Natural Orbital Occupation Numbers Research Articles

Self-Consistent-Field Method for Correlated Many-Electron Systems with an Entropic Cumulant Energy

A self-consistent field method is presented within density matrix functional theory. The computational cost for a correlated many-electron calculation is reduced to that of the self-consistent-field Hartree-Fock method, while the accuracy still reaches that of sophisticated configuration interaction based methods. In this method, the two-electron cumulant energy is measured with an information entropy associated with the Fermi-Dirac distribution of the occupation numbers. An eigenvalue equation for the orbitals is obtained, with the eigenvalues (orbital energies) connected to the occupation numbers through the Fermi-Dirac distribution. The occupation numbers for the strongly occupied orbitals are very close to the natural orbital occupation numbers from wave function methods. It covers in a single scheme the nondynamical correlation in weak or breaking bonds as well as the dynamical correlation at all distances. The method is well suited to large-scale potential energy surface calculation and molecular dynamics simulation.

The thermal isomerization of benzvalyne to benzyne.

The isomerization of the highly strained benzvalyne structure to o-benzyne has been investigated using MCSCF and CCSD(T) levels of theory. Two reaction channels were modeled: the disrotatory one which leads directly to the benzyne product, and the conrotatory one which leads to an intermediate which can subsequently lead to the benzyne product. Energies at the MRMP2 level give 22.9 kcal mol-1 for the disrotatory channel and 21.7 and 1.4 kcal mol-1 for the two steps in the conrotatory one. However, the CCSD(T) energies give 19.3 and 14.2 kcal mol-1 for the two conrotatory steps. The first step of the conrotatory channel is significantly higher than the second so is rate determining for this channel. Comparison of the two separate channels shows that the conrotatory has just a slight energetic edge of 1.2 kcal mol-1 at the MRMP2 level. We did not compute the disrotatory channel at the CCSD(T) level due to the significant biradical nature of the wavefunction with natural orbital occupation numbers of 1.2 and 0.8 in the active space.

A Valence-Bond-Based Multiconfigurational Density Functional Theory: The λ-DFVB Method Revisited.

A recently developed valence-bond-based multireference density functional theory, named λ-DFVB, is revisited in this paper. λ-DFVB remedies the double-counting error of electron correlation by decomposing the electron–electron interactions into the wave function term and density functional term with a variable parameter λ. The λ value is defined as a function of the free valence index in our previous scheme, denoted as λ-DFVB(K) in this paper. Here we revisit the λ-DFVB method and present a new scheme based on natural orbital occupation numbers (NOONs) for parameter λ, named λ-DFVB(IS), to simplify the process of λ-DFVB calculation. In λ-DFVB(IS), the parameter λ is defined as a function of NOONs, which are straightforwardly determined from the many-electron wave function of the molecule. Furthermore, λ-DFVB(IS) does not involve further self-consistent field calculation after performing the valence bond self-consistent field (VBSCF) calculation, and thus, the computational effort in λ-DFVB(IS) is approximately the same as the VBSCF method, greatly reduced from λ-DFVB(K). The performance of λ-DFVB(IS) was investigated on a broader range of molecular properties, including equilibrium bond lengths and dissociation energies, atomization energies, atomic excitation energies, and chemical reaction barriers. The computational results show that λ-DFVB(IS) is more robust without losing accuracy and comparable in accuracy to high-level multireference wave function methods, such as CASPT2.

Automatic Selection of Active Orbitals from Generalized Valence Bond Orbitals.

The accurate multireference (MR) calculation of a strongly correlated chemical system usually relies on a correct preselection of a small number of active orbitals from numerous molecular orbitals. Currently, the active orbitals are generally determined by using a trial-and-error method. Such a preselection by chemical intuition and personal experience may be tedious or unreliable, especially for large complicated systems, and accordingly, the construction of active space becomes a bottleneck for large-scale MR calculations. In this work, we propose to automatically select the active orbitals according to the natural orbital occupation numbers by performing black box generalized valence bond calculations. We demonstrate the accuracy of this method through testing calculations of the ground states in various systems, ranging from bond dissociation of diatomic molecules (N2, C2, Cr2) to conjugated molecules (pentacene, hexacene, and heptacene) as well as a binuclear transition-metal complex [Mn2O2(H2O)2(terpy)2]3+ (terpy = 2,2':6,2″-terpyridine) with active spaces up to (30e, 30o) and comparing with the complete active space self-consistent field (CASSCF), density matrix renormalization group (DMRG)-CASSCF references, and other recently proposed inexpensive strategies for constructing active spaces. The results indicate that our method is among the most successful ones within our comparison, providing high-quality initial active orbitals very close to the finally optimized (DMRG-)CASSCF orbitals. The method proposed here is expected to greatly benefit the practical implementation of large active space ground-state MR calculations, for example, large-scale DMRG calculations.

