Multiconfigurational Wave Functions Research Articles

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory.

State-specific electronic structure theory provides a route toward balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy. Multiconfigurational wave function approximations can describe both closed- and open-shell excited states and avoid the issues associated with state-averaged approaches. We investigate the existence of higher-energy solutions in complete active space self-consistent field (CASSCF) theory and characterize their topological properties. We demonstrate that state-specific approximations can provide accurate higher-energy excited states in H2 (6-31G) with more compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large or symmetry breaking when the active space is too small. Furthermore, we investigate the singlet-triplet crossing in CH2 (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.

A Multiconfigurational Wave Function Implementation of the Frenkel Exciton Model for Molecular Aggregates.

We present an implementation of the Frenkel exciton model into the OpenMolcas program package enabling calculations of collective electronic excited states of molecular aggregates based on a multiconfigurational wave function description of the individual monomers. The computational protocol avoids using diabatization schemes and, thus, supermolecule calculations. Additionally, the use of the Cholesky decomposition of the two-electron integrals entering pair interactions enhances the efficiency of the computational scheme. The application of the method is exemplified for two test systems, that is, a formaldehyde oxime and a bacteriochlorophyll-like dimer. For the sake of comparison with the dipole approximation, we restrict our considerations to situations where intermonomer exchange can be neglected. The protocol is expected to be beneficial for aggregates composed of molecules with extended π systems, unpaired electrons such as radicals or transition metal centers, where it should outperform widely used methods based on time-dependent density functional theory.

Ab initio Valence Bond Theory with Density Functional

Abstract: The accurate description of strongly correlated systems, also known as multireference systems, requires a balanced treatment of static and dynamic correlations and is an important target for developing quantum chemical methods. An appealing treatment to economically describe strongly correlated systems is the multireference density function theory (MRDFT) approach, in which the static correlation is included in the multiconfigurational wave function, while the density functional includes the dynamic correlation. This mini-review focuses on the recent progress and applications of the density functional methods based on valence bond theory. A series of density functional valence bond (DFVB) methods are surveyed, including the dynamic correlation correction- based and Hamiltonian matrix correction-based DFVB methods, the hybrid one-parameter DFVB methods, the block-localized density functional theory and the multistate density functional theory. These methods have been applied to various chemical and physical property calculations of strongly correlated systems, including resonance energies, potential energy curves, spectroscopic constants, atomization energies, spin state energy gaps, excitation energies, and reaction barriers. Most of the test results show that the density functional methods based on VB theory give comparable accuracy but require lower computational cost than high-level quantum computational methods and thus provide a promising strategy for studying strongly correlated systems.

Sensitivity of Kβ mainline X-ray emission to structural dynamics in iron photosensitizer.

Photochemistry and photophysics processes involve structures far from equilibrium. In these reactions, there is often strong coupling between nuclear and electronic degrees of freedom. For first-row transition metals, Kβ X-ray emission spectroscopy (XES) is a sensitive probe of electronic structure due to the direct overlap between the valence orbitals and the 3p hole in the final state. Here the sensitivity of Kβ mainline (Kβ1,3) XES to structural dynamics is analyzed by simulating spectral changes along the excited state dynamics of an iron photosensitizer [FeII(bmip)2]2+ [bmip = 2,6-bis(3-methyl-imidazole-1-ylidine)-pyridine], using both restricted active space (RAS) multiconfigurational wavefunction theory and a one-electron orbital-energy approach in density-functional theory (1-DFT). Both methods predict a spectral blue-shift with increasing metal-ligand distance, which changes the emission intensity for any given detection energy. These results support the suggestion that the [FeII(bmip)2]2+ femtosecond Kβ XES signal shows oscillations due to coherent wavepacket dynamics. Based on the RAS results, the sensitivity to structural dynamics is twice as high for Kβ compared to Kα, with the drawback of a lower signal-to-noise ratio. Kβ sensitivity is favored by a larger spectral blue-shift with increasing metal-ligand distance and larger changes in spectral shape. Comparing the two simulations methods, 1-DFT predicts smaller energy shifts and lower sensitivity, likely due to missing final-state effects. The simulations can be used to design and interpret XES probes of non-equilibrium structures to gain mechanistic insights in photocatalysis.

Application of Multiconfiguration Pair-Density Functional Theory to the Diels-Alder Reaction.

