Multiconfigurational Wave Functions Research Articles

Covalency of Trivalent Actinide Ions with Different Donor Ligands: Do Density Functional and Multiconfigurational Wavefunction Calculations Corroborate the Observed "Breaks"?

A comprehensive ab initio study of periodic actinide-ligand bonding trends for trivalent actinides is performed. Relativistic density functional theory (DFT) and complete active-space (CAS) self-consistent field wavefunction calculations are used to dissect the chemical bonding in the [AnCl6]3-, [An(CN)6]3-, [An(NCS)6]3-, [An(S2PMe2)3], [An(DPA)3]3-, and [An(HOPO)]- series of actinide (An = U-Es) complexes. Except for some differences for the early actinide complexes with DPA, bond orders and excess 5f-shell populations from donation bonding show qualitatively similar trends in 5f n active-space CAS vs DFT calculations. The influence of spin-orbit coupling on donation bonding is small for the tested systems. Along the actinide series, chemically soft vs chemically harder ligands exhibit clear differences in bonding trends. There are pronounced changes in the 5f populations when moving from Pu to Am or Cm, which correlate with previously noted "breaks" in chemical trends. Bonding involving 5f becomes very weak beyond Cm/Bk. We propose that Cm(III) is a borderline case among the trivalent actinides that can be meaningfully considered to be involved in ground-state 5f covalent bonding.

Symmetry-Adapted Perturbation Theory Based on Multiconfigurational Wave Function Description of Monomers.

We present a formulation of the multiconfigurational (MC) wave function symmetry-adapted perturbation theory (SAPT). The method is applicable to noncovalent interactions between monomers which require a multiconfigurational description, in particular when the interacting system is strongly correlated or in an electronically excited state. SAPT(MC) is based on one- and two-particle reduced density matrices of the monomers and assumes the single-exchange approximation for the exchange energy contributions. Second-order terms are expressed through response properties from extended random phase approximation (ERPA). The dispersion components of SAPT(MC) have been introduced in our previous works [HapkaM.J. Chem. Theory Comput.2019, 15, 1016−102730525591; HapkaM.J. Chem. Theory Comput.2019, 15, 6712–672331670950]. SAPT(MC) is applied either with generalized valence bond perfect pairing (GVB) or with complete active space self-consistent field (CASSCF) treatment of the monomers. We discuss two model multireference systems: the H2 ··· H2 dimer in out-of-equilibrium geometries and interaction between the argon atom and excited state of ethylene. Using the C2H4* ··· Ar complex as an example, we examine second-order terms arising from negative transitions in the linear response function of an excited monomer. We demonstrate that the negative-transition terms must be accounted for to ensure qualitative prediction of induction and dispersion energies and develop a procedure allowing for their computation. Factors limiting the accuracy of SAPT(MC) are discussed in comparison with other second-order SAPT schemes on a data set of small single-reference dimers.

Machine-Learned Energy Functionals for Multiconfigurational Wave Functions.

We introduce multiconfiguration data-driven functional methods (MC-DDFMs), a group of methods which aim to correct the total or classical energy of a qualitatively accurate multiconfigurational wave function using a machine-learned functional of some featurization of the wave function such as its density, on-top density, or both. On a data set of carbene singlet-triplet energy splittings, we show that MC-DDFMs are able to achieve near-benchmark performance on systems not used for training with a robust degree of active-space independence. Beyond demonstrating that the density and on-top density hold the information necessary to correct the singlet-triplet energy splittings of multiconfigurational wave functions, this approach shows great promise for the development of functionals for MC-PDFT because corrections to the classical energy appear to be more transferable to types of molecules not included in the training data than corrections to total energies such as those yielded by CASSCF or NEVPT2.

