DOCSIC: A Mean-Field Method for Orbital-by-Orbital Self-Interaction Correction.

We introduce a new method to remove the one-electron self-interaction error in approximate density functional calculations on an orbital-by-orbital basis, as originally proposed by Perdew and Zunger [Phys. Rev. B 1981, 23, 5048]. This method is motivated by a recent proposal by Pederson et al. [J. Chem. Phys. 2014, 140, 121103] to remove self-interaction that employs orbitals derived from the real-space density matrix, known as FLOSIC (Fermi Löwdin orbitals self-interaction correction). However, instead of Fermi Löwdin orbitals, our scheme utilizes columns of the density matrix to determine localized orbitals, like the localization procedure proposed by Fuemmeler et al. [J. Chem. Theory Comput. 2023, 19, 8572]. The new method, dubbed DOCSIC for density matrix as orbital coefficients self-interaction correction, contrasts with traditional Perdew-Zunger or FLOSIC in that it does not incorporate additional optimization parameters, and, unlike the average density self-interaction correction of Ciofini et al. [Chem. Phys. Lett. 2003, 380, 12], it makes use of localized orbitals. Another advantage of DOCSIC is that it can be implemented as a mean-field formalism. We show details of the self-consistent generalized Kohn-Sham implementation, some illustrative results, and we finally highlight its advantages and limitations.

A variance-based optimization for determining ground and excited N-electron wave functions within the doubly occupied configuration interaction scheme.

This work describes optimizations of N-electron system wave functions by means of the simulated annealing technique within the doubly occupied configuration interaction framework. Using that technique, we minimize the energy variance of a Hamiltonian, providing determinations of wave functions corresponding to ground or excited states in an identical manner. The procedure that allows us to determine electronic spectra can be performed using treatments of restricted or unrestricted types. The results found in selected systems, described in terms of energy, spin, and wave function, are analyzed, showing the performance of each method. We also compare these results with those arising from more traditional approaches that minimize the energy, in both restricted and unrestricted versions, and with those obtained from the full configuration interaction treatment.

Determination of electronic excitation energies within the doubly occupied configuration interaction space by means of the Hermitian operator method.

In this work, we formulate the equations of motion corresponding to the Hermitian operator method in the framework of the doubly occupied configuration interaction space. The resulting algorithms turn out to be considerably simpler than the equations provided by that method in more conventional spaces, enabling the determination of excitation energies in N-electron systems under an affordable polynomial computational cost. The implementation of this technique only requires to know the elements of low-order reduced density matrices of an N-electron reference state, which can be obtained from any approximate method. We contrast our procedure against the reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev integrable models, pointing out the reliability of our proposal.

Generalized spin σ-SCF method.

We introduce a generalization of the σ-SCF method to approximate noncollinear spin ground and excited single-reference electronic states by minimizing the Hamiltonian variance. The new method is based on the σ-SCF method, originally proposed by Ye et al. [J. Chem. Phys. 147, 214104 (2017)], and provides a prescription to determine ground and excited noncollinear spin states on an equal footing. Our implementation was carried out utilizing an initial simulated annealing stage followed by a mean-field iterative self-consistent approach to simplify the cumbersome search introduced by generalizing the spin degrees of freedom. The simulated annealing stage ensures a broad exploration of the Hilbert space spanned by the generalized spin single-reference states with random complex element-wise rotations of the generalized density matrix elements in the simulated annealing stage. The mean-field iterative self-consistent stage employs an effective Fockian derived from the variance, which is utilized to converge tightly to the solutions. This process helps us to easily find complex spin structures, avoiding manipulating the initial guess. As proof-of-concept tests, we present results for Hn (n = 3-7) planar rings and polyhedral clusters with geometrical spin frustration. We show that most of these systems have noncollinear spin excited states that can be interpreted in terms of geometric spin frustration. These states are not directly targeted by energy minimization methods, which are meant to converge to the ground state. This stresses the capability of the σ-SCF methodology to find approximate noncollinear spin structures as mean-field excited states.

Variational determination of the two-electron reduced density matrix within the doubly occupied configuration interaction framework: Treatments of triplet N-electron systems.

