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.

The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry.

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

Correlated Dirac-Coulomb-Breit multiconfigurational self-consistent-field methods.

The fully correlated frequency-independent Dirac-Coulomb-Breit Hamiltonian provides the most accurate description of electron-electron interaction before going to a genuine relativistic quantum electrodynamics theory of many-electron systems. In this work, we introduce a correlated Dirac-Coulomb-Breit multiconfigurational self-consistent-field method within the frameworks of complete active space and density matrix renormalization group. In this approach, the Dirac-Coulomb-Breit Hamiltonian is included variationally in both the mean-field and correlated electron treatment. We also analyze the importance of the Breit operator in electron correlation and the rotation between the positive- and negative-orbital space in the no-virtual-pair approximation. Atomic fine-structure splittings and lanthanide contraction in diatomic fluorides are used as benchmark studies to understand the contribution from the Breit correlation.

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.

Phase-Controlled Synthesis and Quasi-Static Dielectric Resonances in Silver Iron Sulfide (AgFeS2 ) Nanocrystals.

Ternary metal-chalcogenide semiconductor nanocrystals are an attractive class of materials due to their tunable optoelectronic properties that result from a wide range of compositional flexibility and structural diversity. Here, the phase-controlled synthesis of colloidal silver iron sulfide (AgFeS2 ) nanocrystals is reported and their resonant light-matter interactions are investigated. The product composition can be shifted selectively from tetragonal to orthorhombic by simply adjusting the coordinating ligand concentration, while keeping the other reaction parameters unchanged. The results show that excess ligands impact precursor reactivity, and consequently the nanocrystal growth rate, thus deterministically dictating the resulting crystal structure. Moreover, it is demonstrated that the strong ultraviolet-visible extinction peak exhibited by AgFeS2 nanocrystals is a consequence of a quasi-static dielectric resonance (DR), analogous to the optical response observed in CuFeS2 nanocrystals. Spectroscopic studies and computational calculations confirm that a negative permittivity at ultraviolet/visible frequencies arises due to the electronic structure of these intermediate-band (IB) semiconductor nanocrystals, resulting in a DR consisting of resonant valence-band-to-intermediate-band excitations, as opposed to the well-known localized surface plasmon resonance response typically observed in metallic nanostructures. Overall, these results expand the current library of an underexplored class of IB semiconductors with unique optical properties, and also enrich the understanding of DRs in ternary metal-iron-sulfide nanomaterials.

Relativistic two-component projection-based quantum embedding for open-shell systems.

In this work, we present a relativistic quantum embedding formalism capable of variationally treating relativistic effects, including scalar-relativity and spin-orbit coupling. We extend density functional theory (DFT)-in-DFT projection-based quantum embedding to a relativistic two-component formalism, where the full spin magnetization vector form is retained throughout the embedding treatment. To benchmark various relativistic embedding schemes, spin-orbit splitting of the nominally t2g valence manifold of W(CO)6, exchange coupling of [(H3N)4Cr(OH)2Cr(NH3)4]4+, and the dissociation potential curve of WF6 are investigated. The relativistic embedding formalism introduced in this work is well suited for efficient modeling of open-shell systems containing late transition metal, lanthanide, and actinide molecular complexes.

OpenMolcas: From Source Code to Insight.

In this Article we describe the OpenMolcas environment and invite the computational chemistry community to collaborate. The open-source project already includes a large number of new developments realized during the transition from the commercial MOLCAS product to the open-source platform. The paper initially describes the technical details of the new software development platform. This is followed by brief presentations of many new methods, implementations, and features of the OpenMolcas program suite. These developments include novel wave function methods such as stochastic complete active space self-consistent field, density matrix renormalization group (DMRG) methods, and hybrid multiconfigurational wave function and density functional theory models. Some of these implementations include an array of additional options and functionalities. The paper proceeds and describes developments related to explorations of potential energy surfaces. Here we present methods for the optimization of conical intersections, the simulation of adiabatic and nonadiabatic molecular dynamics, and interfaces to tools for semiclassical and quantum mechanical nuclear dynamics. Furthermore, the Article describes features unique to simulations of spectroscopic and magnetic phenomena such as the exact semiclassical description of the interaction between light and matter, various X-ray processes, magnetic circular dichroism, and properties. Finally, the paper describes a number of built-in and add-on features to support the OpenMolcas platform with postcalculation analysis and visualization, a multiscale simulation option using frozen-density embedding theory, and new electronic and muonic basis sets.

Embedding non-collinear two-component electronic structure in a collinear quantum environment.

