Erratum to: (Hyper)polarizability density analysis for open-shell molecular systems based on natural orbitals and occupation numbers

(Hyper)polarizability density analysis for open-shell molecular systems based on natural orbitals and occupation numbers

We propose a procedure that combines multiconfigurational (MC) wave functions with two-body density correlation functionals by transforming the latter into functionals of the MC natural orbitals and occupation numbers. The method is tested with the spectroscopic constants of a set of 11 diatomics, the diradical-involved automerization barrier of cyclobutadiene, the energy difference between triplet and open-shell singlet states in He and the methylene molecule, and the magnetic coupling constants of several systems, such as NiO, KNiF(3), K(2)NiF(4), La(2)CuO(4), alpha-4-dehydrotoluene, 1,1('),5,5(')-tetramethyl-6,6(')-dioxo-3,3(')-biverdazyl, [Cu(2)Cl(6)](-2), copper(II) acetate monohidrate and H-He-H. The procedure is applied to the Colle-Salvetti [Theor. Chim. Acta 37, 329 (1975); 53, 55 (1979)], functional and to a size-consistent functional depending on the on-top pair density (F1-5-N(eff)). On average, the best results are provided by the transformed F1-5-N(eff) [J. Chem. Phys. 114, 2022 (2001)] functional.

Natural orbital functional (NOF) theory in its PNOF3 and PNOF4 implementations is used to investigate the electronic structure and energetics of selected diradicals and diradicaloids: the trimethylenmethane (TMM) diradical and the imino-allyl (IMA) and oxyallyl (OXA) diradicaloids. These compounds are taken as paradigmatic cases of molecules with full and partial diradical character, and are considered as a case study to illustrate the potentiality of Piris NOF (PNOF) to yield the correct treatment of near-degeneracy effects in diradicals. The degree of diradical character is determined through the inspection of the corresponding natural orbital occupation numbers. In addition, the energetics of these compounds with respect to their cyclic isomers (methylenecyclopropane, MCP, iminocyclopropane, ICP, and cyclopropanone, OCP) is investigated. We have found that PNOF4 shows a promising description of these species, and yield the correct trends in occupation numbers as compared with wave-function methods such as CASSCF. Therefore, PNOF4 can give a correct description of near-degeneracy effects, and hence, is a promising method to treat diradicals and diradicaloids in chemistry, a delicate type of molecular systems to simulate by theoretical methods.

Two theorems on the eigenvalues of differences of idempotent matrices determine the natural occupation numbers and orbitals of electronic detachment, attachment, or excitation that pertain to transitions between wavefunctions that each consist of a single Slater determinant. They are also applicable to spin density matrices associated with Slater determinants. When the ranks of the matrices differ, unit eigenvalues occur. In addition, there are ±w pairs of eigenvalues where |w| ≤ 1, whose values are related to overlaps, t, between the corresponding orbitals of Amos and Hall, and Löwdin by the formula w=±1-t2 12. Generalized overlap amplitudes, including Dyson orbitals and their probability factors, may be inferred from these eigenvalues, which provide numerical criteria for: classifying transitions according to the number of holes and particles in final states with respect to initial states, identifying the most important effects of orbital relaxation produced by self-consistent fields, and the analysis of Fukui functions. Two similar theorems that apply to sums of idempotent matrices regenerate formulae for the natural orbitals and occupation numbers of an unrestricted Slater determinant that were published first by Amos and Hall.

A recently proposed tailored approach based on the distinguishable cluster method and the stochastic FCI solver, FCIQMC [J. Chem. Theory Comput. 2020, 16, 5621], is extended to open-shell molecular systems. The method is employed to calculate spin gaps of various Fe(II) complexes, including a Fe(II) porphyrin model system. Both distinguishable cluster and fully relaxed CASSCF natural orbitals were used in this work as reference for the subsequent tailored distinguishable cluster calculations. The distinguishable cluster natural orbitals occupation numbers were also used as an aid to the selection of the active space. The effect of the active space sizes and of the explicit correlation correction (F12) onto the predicted spin gaps is investigated. The tailored distinguishable cluster with singles and doubles yields consistently more accurate results compared to the tailored coupled cluster with singles and doubles.