Developing new and understanding old approximations in TDDFT.

When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange-correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn-Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange-correlation potential from a coupling-constant path integration.

The ANO-R Basis Set.

In this work, the new ANO-R basis set for all elements of the first six periods is introduced. The ANO-R basis set is an all-electron basis set that was constructed including scalar-relativistic effects of the exact-two component (X2C) Hamiltonian and modeling the atomic nucleus by a Gaussian charge distribution, which makes the basis set suitable for calculations of both light and heavy elements. For high accuracy, it takes advantage of the general contraction scheme and was developed at the CASSCF/CASPT2 level of theory. The distinguishing feature of the ANO-R basis set is its compactness in terms of both primitive and contracted basis functions, thus containing no superfluous functions for a given quality. An optimum number of primitive basis functions was selected based on studying the convergence toward the complete basis set limit for each element individually. The primitive basis sets were then contracted using the density-averaged atomic-natural-orbital (ANO) scheme, and suitable contraction levels were determined solely based on the natural orbital occupation numbers that describe the contribution of each natural orbital to the one-particle density matrix. Rather than following the common "split-valence n-tuple zeta plus polarization functions" structure, the resulting basis sets ANO-R0 to ANO-R3 possess a unique composition for each element, ensuring that no unnecessary functions are included while the basis sets are still balanced across the first six periods (H-Rn).

Sparse non-orthogonal wave function expansions from the extension of the generalized Pauli constraints to the two-electron reduced density matrix.

Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed "pinning" and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on "pinned" solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.

Curve Effect on Singlet Diradical Contribution in Kekulé-type Diradicals: A Sensitive Probe for Quinoidal Structure in Curved π-Conjugated Molecules.

Curved (non-planar) aromatic compounds have attracted significant research attention in the fields of basic chemistry and materials science. The contribution of the quinoidal structure in the curved π-conjugated structures has been proposed to be the key for materials functions. In this study, the curve effect on the quinoidal contribution was investigated in Kekulé-type singlet diradicals (S-DR1-4) as a sensitive probe for quinoidal structures in curved π-conjugated molecules. The quinoidal contribution in S-DR1-4 was found to increase with increasing the curvature of the curved structure, which was quantitatively analyzed using NBO analysis and the natural orbital occupation numbers computed by the CASSCF method. The curve effect on the singlet-triplet energy gap was examined by the CASPT2 method. The singlet-triplet energy gaps for the highly π-conjugated diradicals were determined for the first time using the CASPT2 method. Substantial quinoidal contribution was found in the curved structures of the delocalized singlet diradicals S-DR1-4, in contrast to its absence in the corresponding triplet states T-DR1-4.

Distinguishing artificial and essential symmetry breaking in a single determinant: approach and application to the C60, C36, and C20 fullerenes.

We present a thorough analysis of symmetry breaking observed in Hartree-Fock (HF) solutions of the C60, C36, and C20 fullerenes in order to characterize the nature of electron correlation in them. Our analysis is based on (1) the critical regularization strength to restore symmetry breaking in the recently developed regularized orbital optimized second-order Møller-Plesset perturbation theory (κ-OOMP2), (2) singlet-triplet gaps from various MP2 methods, and (3) natural orbital occupation numbers from restricted coupled-cluster with singles and doubles (RCCSD) and coupled-cluster valence bond with singles and doubles (CCVB-SD). Based on these three independent probes, we conclude that C36 (D6h) exhibits genuine strong correlation and symmetry breaking whereas C60 exhibits artificial HF symmetry breaking and is not strongly correlated. Investigating the critical regularization strength, we discuss strong correlation in C20 at the Jahn-Teller distorted geometries (C2h, D2h, Ci, and D3h) and the Ih geometry. Only C20 (Ih) was found to be strongly correlated while others exhibit artificial HF symmetry breaking. This analysis highlights a useful feature of the recommended κ-OOMP2 method. It is an electronic structure method that describes dynamic correlation, and attenuates strong correlation in MP2 towards zero by regularization. Therefore, κ-OOMP2 will exhibit symmetry breaking in its reference determinant only when correlation is strong (i.e., essential symmetry breaking). Artificial symmetry breaking (arising in HF due to neglect of dynamic correlation) thus appears to be removed in κ-OOMP2.