In this work, we perform variational calculations of two-electron reduced density matrices corresponding to open-shell N-electron systems within the framework of the doubly occupied configuration interaction treatment, traditionally limited to studies of closed-shell systems. This has allowed us to provide a satisfactory description of molecular systems in triplet states following two methods. One of them adds hydrogen atoms at an infinite distance of the triplet system studied, constituting a singlet supersystem. The energies and reduced density matrices of the triplet system are obtained by removing the contributions of the added atoms from the singlet supersystem results. The second procedure involves variational determination of the two-electron reduced density matrices corresponding to the triplet systems by means of adequate couplings of basis-set functions. Both models have been studied by imposing N-representability conditions on the reduced density matrix calculations. Results obtained from these methods for molecular systems in triplet ground states are reported and compared with those provided by benchmark methods.

Influence of the N-representability conditions on the variational determination of two-electron reduced density matrices for ground and excited N-electron states in the doubly occupied configuration interaction space.

In this work, we analyze the effectiveness of different sets of well-known necessary N-representability conditions that are imposed on the variational determination of two-electron reduced density matrices corresponding to the ground and excited states of N-electron Hamiltonians within doubly occupied configuration interaction subspaces. The two-electron reduced density matrix elements are determined so that they are suitably related to those of higher-order reduced density matrices and other associated matrices, up to a given order, constituting positive semidefinite matrices. These conditions up to two-, three-, and four-order have been sequentially imposed on the variational calculation of the expectation values of the dispersion operator corresponding to several Hamiltonian models, obtaining the electronic spectra. These results have been contrasted with those arising from the full configuration interaction method, illuminating the sufficiency/insufficiency of the variational N-representability conditions applied.

Variational determination of ground and excited-state two-electron reduced density matrices in the doubly occupied configuration space: A dispersion operator approach.

This work implements a variational determination of the elements of two-electron reduced density matrices corresponding to the ground and excited states of N-electron interacting systems based on the dispersion operator technique. The procedure extends the previously reported proposal [Nakata et al., J. Chem. Phys. 125, 244109 (2006)] to two-particle interaction Hamiltonians and N-representability conditions for the two-, three-, and four-particle reduced density matrices in the doubly occupied configuration interaction space. The treatment has been applied to describe electronic spectra using two benchmark exactly solvable pairing models: reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev Hamiltonians. The dispersion operator combined with N-representability conditions up to the four-particle reduced density matrices provides excellent results.

Variational determination of the two-electron reduced density matrix within the doubly occupied configuration interaction scheme: An extension to the study of open-shell systems.

This work proposes to describe open-shell molecules or radicals using the framework of the doubly occupied configuration interaction (DOCI) treatments, so far limited to closed-shell system studies. The proposal is based on considering molecular systems in singlet states generated by adding extra hydrogen atoms located at infinite distance from the target radical system. The energy of this radical is obtained by subtracting the energies of the dissociated hydrogen atoms from that provided by the two-electron reduced density matrix corresponding to the singlet state system in the DOCI space, which is variationally calculated by imposing a set of N-representability conditions. This method is numerically assessed by describing potential energy curves and reduced density matrices in selected ionic and neutral open-shell systems in the doublet spin symmetry ground state.

Magnetic Properties of Co(II) Complexes with Polyhedral Carborane Ligands.

In this work we present a computational analysis of a new family of magnetic Co(II) single-ion complexes with large magnetic anisotropy based on icosahedral and octahedral carborane ligands. In particular, we extend our previous computational work on mononuclear Co(II) complexes with 1,2-(HS)2-1,2-C2B10H10 and 9,12-(HS)2-1,2-C2B10H10 icosahedral o-carborane ligands to a larger set of complexes where the Co(II) ion is doubly chelated by those ligands and by other two positional isomers belonging to the 1,2-dicarba- closo-dodecaborane family. We also describe Co(II) complexes with octahedral ligands derived from 1,2-dicarba- closo-hexaborane and study the effects of replacing a thiol group by a hydroxy group in both polyhedral geometries, as well as the influence of the position of the carbon atoms. On analysis of the results for a total of 20 complexes, our results show that carborane-based Co(II) single-ion compounds present a distorted-tetrahedral geometry, high-spin ground states, and high values for the magnetic anisotropy parameters. We point out which of these would be suitable candidates to be synthesized and used as molecular magnets.

Unrestricted treatment for the direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions.

This work extends to the unrestricted orbital approach the procedure described in our previous report [Alcoba et al., J. Chem. Phys. 148, 024105 (2018)] for determining variationally the two-electron reduced density matrix arising from doubly occupied-configuration-interaction wave functions by imposing two- and three-index N-representability conditions. An analysis of the numerical results obtained in selected systems, from both restricted and unrestricted treatments, allows one to assess the performance of these methodologies as well as to show the influence of the P, Q, G, T1, and T2 positivity conditions. We highlight the satisfactory results obtained within the unrestricted scheme.

