ray absorption near-edge structure (HR-XANES) spectra of actinyls offer valuable insights into the electronic structure and bonding properties of heavy-element complexes. To conduct a comprehensive spectral analysis, it is essential to employ computational methods that accurately account for relativistic effects and electron correlation. In this work, we utilize variational relativistic multireference configurational interaction methods to compute and analyze the X-ray M4-edge absorption spectrum of uranyl. By employing these advanced computational techniques, we achieve excellent agreement between the calculated spectral features and experimental observations. Moreover, the calculations unveil significant shake-up features, which arise from the intricate interplay between strongly correlated 3d core-electron and ligand excitations. This research provides important theoretical insights into the spectral characteristics of heavy-element complexes. Furthermore, it establishes the foundation for utilizing M4,5-edge spectroscopy as a means to investigate the chemical activities of such complexes. By leveraging this technique, we can gain a deeper understanding of the bonding behavior and reactivity of heavy-element compounds.

Comparison of Variational and Perturbative Spin-Orbit Coupling within Two-Component CASSCF

The modeling of spin-orbit coupling (SOC) remains a challenge in computational chemistry due to the high computational cost. With the rising popularity of spin-driven processes and f-block metals in chemistry and materials science, it is incumbent on the community to develop accurate multiconfigurational SOC methods that scale to large systems and understand the limits of different treatments of SOC. Herein, we introduce an implementation of perturbative SOC in scalar-relativistic two-component CASSCF (srX2C-CASSCF-SO). Perspectives on the limitations and accuracy of srX2C-CASSCF-SO are presented via benchmark calculations.

Multiconfiguration Pair-Density Functional Theory for Vertical Excitation Energies in Actinide Molecules

Modeling actinides with electronic structure theories is challenging because these systems present a strong ligand field and metal-ligand covalency. We systematically investigate the effectiveness of pair-density functional theory (PDFT) for the calculation of vertical excitation energies in An(III), [AnIIICl6]3-, and [AnVIO2]2+ (An = U, Np, Pu, and Am). We compare the performance of PDFT, hybrid PDFT, and multistate PDFT with traditional active-space methods followed by perturbation theory, like multistate CASPT2, and with experimental data. Overall, multistate PDFT gives quantitative agreement with multistate CASPT2 at a significantly reduced computational cost.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Improving One-Electron Exact-Two-Component Relativistic Methods with the Dirac-Coulomb-Breit-Parameterized Effective Spin-Orbit Coupling

In photochemical processes, spin-orbit coupling plays a crucial role in determining the outcome of the reaction. However, the exact treatment of the Dirac-Coulomb-Breit two-electron operator required for rigorous inclusion of spin-orbit coupling is computationally prohibitive. To address this challenge, we present a Dirac-Coulomb-Breit-parameterized screened-nuclear spin-orbit factor to approximate two-electron spin-orbit couplings in the effective one-electron spin-orbit Hamiltonian. We propose two schemes, the universal and row-dependent parameterizations, to further improve the accuracy of the method. Benchmark calculations on both atomic and molecular systems are performed and compared to results from the computationally expensive four-component Dirac-Coulomb-Breit method. The Dirac-Coulomb-Breit-parameterized approach offers a more computationally feasible method for accurate spin-orbit coupling calculations.

Relativistic Kramers-Unrestricted Exact-Two-Component Density Matrix Renormalization Group

In this work we develop a variational relativistic density matrix renormalization group (DMRG) approach within the exact-two-component (X2C) framework (X2C-DMRG), using spinor orbitals optimized with the two-component relativistic complete active space self-consistent field. We investigate fine-structure splittings of p- (Ga, In, Tl) and d-block (Sc, Y, La) atoms and excitation energies of monohydride molecules (GeH, SnH, and TlH) with X2C-DMRG calculations using an all-electron relativistic Hamiltonian in a Kramers-unrestricted basis. We find that X2C-DMRG yields accurate ²P and ²D splittings compared to multireference configuration interaction with singles and doubles (MRCISD). We also investigated the degree of symmetry breaking in the atomic multiplets and convergence of electron correlation in the total energies. Symmetry breaking can be large in some cases (∼30 meV); however, increasing the number of renormalized block states m for the DMRG optimization recovers the symmetry breaking by several orders of magnitude. Encouragingly, we find the convergence of electron correlation to be close to MRCISDTQ5 quality. Relativistic X2C-DMRG approaches are important for cases where spin-orbit coupling is significant and the underlying reference wave function requires a large determinantal space. We are able to obtain quantitatively correct fine-structure splittings for systems up to 10¹⁹ number of determinants with traditional CI approaches, which are currently unfeasible to converge for the field.