A dynamic correlation dressed complete active space method: Theory, implementation, and preliminary applications

Perturbation theory in the complete degenerate active space (CDAS-PT2)

Methods based on the multireference perturbation theory (MRPT) with the one-electron zeroth-order Hamiltonian are widely used for the description of excited states, for example, due to their relatively low computational cost. However, current methods have a common drawback-use of a model space with low size. In this article, we propose the MRPT method with the model space extended to the complete active space. The one-electron zeroth-order Hamiltonian suitable for this extension is formulated. The proposed method was applied to common models, such as LiF, ethylene, and trans-butadiene. It was shown to have accuracy superior to XMCQDPT2 in most cases, especially in the case of the small active space.

Ab initio calculation of magnetic anisotropy and thermal spin transition in the variable temperature crystal conformations of [Co(terpy)2]2

The structure-property correlation of [Co(terpy)2]2+, which shows a spin crossover at 270 K, has been computationally investigated based on its variable temperature crystal structures. Among the employed DFT functionals, only the re-parametrized hybrid B3LYP* functional could describe the correct spin transition temperature. Explicit consideration of metal-ligand sigma bonding with dynamic electron correlation is found to be necessary for an accurate determination of the SCO temperature with multi-reference calculations. The metal-ligand axial bond distances are found to be the most significant internal coordinates in deciding SCO. A small structural change along the axial distance causes a change in the t2g orbital splitting pattern and a reorientation of the magnetization axes at the SCO temperature. The complex shows an unusual triaxial magnetic anisotropy, with an easy axis of magnetization developing at higher temperatures. The strong coupling of low-frequency wagging motion of the two terpyridine ligands with the spin states of the complex provides an effective pathway for the relaxation of magnetization, resulting in a small magnetic anisotropy barrier.

Synthesis and Photophysical Properties of Lanthanide Pyridinylphosphonic Tacn and Pyclen Derivatives: From Mononuclear Complexes to Supramolecular Heteronuclear Assemblies

Synthetic methodologies were developed to achieve the preparation of ligands L1 and L2 consisting of tacn- and pyclen-based chelators decorated with pyridinylphosphonic pendant arms combined with ethylpicolinamide or acetate coordinating functions, respectively. Phosphonate functions have been selected for their high affinity toward Ln3+ ions compared to their carboxylated counterparts and for their steric hindrance that favors the formation of less-hydrated complexes. Thanks to regiospecific N-functionalization of the macrocyclic backbones, the two ligands were isolated with good yields and implicated in a comprehensive photophysical study for the complexation of Eu3+, Tb3+, and Yb3+. The coordination behavior of L1 and L2 with these cations has been first investigated by means of a combination of UV-vis absorption spectroscopy, steady-state and time-resolved luminescence spectroscopy, and 1H and 31P NMR titration experiments. Structural characterization in solution was assessed by NMR spectroscopy, corroborated by theoretical calculations. Spectroscopic characterization of the Ln3+ complexes of L1 and L2 was done in water and D2O and showed the effective sensitization of the lanthanide metal-centered emission spectra, each exhibiting typical lanthanide emission bands. The results obtained for the phosphonated ligands were compared with those reported previously for the corresponding carboxylated analogues.

Perturbative Expansion of Nonorthogonal Product Approach for Charge Transfer States

Modeling of the excited states of multichromophoric systems is crucial for the understanding of photosynthesis functioning. The excitonic Hamiltonian method is widely used for such calculations. Excited states of the combined system are constructed from the wave functions of individual chromophores while their interactions are described by excitonic couplings. In the current study, we enhance a previously proposed nonorthogonal product approach to incorporate dynamic correlation effects accounted for by the multireference perturbation theory. We discuss the problems of constructing the excitonic Hamiltonian including charge transfer states for the molecular systems where the overlap contribution to the excitonic couplings is non-negligible. The benchmark calculations were performed for a model system. It was shown that the overlap component of the excitonic coupling is of great importance. The enhanced method provides an accurate description of the excited state energies and other properties.