Synthesis and Characterization of a Cobalt(III) Corrole with an S‐Bound DMSO Ligand

AbstractWe have synthesized a low‐spin Co(III) complex of 5,15‐bis(4‐nitrophenyl)‐10‐(2‐methylcarboxyphenyl)corrole with an S‐bound dimethylsulfoxide (DMSO) ligand (1DMSO) and determined the coordination mode through X‐ray diffraction for the first time. UV‐vis‐NIR spectral data show that the DMSO ligand does not dissociate in MeCN solution, and EPR results indicate that the first oxidation is ligand centered and suggest that not only DMSO remains bound but a second apical ligand, possibly MeCN, binds to the cobalt center. Multiconfigurational wavefunction electronic structure methods (CASSCF/NEVPT2) allowed us to determine that in the neutral complex the corrole behaves as an innocent trianionic ligand and that the σ‐donor effect of the S atom determines the low‐spin configuration by raising the energy of the Co 3dz2 orbital. While DFT calculations predict a ground open‐shell singlet for both S‐bound and O‐bound DMSO variants, CASSCF/NEVPT2 calculations predict a closed‐shell singlet ground state in both cases. These calculations reproduce considerably well the UV‐vis‐NIR spectrum of 1DMSO in solution, validating the closed‐shell singlet description.

Metal Oxo-Fluoride Molecules OnMF2 (M = Mn and Fe; n = 1-4) and O2MnF: Matrix Infrared Spectra and Quantum Chemistry.

On reacting laser-ablated manganese or iron difluorides with O2 or O3 during codeposition in solid neon or argon, infrared absorptions of several new metal oxo-fluoride molecules, including OMF2, (η1-O2)MF2, (η2-O3)MF2, (η1-O2)2MF2 (M = Mn and Fe), and O2MnF, have been observed. Quantum chemical density functional and multiconfiguration wavefunction calculations have been applied to characterize these new products by their geometric and electronic structures, vibrations, charges, and bonding. The assignment of the main vibrational absorptions as dominant symmetric or antisymmetric M-F or M-O stretching modes is confirmed by oxygen isotopic shifts and quantum chemical calculations of frequencies and thermal stabilities. The tendency of Fe to form polyoxygen complexes in lower oxidation states than the preceding element Mn is affirmed experimentally and supported theoretically. The M-F stretching frequencies of the isolated metal oxo-fluorides may provide a scale for the local charge on the MF2 sites in active energy conversion systems. The study of these species provides insights for understanding the trend of oxidation state changes across the transition-metal series.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation.

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

Localized Active Space Pair-Density Functional Theory

Accurate quantum chemical methods for the prediction of spin-state energy gaps for strongly correlated systems are computationally expensive and scale poorly with the size of the system. This makes calculations for many experimentally interesting molecules impractical even with abundant computational resources. Previous work has shown that the localized active space (LAS) self-consistent field (SCF) method can be an efficient way to obtain multiconfiguration SCF wave functions of comparable quality to the corresponding complete active space (CAS) ones. To obtain quantitative results, a post-SCF method is needed to estimate the complete correlation energy. One such method is multiconfiguration pair-density functional theory (PDFT), which calculates the energy based on the density and on-top pair density obtained from a multiconfiguration wave function. In this work, we introduce localized-active-space PDFT, which uses a LAS wave function for subsequent PDFT calculations. The method is tested by computing spin-state energies and gaps in conjugated organic molecules and a bimetallic compound and comparing to the corresponding CAS-PDFT values.

Multiconfiguration Pair-Density Functional Theory.

Kohn-Sham density functional theory with the available exchange-correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.

Magnetic Anisotropy in a Cubane-like Ni4 Complex: An Ab Initio Perspective.

Magnetic anisotropy, in the absence of an external magnetic field, relates to the degeneracy lift of energy levels. In the standard case of transition metal complexes, this property is usually modeled by an anisotropic spin Hamiltonian and one speaks of "zero-field splitting" (ZFS) of spin states. While the case of mononuclear complexes has been extensively described by means of ab initio quantum mechanical calculations, the literature on polynuclear complexes studied with these methodologies is rather scarce. In this work, advanced multiconfigurational wave function theory methods are applied to compute the ZFS of the ground S = 4 state of an actual tetranickel(II) complex, displaying a magnet behavior below 0.5 K. First, the isotropic couplings are computed in the absence of the spin-orbit coupling operator, in the full complex and also in clusters with only two active nickel(II) centers, confirming the occurrence of weak ferromagnetic couplings in this system. Second, the single-site magnetic anisotropies are computed on a cluster bearing only one active nickel(II) site, showing that the single-site anisotropy axes are not oriented in an optimal fashion for generating a large uniaxial molecular anisotropy. Furthermore, the possibility for involving only a few local orbital excited states in the calculation is assessed, actually opening the way for a consistent and manageable treatment of the ZFS of the ground S = 4 state. Third, multiconfigurational calculations are performed on the full complex, confirming the weak uniaxial anisotropy occurring for this state and also, interestingly, revealing a significant contribution of the lowest-lying orbitally excited S = 3 states. Overall, by comparison with the experiment, the reported results question the common habit of using only one structure, in particular derived from a crystallography experiment, to compute magnetic anisotropy parameters.

