Multireference Methods Research Articles

Fine structure of the doublet P levels of boron

We report high-accuracy calculations of the ground and the lowest eight excited Po2 states of the two stable isotopes of the boron atom, B10 and B11, as well as of the boron atom with an infinite nuclear mass ∞B. The nonrelativistic wave function of each of the states is generated in an independent variational calculation by expanding it in terms of a large number, 12000–17000, of all-electron explicitly correlated Gaussian (ECG) functions whose nonlinear parameters are extensively optimized with a procedure that employs analytic energy gradient determined with respect to these parameters. These highly accurate wave functions are used to compute the fine-structure splittings using the first order of the perturbation theory (∼α2), where α is the fine-structure constant, which are then corrected for the electron magnetic moment anomaly (∼α3). As the nonrelativistic Hamiltonian explicitly depends on the mass of the nucleus, the recoil corrections up to the order of α2 are automatically accounted for in the fine-structure calculations. Furthermore, the off-diagonal corrections to the fine structure (∼α4) are estimated using the multireference methods based on one-electron Gaussian orbitals. The results obtained in this paper are considerably more accurate than those available in the literature. Moreover, we report accurate splittings for a number of excited Po2 states, for which there have been no reliable experimental or theoretical data at all. The calculated values presented in this paper may serve as a valuable guide for future experimental measurements of the fine structure of the boron atom. As the fine structure of an atom provides a spectral signature that can facilitate atom's detection, our data can also aid the search for trace amounts of boron in the interstellar medium. Published by the American Physical Society 2024

Converting Second-Order Saddle Points to Transition States: New Principles for the Design of 4π Photoswitches.

Molecular photoswitches have demonstrated potential for storing solar energy at the molecular level, with power densities comparable to commercial batteries and hydroelectric energy storage. However, development of efficient photoswitches is hindered by limitations in cyclability and optical properties of existing materials. We here demonstrate that certain limitations in photoswitches based on electrocyclizations stem from the issue of controlling competition between Woodward-Hoffmann allowed and forbidden pathways. Our approach moves beyond the traditional view of activation barriers and reveals that second-order saddle points are crucial in dictating the competition between disrotatory and conrotatory pathways. These insights suggest new opportunities to manipulate the competition between these pathways through geometric constraints, fundamentally altering the connectivity of the potential energy surface. Our study also emphasizes the necessity of multi-reference methods and the need to conduct higher-dimensional explorations for competing pathways beyond photoswitch design.

Force-Assisted Orbital Crossing in Mechanochemical Oxirane Ring Opening.

Polymer mechanochemistry induces chemical reactivity by applying a directed force, which can lead to unexpected reaction mechanisms. Strained cyclic molecules are often used in force-sensitive motifs because of the strong force coupling of ring-opening reactions. In this computational study, the force dependence of the ring-opening reactions of oxirane will be investigated. Density functional theory and multireference methods were used to investigate the electronic character of both symmetry-allowed and symmetry-forbidden reactions. In the latter case, an orbital crossing occurs during the reaction course, forcing the Woodward-Hoffmann-forbidden reaction to proceed via a diradical pathway. The performance of broken-symmetry density functional theory is evaluated and compares well to high-accuracy CASPT2, MRCI, and ic-MRCC computations. Due to the high ring strain, the barrier heights of both ring-opening reactions are steeply reduced by the application of an external force. Furthermore, the use of unsaturated linkers was shown to yield a significant reduction of the barrier heights, explaining previous experimental findings. Finally, we show through analysis of the PES topology how the external force transforms characteristic points such as saddle points and bifurcations, providing insights into force-dependent mechanism changes.

Accurate Thermochemistry with Multireference Methods: A Stress Test for Internally Contracted Multireference Coupled-Cluster Theory.