Spin-containing materials are important for spintronic applications. In this work, we present a computational framework to embed nonrelativistic, two-component calculations in a one-component environment. In this framework, both embedding scalar potential and magnetic field can be included to describe the interaction between quantum subsystems. In this current development, a generalized Kohn-Sham density functional theory electronic structure is embedded in unrestricted Kohn-Sham density functional theory. Two test systems are studied: a Li3 on a closed-shell He lattice and a Li3 on a He lattice containing a Li atom defect. The noncollinearity of Li3 is unaffected upon embedding in a closed-shell environment through the scalar potential embedding. However, the open-shell nature of the Li atom defect introduces an effective magnetic field that couples to the magnetic components of the generalized Kohn-Sham Hamiltonian. These results show that noncollinear quantum embedding in an open-shell collinear environment may modify the spin structure of the embedded system. The formalism developed herein may serve as a useful tool in the modeling of inhomogeneous magnetic fields in two-component calculations.

State-interaction pair-density functional theory.

The accurate description of ground- and excited-state potential energy surfaces poses a challenge for many electronic structure methods, especially in regions where strong electronic state interaction occurs. Here we introduce a new methodology, state-interaction pair-density functional theory (SI-PDFT), to target molecular systems exhibiting strong interaction of electronic states. SI-PDFT is an extension of multiconfiguration pair-density functional theory in which a set of N electronic states is generated through the diagonalization of an N × N effective Hamiltonian. We demonstrate the accuracy of the method by performing calculations on the ionic-neutral avoided crossing in lithium fluoride and the 1ππ-1πσ* avoided crossing in the H-O bond photodissociation in phenol. We show that SI-PDFT can be a useful tool in the study of photochemistry and nonadiabatic dynamics.

Analytic Gradients for Complete Active Space Pair-Density Functional Theory.

Analytic gradient routines are a desirable feature for quantum mechanical methods, allowing for efficient determination of equilibrium and transition state structures and several other molecular properties. In this work, we present analytical gradients for multiconfiguration pair-density functional theory (MC-PDFT) when used with a state-specific complete active space self-consistent field reference wave function. Our approach constructs a Lagrangian that is variational in all wave function parameters. We find that MC-PDFT locates equilibrium geometries for several small- to medium-sized organic molecules that are similar to those located by complete active space second-order perturbation theory but that are obtained with decreased computational cost.

Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems.

The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus determines molecular structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calculate the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chemistry, systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order reference wave function. This Account describes multiconfiguration pair-density functional theory (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and density functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as reference wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. It is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calculated from a reference wave function, while the rest of the energy is obtained from a density functional. However, there are two main differences with respent to Kohn-Sham DFT: (i) The reference wave function is multiconfigurational rather than being a single Slater determinant. (ii) The density functional is a function of the total density and the on-top pair density rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the reference wave function has the correct spatial and spin symmetry and can describe bond dissociation (of both single and multiple bonds) and electronic excitations in a formally and physically correct way. We then review the formulation of density functionals in terms of the on-top pair density. Finally we review successful applications of the theory to bond energies and bond dissociation potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of molecules, and to singlet-triplet splittings of radicals. We find that that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions but also with generalized-active-space multiconfiguration wave functions, which are practical for larger numbers of active electrons and active orbitals than are complete-active-space wave functions. The separated-pair approximation, which is a special case of generalized active space self-consistent-field theory, is especially promising. MC-PDFT, because it requires much less computer time and storage than pure WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.

The DQ and DQΦ electronic structure diabatization methods: Validation for general applications.

We recently proposed the dipole-quadrupole (DQ) method for transforming adiabatic electronic states to diabatic states by using matrix elements of the dipole and quadrupole operators, and we applied the method to 3-state diabatizations of LiH and phenol. Here we extend the method to also include the electrostatic potential, and we call the resulting method the DQΦ method, which denotes the dipole-quadrupole-electrostatic-potential diabatization method. The electrostatic potential provides extra flexibility, and the goal of the present work is to test and illustrate the robustness of the methods for producing diabatic potential energy curves that tend to the adiabatic curves away from crossings and avoided crossings and are smooth in regions of crossings and avoided crossings. We illustrate the generality of the methods by an application to LiH with four states and by two-state diabatizations of HCl, (H2)2, O3, and the reaction Li + HF → LiF + H. We find that-if enough states are included-the DQ method does not have a significant dependence on the parameter weighting the quadrupole moment, and a geometry-independent value of 10 a0 (-2) is adequate in all cases tested. We also find that the addition of the electrostatic potential improves the diabatic potentials in some cases and provides an additional property useful for increasing the generality of the method for diabatization.

Comment on "Fe2: As simple as a Herculean labour. Neutral (Fe2), cationic (Fe2(+)), and anionic (Fe2(-)) species" [J. Chem. Phys. 142, 244304 (2015)].

A recent paper on Fe2 [A. Kalemos, J. Chem. Phys. 142, 244304 (2015)] critiqued our previous work on the system [Hoyer et al., J. Chem. Phys. 141, 204309 (2014)]. In this comment, we explain the nature of our previously reported potential energy curve for Fe2 and we discuss our computed properties for Fe2. Additionally, we fix a labeling error that was present in our previous work, although this error is unrelated to the main point of discussion.