Transition states for Diels-Alder reactions are strongly correlated, as evidenced by high-to-very-high M diagnostics, and therefore they require treatment by multireference methods. Multiconfiguration pair-density functional theory (MC-PDFT) combines a multiconfiguration wave function with a functional of the electron density and the on-top pair density to calculate the electronic energy for strongly correlated systems at a much lower cost than wave function methods that do not employ density functionals. Here we apply MC-PDFT to the Diels-Alder cycloaddition reaction of 1,3-butadiene with ethylene, where two kinds of reaction paths have been widely studied: concerted synchronous paths and diradical stepwise paths. The lowest-energy reaction path is now known to be a concerted synchronous one, and a method's ability to predict this is an important test. By comparison to the best available theoretical results in the literature, we test the accuracy of MC-PDFT with several choices of on-top functional for geometries and enthalpies of stable structures along both paths and for the transition state geometries. We also calculate the Arrhenius activation energies for both paths and compare these to experiment. We also compare to Kohn-Sham density functional theory (KS-DFT) with selected exchange-correlation functionals. CAS-PDFT gives consistently good energies and geometries for both the concerted and stepwise mechanisms, but none of the KS-DFT functionals gives accurate activation energies for both. The stepwise transition state is very strongly correlated, and MC-PDFT can treat it, but KS-DFT (which involves a single-configuration treatment) has larger errors. The results confirm that using a multiconfigurational reference function for strongly correlated transition states can significantly improve the reliability and that MC-PDFT can provide good accuracy at a much lower computational cost than competing multireference methods.

Active Spaces and Non-Orthogonal Configuration Interaction Approaches for Investigating Molecules on Metal Surfaces.

We test a set of multiconfigurational wavefunction approaches for calculating the ground state electron population for a two-site Anderson model representing a molecule on a metal surface. In particular, we compare (i) a Hartree Fock like wavefunction where frontier orbitals are allowed to be nonorthogonal versus (ii) a fully non-orthogonal configuration interaction wavefunction based on constrained Hartree-Fock states. We test both the strong and weak metal-molecule hybridization (Γ) limits as well as the strong and weak electron-electron repulsion (U) limits. We obtain accurate results as compared with exact numerical renormalization group theory, recovering charge transfer states where appropriate. The current framework should open a path to run molecular non-adiabatic dynamics on metal surfaces.

Electrochemical Hydrogenation of CO on Cu(100): Insights from Accurate Multiconfigurational Wavefunction Methods.

Copper (Cu) remains the most efficacious electrocatalyst for electrochemical CO2 reduction (CO2R). Its activity and selectivity are highly facet-dependent. We recently examined the commonly proposed rate-limiting CO hydrogenation step on Cu(111) via embedded correlated wavefunction (ECW) theory and demonstrated that only this higher-level theory yields predictions consistent with potential-dependent experimental kinetics. Here, to understand the differing activities of Cu(111) and Cu(100) in catalyzing CO2R, we explore CO hydrogenation on Cu(100) using ECW theory. We predict that the preferred pathway involves the reduction of adsorbed CO (*CO) to *COH via proton-coupled electron transfer (PCET) at working potentials, although *CHO also may form with a kinetically accessible but higher barrier. In contrast, our earlier work on Cu(111) concluded that *COH and *CHO formation via PCET are equally feasible. This work illustrates one possible origin of the facet dependence of CO2R mechanisms and products on Cu electrodes and sheds light on how the selectivity of CO2R electrocatalysts can be controlled by the surface morphology.

Extension of natural reaction orbital approach to multiconfigurational wavefunctions.

Recently, we proposed a new orbital analysis method, natural reaction orbital (NRO), which automatically extracts orbital pairs that characterize electron transfer in reaction processes by singular value decomposition of the first-order orbital response matrix to the nuclear coordinate displacements [Ebisawa et al., Phys. Chem. Chem. Phys. 24, 3532 (2022)]. NRO analysis along the intrinsic reaction coordinate (IRC) for several typical chemical reactions demonstrated that electron transfer occurs mainly in the vicinity of transition states and in regions where the energy profile along the IRC shows shoulder features, allowing the reaction mechanism to be explained in terms of electron motion. However, its application has been limited to single configuration theories such as Hartree-Fock theory and density functional theory. In this work, the concept of NRO is extended to multiconfigurational wavefunctions and formulated as the multiconfiguration NRO (MC-NRO). The MC-NRO method is applicable to various types of electronic structure theories, including multiconfigurational theory and linear response theory, and is expected to be a practical tool for extracting the essential qualitative features of a broad range of chemical reactions, including covalent bond dissociation and chemical reactions in electronically excited states. In this paper, we calculate the IRC for five basic chemical reaction processes at the level of the complete active space self-consistent field theory and discuss the phenomenon of electron transfer by performing MC-NRO analysis along each IRC. Finally, issues and future prospects of the MC-NRO method are discussed.