GronOR: Scalable and Accelerated Nonorthogonal Configuration Interaction for Molecular Fragment Wave Functions

GronOR is a program package for nonorthogonal configuration interaction calculations. Electronic wave functions are constructed in terms of antisymmetrized products of multiconfiguration molecular fragment wave functions. The computational complexity of the nonorthogonal methodologies implemented in GronOR applied to large molecular assemblies requires a design that takes full advantage of massively parallel supercomputer architectures and accelerator technologies. This work describes the implementation strategy and resulting performance characteristics. In addition to parallelization and acceleration, the software development strategy includes aspects of fault resiliency and heterogeneous computing. The program was designed for large-scale supercomputers but also runs effectively on small clusters and workstations for small molecular systems. GronOR is available as open source to the scientific community.

Elucidating the molecular orbital dependence of the total electronic energy in multireference problems

The accurate resolution of the chemical properties of strongly correlated systems, such as biradicals, requires the use of electronic structure theories that account for both multi-reference and dynamic correlation effects. A variety of methods exist that aim to resolve the dynamic correlation in multi-reference problems, commonly relying on an exponentially scaling complete-active-space self-consistent-field (CASSCF) calculation to generate reference molecular orbitals (MOs). However, while CASSCF orbitals provide the optimal solution for a selected set of correlated (active) orbitals, their suitability in the quest for the resolution of the total correlation energy has not been thoroughly investigated. Recent research has shown the ability of Kohn-Shan density functional theory to provide improved orbitals for coupled cluster (CC) and Møller-Plesset perturbation theory (MP) calculations. Here, we extend the search for optimal and more cost effective MOs to post-configuration-interaction [post-(CI)] methods, surveying the ability of the MOs obtained with various density functional theory (DFT) functionals, as well as Hartree-Fock and CC and MP calculations to accurately capture the total electronic correlation energy. Applying the anti-Hermitian contracted Schrödinger equation to the dissociation of N₂, the calculation of biradical singlet-triplet gaps, and the transition states of bicylobutane isomerization, we demonstrate that DFT provides a cost-effective alternative to CASSCF in providing reference orbitals for post-CI dynamic correlation calculations.

On the role of dynamic electron correlation in non-orthogonal configuration interaction with fragments

Two different approaches have been implemented to include the effect of dynamic electron correlation in the Non-Orthogonal Configuration Interaction for Fragments (NOCI-F) method. The first is based on shifting the diagonal matrix elements of the NOCI matrix, while the second incorporates the dynamic correlation explicitly in the fragment wave functions used to construct the many-electron basis functions of the NOCI. The two approaches are illustrated for the calculation of the electronic coupling relevant in singlet fission and the coupling of spin moments in organic radicals. Comparison of the calculated diabatic couplings, the NOCI energies and wave functions shows that dynamic electron correlation is not only efficiently but also effectively incorporated by the shifting approach and can largely affect the coupling between electronic states. Also, it brings the NOCI coupling of the spin moments in close agreement with benchmark calculations.

LFDFT—A Practical Tool for Coordination Chemistry

The electronic structure of coordination compounds with lanthanide ions is studied by means of density functional theory (DFT) calculations. This work deals with the electronic structure and properties of open-shell systems based on the calculation of multiplet structure and ligand-field interaction, within the framework of the Ligand–Field Density-Functional Theory (LFDFT) method. Using effective Hamiltonian in conjunction with the DFT, we are able to reasonably calculate the low-lying excited states of the molecular [Eu(NO3)3(phenanthroline)2] complex, subjected to the Eu3+ configuration 4f6. The results are compared with available experimental data, revealing relative uncertainties of less than 5% for many energy levels. We also demonstrate the ability of the LFDFT method to simulate absorption spectrum, considering cerocene as an example. Ce M4,5 X-ray absorption spectra are simulated for the complexes [Ce(η8−C8H8)2] and [Ce(η8−C8H8)2][Li(tetrahydrofurane)4], which are approximated by the Ce oxidation states 4+ and 3+, respectively. The results showed a very good agreement with the experimental data for the Ce3+ compound, unlike for the Ce4+ one, where charge transfer electronic structure is still missing in the theoretical model. Therefore this presentation reports the benefits of having a theoretical method that is primarily dedicated to coordination chemistry, but it also outlines limitations and places the ongoing developmental efforts in the broader context of treating complex molecular systems.

