Natural Orbital Occupation Research Articles

Effective Josephson dynamics in resonantly driven Bose–Einstein condensates

We show that the orbital Josephson effect appears in a wide range of driven atomic Bose–Einstein condensed systems, including quantum ratchets, double wells and box potentials. We use three separate numerical methods: the Gross–Pitaevskii equation, exact diagonalization of the few-mode problem and the multi-configurational time-dependent Hartree for bosons algorithm. We establish the limits of mean-field and few-mode descriptions, demonstrating that the few-mode approximation represents the full many-body dynamics to high accuracy in the weak driving limit. Among other quantum measures, we compute the instantaneous particle current and the occupation of natural orbitals. We explore four separate dynamical regimes, the Rabi limit, chaos, the critical point and self-trapping; a favorable comparison is found even in the regimes of dynamical instabilities or macroscopic quantum self-trapping. Finally, we present an extension of the (t,t′)-formalism to general time-periodic equations of motion, which permits a systematic description of the long-time dynamics of resonantly driven many-body systems, including those relevant to the orbital Josephson effect.

Natural occupation numbers: When do they vanish?

The non-vanishing of the natural orbital (NO) occupation numbers of the one-particle density matrix of many-body systems has important consequences for the existence of a density matrix-potential mapping for nonlocal potentials in reduced density matrix functional theory and for the validity of the extended Koopmans' theorem. On the basis of Weyl's theorem we give a connection between the differentiability properties of the ground state wavefunction and the rate at which the natural occupations approach zero when ordered as a descending series. We show, in particular, that the presence of a Coulomb cusp in the wavefunction leads, in general, to a power law decay of the natural occupations, whereas infinitely differentiable wavefunctions typically have natural occupations that decay exponentially. We analyze for a number of explicit examples of two-particle systems that in case the wavefunction is non-analytic at its spatial diagonal (for instance, due to the presence of a Coulomb cusp) the natural orbital occupations are non-vanishing. We further derive a more general criterium for the non-vanishing of NO occupations for two-particle wavefunctions with a certain separability structure. On the basis of this criterium we show that for a two-particle system of harmonically confined electrons with a Coulombic interaction (the so-called Hookium) the natural orbital occupations never vanish.

A natural orbital analysis of the long range behavior of chemical bonding and van der Waals interaction in singlet H2: The issue of zero natural orbital occupation numbers

This paper gives a natural orbital (NO) based analysis of the van der Waals interaction in (singlet) H2 at long distance. The van der Waals interaction, even if not leading to a distinct van der Waals well, affects the shape of the interaction potential in the van der Waals distance range of 5-9 bohrs and can be clearly distinguished from chemical bonding effects. In the NO basis the van der Waals interaction can be quantitatively covered with, apart from the ground state configurations (1σ(g))(2) and (1σ(u))(2), just the 4 configurations (2σ(g))(2) and (2σ(u))(2), and (1π(u))(2) and (1π(g))(2). The physics of the dispersion interaction requires and explains the peculiar relatively large positive CI coefficients of the doubly excited electron configurations (2σ(u))(2) and (1π(g))(2) (the occupancy amplitudes of the 2σ(u) and 1π(gx, y) NOs) in the distance range 5-9 bohrs, which have been observed before by Cioslowski and Pernal [Chem. Phys. Lett. 430, 188 (2006)]. We show that such positive occupancy amplitudes do not necessarily lead to the existence of zero occupation numbers at some H-H distances.

A theoretical study of the adsorption behavior of N2O on single-walled AlN and AlP nanotubes

We have performed first-principles computations to investigate the adsorption properties of the N2O on the outer surfaces of H-capped single-walled AlN and AlP nanotubes (SWAlNNTs and SWAlPNTs). Binding energy corresponding to the most stable configuration (O-side) of N2O on the AlNNTs is found to be −25.37kJmol−1, which is not typical for the chemisorption process. For the N2O/AlNNTs complexes, the energy gaps, dipole moments, natural atomic orbital occupancies and global indices are calculated. The computed density of states (DOSs) reveals that there is a significant orbital hybridization between two species in adsorption process being an evidence of strong interaction. Finally, we clarify that AlNNTs plays an important role as a suitable sensor.

