FCIQMC-Tailored Distinguishable Cluster Approach: Open-Shell Systems.

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.

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.

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

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.

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.

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 CASPT2+δMRCI composite approach reported in a companion paper has been extended and used to provide high-quality reference data for a series of adiabatic spin gaps (defined as ΔE = Equintet - Esinglet) of [FeIIL6]2+ complexes (L = CNH, CO, NCH, NH3, H2O), either at nonrelativistic level or including scalar relativistic effects. These highly accurate data have been used to evaluate the performance of various more approximate methods. Coupled-cluster theory with singles, doubles, and perturbative triples, CCSD(T), is found to agree well with the new reference data for Werner-type complexes but exhibits larger underestimates by up to 70 kJ/mol for the π-acceptor ligands, due to appreciable static correlation in the low-spin states of these systems. Widely used domain-based local CCSD(T) calculations, DLPNO-CCSD(T), are shown to depend very sensitively on the cutoff values used to construct the localized domains, and standard values are not sufficient. A large number of density functional approximations have been evaluated against the new reference data. The B2PLYP double hybrid gives the smallest deviations, but several functionals from different rungs of the usual ladder hierarchy give mean absolute deviations below 20 kJ/mol. This includes the B97-D semilocal functional, the PBE0* global hybrid with 15% exact-exchange admixture, as well as the local hybrids LH07s-SVWN and LH07t-SVWN. Several further functionals achieve mean absolute errors below 30 kJ/mol (M06L-D4, SSB-D, B97-1-D4, LC-ωPBE-D4, LH12ct-SsirPW92-D4, LH12ct-SsifPW92-D4, LH14t-calPBE-D4, LHJ-HFcal-D4, and several further double hybrids) and thereby also still overall outperform CCSD(T) or uncorrected CASPT2. While exact-exchange admixture is a crucial factor in favoring high-spin states, the present evaluations confirm that other aspects can be important as well. A number of the better-performing functionals underestimate the spin gaps for the π-acceptor ligands but overestimate them for L = NH3, H2O. In contrast to a previous suggestion, non-self-consistent density functional theory (DFT) computations on top of Hartree-Fock orbitals are not a promising path to produce accurate spin gaps in such complexes.

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.

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.

Theoretical structural and vibrational properties of the artificial sweetener sucralose

ABSTRACTIn this work, we will express a comprehensive theoretical study on C32H12 as a new bowl-shaped structure and a sub-fullerene structure. The specific structural properties of title compound were evaluated by density functional theory (DFT) and time-dependent density functional theory (TD-DFT) computations. A wide study of vibrational spectral analysis has been done and assignments of the fundamental modes have been proposed on the basis of peak positions. Relative frequencies, natural bond orbital (NBO) charges, HOMO–LUMO energies and several thermodynamic properties in the ground state were calculated using B3LYP/6–311++G(d,p) level of theory. The C32H12 was subjected to the analysis of simulated properties include geometries, charges, 13C NMR, 1H NMR and UV/visible spectra. Results showed that most of the modes have wavenumbers in the expected range. Global and local reactivity descriptors of the C32H12 have been computed. The theoretical IR and Raman spectra have also been constructed. NBO study explains charge delocalization of the title molecule.

A well-parallelized local singles and doubles coupled-cluster (LCCSD) method using pair natural virtual orbitals (PNOs) is presented. The PNOs are constructed using large domains of projected atomic orbitals (PAOs) and orbital specific virtual orbitals (OSVs). We introduce a hierarchy of close, weak, and distant pairs, based on pair energies evaluated by local Møller-Plesset perturbation theory (LMP2). In contrast to most previous implementations of LCCSD methods, the close and weak pairs are not approximated by LMP2 but treated by higher-order methods. This leads to greatly improved accuracy for large systems, in particular when long-range dispersion interactions are important. Close pair amplitudes are optimized using approximate LCCSD equations, in which slowly decaying terms that mutually cancel at long-range are neglected. For weak pairs, the same approximations are used, but in addition, the nonlinear terms are neglected (coupled electron pair approximation). Distant pairs are treated by spin-component scaled (SCS)-LMP2 using multipole approximations. For efficiency reasons, various projection approximations are also introduced. The impact of these approximations on absolute and relative energies depends on the PNO domain sizes. The errors are found to be negligible, provided that sufficiently large PNO domains are used for close and weak pairs. For the selection of these domains the usual natural orbital occupation number criterion is found to be insufficient, and an additional energy criterion is used. For extended one-dimensional systems, the computational effort of the method scales nearly linearly with the number of correlated electrons, but the linear scaling regime is usually not reached in real-life applications for three-dimensional systems. Nevertheless, due to the parallelization that is efficient up to about 100-200 CPU cores (dependent on the molecular size), accurate calculations for three-dimensional molecules with about 100 atoms and augmented triple-ζ basis sets (e.g., cc-pVTZ-F12) can be carried out in 1-3 h of elapsed time (depending on the molecular structure and the number of CPU cores, excluding the time for Hartree-Fock). The convergence of the results with respect to the thresholds and options that control the domain, pair, and projection approximations is extensively tested. Benchmark examples for several difficult and large cases are presented, which demonstrate that the errors of relative energies (e.g., reaction energies, barrier heights) caused by the pair and projection approximations can be reduced to below 1 kJ mol-1. The remaining errors are mainly caused by the finite PAO domains. The larger these are made, the more intramolecular or intermolecular basis set superposition errors are introduced, and the canonical results are approached only very slowly. This problem is eliminated in the explicitly correlated variant of our method (PNO-LCCSD-F12), which will be described in a separate paper, along with a larger set of benchmark calculations.

For translationally invariant one-band lattice models, we exploit the abinitio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered. First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P_{N}^{1} and E_{N}^{1} of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope E_{N}^{1}≡P_{N}^{1}, described by linear constraints D^{(j)}(n)≥0. For smaller systems, it follows as F[n]=[under ∑]i,i^{'}V[over ¯]_{i,i^{'}}sqrt[D^{(i)}(n)D^{(i^{'})}(n)]. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a nonanalytical term involving the interaction V[over ^]. Third, the gradient dF/dn is shown to diverge on the boundary ∂E_{N}^{1}, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an "exchange force." All findings hold for systems with a nonfixed particle number as well and V[over ^] can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.

Fruits, vegetables, oils, coffee, nuts, and tea contain coumarin. The scientific attraction of coumarins is attributed to their pharmacological activity. In this study, density functional theory (DFT) calculations were performed in the B3LYP/6-31G(d) level of theory to evaluate the potentialities of single-walled carbon nanotubes (SWCNT) as a carrier for 3-acetyl-6-methyl-4-hydroxy-coumarin. The natural bond orbital (NBO) analysis suggested that the functionalized nanotube plays the role of an electron donor and 3-acetyl-6-methyl -4-hydroxy- coumarin molecule acts as an electron acceptor at the SWCNT -coumarin complex. Variations of density and dipole moment values of coumarin after interaction were studied. Also, the frontier orbital energies, HOMO-LUMO gap, chemical hardness (η), electrophilicity index (ω), charge transferred (ΔNmax) and chemical potential (μ) were computed. Pure coumarin is less conductive and smoother than SWCNT -coumarin. Our results show efficient coumarin loading with the interaction energy of -1.19 eV. NBO analysis shows that the occupied orbitals in the coumarin and the unoccupied orbitals in the nanotube overlap.