The natural orbitals and their corresponding occupation numbers are constructed for several interesting problems to demonstrate that the existence of negative natural orbital occupation numbers for single reference correlation methods provides a simple diagnostic for the need for a multiconfigurational description of the wave function.

Natural orbitals (NOs) are central constituents for evaluating correlation energies through efficient approximations. Here, we report the closed-form expression of the NOs of two-electron quantum rings, which are prototypical finite-extension systems and new starting points for the development of exchange-correlation functionals in density functional theory. We also show that the natural occupation numbers for these two-electron paradigms are in general non-vanishing and follow the same power law decay as atomic and molecular two-electron systems.

The occupation numbers and natural orbitals of several model π-electron molecules are presented and analyzed, which are directly calculated by employing a theory developed earlier by one of the authors. Computational details are discussed briefly, and the errors which arise in our calculations are estimated through a comparison with the density matrix of a full configuration interaction wave function for the pyridine molecule.

Whereas a density functional that incorporates dispersion interaction has remained elusive to date, we demonstrate that in principle the dispersion energy can be obtained from a density matrix functional. In density matrix functional theory one tries to find suitable approximations to the two-particle reduced density matrix (2RDM) in terms of natural orbitals (NOs) and natural orbital occupation numbers (ONs). The total energy is then given as a function(al) of the NOs and ONs, i.e., as an implicit functional of the 1RDM. The left-right correlation in a (dissociating) bond, as well as various types of dynamical correlation, can be described accurately with a NO functional employing only J and K integrals (JK-only functional). We give a detailed analysis of the full CI wavefunction of the He(2) dimer, from which the dispersion part of the two-particle density matrix is obtained. It emerges that the entirely different physics embodied in the dispersion interaction leads to an essentially different type of exchange-correlation orbital functional for the dispersion energy (non-JK). The distinct NO functionals for the different types of correlation imply that they can be used in conjunction without problems of double counting. Requirements on the (primitive) basis set for Van der Waals bonding appear to be more modest than for other types of correlation.

By examining the natural orbital occupation numbers, it was discovered that unrestricted Møller–Plesset perturbation theory (MP2) can violate N-representability by having natural occupation numbers greater than 2 or less than 0, even though the energy appears well behaved. Analytically, this problem stems from the fact that the MP2 effective one-particle density matrix is dependent on the inverse of the orbital hessian matrix (A′). When a molecular system goes through the point where spin-restricted orbitals become unstable to unrestriction, one of the eigenvalues of the A′ matrix will become 0, making it a singular matrix, thereby causing problems in the relaxed one-particle density matrix and discontinuities in the first order properties such as nuclear forces and dipole moments. Since the new ‘double-hybrid’ density functionals also involve a second-order perturbation expression for correlation using Kohn–Sham orbitals, these functionals also exhibit similar issues near the unrestriction point. The unphysical orbital occupations can be eliminated and the first derivative discontinuities removed by optimizing orbitals with inclusion of the second order correlation energy. Connections to the second derivative discontinuities in Hartree–Fock and Kohn–Sham density functional theory are also discussed.

The natural orbital functional theory provides two complementary representations of the one-electron picture in molecules, namely, the natural orbital (NO) representation and the canonical orbital (CO) representation. The former arises directly from the optimization process solving the corresponding Euler equations, whereas the latter is attained from the diagonalization of the matrix of Lagrange multipliers obtained in the NO representation. In general, the one-particle reduced-density matrix (1-RDM) and the Lagrangian cannot be simultaneously brought to the diagonal form, except for the special Hartree-Fock case. The 1-RDM is diagonal in the NO representation, but not the Lagrangian, which is only a Hermitian matrix. Conversely, in the CO representation, the Lagrangian is diagonal, but not the 1-RDM. Combining both representations we have the whole picture concerning the occupation numbers and the orbital energies. The Piris natural orbital functional 5 leads generally to the localization of the molecular orbitals in the NO representation. Accordingly, it provides an orbital picture that agrees closely with the empirical valence shell electron pair repulsion theory and the Bent’s rule, along with the theoretical valence bond method. On the other hand, the equivalent CO representation can afford delocalized molecular orbitals adapted to the symmetry of the molecule. We show by means of the extended Koopmans’ theorem that the one-particle energies associated with the COs can yield reasonable principal ionization potentials when the 1-RDM remains close to the diagonal form. The relationship between NOs and COs is illustrated by several examples, showing that both orbital representations complement each other.