Strong Electron Correlation in Nitrogenase Cofactor, FeMoco.

FeMoco, MoFe7S9C, has been shown to be the active catalytic site for the reduction of nitrogen to ammonia in the nitrogenase protein. An understanding of its electronic structure including strong electron correlation is key to designing mimic catalysts capable of ambient nitrogen fixation. Active spaces ranging from [54, 54] to [65, 57] have been predicted for a quantitative description of FeMoco's electronic structure. However, a wave function approach for a singlet state using a [54, 54] active space would require 1029 variables. In this work, we systematically explore the active-space size necessary to qualitatively capture strong correlation in FeMoco and two related moieties, MoFe3S7 and Fe4S7. Using CASSCF and 2-RDM methods, we consider active-space sizes up to [14, 14] and [30, 30], respectively, with STO-3G, 3-21G, and DZP basis sets and use fractional natural-orbital occupation numbers to assess the level of multireference electron correlation, an examination of which reveals a competition between single-reference and multireference solutions to the electronic Schrödinger equation for smaller active spaces and a consistent multireference solution for larger active spaces.

A crucial role of fractional occupation numbers of natural orbitals (NOs) in the description of double excitations in response time-dependent NO functional theory

We demonstrate a crucial role of fractional occupation numbers (FONs) of natural orbitals (NOs) in the description of double excitations in time-dependent NO functional theory (TDNOFT). An analytical dependence of the double excitation energy ωα on the ratio of the FONs is derived in a model from the matrix diagonalization problem. In the large ratio Heitler-London limit the derived formula reproduces the correct asymptotics of ωα of the ionic state of double excitation character. In the small ratio Møller-Plesset, MP limit the reverse relation of static MP perturbation theory emerges in the dynamical response theory to provide ωα.

Coupled-Cluster Valence-Bond Singles and Doubles for Strongly Correlated Systems: Block-Tensor Based Implementation and Application to Oligoacenes.

We demonstrate a block-tensor based implementation of coupled-cluster valence-bond singles and doubles (CCVB-SD) [Small, D. W.; Head-Gordon M. J. Chem. Phys. 2012, 137, 114103] which is a simple modification to restricted CCSD (RCCSD) that provides a qualitatively correct description of valence correlations even in strongly correlated systems. We derive the Λ-equation of CCVB-SD and the corresponding unrelaxed density matrices. The resulting production-level implementation is applied to oligoacenes, correlating up to 318 electrons in 318 orbitals. CCVB-SD shows a qualitative agreement with exact methods for short acenes and reaches the bulk limit of oligoacenes in terms of natural orbital occupation numbers, whereas RCCSD shows nonvariational behavior even for relatively short acenes. A significant reduction in polyradicaloid character is found when correlating all valence electrons instead of only the π-electrons.

Density‐matrix propagation driven by semiclassical correlation

Methods based on propagation of the one‐body reduced density‐matrix hold much promise for the simulation of correlated many‐electron dynamics far from equilibrium, but difficulties with finding good approximations for the interaction term in its equation of motion have so far impeded their application. These difficulties include the violation of fundamental physical principles such as energy conservation, positivity conditions on the density, or unchanging natural orbital occupation numbers. We review some of the recent efforts to confront these problems, and explore a semiclassical approximation for electron correlation coupled to time‐dependent Hartree‐Fock propagation. We find that this approach captures changing occupation numbers, and excitations to doubly‐excited states, improving over TDHF and adiabatic approximations in density‐matrix propagation. However, it does not guarantee N‐representability of the density‐matrix, consequently resulting sometimes in violation of positivity conditions, even though a purely semiclassical treatment preserves these conditions. © 2016 Wiley Periodicals, Inc.