Zero-Field Splitting Calculations by Multiconfiguration Pair-Density Functional Theory

Zero-field splitting (ZFS) is a fundamental molecular property that is especially relevant for single-molecule magnets (SMMs), electron paramagnetic resonance spectra, and quantum computing. Developing a method that can accurately predict ZFS parameters can be very powerful for designing new SMMs. One of the challenges is to include external correlation in an inherently multiconfigurational open-shell species for the accurate prediction of magnetic properties. Previously available methods depend on expensive multireference perturbation theory calculations to include external correlation. In this paper, we present spin-orbit-inclusive multiconfiguration and multistate pair-density functional theory (MC-PDFT) calculations of ZFSs; these calculations have a cost comparable to complete-active-space self-consistent field (CASSCF) theory, but they include correlation external to the active space. We found that combining a multistate formulation of MC-PDFT, namely, compressed-state multistate pair-density functional theory, with orbitals optimized by weighted-state-averaged CASSCF, yields reasonably accurate ZFS results.

The Adsorption of Small Molecules on the Copper Paddle-Wheel: Influence of the Multi-Reference Ground State

We report a theoretical study of the adsorption of a set of small molecules (C2H2, CO, CO2, O2, H2O, CH3OH, C2H5OH) on the metal centers of the "copper paddle-wheel"-a key structural motif of many MOFs. A systematic comparison between DFT of different rungs, single-reference post-HF methods (MP2, SOS-MP2, MP3, DLPNO-CCSD(T)), and multi-reference approaches (CASSCF, DCD-CAS(2), NEVPT2) is performed in order to find a methodology that correctly describes the complicated electronic structure of paddle-wheel structure together with a reasonable description of non-covalent interactions. Apart from comparison with literature data (experimental values wherever possible), benchmark calculations with DLPNO-MR-CCSD were also performed. Despite tested methods show qualitative agreement in the majority of cases, we showed and discussed reasons for quantitative differences as well as more fundamental problems of specific cases.

First-Principles Study of Optical Absorption Energies, Ligand Field and Spin-Hamiltonian Parameters of Cr3+ Ions in Emeralds

Herein, we study the electronic structure, energies, and vibronic structure of optical d-d transitions of Cr³⁺ ions doped in beryl (Be₃Si₆Al₂O₁₈:Cr³⁺, emerald). A computational protocol is developed that combines periodic density functional theory (for modeling of the bulk crystalline lattice of emerald) and the multireference configuration interaction complete active space self-consistent field method supplemented with n-electron valence second-order perturbation theory (for the calculation of the energy levels, wave functions, and spin-Hamiltonian and ligand-field parameters of the trigonal Cr³⁺ centers in the [CrO₆]^9- clusters embedded in an extended point charge field). Ligand-field parameters were extracted from mapping the effective ligand-field Hamiltonian onto the full many-particle Hamiltonian from one side and from a direct fit to energies of computed d-d transitions on the other side. These have been analyzed using ab initio ligand-field theory. The quality of the theoretical predictions is critically assessed through a detailed comparison with the available experimental data.

The Static-Dynamic-Static Family of Methods for Strongly Correlated Electrons: Methodology and Benchmarking

A series of methods (SDSCI, SDSPT2, iCI, iCIPT2, iCISCF(2), iVI, and iCAS) is introduced to accurately describe strongly correlated systems of electrons. Born from the (restricted) static-dynamic-static (SDS) framework for designing many-electron wave functions, SDSCI is a minimal multireference (MR) configuration interaction (CI) approach that constructs and diagonalizes a [Formula: see text] matrix for [Formula: see text] states, regardless of the numbers of orbitals and electrons to be correlated. If the full molecular Hamiltonian H in the QHQ block (which describes couplings between functions of the first-order interaction space Q) of the SDSCI CI matrix is replaced with a zeroth-order Hamiltonian [Formula: see text] before the diagonalization is taken, we obtain SDSPT2, a CI-like second-order perturbation theory (PT2). Unlike most variants of MRPT2, SDSPT2 treats single and multiple states in the same way and is particularly advantageous in the presence of near degeneracy. On the other hand, if the SDSCI procedure is repeated until convergence, we will have iterative CI (iCI), which can converge quickly from the above to the exact solutions (full CI) even when starting with a poor guess. When further combined with the selection of important configurations followed by a PT2 treatment of dynamic correlation, iCI becomes iCIPT2, which is a near-exact theory for medium-sized systems. The microiterations of iCI for relaxing the coefficients of contracted many-electron functions can be generalized to an iterative vector interaction (iVI) approach for finding exterior or interior roots of a given matrix, in which the dimension of the search subspace is fixed by either the number of target roots or the user-specified energy window. Naturally, iCIPT2 can be employed as the active space solver of the complete active space (CAS) self-consistent field, leading to iCISCF(2), which can further be combined with iCAS for automated selection of active orbitals and assurance of the same CAS for all states and all geometries. The methods are calibrated by taking the Thiel set of benchmark systems as examples. Results for the corresponding cations, a new set of benchmark systems, are also reported.