The internally contracted multireference coupled-cluster method with single, double and perturbative triple excitations, icMRCCSD(T), was tested for its performance in the context of computational high-accuracy thermochemistry. The results were gauged against the standard single-reference coupled-cluster hierarchy with up to 5-fold excitations. The test set comprised of a selection of first-row dinuclear compounds and the three 3d-transition metal compounds MnH, FeH, and CoH. The results revealed two problems with the current formulation of icMRCCSD(T). First, the choice of the Dyall Hamiltonian as the zeroth-order Hamiltonian, which leads to a biased description of the different orbital subspaces and particularly poor results for the atomic correlation energies, and second, the tendency to overestimate the perturbative correction for triply excited clusters, in particular in the presence of open shells and correspondingly low orbital-energy gaps. The two problems could be solved by resorting to the effective Fock operator as zeroth-order Hamiltonian and by adopting a modified amplitude equation that includes terms quadratic in the pair clusters. A similar modification was recently proposed by Masios et al. (Phys. Rev. Lett. 2023, 131, 186401) in the context of applying single-reference coupled-cluster theory to systems with small or vanishing band gaps and we chose the acronym '(cT*) correction' in analogy to that work. In contrast to the work of Masios et al., additional terms including single excitation clusters were omitted, as these again lead to an overestimation of correlation effects in more difficult cases. We also tested another alternative for the zeroth-order Hamiltonian and additional higher-order corrections for the correlation energy. These extensions did not significantly improve the results and were also computationally more demanding. The improved icMRCCSD(cT*)F method yields very accurate results with errors, relative to accurate benchmarks, better than 2 kJ/mol for total energies and atomization energies for the entire set of examples considered in this work.

Orbital entanglement and the double d-shell effect in binary transition metal molecules.

Accurate modeling of transition metal-containing compounds is of great interest due to their wide-ranging and significant applications. These systems present several challenges from an electronic structure perspective, including significant multi-reference characters and many chemically relevant orbitals. A further complication arises from the so-called double d-shell effect, which is known to cause a myriad of issues in the treatment of first-row transition metals with both single- and multi-reference methods. While this effect has been well documented for several decades, a comprehensive understanding of its consequences and underlying causes is still evolving. Here, we characterize the second d-shell effect by analyzing the information entropy of correlated wavefunctions in a periodic series of 3d and 4d transition metal molecular hydrides and oxides. These quantum information techniques provide unique insight into the nuanced electronic structure of these species and are powerful tools for the study of weak and strong correlations in the transition metal d manifold.

Photophysics of Nitro-Substituted Unnatural Nucleic Acid Base.

The unnatural nucleic acid base (uNAB), 6-amino-3-methyl-5-nitropyridin-2(1H)one, often referred to as Z can form a base pair with the uNAB 2-aminoimidazo[1,2-a]-1,3,5-triazin-4(8H)-one (referred to as P) and is analogous to a guanine-cytosine (G-C) pair. However, it is well-known that the nonradiative decay pathway of the P-Z pair is significantly different from that of the G-C pair (Cui et al., Front. Chem. 2020, 8, 605117-605125). In this work, we study the excited state processes in Z using state-of-the-art multireference methods and dynamical techniques to ascertain the predominant nonradiative channels. We find that unlike in the natural NABs, the excited state processes in Z are driven primarily by the -NO2 group rotation. The electron-withdrawing effect of the -NO2 substituent plays a crucial role. We further ascertained that ultrafast deactivation channels are possible in Z and identified the stationary point geometries that are responsible for these channels.

Unveiling the Redox Noninnocence of Metallocorroles: Exploring K-Edge X-ray Absorption Near-Edge Spectroscopy with a Multiconfigurational Wave Function Approach.

X-ray absorption near-edge spectroscopy (XANES) is an advanced technique for probing the local electronic structure of catalysts, effectively identifying the noninnocent nature of ligands in transition-metal complexes. Metallocorroles with noninnocent corrole rings exhibit unusual electronic structures that challenge traditional density functional theory (DFT) methods, necessitating more rigorous approaches to describe electron correlation accurately. We explored K-edge XANES spectra of Fe, Mn, and Co metallocorroles using TDDFT and wave function-based methods. This is the first investigation employing multireference methods, specifically RASSCF, RASPT2, and MC-PDFT, to analyze the redox noninnocent nature of metallocorroles reflected in their XANES spectra. We quantified the noninnocent character of the corrole and the oxidation states of the metals, capturing more than singly excited excitations responsible for the pre-edge peak. Our findings demonstrate the importance of these advanced computational techniques for accurately predicting XANES spectra, providing a reliable understanding of the electronic properties of such complexes. This study offers a new strategy for investigating ligand redox noninnocence via integrated experimental and computational XANES.