DENSITY FUNCTIONAL AND GREEN’S FUNCTIONS METHOD TO COMPUTING SPECTRAL PARAMETERS OF DIATOMIC MOLECULES

It is presented a new effective method of calculating the energy and spectral parameters of diatomic molecules based on the hybrid theory of the quasi-particle density functional and the theory of Green's functions. As an illustration, the data of the calculation of the vertical ionization potentials and the coupling constants (vibrational structure) of the photoelectron spectra of a number of diatomic molecules, in particular, N2, are received. It is presented a detailed comparison of the received results with the data of standard theories of the Hartree-Fock type, the multi-configuration electron propagator method, and the extended theory based on Koopmans' theorem using multi-configurational self-consistent field wave functions with different sets of basis functions. Iit is shown that the consistent, maximally precise consideration of exchange-correlation effects and reorganization effects within the framework of the combined theory leads to a rather significant improvement in the agreement of theoretical and experimental data both in terms of ionization potentials and photoelectron spectra in general.

On the three lowest spin states of Na13+: Hybrid Density Functional Theory and benchmark CASSCF(12,12)+CASPT2 studies

AbstractThe three lowest spin states (S = 0,1,2) of 12 representative isomers have been estudied using both, KS‐DFT via three hybrid density functionals, and benchmark multireference CASSCF and CASPT2 methods with a couple of Dunning's correlation consistent basis sets. CASSCF(12,12) geometry optimizations were carried out. Since 12 electrons in 12 active orbitals span the chemically‐significant complete valence space, the results of the present study provide benchmarks for . The CASPT2(12,12)/cc‐pVTZ* lowest energy structures are three nearly degenerate singlets (S = 0): an isomer formed from two pentagonal bipyramids fused together (PBPb), a capped centered‐squared antiprism [CSAP‐(1,3)] and an optimum tetrahedral OPTET(II) structure, the last two lying 0.88 and 1.63 kcal/mol above the first, respectively. The lowest triplet (S = 1) and quintet (S = 2) states lie 4.33 and 3.77 kcal/mol above the singlet global minimum, respectively. The latter is a deformed icosahedron while the former is a CSAP‐(1,3). The flatness of the potential energy surface of this cluster suggests a rather strong dynamical character at finite temperature. Prediction of the lowest energy structures and electronic properties is crucially sensitive both to non‐dynamical and dynamical electron correlation treatment. The CASPT2 vertical ionization energy is 3.66 eV, in excellent agreement with the 3.6 ± 0.1 eV experimental figure. All the isomers are found to have a strong multireference character, thus making KS density functional theory fundamentally inappropriate for these systems. Only large multiconfigurational CASSCF wavefunctions provide a reliable zeroth‐order description; then the dynamic correlation effects must be properly taken into account for a truly accurate account of the structural and energetic features of alkali‐metal clusters.

Theoretical Evaluation of Metal-Ligand Bonding in Neptunium Compounds in Relation to 237Np Mössbauer Spectroscopy.

The 237Np Mössbauer isomer shift and quadrupole splitting (QS) are powerful probes for the metal-ligand bonding of neptunium, a 5f-element of vital importance in the nuclear fuel cycle. A large set of Np compounds with different oxidation states (III) to (VII) is studied to investigate, by first-principles calculations, isomer shifts and the QS trends in relation to the Np oxidation state. Natural Bond Orbital analysis reveals that in addition to donation bonding to the 5f shell, participation of the 6d and 7s neptunium shells in covalent (donation) bonding substantially impacts the isomer shifts. The isomer shift cannot be interpreted solely by the 5f shell electron count. The isomer shift for Np(II) compounds is estimated to be in the range of 31-34 mm/s, less positive than for Np(III) compounds. For the QS, density functional calculations fail to reproduce the quadrupole splitting for some Np(VI) ionic solids. A multiconfigurational wave function approach reproduces the observed QS trends. The calculations give a semiquantitative interpretation of the trends for Np oxidation states (V) to (VII). The contrasting QS for standard and "reverse" neptunyl(VI), at the opposite extremes of the observed QS scale, arises predominantly from the different crystal environments.