Exploration of interlacing and avoided crossings in a manifold of potential energy curves by a unitary group adapted state specific multi-reference perturbation theory (UGA-SSMRPT)

The Unitary Group Adapted State-Specific Multi-Reference Perturbation Theory (UGA-SSMRPT2) developed by Mukherjee et al. [J. Comput. Chem. 36, 670 (2015)] has successfully realized the goal of studying bond dissociation in a numerically stable, spin-preserving, and size-consistent manner. We explore and analyze here the efficacy of the UGA-SSMRPT2 theory in the description of the avoided crossings and interlacings between a manifold of potential energy curves for states belonging to the same space-spin symmetry. Three different aspects of UGA-SSMRPT2 have been studied: (a) We introduce and develop the most rigorous version of UGA-SSMRPT2 that emerges from the rigorous version of UGA-SSMRCC utilizing a linearly independent virtual manifold; we call this the "projection" version of UGA-SSMRPT2 (UGA-SSMRPT2 scheme P). We compare and contrast this approach with our earlier formulation that used extra sufficiency conditions via amplitude equations (UGA-SSMRPT2 scheme A). (b) We present the results for a variety of electronic states of a set of molecules, which display the striking accuracy of both the two versions of UGA-SSMRPT2 with respect to three different situations involving weakly avoided crossings, moderate/strongly avoided crossings, and interlacing in a manifold of potential energy curves (PECs) of the same symmetry. Accuracy of our results has been benchmarked against IC-MRCISD + Q.

Photodesorption mechanism of water on WO3(001) - a combined embedded cluster, computational intelligence and wave packet approach

In this work we investigate the mechanism of photodesorption of water from a WO3(001) surface by theoretical calculations, applying an embedded cluster model. Using the CASSCF method, we have calculated both the ground state as well as the energetically preferred charge-transfer state in three degrees of freedom of the water molecule on the surface. The calculated potential energy surfaces were afterwards fitted with a neural network optimized by a genetic algorithm. A final quantum dynamic wave packet study provided insight into the photodesorption mechanism.

Late first-row transition metal complexes of a 17-membered piperazine-based macrocyclic ligand: structures and magnetism

A 17-membered piperazine-based macrocyclic ligand LdiProp (1,5,13,17,22-pentaazatricyclo[15.2.2.17,11]docosa-7,9,11(22)-triene) was resynthesized in high yield by using a linear pump. Its Mn(ii), Fe(ii), Co(ii) and Ni(ii) complexes of the general formula [MnLdiProp(ClO4)2] (1), [FeLdiProp(CH3CN)](ClO4)2 (2), [CoLdiProp(CH3CN)](ClO4)2 (3), [NiLdiProp](ClO4)2 (4) were prepared and thoroughly characterized. X-ray diffraction analysis confirmed that Mn(ii) complex 1 has capped trigonal prismatic geometry with a coordination number of seven, Fe(ii) and Co(ii) complexes 2 and 3 are trigonal prismatic with a coordination number of six and Ni(ii) complex 4 has square pyramidal geometry with a coordination number of five. The decrease of the coordination number is accompanied by a shortening of M-N distances and an increase of torsion of the piperazine ring from the equatorial plane. Magnetic measurement reveals moderate anisotropy for 4 and rather large magnetic anisotropy for 2 and 3 (axial zero-field splitting parameter D(Ni) = 9.0 cm-1, D(Fe) = -14.4 cm-1, D(Co) = -25.8 cm-1, together with rather high rhombicity). Co(ii) complex 3 behaves as a field-induced SMM with a combination of Raman and direct or Orbach and direct relaxation mechanisms. Obtained magnetic data were extensively supported by theoretical CASSCF calculations. The flexibility and rather large 17-membered macrocyclic cavity of ligand LdiProp could be responsible for the variation of coordination numbers and geometries for the investigated late-first row transition metals.