Multiconfiguration Density-Coherence Functional Theory.

This paper presents a new theory called multiconfiguration density coherence functional theory (MC-DCFT). This theory provides a new route to define density functionals for multiconfiguration wave functions, in particular by using the one-particle density matrix in the coordinate representation. The theory is illustrated by calculating the dissociation curve of four heteronuclear and homonuclear diatomic molecules, namely, H2, F2, N2, and HF, using density coherence functionals converted from PBE, BLYP, and PW91. By introducing two parameters in the converted density functionals, we are able to calculate bond dissociation energies of comparable accuracy as those calculated by multiconfiguration pair-density functional theory (MC-PDFT) and complete active space second-order perturbation theory (CASPT2). This demonstrates that it would be possible to build a successful multiconfiguration density functional theory based on density coherence.

Tunneling and Nonadiabatic Effects on a Proton-Coupled Electron Transfer Model for the Qo Site in Cytochrome bc1.

Cytochrome bc1 is a fundamental enzyme for cellular respiration and photosynthesis. This dimeric protein complex catalyzes a proton-coupled electron transfer (PCET) from the reduced coenzyme-Q substrate (Q) to a bimetallic iron-sulfur cluster in the Qo active site. Herein, we combine molecular dynamics simulations of the complete cytochrome bc1 protein with electronic-structure calculations of truncated models and a semiclassical tunneling theory to investigate the electron-proton adiabaticity of the initial reaction catalyzed in the Qo site. After sampling possible orientations between the Q substrate and a histidine side chain that functions as hydrogen acceptor, we find that a truncated model composed by ubiquinol-methyl and imidazole-iron(III)-sulfide captures the expected changes in oxidation and spin states of the electron donor and acceptor. Diabatic electronic surfaces obtained for this model with multiconfigurational wave function calculations demonstrate that this reaction is electronic nonadiabatic, and proton tunneling is faster than mixing of electronic configurations. These results indicate the formalism that should be used to calculate vibronic couplings and kinetic parameters for the initial reaction in the Qo site of cytochrome bc1. This framework for molecular simulation may also be applied to investigate other PCET reactions in the Q-cycle or in various metalloproteins that catalyze proton translocation coupled to redox processes.

Short-range DFT energy correction to multiconfigurational wave functions for open-shell systems.

Electronic structure methods emerging from the combination of multiconfigurational wave functions and density functional theory (DFT) aim to take advantage of the strengths of the two nearly antagonistic theories. One of the common strategies employed to merge wave function theory (WFT) with DFT relies on the range separation of the Coulomb operator in which DFT functionals take care of the short-distance part, while long-range inter-electronic interactions are evaluated by using the chosen wave function method (WFT-srDFT). In this work, we uncover the limitations of WFT-srDFT in the characterization of open-shell systems. We show that spin polarization effects have a major impact on the (short-range) DFT exchange energy and are of vital importance in order to provide a balanced description between closed and open-shell configurations. We introduce different strategies to account for spin polarization in the short range based on the definition of a spin polarized electron density and with the use of short-range exact exchange. We test the performance of these approaches in the dissociation of the hydrogen molecule, the calculation of energy gaps in spin-triplet atoms and molecular diradicals, and the characterization of low-lying states of the gallium dimer. Our results indicate that the use of short-range DFT correlation in combination with a (full-range) multiconfigurational wave function might be an excellent approach for the study of open-shell molecules and largely improves the performance of WFT and WFT-srDFT.

Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations.