Modelling ultrafast dynamics at a conical intersection with regularized diabatic states: An approach based on multiplicative neural networks

• Neural network (NN) potentials for quantum dynamics of cis–trans photoisomerization. • Multiplicative NNs adapted to regularized diabatic states at a conical intersection. • NN potentials match a sum-of-products form required by MCTDH wavefunction propagation. • Good agreement found for highly correlated dynamics with up to 13 degrees of freedom. Neural-network (NN) potentials are employed in conjunction with the Multiconfiguration Time-Dependent Hartree (MCTDH) method in order to simulate an ultrafast photoinduced cis–trans type isomerization process induced by a conical intersection. To this end, NN potentials are fitted to a diabatic potential of regularized diabatic states type [Köppel et al., J. Chem. Phys. 115, 2377 (2001)], which entirely relies on adiabatic potential information. Multiplicative NNs are employed which match the sum-of-products form of the multiconfigurational MCTDH wavefunction. Good agreement with the reference dynamics is obtained for simulations of the highly correlated dynamics with up to 13 degrees of freedom. This study contributes to developing NN methodologies suitable for photochemical dynamics at complex excited-state topologies.

237Np Mössbauer Isomer Shifts: A Lesson About the Balance of Static and Dynamic Electron Correlation in Heavy Element Complexes.

A large set of neptunium compounds with different oxidation states (III to VII) was assembled to study the Mössbauer isomer shift by wave function calculations and better understand covalency in f-elements complexes. The contact density approach was used to calculate the isomer shift using complete active space self-consistent field (CASSCF) multiconfiguration wave functions, as well as matrix product states [from Density Matrix Renormalization Group (DMRG) algorithms] for large active spaces. Dynamic correlation effects for the isomer shifts were treated via CASPT2 energy derivatives with respect to the nuclear radius. The CASSCF calculations appear to produce different orbital overlocalization errors for low and high Np oxidation states. For compounds with low Np oxidation numbers, the errors can be attributed to the overlocalization of the 5f orbitals. For the compounds with high Np oxidation numbers, the main errors arise from an overlocalization of ligand orbitals and concomitant to weak donation bonding. Attempts to mitigate the overlocalization errors with large active spaces using DMRG were only partially successful, showing that explicit treatment of dynamic correlation is necessary for accurate predictions of Mössbauer isomer shifts. The CASPT2 calculations perform very satisfactorily. For a subset of Np compounds, both static and dynamic correlation effects were substantial. A rational active space selection based on orbital entanglement diagrams proved beneficial for determining the optimal reference wave function.

Benchmarking ANO-R basis set for multiconfigurational calculations

The selection of basis sets is very important for multiconfigurational wave function calculation, due to a balance between a desired accuracy and computational costs. Recently, the atomic natural orbital-relativistic (ANO-R) basis set was published as a suggested replacement for the ANO-RCC basis set for scalar-relativistic correlated calculations Zobel et al (2021 J. Chem. Theory Comput. 16 278–294). Benchmarking ANO-R basis set against ANO-RCC for atoms (from H to Rn) and their compounds is the goal of this study. Many of these compounds (for instance, diatomic molecules containing transition metals) have open shells, for which reason a multiconfigurational approach is necessary and was primarily used throughout this project. Performance of the ANO-R basis set in multiconfigurational calculations is similar to the ANO-RCC basis set for the ionisation potential of atoms, and the bond distance in diatomic molecules. Deficiencies are noted for atomic electron affinities and dissociation energies of fluoride diatomic molecules. ANO-R basis sets are more compact in comparison to the corresponding ANO-RCC basis sets leading to smaller computational costs, which was demonstrated by chloroiron corrole molecule as an example.

Soft X-ray Spectroscopy Simulations with Multiconfigurational Wave Function Theory: Spectrum Completeness, Sub-eV Accuracy, and Quantitative Reproduction of Line Shapes.

Multireference methods are known for their ability to accurately treat states of very different nature in many molecular systems, facilitating high-quality simulations of a large variety of spectroscopic techniques. Here, we couple the multiconfigurational restricted active space self-consistent field RASSCF/RASPT2 method (of the CASSCF/CASPT2 methods family) to the displaced harmonic oscillator (DHO) model, to simulate soft X-ray spectroscopy. We applied such an RASSCF/RASPT2+DHO approach at the K-edges of various second-row elements for a set of small organic molecules that have been recently investigated at other levels of theory. X-ray absorption near-edge structure (XANES) and X-ray photoelectron spectroscopy (XPS) are simulated with a sub-eV accuracy and a correct description of the spectral line shapes. The method is extremely sensitive to the observed spectral shifts on a series of differently fluorinated ethylene systems, provides spectral fingerprints to distinguish between stable conformers of the glycine molecule, and accurately captures the vibrationally resolved carbon K-edge spectrum of formaldehyde. Differences with other theoretical methods are demonstrated, which show the advantages of employing a multireference/multiconfigurational approach. A protocol to systematically increase the number of core-excited states considered while maintaining a contained computational cost is presented. Insight is eventually provided for the effects caused by removing core–electrons from a given atom in terms of bond rearrangement and influence on the resulting spectral shapes within a unitary orbital-based framework for both XPS and XANES spectra.

Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory.

Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.

X-ray absorption spectra of f-element complexes: insight from relativistic multiconfigurational wavefunction theory.

X-ray absorption near edge structure (XANES) spectroscopy, coupled with ab initio calculations, has emerged as the state-of-the-art tool for elucidating the metal-ligand bonding in f-element complexes. This highlight presents recent efforts in calculating XANES spectra of lanthanide and actinide compounds with relativistic multiconfiguration wavefunction approaches that account for differences in donation bonding in the ground state (GS) versus a core-excited state (ES), multiplet effects, and spin-orbit-coupling. With the GS and ES wavefunctions available, including spin-orbit effects, an arsenal of chemical bonding tools that are popular among chemists can be applied to rationalize the observed intensities in terms of covalent bonding.

Covalency in actinide(iv) hexachlorides in relation to the chlorine K-edge X-ray absorption structure.

Chlorine K-edge X-ray absorption near edge structure (XANES) in actinideIV hexachlorides, [AnCl6]2− (An = Th–Pu), is calculated with relativistic multiconfiguration wavefunction theory (WFT). Of particular focus is a 3-peak feature emerging from U toward Pu, and its assignment in terms of donation bonding to the An 5f vs. 6d shells. With or without spin–orbit coupling, the calculated and previously measured XANES spectra are in excellent agreement with respect to relative peak positions, relative peak intensities, and peak assignments. Metal–ligand bonding analyses from WFT and Kohn–Sham theory (KST) predict comparable An 5f and 6d covalency from U to Np and Pu. Although some frontier molecular orbitals in the KST calculations display increasing An 5f–Cl 3p mixing from Th to Pu, because of energetic stabilization of 5f relative to the Cl 3p combinations of the matching symmetry, increasing hybridization is neither seen in the WFT natural orbitals, nor is it reflected in the calculated bond orders. The appearance of the pre-edge peaks from U to Pu and their relative intensities are rationalized simply by the energetic separation of transitions to 6d t2gversus transitions to weakly-bonded and strongly stabilized a2u, t2u and t1u orbitals with 5f character. The study highlights potential pitfalls when interpreting XANES spectra based on ground state Kohn–Sham molecular orbitals.

Near-infrared circular dichroism of the ytterbium DOTMA complex: an ab initio investigation.

The electronic structure and circular dichroism spectra of the ytterbium(III) complex [Yb(DOTMA)]- are calculated using complete and restricted active space self-consistent field wavefunction methods with the spin-orbit coupling treated by the state interaction approach. The influence of the dynamical correlation effect is then included via the 2nd order perturbation method. The experimental circular dichroism spectrum is well reproduced by calculations, both in terms of relative energy excitations and in terms of rotatory strength intensities. The results allow highlighting the mechanism that drives the chiroptical properties in Yb(III) complexes and reveal the importance of taking into account the 4f125d1 electronic configurations in the calculated wavefunctions to properly describe the chiroptical properties of the 4f-4f transitions.

Evaluating the Electronic Structure of Coexisting Excitonic and Multiexcitonic States in Periodic Systems: Significance for Singlet Fission.

Singlet fission (SF) in organic molecular solids is an example of a process that is challenging to describe with the most common electronic structure approaches. It involves optically bright singlet excited states delocalized over many molecules, which could be efficiently treated by density functional theory, and multiexcitonic localized states that have to be studied with wavefunction methods, usually with small clusters considering their expensive computational costs. In this work, we propose a methodology to combine multiconfigurational wavefunction calculations with reduced Hamiltonian to investigate the electronic structure of large clusters or fully periodic systems. The method is applied to the prototypical SF materials tetracene and pentacene. The results allow one to study how states of different natures (excitonic, charge-transfer, and multiexcitonic) coexist and are contaminated by their couplings in large or periodic systems. Novel insights are therefore possible. For example, because the excitonic bands are relatively broad with respect to the multiexcitonic states, there are limited regions of the crystal momentum space where the transition between the two is more likely.