Fractional Occupation Numbers Research Articles

Ab Initio Nonadiabatic Molecular Dynamics with Hole-Hole Tamm-Dancoff Approximated Density Functional Theory.

The study of photoinduced dynamics in chemical systems necessitates accurate and computationally efficient electronic structure methods, especially as the systems of interest grow larger. The linear response hole-hole Tamm-Dancoff approximated (hh-TDA) density functional theory method was recently proposed to satisfy such demands. The N-electron electronic states are obtained by means of double annihilations on a doubly anionic (N + 2)-electron reference state, allowing for the ground and excited states to be formed on the same footing and thus enabling the correct description of conical intersections. Dynamic electron correlation effects are incorporated by means of the exchange-correlation functional. The accuracy afforded by the simultaneous treatment of static and dynamic correlation in addition to the relatively low computational cost, comparable to that of time-dependent density functional theory (TDDFT), makes it a promising ab initio electronic structure method for on-the-fly generation of potential energy surfaces in nonadiabatic dynamics simulations of photochemical systems, particularly those for which the nπ* and ππ* electronic excitations are most relevant. Here, we apply the hh-TDA method to nonadiabatic dynamics simulations of prototypical photochemical processes. First, we demonstrate the ability of hh-TDA to adequately describe conical intersection geometries. We next examine its ability to describe the ultrafast excited state dynamics of photoexcited ethylene through an ab initio multiple spawning (AIMS) dynamics simulation. Finally, we present an alternative variant of the hh-TDA method, which uses orbitals from a fractional occupation number Kohn-Sham (FON-KS) calculation applied to an ensemble with N-electrons. The resulting method is termed floating occupation molecular orbital hh-TDA (FOMO-hh-TDA). This scheme allows us to combine hh-TDA with global hybrid functionals and allows us to avoid unbound valence orbitals that may result from an (N + 2)-electron self-consistent field (SCF) procedure. FOMO-hh-TDA-BHLYP faithfully reproduces the nonadiabatic dynamics of trans-azobenzene (TAB) and is used to characterize the excited state decay pathways from the first (nπ*) excited state.

Finite-temperature-based time-dependent density-functional theory method for static electron correlation systems.

In this study, we developed a time-dependent density-functional theory (TDDFT) with a finite-temperature (FT) scheme, denoted as FT-TDDFT. We introduced the concept of fractional occupation numbers for random phase approximation equation and evaluated the excited-state electronic entropy terms with excited-state occupation number. The orbital occupation numbers for the excited state were evaluated from the change in the ground-state electron configuration with excitation and deexcitation coefficients. Furthermore, we extended the FT formulation to the time-dependent density-functional tight-binding (TDDFTB) method for larger systems, denoted as FT-TDDFTB. Numerical assessment for the FT-(TD)DFT method showed smooth potential curves for double-bond rotation of ethylene in both ground and excited states. Excited-state calculations based on the FT-TDDFTB method were applied to the uniform π-stacking columns composed of trioxotriangulene, possessing neutral radicals in strong correlation systems.

Effects on the aromaticity and on the biradicaloid nature of acenes by the inclusion of a cyclobutadiene linkage

Biphenylene is a well-characterized compound that includes the union of both aromatic and antiaromatic units: benzene and cyclobutadiene, respectively. In this work, we extend this structural theme to acene analogues with a cyclobutadiene linkage in a central position. At the single reference domain, perturbation theory and DFT calculations were applied to characterize the electronic structure of the extended biphenylene systems, as well as of some acenes of interest for comparison. Multireference calculations were also used to provide more accurate information about the energetic stability and insights into the radical character. Using descriptors such as the singlet–triplet splitting, the number of effectively unpaired electrons, fractional occupation number weighted density analysis and aromaticity indexes, the different compounds have been compared with respect to biradicaloid character and chemical stability through structural and energetic approaches. The results indicate that biphenylene derivatives should possess similar reactivities and stabilities as their half acenes fragments.

Density-Matrix Based Extended Lagrangian Born-Oppenheimer Molecular Dynamics.