A computational approach to the free radical polymerization kinetics of alkyl α‐hydroxymethacrylate monomers: A structure–reactivity relationship

ABSTRACTDensity functional theory calculations at the B3LYP/6–31+G(d) level were carried out for 16 alkyl α‐hydroxymethacrylate (RHMA) monomers, and the calculated quantum chemical descriptors were used to construct a quantitative structure–activity relationship model of the reactivity parameters of those monomers. A multiple regression analysis was adopted to predict the polymerization kinetics. The molecular descriptors used in this study are the Mulliken and APT charges of the backbone C atoms in proximity of the reactive center, the total dipole moment (µ), the energy of the highest occupied molecular orbital (EHOMO), the energy of the lowest unoccupied molecular orbital (ELUMO), the isotropic polarizability (α), the Mulliken atomic spin density (ρs), the ionization energy (IE), the electron affinity (EA), the resonance parameter (Res), the natural atomic orbital occupancies, and the bond distances of the bonds in the proximity to the reactive center of the RHMA monomers and their corresponding radicals. Among the various descriptors considered, the IE and EA of the monomers, the bond order of carbonyl double bond of monomer (BOCO), the resonance stabilization parameter of the monomeric radical, and the Mulliken atomic spin density (ρs) of C5 of the radical were found to be correlated (R2 = 0.9416) with the experimental rates of photopolymerization. The correlation with experiment is also satisfactory when additional monomers are included in the data set; a new correlation is thus generated (R2 = 0.9186). This study is expected to shed light on predicting the reactivities of RHMA monomers prior to their synthesis and to rank their photopolymerization rates based on their quantitative structural properties. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 2375–2384

Relative Energies and Geometries of the cis- and trans-HO3 Radicals from the Parametric 2-Electron Density Matrix Method

The parametric 2-electron reduced density matrix (2-RDM) method employing the M functional [Mazziotti, D. A. Phys. Rev. Lett. 2008, 101, 253002], also known as the 2-RDM(M) method, improves on the accuracy of coupled electron-pair theories including coupled cluster with single-double excitations at the computational cost of configuration interaction with single-double excitations. The cis- and trans-HO(3) isomers along with their isomerization transition state were examined using the recent extension of 2-RDM(M) to nonsinglet open-shell states [Schwerdtfeger, C. A.; Mazziotti, D. A. J. Chem. Phys. 2012, 137, 034107] and several coupled cluster methods. We report the calculated energies, geometries, natural-orbital occupation numbers, and reaction barriers for the HO(3) isomers. We find that the 2-RDM(M) method predicts that the trans isomer of HO(3) is lower in energy than the cis isomer by 1.71 kcal/mol in the correlation-consistent polarized valence quadruple-ζ (cc-pVQZ) basis set and 1.84 kcal/mol in the augmented correlation-consistent polarized valence quadruple-ζ (aug-cc-pVQZ) basis set. Results include the harmonic zero-point vibrational energies calculated in the correlation-consistent polarized valence double-ζ basis set. On the basis of the results of a geometry optimization in the augmented correlation consistent polarized valence triple-ζ basis set, the parametric 2-RDM(M) method predicts a central oxygen-oxygen bond of 1.6187 Å. We compare these energies and geometries to those predicted by three single-reference coupled cluster methods and experimental results and find that the inclusion of multireference correlation is important to describe properly the relative energies of the cis- and trans-HO(3) isomers and improve agreement with experimental geometries.

A first-principles study of the adsorption behavior of CO on Al- and Ga-doped single-walled BN nanotubes

First-principles computations have been applied to scrutinize the adsorption behavior of CO molecule on the external surface of H-capped aluminum- and gallium-doped (6, 0), (8, 0) zigzag and (5, 5) armchair single-walled BN nanotubes (SWBNNTs). Binding energy corresponding to the most stable configuration of CO on the gallium-doped (6, 0) BNNT is found to be −0.83eV, which is high typical sensitivity to CO molecule. Our results indicate that both Al- and Ga-doping can notably enhance the adsorption energy of CO/BNNTs complexes. Our electronic results reveal that there is a notable orbital hybridization among two species in adsorption process being an evidence of strong interaction. For the CO/BNNTs complexes, the energy gaps, NBO, dipole moments, natural atomic orbital occupancies and global indices are computed. Finally, we reported a novel type of toxic gas sensor that can be used for detecting the presence of CO molecule.

Accurate Noncovalent Interaction Energies Using Truncated Basis Sets Based on Frozen Natural Orbitals.