Magnetic Properties of Mononuclear Co(II) Complexes with Carborane Ligands.

We analyze the magnetic properties of three mononuclear Co(II) coordination complexes using quantum chemical complete active space self-consistent field and N-electron valence perturbation theory approaches. The complexes are characterized by a distorted tetrahedral geometry in which the central ion is doubly chelated by the icosahedral ligands derived from 1,2-(HS)2-1,2-C2B10H10 (complex I), from 1,2-(HS)2-1,2-C2B10H10 and 9,12-(HS)2-1,2-C2B10H10 (complex II), and from 9,12-(HS)2-1,2-C2B10H10 (complex III), which are two positional isomers of dithiolated 1,2-dicarba- closo-dodecaborane (complex I). Complex I was realized experimentally recently (Tu, D.; Shao, D.; Yan, H.; Lu, C. Chem. Commun. 2016, 52, 14326) and served to validate the computational protocol employed in this work, while the remaining two proposed complexes can be considered positional isomers of I. Our calculations show that these complexes present different axial and rhombic zero-field splitting anisotropy parameters and different values of the most significant components of the g tensor. The predicted axial anisotropy D = -147.2 cm-1 for complex II is twice that observed experimentally for complex I, D = -72.8 cm-1, suggesting that this complex may be of interest for practical applications. We also analyze the temperature dependence of the magnetic susceptibility and molar magnetization for these complexes when subject to an external magnetic field. Overall, our results suggest that o-carborane-incorporated Co(II) complexes are worthwhile candidates for experimental exploration as single-ion molecular magnets.

Variational reduced density matrix method in the doubly occupied configuration interaction space using three-particle N-representability conditions.

Ground-state energies and two-particle reduced density matrices (2-RDMs) corresponding to N-particle systems are computed variationally within the doubly occupied configuration interaction (DOCI) space by constraining the 2-RDM to satisfy a complete set of three-particle N-representability conditions known as three-positivity conditions. These conditions are derived and implemented in the variational calculation of the 2-RDM with standard semidefinite programming algorithms. Ground state energies and 2-RDMs are computed for N2, CO, CN-, and NO+ molecules at both equilibrium and nonequilibrium geometries as well as for pairing models at different repulsive interaction strengths. The results from the full three-positivity conditions are compared with those from the exact DOCI method and with approximated 2-RDM variational ones obtained within two-positivity and two-positivity plus a subset of three-positivity conditions, as recently reported [D. R. Alcoba et al., J. Chem. Phys. 148, 024105 (2018) and A. Rubio-García et al., J. Chem. Theory Comput. 14, 4183 (2018)]. The accuracy of these numerical determinations and their low computational cost demonstrate the usefulness of the three-particle variational constraints within the DOCI framework.

Direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions: The influence of three-index N-representability conditions.

This work proposes the variational determination of two-electron reduced density matrices corresponding to the ground state of N-electron systems within the doubly occupied-configuration-interaction methodology. The P, Q, and G two-index N-representability conditions have been extended to the T1 and T2 (T2') three-index ones and the resulting optimization problem has been addressed using a standard semidefinite program. We report results obtained from the doubly occupied-configuration-interaction method, from the two-index constraint variational procedure and from the two- and three-index constraint variational treatment. The discussion of these results along with a study of the computational cost demanded shows the usefulness of our proposal.

Atom and Bond Fukui Functions and Matrices: A Hirshfeld-I Atoms-in-Molecule Approach.

The Fukui function is often used in its atom-condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms-in-molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld-I atoms-in-molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld-I approach to obtain atom- and bond-condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms-in-molecule approach shows low correlation between the two partitioning schemes.

Spin contamination-free N-electron wave functions in the excitation-based configuration interaction treatment.

This work deals with the spin contamination in N-electron wave functions provided by the excitation-based configuration interaction methods. We propose a procedure to ensure a suitable selection of excited N-electron Slater determinants with respect to a given reference determinant, required in these schemes. The procedure guarantees the construction of N-electron wave functions which are eigenfunctions of the spin-squared operator Sˆ(2), avoiding any spin contamination. Our treatment is based on the evaluation of the excitation level of the determinants by means of the expectation value of an excitation operator formulated in terms of spin-free replacement operators. We report numerical determinations of energies and 〈Sˆ(2)〉 expectation values, arising from our proposal as well as from traditional configuration interaction methods, in selected open-shell systems, in order to compare the behavior of these procedures and their computational costs.