Exploring electronic resonances in pyridine: Insights from orbital stabilization techniques.

Electron attachment to pyridine results in electronic resonances, metastable states that can decay through electronic or nuclear degrees of freedom. This study uses orbital stabilization techniques combined with bound electronic structure methods, based on equation of motion coupled cluster or multi-reference methods, to calculate positions and widths of electronic resonances in pyridine that exist below 10eV. We report four 2B1 and four 2A2 resonances, including one 2B1 not previously reported experimentally and two 2A2 resonances not reported at all in the literature. The two lower energy resonances are one-particle shape resonances, while the remaining are mixed or primarily core-excited resonances. Multi-reference perturbation theory provides the best description of these resonances, especially when their character is mixed. We describe the character of these resonances qualitatively and calculate Dyson orbitals, which provide information about their decay channels.

Multireference calculations on bond dissociation and biradical polycyclic aromatic hydrocarbons as guidance for fractional occupation number weighted density analysis in DFT calculations

This study explores open-shell biradical and polyradical molecular compounds based on extended multireference (MR) methods (MR-configuration interaction with singles and doubles (CISD) and MR-averaged quadratic coupled cluster (AQCC) approach) using the numbers of unpaired densities NU. These results were used to guide the analysis of the fractional occupation number weighted density (FOD) calculated within the finite temperature (FT) density functional theory (DFT) approach. As critical test examples, the dissociation of carbon–carbon (CC) single, double and triple bonds and a benchmark set of polycyclic aromatic hydrocarbons (PAHs) have been chosen. By examining single, double, and triple bond dissociations, we demonstrate the utility and accuracy but also limitations of the FOD analysis for describing these dissociation processes. In significant extension of previous work (Phys Chem Chem Phys 25: 27380–27393), the assessment of FOD applications for different classes of DFT functionals was performed examining the range-separated functionals ωB97XD, ωB97M-V, CAM-B3LYP, LC-ωPBE, and MN12-SX, the hybrid (M06-2X) functional and the double hybrid (B2P-LYP) functional. In all cases, strong correlations between NFOD and NU values are found. The major task was to develop a new linear regression formula for range-separated functionals allowing a convenient determination of the optimal electronic temperature Tel for the FT-DFT calculation. We also established an optimal temperature for the semiempirical extended tight-binding GFN2-xTB method. These findings significantly broaden the applicability of FOD analysis across various DFT functionals and semiempirical methods.

Investigation of the Nonradiative Photoprocesses of Unnatural DNA Base: 7-(2-Thienyl)-imidazo[4,5-b]pyridine (Ds)─A Computational Study.

7-(2-Thienyl)-imidazo[4,5-b]pyridine (Ds) is an unnatural nucleic acid that forms a stable pair with pyrrole-2-carbaldehyde (Pa) in DNA. This Ds-Pa pair gets stabilized via van der Waals interaction and shape fitting. In our previous study [Ghosh, P. J. Phys. Chem. A 2021, 125, 5556-5561], we investigated the nonradiative photoprocesses of the unnatural DNA base Pa, and also there are some studies on its stability and reactivity in the ground state. But, to consider it as a good unnatural base pair, one has to understand its stability not only in the ground state but also in the excited states after absorbing ultraviolet (UV) radiation. Therefore, in this study, the excited-state photoprocesses of Ds on UV irradiation and its nonradiative decay channels have been investigated using state-of-the-art multireference methods, and this investigation finally leads the molecule to access the minimum energy crossing point (MECP) via a downhill pathway.

Theoretical Kinetics of Radical-Radical Reaction NH2ṄH + ṄH2 and Its Implications for Monomethylhydrazine Pyrolysis Mechanism.