Structural, magnetic, redox and theoretical characterization of seven-coordinate first-row transition metal complexes with a macrocyclic ligand containing two benzimidazolyl N-pendant arms

A structurally new heptadentate derivative of a 15-membered pyridine-based macrocycle containing two benzimidazol-2-yl-methyl N-pendant arms (L = 3,12-bis((1H-benzimidazol-2-yl)methyl)-6,9-dioxa-3,12,18-triazabicyclo[12.3.1]octadeca-1(18),14,16-triene) was synthesized and its complexes with the general formula [M(L)](ClO4)2·1.5CH3NO2 (M = MnII (1), FeII (2), CoII (3) and NiII (4)) were thoroughly investigated. X-ray crystal structures confirmed that all complexes are seven-coordinate with axially compressed pentagonal bipyramidal geometry having the largest distortion for NiII complex 4. FeII, CoII and NiII complexes 2, 3 and 4 show rather large magnetic anisotropy manifested by moderate to high obtained values of the axial zero-field splitting parameter D (7.9, 40.3, and -17.2 cm-1, respectively). Magneto-structural correlation of the FeII, CoII and NiII complexes with L and with previously studied structurally similar ligands revealed a significant impact of the functional group in pendant arms on the magnetic anisotropy especially that of the CoII and NiII complexes and some recommendations concerning the ligand-field design important for anisotropy tuning in future. Furthermore, complex 3 showed field-induced single-molecule magnet behavior described with the Raman (C = 507 K-n s-1 for n = 2.58) relaxation process. The magnetic properties of the studied complexes were supported by theoretical calculations, which very well correspond with the experimental data of magnetic anisotropy. Electrochemical measurements revealed high positive redox potentials for M3+/2+ couples and high negative redox potentials for M2+/+ couples, which indicate the stabilization of the oxidation state +ii expected for the σ-donor/π-acceptor ability of benzimidazole functional groups.

GronOR: Massively parallel and GPU-accelerated non-orthogonal configuration interaction for large molecular systems

GronOR is a program package for non-orthogonal configuration interaction calculations for an electronic wave function built in terms of anti-symmetrized products of multi-configuration molecular fragment wave functions. The two-electron integrals that have to be processed may be expressed in terms of atomic orbitals or in terms of an orbital basis determined from the molecular orbitals of the fragments. The code has been specifically designed for execution on distributed memory massively parallel and Graphics Processing Unit (GPU)-accelerated computer architectures, using an MPI+OpenACC/OpenMP programming approach. The task-based execution model used in the implementation allows for linear scaling with the number of nodes on the largest pre-exascale architectures available, provides hardware fault resiliency, and enables effective execution on systems with distinct central processing unit-only and GPU-accelerated partitions. The code interfaces with existing multi-configuration electronic structure codes that provide optimized molecular fragment orbitals, configuration interaction coefficients, and the required integrals. Algorithm and implementation details, parallel and accelerated performance benchmarks, and an analysis of the sensitivity of the accuracy of results and computational performance to thresholds used in the calculations are presented.

