Natural occupation numbers in two-electron quantum rings.

We have developed a method for analyzing the (hyper)polarizabilities of open-shell molecular systems. This method employs the (hyper)polarizability densities based on the natural orbitals and occupation numbers, which enables us to analyze the contributions of odd electrons having various open-shell (diradical) characters. Within broken-symmetry, i.e., spin-unrestricted, single-determinant molecular orbital and density functional theory approaches, we can also remove the spin contamination effects on these quantities through spin projection. To do that, an approximate spin projected method has been elaborated and applied to the analysis of the (hyper)polarizability of multi-radical systems. As examples, typical open-shell singlet systems, 1,3-dipoles and rectangular graphene nanoflakes, are examined.

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.

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.

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.

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.

The one-electron description of excited states: Natural excitation orbitals of density matrix theory and Kohn–Sham orbitals of density functional theory as ideal orbitals

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.

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

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.

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.

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.

An overview of recent studies, which show the important role played by computational chemistry in explaining and predicting the behavior of a great variety of systems is presented. The first example illustrates the useful interplay experiment-theory which allowed to unambiguously characterize the X2(CH3)7 + (X = Si, Ge, Sn, Pb) cations as paradigmatic examples of molecular systems involving pentacovalently bound carbon atoms. Also, the possibility of deeply analyzing the bonding in molecular systems, led to a rationalization of the quite different behavior of acetylene and iminoborane derivatives, in spite of being isoelectronic species. Similarly, the analysis of the bonding perturbations usually associated with ion-molecule interactions in the gas phase, or with protonation and deprotonation processes, has made possible the characterization of non-conventional complexes as the key structures in the interactions of unsaturated and aromatic Si and Ge containing compounds with transition metal ions or to explain the dramatic enhanced acidity of alkylboranes and α,β- unsaturated borane derivatives. Keywords: Molecular orbital theory, density functional theory, bonding, gas-phase ion chemistry, non-conventional structures, intrinsic reactivity, Computational Chemistry, Gas-phase basicity or acidity scales, Uracil, Density functional theory (DFT), MP2 or DFT calculations, Atoms in molecules (AIM), The electron localization function (ELF) theory, Natural bond orbital (NBO) method, Bond critical points (BCP), Becke and Edgecombe electron localization function, Pentacovalent Carbon Atoms, Methylborane, Dysinaptic V(C,C) basinHole effect

Density functional theory (DFT) (B3LYP and B3PW91) calculations have been carried out for 2,6-dimethyl-2,5-heptadien-4-one (DMHD4O) using 6–311++ G** basis set. Complete vibrational assignment and analysis of the fundamental modes of the compound were carried out from the FTIR and FT-Raman spectral data. The theoretical electronic absorption has been calculated by using time-dependent DFT (TD-DFT) methods and compared with the experimental spectra. The theoretically computed Frontier energy gaps and TD-DFT calculations are in good agreement with the experimental UV–vis spectral absorption. The chemical hardness measured from the Frontier molecular orbital energies of DMHD4O is 0.0693 eV. Electronic stability of the compound arising from hyperconjugative interactions and charge delocalisation were also investigated based on the natural bond orbital (NBO) analysis. Effective stabilisation energy E (2) associated with the interactions of the π and the lone pair of electrons was determined by the NBO analysis. 13C and 1H NMR chemical shifts of the compound have been calculated by means of Gauge-Invariant Atomic Orbital using B3LYP/6–311++ G** method. The partial ionic character of the carbonyl group due to resonance render a partially positive charge to the carbonyl carbon, and thus C4 chemical shift lie in the very downfield 191.6 ppm. Comparison between the experimental and the theoretical results indicates that B3LYP method is able to provide satisfactory results for predicting vibrational, electronic and NMR properties.

Density functional theory (DFT) and natural bond orbital (NBO) study of vibrational spectra and intramolecular hydrogen bond interaction of l-ornithine–l-aspartate

Investigation of structure, vibrational, electronic, NBO and NMR analyses of 2-chloro-4-nitropyridine (CNP), 2-chloro-4-methyl-5-nitropyridine (CMNP) and 3-amino-2-chloro-4-methylpyridine (ACMP) by experimental and theoretical approach