We assess the accuracy of basis set truncations based on natural orbitals determined by second-order perturbation theory for computing noncovalent interaction energies with coupled cluster through perturbative triples [CCSD(T)]. We consider two methods for truncation: (i) the usual frozen natural orbital approach (FNO) in which the basis set truncation occurs before the iterative CCSD computation [FNO CCSD(T)] and (ii) an approach in which the truncation occurs only for the perturbative triples contribution [CCSD+FNO(T)]. The errors incurred are comparable for both methods and are small enough for the methods to be used for benchmark-quality studies of noncovalent interactions. For the FNO CCSD(T) method with a modest natural orbital occupation tolerance of 10(-5), the mean absolute error in the interaction energies for the S22 data set in an aug-cc-pVDZ basis set is only 0.012 kcal mol(-1) versus canonical CCSD(T) values. The same method exhibits a mean absolute error of 0.020 kcal mol(-1) for the S11 data set in the aug-cc-pVTZ basis set versus canonical CCSD(T) values.

On the formulation of a density matrix functional for Van der Waals interaction of like- and opposite-spin electrons in the helium dimer

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.

Quenches in a quasidisordered integrable lattice system: Dynamics and statistical description of observables after relaxation

We study the dynamics and the resulting state after relaxation in a quasidisordered integrable lattice system after a sudden quench. Specifically, we consider hard-core bosons in an isolated one-dimensional geometry in the presence of a quasiperiodic potential whose strength is abruptly changed to take the system out of equilibrium. In the delocalized regime, we find that the relaxation dynamics of one-body observables, such as the density, the momentum distribution function, and the occupation of the natural orbitals, follow, to a good approximation, power laws. In that regime, we also show that the observables after relaxation can be described by the generalized Gibbs ensemble, while such a description fails for the momentum distribution and the natural orbital occupations in the presence of localization. At the critical point, the relaxation dynamics is found to be slower than in the delocalized phase.

Parametric two-electron reduced-density-matrix method with application to diradical rectangular H4

Parameterization of the two-electron reduced density matrix (2-RDM) has made possible the determination of electronic energies with greater accuracy and reduced computational cost compared to traditional electron-pair theories, including coupled cluster with single and double excitations [D.A. Mazziotti, Phys. Rev. Lett. 101 (2008) 253002]. We apply the method to an H4 model system, a rectangular arrangement of two H2 monomers (P4), which is often used for benchmark calculations of multireference methods. At the square geometry, H4 becomes a diradical. We find that the parametric 2-RDM method obtains occupation numbers of 0.5471 and 0.4489 for the Nth and (N+1)th natural orbitals, respectively, which indicate diradical character. Energies and orbital occupation numbers obtained from the parametric 2-RDM method are found to be more accurate than single-reference wavefunction methods of comparable computational cost. We report energies and natural orbital occupation numbers for several geometries in the rectangular H4 system.

Avoided crossings, conical intersections, and low-lying excited states with a single reference method: The restricted active space spin-flip configuration interaction approach

The restricted active space spin-flip CI (RASCI-SF) performance is tested in the electronic structure computation of the ground and the lowest electronically excited states in the presence of near-degeneracies. The feasibility of the method is demonstrated by analyzing the avoided crossing between the ionic and neutral singlet states of LiF along the molecular dissociation. The two potential energy surfaces (PESs) are explored by means of the energies of computed adiabatic and approximated diabatic states, dipole moments, and natural orbital electronic occupancies of both states. The RASCI-SF methodology is also used to study the ground and first excited singlet surface crossing involved in the double bond isomerization of ethylene, as a model case. The two-dimensional PESs of the ground (S(0)) and excited (S(1)) states are calculated for the complete configuration space of torsion and pyramidalization molecular distortions. The parameters that define the state energetics in the vicinity of the S(0)/S(1) conical intersection region are compared to complete active space self-consistent field (CASSCF) results. These examples show that it is possible to describe strongly correlated electronic states using a single reference methodology without the need to expand the wavefunction to high levels of collective excitations. Finally, RASCI is also examined in the electronic structure characterization of the ground and 2(1)A(g)(-), 1(1)B(u)(+), 1(1)B(u)(-), and 1(3)B(u)(-) states of all-trans polyenes with two to seven double bonds and beyond. Transition energies are compared to configuration interaction singles, time-dependent density functional theory (TDDFT), CASSCF, and its second-order perturbation correction calculations, and to experimental data. The capability of RASCI-SF to describe the nature and properties of each electronic state is discussed in detail. This example is also used to expose the properties of different truncations of the RASCI wavefunction and to show the possibility to use an excitation operator with any number of α-to-β electronic promotions.