Significant discrepancies were observed between the experiments and the simulations for ṄH2 time-histories in monomethylhydrazine pyrolysis with the robust mechanism proposed by Pascal and Catoire. The rate of formation analyses for ṄH2 indicated the significance of the reaction NH2ṄH + ṄH2 = H2NN + NH3, which has not been well-defined. In this study, ab initio calculations were performed for the theoretical description of the NH2ṄH + ṄH2 chemistry. Most stationary points on the potential energy surface were quantified at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level, and the multireference methods were employed for barrier-less reaction and some transition states. The temperature- and pressure-dependent rate coefficients were determined using classical and microcanonical variational transition state theories. Four primary reaction channels were identified as competitive: 1) The H atom abstraction reaction yielding N2H2(T) + NH3, dominating at 1350-3000 K across the 0.001-100 atm pressure range. 2) The H atom abstraction reaction forming N2H2(S) + NH3, dominating at 800-1350 K and competing with the processes of chemical activation and collisional stabilization below 800 K. 3) The chemical-activated reaction resulting in H2NN(S) + NH3, dominating below 800 K at 0.001 atm. 4) The collisional-stabilized recombination reaction leading to N3H5, becoming significant as pressure increases and dominating below 600 and 650 K at 1 and 100 atm, respectively. The implications of newly calculated NH2ṄH + ṄH2 kinetics for the monomethylhydrazine pyrolysis mechanism were evaluated, and updates were implemented. Sensitivity analyses indicated the necessity of additional research efforts to comprehend the dynamics of CH3NH2 unimolecular and N2H2 + ṄH2 reaction systems. The rate coefficients presented in this study can be employed to develop the chemical kinetic model of nitryl-containing systems.

Efficient random phase approximation for diradicals.

We apply the analytically solvable model of two electrons in two orbitals to diradical molecules, characterized by two unpaired electrons. The effect of doubly occupied and empty orbitals is taken into account by means of random phase approximation (RPA). We show that in the static limit, the direct RPA leads to the renormalization of the parameters of the two-orbital model. We test our model by comparing its predictions for singlet-triplet splitting with the results of several multi-reference methods for a set of thirteen molecules. We find that for this set, the static RPA results are close to those of the NEVPT2 method with two orbitals and two electrons in the active space.

Renormalized Internally Contracted Multireference Coupled Cluster with Perturbative Triples.

In this work, we combine the many-body formulation of the internally contracted multireference coupled cluster (ic-MRCC) method with Evangelista's multireference formulation of the driven similarity renormalization group (DSRG). The DSRG method can be viewed as a unitary multireference coupled cluster theory, which renormalizes the amplitudes based on a flow equation approach to eliminate numerical instabilities. We extend this approach by demonstrating that the unitary flow equation approach can be adapted for nonunitary transformations, rationalizing the renormalization of ic-MRCC amplitudes. We denote the new approach, the renormalized ic-MRCC (ric-MRCC) method. To achieve high accuracy with a reasonable computational cost, we introduce a new approximation to the Baker-Campbell-Hausdorff expansion. We fully consider the linear commutator while approximating the quadratic commutator, for which we neglect specific contractions involving amplitudes with active indices. Moreover, we introduce approximate perturbative triples to obtain the ric-MRCCSD[T] method. We demonstrate the accuracy of our approaches in comparison to advanced multireference methods for the potential energy curves of H8, F2, H2O, N2, and Cr2. Additionally, we show that ric-MRCCSD and ric-MRCSSD[T] match the accuracy of CCSD(T) for evaluating spectroscopic constants and of full configuration interaction energies for a set of small molecules.

Electronic Structure Methods for Simulating Flavin's Spectroscopy and Photophysics: Comparison of Multi-reference, TD-DFT, and Single-Reference Wave Function Methods.