Multiconfiguration Pair-Density Functional Theory Is as Accurate as CASPT2 for Electronic Excitation.

A correct description of electronically excited states is critical to the interpretation of visible-ultraviolet spectra, photochemical reactions, and excited-state charge-transfer processes in chemical systems. We have recently proposed a theory called multiconfiguration pair-density functional theory (MC-PDFT), which is based on a combination of multiconfiguration wave function theory and a new kind of density functional called an on-top density functional. Here, we show that MC-PDFT with a first-generation on-top density functional performs as well as CASPT2 for an organic chemistry database including valence, Rydberg, and charge-transfer excitations. The results are very encouraging for practical applications.

Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table.

In this report, we summarize and describe the recent unique updates and additions to the Molcas quantum chemistry program suite as contained in release version 8. These updates include natural and spin orbitals for studies of magnetic properties, local and linear scaling methods for the Douglas-Kroll-Hess transformation, the generalized active space concept in MCSCF methods, a combination of multiconfigurational wave functions with density functional theory in the MC-PDFT method, additional methods for computation of magnetic properties, methods for diabatization, analytical gradients of state average complete active space SCF in association with density fitting, methods for constrained fragment optimization, large-scale parallel multireference configuration interaction including analytic gradients via the interface to the Columbus package, and approximations of the CASPT2 method to be used for computations of large systems. In addition, the report includes the description of a computational machinery for nonlinear optical spectroscopy through an interface to the QM/MM package Cobramm. Further, a module to run molecular dynamics simulations is added, two surface hopping algorithms are included to enable nonadiabatic calculations, and the DQ method for diabatization is added. Finally, we report on the subject of improvements with respects to alternative file options and parallelization.

Multiconfiguration Pair-Density Functional Theory Outperforms Kohn-Sham Density Functional Theory and Multireference Perturbation Theory for Ground-State and Excited-State Charge Transfer.

The correct description of charge transfer in ground and excited states is very important for molecular interactions, photochemistry, electrochemistry, and charge transport, but it is very challenging for Kohn-Sham (KS) density functional theory (DFT). KS-DFT exchange-correlation functionals without nonlocal exchange fail to describe both ground- and excited-state charge transfer properly. We have recently proposed a theory called multiconfiguration pair-density functional theory (MC-PDFT), which is based on a combination of multiconfiguration wave function theory with a new type of density functional called an on-top density functional. Here we have used MC-PDFT to study challenging ground- and excited-state charge-transfer processes by using on-top density functionals obtained by translating KS exchange-correlation functionals. For ground-state charge transfer, MC-PDFT performs better than either the PBE exchange-correlation functional or CASPT2 wave function theory. For excited-state charge transfer, MC-PDFT (unlike KS-DFT) shows qualitatively correct behavior at long-range with great improvement in predicted excitation energies.

Nonintuitive Diabatic Potential Energy Surfaces for Thioanisole.

Diabatization of potential energy surfaces is a technique that enables convenient molecular dynamics simulations of electronically nonadiabatic processes, but diabatization itself is nonunique and can be inconvenient; the best methods to achieve diabatization are still under study. Here, we present the diabatization of two electronic states of thioanisole in the S-CH3 bond stretching and C-C-S-C torsion two-dimensional nuclear coordinate space containing a conical intersection. We use two systematic methods: the (orbital-dependent) 4-fold way and the (orbital-free) Boys localization diabatization method. These very different methods yield strikingly similar diabatic potential energy surfaces that cross at geometries where the adiabatic surfaces are well separated and do not exhibit avoided crossings, and the contours of the diabatic gap and diabatic coupling are similar for the two methods. The validity of the diabatization is supported by comparing the nonadiabatic couplings calculated from the diabatic matrix elements to those calculated by direct differentiation of the adiabatic states.

Multiconfiguration Pair-Density Functional Theory Spectral Calculations Are Stable to Adding Diffuse Basis Functions.

Time-dependent Kohn-Sham density functional theory (TD-KS-DFT) is useful for calculating electronic excitation spectra of large systems, but the low-energy spectra are often complicated by artificially lowered higher-energy states. This affects even the lowest energy excited states. Here, by calculating the lowest energy spin-conserving excited state for atoms from H to K and for formaldehyde, we show that this problem does not occur in multiconfiguration pair-density functional theory (MC-PDFT). We use the tPBE on-top density functional, which is a translation of the PBE exchange-correlation functional. We compare to a robust multireference method, namely, complete active space second-order perturbation theory (CASPT2), and to TD-KS-DFT with two popular exchange-correlation functionals, PBE and PBE0. We find for atoms that the mean unsigned error (MUE) of MC-PDFT with the tPBE functional improves from 0.42 to 0.40 eV with a double set of diffuse functions, whereas the MUEs for PBE and PBE0 drastically increase from 0.74 to 2.49 eV and from 0.45 to 1.47 eV, respectively.