Theoretical investigation of OCN− adsorption onto boron nitride nanotubes

First-principles calculations based on density functional theory (DFT) method are used to investigate the adsorption properties of OCN− on H-capped zigzag and armchair single-walled BN nanotubes (BNNTs). The results indicate that OCN− is strongly bound to the outer surface of zigzag (6,0) BNNTs in comparison with armchair (5,5) BNNT. Binding energy and equilibrium distance corresponding to the most stable configuration are found to be −486.79kJmol−1 and 1.526Å, respectively being typical for the chemisorptions. Energy gap, dipole moment, natural atomic orbital occupancies and global indices for most stable configuration are calculated. Furthermore, the effect of the OCN− adsorption on the geometries and electronic properties of related BNNT is also studied. The calculated density of states (DOS) reveals that there is a significant orbital hybridization between two species in adsorption process being an evidence of strong interaction. Therefore, one can conclude that BNNTs play an important role as suitable sensor.

Performance of PNOF5 Natural Orbital Functional for Radical Formation Reactions: Hydrogen Atom Abstraction and C-C and O-O Homolytic Bond Cleavage in Selected Molecules.

Radical formation through hydrogen abstraction and C-C and O-O homolytic bond cleavage from selected molecules is investigated by use of natural orbital functional theory in its PNOF5 natural orbital functional implementation, and the results are compared to high-level ab initio complete active space self-consistent field (CASSCF) and complete active space with second-order perturbation theory (CASPT2) methods and experimental data. It is observed that PNOF5 is able to treat the strong electron correlation effects along the homolysis of X-H (X = C, N, O) and X-X (X = C, O) bonds, leading, in general, to the correct trends in the corresponding bond strengths and a good description of the resultant electronic structure for these radicals. In general, PNOF5 bond energies are lower than the experimental ones, because of partial lack of dynamical electron correlation. However, the part of dynamical electron correlation recovered by PNOF5 allows it to give more accurate results than CASSCF methods with a minimum window required to treat near-degeneracy effects. In addition, inspection of the natural orbital occupancies with respect to the CASSCF ones shows an outstanding performance of PNOF5 in treating degenerate and quasidegenerate states, giving a correct description of diradicals and diradicaloids formed upon C-C cleavage in cyclopropane and derivatives.

Three strongly correlated charged bosons in a one-dimensional harmonic trap: natural orbital occupancies

We study a one-dimensional system composed of three charged bosons confined in an external harmonic potential. More precisely, we investigate the ground-state correlation properties of the system, paying particular attention to the strong-interaction limit. We explain for the first time the nature of the degeneracies appearing in this limit in the spectrum of the reduced density matrix. An explicit representation of the asymptotic natural orbitals and their occupancies is given in terms of some integral equations.

Exploration of the π-Electronic Structure of Singlet, Triplet, and Quintet States of Fulvenes and Fulvalenes Using the Electron Localization Function

The singlet ground states and lowest triplet states of penta- and heptafulvene, their benzannulated derivatives, as well as the lowest quintet states of pentaheptafulvalenes, either the parent compound or compounds in which the two rings are intercepted by either an alkynyl or a phenyl segment, were investigated at the (U)OLYP/6-311G(d,p) density functional theory level. The influence of (anti)aromaticity was analyzed by the structure-based aromaticity index HOMA, the harmonic oscillator model of aromaticity. The extent of (anti)aromatic character was also evaluated in terms of the π-electron (de)localization as measured by the π component of the electron localization function (ELF(π)). The natural atomic orbital (NAO) occupancies were calculated in order to evaluate the degree of π-electron shift caused by the opposing electron-counting rules for aromaticity in the electronic ground state (S(0); Hückel's rule) and the first ππ* excited triplet state (T(1); Baird's rule). Pentaheptafulvalene (5) shows a shift of 0.5 π electrons from the 5-ring to the 7-ring when going from the S(0) state to the lowest quintet state (Qu(1)). The pentaheptafulvalene 5 and [5.6.7]quinarene 7 were also investigated in their 90° twisted conformations. From our study it is apparent that excitation localization in fulvalenes, but not in fulvenes, to a substantial degree is determined by aromaticity localization to triplet biradical 4n π-electron cycles. Isolated benzene rings in these compounds tend to remain as closed-shell 6π-electron cycles.