The use of flavins and flavoproteins in photocatalytic, sensing, and biotechnological applications has led to a growing interest in computationally modeling the excited-state electronic structure and photophysics of flavin. However, there is limited consensus regarding which computational methods are appropriate for modeling flavin's photophysics. We compare the energies of low-lying excited states of flavin computed with time-dependent density functional theory (TD-DFT), equation-of-motion coupled cluster (EOM-EE-CCSD), scaled opposite-spin configuration interaction [SOS-CIS(D)], multiconfiguration pair-density functional theory (MC-PDFT), and several multireference perturbation theory (MR-PT2) methods. In the first part, we focus on excitation energies of the first singlet excited state (S1) of five different redox and protonation states of flavin, with the goal of finding a suitable active space for MR-PT2 calculations. In the second part, we construct two sets of one-dimensional potential energy surfaces connecting the S0 and S1 equilibrium geometries (S0-S1 path) and the S1 (π,π*) and S2 (n,π*) equilibrium geometries (S1-S2 path). The first path therefore follows a Franck-Condon active mode of flavin while the second path maps crossings points between low-lying singlet and triplet states in flavin. We discuss the similarities and differences in the TD-DFT, EOM-EE-CCSD, SOS-CIS(D), MC-PDFT and MR-PT2 energy profiles along these paths. We find that (TD-)DFT methods are suitable for applications such as simulating the spectra of flavins but are inconsistent with several other methods when used for some geometry optimizations and when describing the energetics of dark (n,π*) states. MR-PT2 methods show promise for the simulation of flavin's low-lying excited states, but the selection of orbitals for the active space and the number of roots used for state averaging must be done carefully to avoid artifacts. Some properties, such as the intersystem crossing geometry and energy between the S1 (π,π*) and T2 (n,π*) states, may require additional benchmarking before they can be determined quantitatively.

Distinguishing homolytic vs heterolytic bond dissociation of phenylsulfonium cations with localized active space methods

Modeling chemical reactions with quantum chemical methods is challenging when the electronic structure varies significantly throughout the reaction and when electronic excited states are involved. Multireference methods, such as complete active space self-consistent field (CASSCF), can handle these multiconfigurational situations. However, even if the size of the needed active space is affordable, in many cases, the active space does not change consistently from reactant to product, causing discontinuities in the potential energy surface. The localized active space SCF (LASSCF) is a cheaper alternative to CASSCF for strongly correlated systems with weakly correlated fragments. The method is used for the first time to study a chemical reaction, namely the bond dissociation of a mono-, di-, and triphenylsulfonium cation. LASSCF calculations generate smooth potential energy scans more easily than the corresponding, more computationally expensive CASSCF calculations while predicting similar bond dissociation energies. Our calculations suggest a homolytic bond cleavage for di- and triphenylsulfonium and a heterolytic pathway for monophenylsulfonium.

Ultrafast Photoinduced Dynamics in 1,3-Cyclohexadiene: A Comparison of Trajectory Surface Hopping Schemes†.

Photoinduced nonadiabatic processes play a crucial role in a wide range of disciplines, from fundamental steps in biology to modern applications in advanced materials science. A theoretical understanding of these processes is highly desirable, and trajectory surface hopping (TSH) has proven to be a well-suited framework for a wide range of systems. In this work, we present a comprehensive comparison between two TSH algorithms, the conventional Tully's fewest switches surface hopping (FSSH) scheme and the Landau-Zener surface hopping (LZSH), to study the photoinduced ring-opening of 1,3-cyclohexadiene (CHD) to 1,3,5-hexatriene at the spin-flip time-dependent density functional theory (SF-TDDFT) level of theory. Additionally, we compare our results with a literature study at the extended multistate complete active space second-order perturbation theory method (XMS-CASPT2) level of theory. Our results show that the average population and lifetimes estimated with LZSH using SF-TDDFT are closer to the literature (using multireference methods) than those estimated with FSSH using SF-TDDFT. The latter speaks in favor of applying LZSH in combination with the SF-TDDFT method to study larger and more complex systems such as molecular photoswitches where the CHD molecule acts as a backbone. In addition, we present an implementation of Tully's FSSH algorithm as an extension to the PySurf software package.

Understanding the Photoprocesses in Biological Systems: Need for Accurate Multireference Treatment.

Light-matter interaction is crucial to life itself and revolves around many of the central processes in biology. The need for understanding these photochemical and photophysical processes cannot be overemphasized. Interaction of light with biological systems starts with the absorption of light and subsequent phenomena that occur in the excited states of the system. However, excited states are typically difficult to understand within the mean field approximation of quantum chemical methods. Therefore, suitable multireference methods and methodologies have been developed to understand these phenomena. In this Perspective, we will describe a few methods and methodologies suitable for these descriptions and discuss some persisting difficulties.