Choosing RASSCF orbital active spaces for multiple electronic states

A systematic approach is introduced for defining Restricted Active Space Self-Consistent-Field (RASSCF) orbitals for treating multiple electronic states in a balanced way, based on an SCF state-averaging procedure (Thom and Head-Gordon, 2008). We start from an SCF density matrix averaged over the electronic states of interest and choose a RAS partition based on the corresponding natural orbital occupation numbers. In order to test this procedure, we apply it to naphthalene cation, previously studied as a potential interstellar Polycyclic Aromatic Hydrocarbon (PAH) because of its photo stability following electronic excitation (Hall et al., 2006). The results obtained with this method in terms of energies and bond lengths for the first three electronic states and the corresponding minimum energy conical intersections between potential energy surfaces compare very well with CASSCF results and suggest further testing of the method for RASSCF calculations.

Natural orbital functional theory and pairing correlation effects in electron momentum density

Occupation numbers of natural orbitals capture the physics of strong electron correlations in momentum space. A natural orbital density functional theory based on the antisymmetrized geminal product provides these occupation numbers and the corresponding electron momentum density. A practical implementation of this theory approximates the natural orbitals by the Kohn–Sham orbitals and uses a mean-field approach to estimate pairing amplitudes leading to corrections for the independent particle model. The method is applied to weakly doped La2CuO4.

Kinetic and interaction components of the exact time-dependent correlation potential.

The exact exchange-correlation (xc) potential of time-dependent density functional theory has been shown to have striking features. For example, step and peak features are generically found when the system is far from its ground-state, and these depend nonlocally on the density in space and time. We analyze the xc potential by decomposing it into kinetic and interaction components and comparing each with their exact-adiabatic counterparts, for a range of dynamical situations in model one-dimensional two-electron systems. We find that often, but not always, the kinetic contribution is largely responsible for these features that are missed by the adiabatic approximation. The adiabatic approximation often makes a smaller error for the interaction component, which we write in two parts, one being the Coulomb potential due to the time-dependent xc hole. Non-adiabatic features of the kinetic component were also larger than those of the interaction component in cases that we studied when there is negligible step structure. In ground-state situations, step and peak structures arise in cases of static correlation, when more than one determinant is essential to describe the interacting state. We investigate the time-dependent natural orbital occupation numbers and find the corresponding relation between these and the dynamical step is more complex than for the ground-state case.

Theoretical Study of Mononuclear Nickel(I), Nickel(0), Copper(I), and Cobalt(I) Dioxygen Complexes: New Insight into Differences and Similarities in Geometry and Bonding Nature

Geometries, bonding nature, and electronic structures of (N^N)Ni(O2) (N^N = β-diketiminate), its cobalt(I) and copper(I) analogues, and (Ph3P)2Ni(O2) were investigated by density functional theory (DFT) and multistate restricted active space multiconfigurational second-order perturbation (MS-RASPT2) methods. Only (N^N)Ni(O2) takes a C(S) symmetry structure, because of the pseudo-Jahn-Teller effect, while all other complexes take a C(2V) structure. The symmetry lowering in (N^N)Ni(O2) is induced by the presence of the singly occupied δ(d(xy)-π(x)*) orbital. In all of these complexes, significant superoxo (O2-) character is found from the occupation numbers of natural orbitals and the O-O π* bond order, which is independent of the number of d electrons and the oxidation state of metal center. However, this is not a typical superoxo species, because the spin density is not found on the O2 moiety, even in open-shell complexes, (N^N)Ni(O2) and (N^N)Co(O2). The M-O and O-O distances are considerably different from each other, despite the similar superoxo character. The M-O distance and the interaction energy between the metal and O2 moieties are determined by the d(yz) orbital energy of the metal moiety taking the valence state. The binding energy of the O2 moiety is understood in terms of the d(yz) orbital energy in the valence state and the promotion energy of the metal moiety from the ground state to the valence state. Because of the participations of various charge transfer (CT) interactions between the metal and O2 moieties, neither the d(yz) orbital energy nor the electron population of the O2 moiety are clearly related to the O-O bond length. Here, the π bond order of the O2 moiety is proposed as a good measure for discussing the O-O bond length. Because the d electron configuration is different among these complexes, the CT interactions are different, leading to the differences in the π bond order and, hence, the O-O distance among these complexes. The reactivity of dioxygen complex is discussed with the d(yz) orbital energy.

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.

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.