Extended Lagrangian Born-Oppenheimer molecular dynamics [ Phys. Rev. Lett. 2008, 100, 123004] is presented for Hartree-Fock theory, where the extended electronic degrees of freedom are represented by a density matrix, including fractional occupation numbers at elevated electronic temperatures. In contrast to regular direct Born-Oppenheimer molecular dynamics simulations, no iterative self-consistent field optimization is required prior to the force evaluations. To sample regions of the potential energy landscape where the gap is small or vanishing, which leads to particular convergence problems in regular direct Born-Oppenheimer molecular dynamics simulations, an adaptive integration scheme for the extended electronic degrees of freedom is presented. The integration scheme is based on a tunable, low-rank approximation of a fourth-order kernel, [Formula: see text], that determines the metric tensor, [Formula: see text], used in the extended harmonic oscillator of the Lagrangian that generates the dynamics of the electronic degrees of freedom. The formulation and algorithms provide a general guide to implement extended Lagrangian Born-Oppenheimer molecular dynamics for quantum chemistry, density functional theory, and semiempirical methods using a density matrix formalism.

Excitation energies expressed as orbital energies of Kohn-Sham density functional theory with long-range corrected functionals.

A new simple and conceptual theoretical scheme is proposed for estimating one-electron excitation energies using Kohn-Sham (KS) solutions. One-electron transitions that are dominated by the promotion from one initially occupied orbital to one unoccupied orbital of a molecular system can be expressed in a two-step process, ionization, and electron attachment. KS with long-range corrected (LC) functionals satisfies Janak's theorem and LC total energy varies almost linearly as a function of its fractional occupation number between the integer electron points. Thus, LC reproduces ionization energies (IPs) and electron affinities (EAs) with high accuracy and one-electron excitation energies are expressed as the difference between the occupied orbital energy of a neutral molecule and the corresponding unoccupied orbital energy of its cation. Two such expressions can be used, with one employing the orbital energies for the neutral and cationic systems, while the other utilizes orbital energies of just the cation. Because the EA of a molecule is the IP of its anion, if we utilize this identity, the two expressions coincide and give the same excitation energies. Reasonable results are obtained for valence and core excitations using only orbital energies.

UNO(ULO) active space for multireference calculations on classical and quantum computers. Revisit to the iron-sulfur complexes

The seamless continuation from single reference (SR) broken-symmetry (BS) to multi-reference (MR) methods such as the density matrix renormalization group (DMRG) was revisited to elucidate natural orbitals (UNO) and their occupation numbers for iron-sulfur clusters such as P- and FeMoco clusters in nitrogenase. Fractional occupation numbers of UNO for these clusters by SR-BS were consistent with those of open-shell orbitals by MR-DMRG, supporting the UNO-CI calculations of large Fe-S clusters on classical and quantum computers.

Fractional occupation numbers and self-interaction correction-scaling methods with the Fermi-Löwdin orbital self-interaction correction approach.

We present a new assessment of the Fermi-Löwdin orbital self-interaction correction (FLO-SIC) approach with an emphasis on its performance for predicting energies as a function of fractional occupation numbers (FONs) for various multielectron systems. Our approach is implemented in the massively parallelized NWChem quantum chemistry software package and has been benchmarked on the prediction of total energies, atomization energies, and ionization potentials of small molecules and relatively large aromatic systems. Within our study, we also derive an alternate expression for the FLO-SIC energy gradient expressed in terms of gradients of the Fermi-orbital eigenvalues and revisit how the FLO-SIC methodology can be seen as a constrained unitary transformation of the canonical Kohn-Sham orbitals. Finally, we conclude with calculations of energies as a function of FONs using various SIC-scaling methods to test the limits of the FLO-SIC formalism on a variety of multielectron systems. We find that these relatively simple scaling methods do improve the prediction of total energies of atomic systems as well as enhance the accuracy of energies as a function of FONs for other multielectron chemical species.

Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals

A recently developed finite element approach for fully numerical atomic structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] is extended to the description of atoms with spherically symmetric densities via fractionally occupied orbitals. Specialized versions of Hartree-Fock as well as local density and generalized gradient approximation density functionals are developed, allowing extremely rapid calculations at the basis set limit on the ground and low-lying excited states even for heavy atoms. The implementation of range-separation based on the Yukawa or complementary error function (erfc) kernels is also described, allowing complete basis set benchmarks of modern range-separated hybrid functionals with either integer or fractional occupation numbers. Finally, computation of atomic effective potentials at the local density or generalized gradient approximation levels for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and efficient way to initialize electronic structure calculations is described. The present numerical approach is shown to afford beyond microhartree accuracy with a small number of numerical basis functions, and to reproduce literature results for the ground states of atoms and their cations for $1 \leq Z \leq 86 $. Our results indicate that the literature values deviate by up to 10 {\mu}Eh from the complete basis set limit. The numerical scheme for the erfc kernel is shown to work by comparison to results from large Gaussian basis set calculations from the literature. Spin-restricted ground states are reported for Hartree-Fock and Hartree-Fock-Slater calculations with fractional occupations for $1 \leq Z \leq 118$.