Several methods of constructing the active orbital space for multiconfigurational wave functions are compared on typical moderately strongly or strongly correlated ground-state molecules. The relative merits of these methods and problems inherent in multiconfigurational calculations are discussed. Strong correlation in the ground electronic state is found typically in larger conjugated and in antiaromatic systems, transition states which involve bond breaking or formation, and transition metal complexes. Our examples include polyenes, polyacenes, the reactant, product and transition state of the Bergman cyclization, and two transition metal complexes: Hieber's anion [(CO)3FeNO]- and ferrocene. For the systems investigated, the simplest and oldest selection method, based on the fractional occupancy of unrestricted Hartree-Fock natural orbitals (the UNO criterion), yields the same active space as much more expensive approximate full CI methods. A disadvantage of this method used to be the difficulty of finding broken spin symmetry UHF solutions. However, our analytical method, accurate to fourth order in the orbital rotation angles (Tóth and Pulay J. Chem. Phys. 2016, 145, 164102.), has solved this problem. Two further advantages of the UNO criterion are that, unlike most other methods, it measures not only the energetic proximity to the Fermi level but also the magnitude of the exchange interaction with strongly occupied orbitals and therefore allows the estimation of the correlation strength for orbital selection in Restricted Active Space methods.

Elucidating the Relationship between Multiradical Character and Predicted Singlet Fission Activity

AbstractThe diradical and multiradical characters of a large and diverse set of 241 predicted singlet fission candidates are determined by multiconfigurational wavefunction methods, specifically natural orbital occupation number (NOON) analysis using the CASSCF method. The ascertained multiradical character strongly supports the current prediction that a good singlet fission candidate, independent of its structure, will tend to be of the weak biradicaloid class as well as having very little tetraradical character. Although the rule remains robust, it is not absolute as a small portion of our potential candidates are predicted to be closed shell. Additionally, a smaller handful of outliers with apparent tetraradical characters are identified, suggesting there may be the possibility, albeit rare, of singlet fission amongst weak tetraradicaloids.

Exact semi-classical light-matter interaction operator applied to two-photon processes with strong relativistic effects.

X-ray processes involve interactions with high-energy photons. For these short wavelengths, the perturbing field cannot be treated as constant, and there is a need to go beyond the electric-dipole approximation. The exact semi-classical light-matter interaction operator offers several advantages compared to the multipole expansion such as improved stability and ease of implementation. Here, the exact operator is used to model x-ray scattering in metal K pre-edges. This is a relativistic two-photon process where absorption is dominated by electric-dipole forbidden transitions. With the restricted active space state-interaction approach, spectra can be calculated even for the multiconfigurational wavefunctions including second-order perturbation. However, as the operator itself depends on the transition energy, the cost for evaluating integrals for hundreds of thousands unique transitions becomes a bottleneck. Here, this is solved by calculating the integrals in a molecular-orbital basis that only runs over the active space, combined with a grouping scheme where the operator is the same for close-lying transitions. This speeds up the calculations of single-photon processes and is critical for the modeling of two-photon scattering processes. The new scheme is used to model Kα resonant inelastic x-ray scattering of iron-porphyrin complexes with relevance to studies of heme enzymes, for which the total computational time is reduced by several orders of magnitude with an effect on transition intensities of 0.1% or less.

Quantum chemical topology at the spin-orbit configuration interaction level: Application to astatine compounds.

We report a methodology that allows the investigation of the consequences of the spin-orbit coupling by means of the QTAIM and ELF topological analyses performed on top of relativistic and multiconfigurational wave functions. In practice, it relies on the "state-specific" natural orbitals (NOs; expressed in a Cartesian Gaussian-type orbital basis) and their occupation numbers (ONs) for the quantum state of interest, arising from a spin-orbit configuration interaction calculation. The ground states of astatine diatomic molecules (AtX with X = AtF) and trihalide anions (IAtI- , BrAtBr- , and IAtBr- ) are studied, at exact two-component relativistic coupled cluster geometries, revealing unusual topological properties as well as a significant role of the spin-orbit coupling on these. In essence, the presented methodology can also be applied to the ground and/or excited states of any compound, with controlled validity up to including elements with active 5d, 6p, and/or 5f shells, and potential limitations starting with active 6d, 7p, and/or 6f shells bearing strong spin-orbit couplings.

Origin of core-to-core x-ray emission spectroscopy sensitivity to structural dynamics.