Ab initio quantum-chemical computations of the electronic states in HgBr2 and IBr: Molecules of interest on the Earth's atmosphere.

The electronic states of atmospheric relevant molecules IBr and HgBr2 are reported, within the UV-Vis spectrum range (170nm≤λphoton≤600 nm) by means of the complete-active-space self-consistent field/multi-state complete-active-space second-order perturbation theory/spin-orbit restricted-active-space state-interaction (CASSCF/MS-CASPT2/SO-RASSI) quantum-chemical approach and atomic-natural-orbital relativistic-correlation-consistent (ANO-RCC) basis sets. Several analyses of the methodology were carried out in order to reach converged results and therefore to establish a highly accurate level of theory. Good agreement is found with the experimental data with errors not higher than around 0.1 eV. The presented analyses shall allow upcoming studies aimed to accurately determine the absorption cross sections of interhalogen molecules and compounds with Hg that are relevant to better comprehend the photochemical processes taking place in the atmosphere.

Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions.

A class of polynomial scaling methods that approximate Doubly Occupied Configuration Interaction (DOCI) wave functions and improve the description of dynamic correlation is introduced. The accuracy of the resulting wave functions is analysed by comparing energies and studying the overlap between the newly developed methods and full configuration interaction wave functions, showing that a low energy does not necessarily entail a good approximation of the exact wave function. Due to the dependence of DOCI wave functions on the single-particle basis chosen, several orbital optimisation algorithms are introduced. An energy-based algorithm using the simulated annealing method is used as a benchmark. As a computationally more affordable alternative, a seniority number minimising algorithm is developed and compared to the energy based one revealing that the seniority minimising orbital set performs well. Given a well-chosen orbital basis, it is shown that the newly developed DOCI based wave functions are especially suitable for the computationally efficient description of static correlation and to lesser extent dynamic correlation.

Configuration interaction wave functions: a seniority number approach.

This work deals with the configuration interaction method when an N-electron Hamiltonian is projected on Slater determinants which are classified according to their seniority number values. We study the spin features of the wave functions and the size of the matrices required to formulate states of any spin symmetry within this treatment. Correlation energies associated with the wave functions arising from the seniority-based configuration interaction procedure are determined for three types of molecular orbital basis: canonical molecular orbitals, natural orbitals, and the orbitals resulting from minimizing the expectation value of the N-electron seniority number operator. The performance of these bases is analyzed by means of numerical results obtained from selected N-electron systems of several spin symmetries. The comparison of the results highlights the efficiency of the molecular orbital basis which minimizes the mean value of the seniority number for a state, yielding energy values closer to those provided by the full configuration interaction procedure.

A hybrid configuration interaction treatment based on seniority number and excitation schemes.

We present a configuration interaction method in which the Hamiltonian of an N-electron system is projected on Slater determinants selected according to the seniority-number criterion along with the traditional excitation-based procedure. This proposed method is especially useful to describe systems which exhibit dynamic (weak) correlation at determined geometric arrangements (where the excitation-based procedure is more suitable) but show static (strong) correlation at other arrangements (where the seniority-number technique is preferred). The hybrid method amends the shortcomings of both individual determinant selection procedures, yielding correct shapes of potential energy curves with results closer to those provided by the full configuration interaction method.

Fukui and dual-descriptor matrices within the framework of spin-polarized density functional theory.

This work deals with the Fukui and dual reactivity descriptors within the framework of the spin-polarized density functional theory. The first and second derivatives of the electron density and the spin density with respect to the total number of electrons N = Nα + Nß and with respect to the spin number NS = Nα-Nß have been formulated by means of reduced density matrices in the representation of the spin-orbitals of a given basis set, providing the matrix extension of those descriptors. The analysis of the eigenvalues and eigenvectors of the Fukui and dual-descriptor matrices yields information on the role played by the molecular orbitals in charge-transfer and spin-polarization processes. This matrix formulation enables determining similarity indices which allows one to evaluate quantitatively the quality of the simple frontier molecular orbital model in conceptual density functional theory. Selected closed- and open-shell systems in different spin symmetries have been studied with this matrix formalism at several levels of electronic correlation. The results confirm the suitability of this approach.