ABSTRACTThe conformational analysis was carried out to predict the most stable, optimized structure of 2-hydroxy-3,5-dinitropyridine among various conformers. The most stable, optimized structure of the molecule was predicted by the density functional theory (DFT) using the B3LYP method with the cc-pVQZ basis set. The vibrational frequencies, natural atomic charge distribution, and thermodynamic properties were calculated. The molecular electrostatic potential surface and contour map were simulated. The experimental and theoretical vibrational frequencies were assigned on the basis of potential energy distribution calculation. The density of states (DOS) spectrum was simulated. The frontier molecular orbital analysis was carried out. The natural bond orbital (NBO) analysis was performed to evaluate the donor–acceptor interactions in the molecule. The structure–activity descriptors were determined by Fukui functions and local reactivity descriptor calculations. The nonlinear optical (NLO) activity of the molecule was studied. The second harmonic generation test confirms the NLO activity of the title molecule.

The Pauli exclusion principle gives an upper bound of 1 on the natural\noccupation numbers. Recently there has been an intriguing amount of theoretical\nevidence that there is a plethora of additional generalized Pauli restrictions\nor (in)equalities, of kinematic nature, satisfied by these numbers. Here for\nthe first time a numerical analysis of the nature of such constraints is\neffected in real atoms. The inequalities are nearly saturated, or quasi-pinned.\nFor rank-six and rank-seven approximations for lithium, the deviation from\nsaturation is smaller than the lowest occupancy number. For a rank-eight\napproximation we find well-defined families of saturation conditions.\n

Benzocyclobutenes (BCBs) are valuable precursors for the synthesis of bioactive molecules such as steroids, alkaloids, and terpenoids. However, both the ring-opening temperature (ROT) of BCBs and the precise electronic nature of the resulting reactive intermediate, ortho-quinodimethane (o-QDM) with varying degrees of diradical character, is crucial for controlling reaction outcomes. In this study, we prepared and investigated the ring-opening behavior of a series of 1-substituted BCBs bearing ether, amine, amide, or carbamate groups. ROT values, determined by differential scanning calorimetry (DSC), were found to be significantly influenced by the electronic nature of the substituent. The natural bond orbital (NBO) method was employed to assess the orbital contribution of the substituent lone pair, confirming its strong impact on ROT. Thermal activation of these compounds in the presence of either a dienophile or a spin trap (TEMPO) led to the formation of the corresponding Diels-Alder adducts or dialkoxyamines, respectively, consistent with the involvement of a reactive o-QDM intermediate exhibiting partial diradical character. Furthermore, the diradical character of the ring-opened species was estimated to be 15%-19% using the Natural Orbital Occupation Number (NOON) computational method. The combination of experimental and computational data provides significant insight into the electronic structure governing the ring-opening process.

We study the performance of different approaches that combine multiconfigurational wavefunctions with correlation functionals for the calculation of magnetic coupling constants of several materials and molecules. The systems under study include four antiferromagnetic materials: NiO, KNiF(3), K(2)NiF(4) and La(2)CuO(4); two biradicals: alpha-4-Dehydrotoluene and 1,1',5,5'-Tetramethyl-6,6'-dioxo-3,3'-biverdazyl; two molecular complexes: [Cu(2)Cl(6)](-2) and Copper(II) acetate monohidrate; and the prototypical H-He-H system. On average, the best results are obtained with a recently proposed method [Phys. Rev. A 75, 012503 (2007)] that estimates the correlation energy of density functionals from a pair of alternative spin densities built from the natural orbitals and occupation numbers of the multiconfigurational wavefunction.