Stereochemistry of coordination polyhedra vs. single ion magnetism in penta- and hexacoordinated Co(ii) complexes with tridentate rigid ligands

A tridentate ligand L (2,6-bis(1-(3,5-di-tert-butylbenzyl)-1H-benzimidazol-2-yl)pyridine) was synthesized and used for the preparation of three pentacoordinated Co(ii) complexes of formula [Co(L)X₂] (where X = NCS^- for 1, X = Cl^- for 2 and X = Br^- for 3) and one ionic compound 4 ([Co(L)₂]Br₂·2CH₃OH·H₂O) containing a hexacoordinated Co(ii) centre. Static magnetic data were analysed with respect to the spin (1-3) or the Griffith-Figgis (4) Hamiltonian. Ab initio calculations enable us to identify the positive axial magnetic anisotropy parameter D accompanied by a significant degree of rhombicity in the reported complexes. Also, magneto-structural correlation was outlined for this class of compounds. Moreover, all four compounds exhibit slow relaxation of magnetisation at an applied static magnetic field with either both low- and high-frequency relaxation channels (3) or a single high-frequency relaxation process (1, 2 and 4). The interplay between the stereochemistry of coordination polyhedra, magnetic anisotropy and the relaxation processes was investigated and discussed in detail.

Improvement of Ab Initio Ligand Field Theory by Means of Multistate Perturbation Theory

Over the last few years, ab initio ligand field theory (AILFT) has evolved into an important tool for the extraction of ligand field models from ab initio calculations. The inclusion of dynamic correlation on top of complete active space self-consistent field (CASSCF) reference functions, which is important for accurate results, was so far realized at the level of second-order N-electron valence state perturbation theory (NEVPT2). In this work, we introduce two alternative methods for the inclusion of dynamic correlation into AILFT calculations, the second-order dynamic correlation dressed complete active space method (DCD-CAS(2)) and the Hermitian quasi-degenerate NEVPT2 (HQD-NEVPT2). These methods belong to the class of multistate perturbation theory approaches, which allow for the mixing of CASSCF states under the effect of dynamic correlation (state-mixing). The two new versions of AILFT were tested for a diverse set of transition-metal complexes. It was found that the multistate methods have, compared to NEVPT2, an AILFT fit with smaller root mean square deviations (rmsds) between ab initio and AILFT energies. A comparison of AILFT excitation energies with the experiment shows that for some systems, the agreement gets better at the multistate level because of the smaller rmsds. However, for some systems, the agreement gets worse, which could be attributed to a cancellation of errors at the NEVPT2 level that is partly removed at the multistate level. An investigation of trends in the extracted ligand field parameters shows that at the multistate level, the ligand field splitting Δ gets larger, whereas the Racah parameters B and C get smaller and larger, respectively. An investigation of the reasons for the observed improvement for octahedral Cr^III halide complexes shows that the possibility of state-mixing relaxes constraints that are present at the NEVPT2 level and that keep Δ and B from following their individual preferences.

Direct observation of the intermediate radical in the photodissociation of 1,3-cyclohexane dinitrite

A two-step photodissociation mechanism was proposed in the literature for dinitrites in the absence of direct evidence of the intermediate species. In this work, photodissociation dynamics of cis and trans 1,3-cyclohexane dinitrites are investigated by laser-induced fluorescence (LIF) spectroscopy and theoretical calculation methods. Observation of the fluorescence spectra of the 3-nitrosooxy cyclohexoxy radical provides direct experimental evidence that the intermediate species exists. The results indicate that photodissociation of dinitrites indeed follows a two-step mechanism, i.e. one of the O-NO bonds of the molecule breaks first upon 355 nm laser photolysis and generates an alkoxy radical (RO) plus NO; the alkoxy radical further dissociates in the secondary dissociation step and produces small fragments such as vinoxy etc.

Polynomial-product states: A symmetry-projection-based factorization of the full coupled cluster wavefunction in terms of polynomials of double excitations

Our goal is to remedy the failure of symmetry-adapted coupled-cluster theory in the presence of strong correlation. Previous work along these lines has taken us from a diagram-level analysis of the coupled-cluster equations to an understanding of the collective modes which can occur in various channels of the coupled-cluster equations to the exploration of non-exponential wavefunctions in efforts to combine coupled-cluster theory with symmetry projection. In this manuscript, we extend these efforts by introducing a new, polynomial product wavefunction ansatz that incorporates information from symmetry projection into standard coupled-cluster theory in a way that attempts to mitigate the effects of the lack of size extensivity and size consistency characteristic of symmetry-projected methods. We describe the new approach in detail within the context of our previous efforts, explore some illustrative calculations, and consider one route for reducing the computational cost of the new method.