Relativistic Douglas–Kroll–Hess calculations of hyperfine interactions within first-principles multireference methods

A relativistic magnetic hyperfine interaction Hamiltonian based on the Douglas–Kroll–Hess (DKH) theory up to the second order is implemented within the ab initio multireference methods, including spin–orbit coupling in the Molcas/OpenMolcas package. This implementation is applied to calculate relativistic hyperfine coupling (HFC) parameters for atomic systems and diatomic radicals with valence s or d orbitals by systematically varying active space size in the restricted active space self-consistent field formalism with restricted active space state interaction for spin–orbit coupling. The DKH relativistic treatment of the hyperfine interaction reduces the Fermi contact contribution to the HFC due to the presence of kinetic factors that regularize the singularity of the Dirac delta function in the nonrelativistic Fermi contact operator. This effect is more prominent for heavier nuclei. As the active space size increases, the relativistic correction of the Fermi contact contribution converges well to the experimental data for light and moderately heavy nuclei. The relativistic correction, however, does not significantly affect the spin-dipole contribution to the hyperfine interaction. In addition to the atomic and molecular systems, the implementation is applied to calculate the relativistic HFC parameters for large trivalent and divalent Tb-based single-molecule magnets (SMMs), such as Tb(III)Pc2 and Tb(II)(CpiPr5)2 without ligand truncation using well-converged basis sets. In particular, for the divalent SMM, which has an unpaired valence 6s/5d hybrid orbital, the relativistic treatment of HFC is crucial for a proper description of the Fermi contact contribution. Even with the relativistic hyperfine Hamiltonian, the divalent SMM is shown to exhibit strong tunability of HFC via an external electric field (i.e., strong hyperfine Stark effect).

Pathway to Polyradicals: A Planar and Fully π-Conjugated Organic Tetraradical(oid).

In this work, we provide a general strategy to stabilize the ground state of polyradical(oid)s and make higher spin states thermally accessible. As a proof of concept, we propose to merge two planar fully π-conjugated diradical(oid)s to obtain a planar and cross-conjugated tetraradical(oid). Using multireference quantum chemistry methods, we show that the designed tetraradical(oid) is stabilized by aromaticity and delozalization in the π-system and has six thermally accessible spin states within 1.72 kcal/mol. Analysis of the electronic structure of these six states of the tetraradical(oid) shows that its frontier π-system consists of two weakly interacting subsystems: aromatic cycles and four unpaired electrons. Conjugation between unpaired electrons, which favors closed-shell structures, is mitigated by delocalization and the aromaticity of the bridging groups, leading to the synergistic cross-coupling between two diradical(oid) subunits to stabilize the tetraradical(oid) electronic structure.

How the Amount of Hartree-Fock Exchange Affects the Observation of the Pseudo Jahn-Teller Effect

Heavy group 14 analogs of planar carbon molecules are well known to prefer bent or pyramidalized geometries. The pseudo Jahn-Teller effect (pJTE) is a general approach to understanding the origin of such molecular geometries. Computationally intensive multi-reference post Hartree-Fock methods are generally required to obtain values relevant to the pJTE. However, an alternative approach, using more computationally efficient density functional theory (DFT) and fitting a model vibronic Hamiltonian, can account for the pJTE parameters. Previous studies have detailed the effect of both integration grid size and exact exchange on various properties including the nature of stationary points (i.e., transition structure v. minimum) on potential energy surfaces but none have addressed the root cause of such discrepancies. In this work we examine two contentious stationary points belonging to planar disilene and 2Si TCNQ from the perspective of the pseudo Jahn-Teller effect (pJTE) using DFT methods. First the planar stationary points are characterized using a variety of model chemistries and integration grids. The effect of the amount of Hartree-Fock exchange is then studied and the usage of DFT for assessing pJT parameters is explored. It is shown that the DFT approach is a viable alternative for determining values relevant to the pJTE. A complementary cusp catastrophe theory approach to the problem is also introduced. Difficulties associated with the DFT-based approach to the study of the pJTE are discussed and changes in the electronic structure are explored using the Quantum Theory of Atoms in Molecules.