Recently, coherent structural dynamics in the excited state of an iron photosensitizer was observed through oscillations in the intensity of Kα x-ray emission spectroscopy (XES). Understanding the origin of the unexpected sensitivity of core-to-core transitions to structural dynamics is important for further development of femtosecond time-resolved XES methods and, we believe, generally necessary for interpretation of XES signals from highly non-equilibrium structures that are ubiquitous in photophysics and photochemistry. Here, we use multiconfigurational wavefunction calculations combined with atomic theory to analyze the emission process in detail. The sensitivity of core-to-core transitions to structural dynamics is due to a shift of the minimum energy metal–ligand bond distance between 1s and 2p core-hole states. A key effect is the additional contraction of the non-bonding 3s and 3p orbitals in 1s core-hole states, which decreases electron–electron repulsion and increases overlap in the metal–ligand bonds. The effect is believed to be general and especially pronounced for systems with strong bonds. The important role of 3s and 3p orbitals is consistent with the analysis of radial charge and spin densities and can be connected to the negative chemical shift observed for many transition metal complexes. The XES sensitivity to structural dynamics can be optimized by tuning the emission energy spectrometer, with oscillations up to ±4% of the maximum intensity for the current system. The theoretical predictions can be used to design experiments that separate electronic and nuclear degrees of freedom in ultrafast excited state dynamics.

Reduced Common Molecular Orbital Basis for Nonorthogonal Configuration Interaction.

Electron and charge transfers are part of many vital processes in nature and technology. Ab initio descriptions of these processes provide useful insights that can be utilized for applications. A combination of the embedded cluster material model and nonorthogonal configuration interaction (NOCI), in which the cluster wave functions are expanded in many-electron basis functions (MEBFs) consisting of spin-adapted, antisymmetrized products of multiconfigurational wave functions of fragments (which are usually molecules) in the cluster, appears to provide a compromise between accuracy and calculation time. Additional advantages of this NOCI–Fragments approach are the chemically convenient interpretation of the wave function in terms of molecular states, and the direct accessibility of electronic coupling between diabatic states to describe energy and electron transfer processes. Bottlenecks in this method are the large number of two-electron integrals that have to be handled for the calculation of an electronic coupling matrix element and the enormous number of matrix elements over determinant pairs that have to be evaluated for the calculation of one matrix element between the MEBFs. We show here how we created a reduced common molecular orbital basis that is utilized to significantly reduce the number of two-electron integrals that need to be handled. The results obtained with this basis do not show any loss of accuracy in relevant quantities like electronic couplings and vertical excitation energies. We also show a significant reduction in computation time without loss in accuracy when matrix elements over determinant pairs with small weights are neglected in the NOCI. These improvements in the methodology render NOCI–Fragments to be also applicable to treat clusters of larger molecular systems with larger atomic basis sets and larger active spaces, as the computation time becomes dependent on the number of occupied orbitals and less dependent on the size of the active space.

Quantifying similarity for spectra with a large number of overlapping transitions: Examples from soft X-ray spectroscopy

Theoretical simulations are frequently used to assign electronic and geometric structure from spectral fingerprints. However, such assignments are prone to expectation bias. Bias can be reduced by using numerical measures of the similarity between calculated and experimental spectra. However, the commonly used point-wise comparisons cannot handle larger deviations in peak position. Here a weighted cross-correlation function is used to evaluate similarity scores for soft X-ray spectra of first-row transition metals. These spectra consist of hundreds of overlapping resonances, which makes spectral decomposition difficult. They are also challenging to model, leading to significant errors in both peak position and intensity. It is first shown how the choice of weight-function width can be related to the modeling errors. The method is then applied to evaluate the sensitivity of multiconfigurational wavefunction and charge-transfer multiplet simulations to model choices. The approach makes it possible to assess the reliability of assignments from spectral fingerprinting.

Semiclassical Dispersion Corrections Efficiently Improve Multiconfigurational Theory with Short-Range Density-Functional Dynamic Correlation.

Multiconfigurational wave functions are known to describe the electronic structure across a Born-Oppenheimer surface qualitatively correct. However, for quantitative reaction energies, dynamic correlation originating from the many configurations involving excitations out of the restricted orbital space, the active space, must be considered. Standard procedures involve approximations that eventually limit the ultimate accuracy achievable (most prominently, multireference perturbation theory). At the same time, the computational cost increases dramatically due to the necessity to obtain higher-order reduced density matrices. It is this disproportion that leads us here to propose an MC-srDFT-D hybrid approach of semiclassical dispersion (D) corrections to cover long-range dynamic correlation in a multiconfigurational (MC) wave function theory, which includes short-range (sr) dynamic correlation by density functional theory (DFT) without double counting. We demonstrate that the reliability of this approach is very good (at negligible cost), especially when considering that standard second-order multireference perturbation theory usually overestimates dispersion interactions.