Isoelectronic analogue of oxywater: a parametric two-electron reduced-density-matrix study of ammonia oxide

A parametrization of the two-electron reduced density matrix (2-RDM) provides energies that improve on the accuracy of coupled electron-pair theories including coupled cluster with single-double excitations at the computational cost of configuration interaction with single-double excitations [Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. This parametric 2-RDM method was recently employed to study the isomerization of oxywater to hydrogen peroxide where it predicted a lower energy barrier from oxywater (2.1 kcal mol−1) than coupled cluster methods (4.2 kcal mol−1). In this paper we study an isoelectronic analogue, the isomerization of ammonia oxide to hydroxylamine. In the extrapolated basis-set limit, using the augmented correlation-consistent polarized valance quadruple-zeta (aug-cc-pVQZ) basis set, the parametric 2-RDM method predicts a 27.5 kcal mol−1 barrier from ammonia oxide to hydroxylamine. We report reaction energies, barriers, geometries, and natural-orbital occupation numbers for the ammonia-oxide reaction and compare them to those from the oxywater reaction. We find that the parametric 2-RDM method agrees with dynamic correlation wavefunction methods when the multi-reference character of the system is small as in the ammonia-oxide isomerization computed here but that it captures additional multi-reference correlation, usually requiring a multi-reference method, when such correlation increases as in the oxywater isomerization.

Strongly correlated barriers to rotation from parametric two-electron reduced-density-matrix methods in application to the isomerization of diazene

Recently, parameterization of the two-electron reduced density matrix (2-RDM) has made possible the determination of electronic energies with greater accuracy and lower cost than traditional electron-pair theories including coupled cluster with single and double excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. We examine the method's performance for strongly correlated barriers to rotation; in particular, we study two distinct pathways in the isomerization of diazene (N(2)H(2)) from cis to trans: (i) a strongly correlated rotational pathway and (ii) a moderately correlated inversion pathway. While single reference wavefunction methods predict that the rotational barrier is higher than the inversional barrier, the parametric 2-RDM method predicts that the rotational barrier is lower than the inversional barrier by 3.1 kcal/mol in the extrapolated basis set limit. The parametric 2-RDM results are in agreement with those from multireference methods including multireference perturbation theory and the solution to the anti-Hermitian contracted Schrödinger equation. We report energies, optimized structures, and natural orbital occupation numbers for three diazene minima and two transition states.

Strong Correlation in Acene Sheets from the Active-Space Variational Two-Electron Reduced Density Matrix Method: Effects of Symmetry and Size

Polyaromatic hydrocarbons (PAHs) are a class of organic molecules with importance in several branches of science, including medicine, combustion chemistry, and materials science. The delocalized π-orbital systems in PAHs require highly accurate electronic structure methods to capture strong electron correlation. Treating correlation in PAHs has been challenging because (i) traditional wave function methods for strong correlation have not been applicable since they scale exponentially in the number of strongly correlated orbitals, and (ii) alternative methods such as the density-matrix renormalization group and variational two-electron reduced density matrix (2-RDM) methods have not been applied beyond linear acene chains. In this paper we extend the earlier results from active-space variational 2-RDM theory [Gidofalvi, G.; Mazziotti, D. A. J. Chem. Phys. 2008, 129, 134108] to the more general two-dimensional arrangement of rings--acene sheets--to study the relationship between geometry and electron correlation in PAHs. The acene-sheet calculations, if performed with conventional wave function methods, would require wave function expansions with as many as 1.5 × 10(17) configuration state functions. To measure electron correlation, we employ several RDM-based metrics: (i) natural-orbital occupation numbers, (ii) the 1-RDM von Neumann entropy, (iii) the correlation energy per carbon atom, and (iv) the squared Frobenius norm of the cumulant 2-RDM. The results confirm a trend of increasing polyradical character with increasing molecular size previously observed in linear PAHs and reveal a corresponding trend in two-dimensional (arch-shaped) PAHs. Furthermore, in PAHs of similar size they show significant variations in correlation with geometry. PAHs with the strictly linear geometry (chains) exhibit more electron correlation than PAHs with nonlinear geometries (sheets).

Diradicals and Diradicaloids in Natural Orbital Functional Theory

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.