Machine-learned electron correlation model based on correlation energy density at complete basis set limit.

We propose a machine-learned correlation model that is built using the regression between density variables such as electron density and correlation energy density. The correlation energy density of coupled cluster singles, doubles, and perturbative triples [CCSD(T)] is derived based on grid-based energy density analysis. The complete basis set (CBS) limit is estimated using the composite method, which has been reported to calculate the total correlation energy. The numerical examination revealed that the correlation energy density of the CCSD(T)/CBS level is appropriate for the response variable of machine learning. In addition to the density variables used in the exchange-correlation functionals of the density functional theory, the Hartree-Fock (HF) exchange energy density and electron density based on the fractional occupation number of molecular orbitals were employed as explanatory variables. Numerical assessments confirmed the accuracy and efficiency of the present correlation model. Consequently, the present protocol, namely, learning the CCSD(T)/CBS correlation energy density using density variables obtained by the HF calculation with a small basis set, yields an efficient correlation model.

Jahn-Teller effect in density-functional theory

Jahn and Teller used degenerate perturbation theory to prove that open shell molecules with symmetry Jahn-Teller distort. In that vein, we have developed a perturbative approach to computing Jahn-Teller distortions in Kohn-Sham density-functional theory, starting from a symmetrized molecule with an electronic shell closed by fractional occupation numbers. The resulting highly symmetric state is an energy extremum but not necessarily a minimum. Using second-order perturbation theory, we find the changes in geometry that occur when the symmetry of the electron density is broken to form a state with integer occupation numbers. This methodology allows us to retain many of the computational benefits of working in higher symmetry. As a demonstration, we solve the resulting equations for ten electrons in a superatomlike harmonic-oscillator potential.

Accurate Formula for the Macroscopic Polarization of Strongly Correlated Materials.

The many-body Berry phase formula for macroscopic polarization is approximated by a sum of natural orbital geometric phases with fractional occupation numbers accounting for the dominant correlation effects. This formula accurately reproduces the exact polarization in the Rice-Mele-Hubbard model across the band insulator-Mott insulator transition. A similar formula based on a reduced Berry curvature accurately predicts the interaction-induced quenching of Thouless topological charge pumping.

Linear Scaling Pseudo Fermi-Operator Expansion for Fractional Occupation.

Recursive Fermi-operator expansion methods for the calculation of the idempotent density matrix are valid only at zero electronic temperature with integer occupation numbers. We show how such methods can be modified to include fractional occupation numbers of an approximate or pseudo Fermi-Dirac distribution and how the corresponding entropy term of the free energy is calculated. The proposed methodology is demonstrated and evaluated for different electronic structure methods, including density functional tight-binding theory, Kohn-Sham density functional theory using numerical orbitals, and quantum chemistry Hartree-Fock theory using Gaussian basis functions.

Density functional theory and ab initio studies on hyperfine coupling constants of phosphinyl radicals

AbstractThe 31P hyperfine coupling constants for various phosphinyl radicals were calculated using density functional theory (DFT) as well as the ab initio orbital‐optimized spin‐component‐scaled MP2 (OO‐SCS‐MP2) and coupled cluster methods (with relaxed and unrelaxed density). However before that, electron correlation in phosphinyl radicals was investigated by means of the fractional occupation number weighted electron density (FOD). The isotropic hyperfine coupling constants for phosphinyl radicals posed a considerable challenge to DFT; among other issues, the well‐known functional B3LYP substantially underperformed, but TPSS0 provided highly accurate results. Two tested double hybrid functionals, namely B2PLYP and mPW2PLYP, and OO‐SCS‐MP2 were found to be reasonably accurate. The unbalanced description of spin polarization in the valance region was identified as the root of the problems of DFT methods with correct prediction of the isotropic hyperfine coupling constants. In calculations at the coupled cluster level, relaxed and unrelaxed density matrices were tested and both were proved to provide highly satisfactory results for phosphinyl radicals. This was confirmed for a set of small paramagnetic species.

Delocalization Errors in Density Functional Theory Are Essentially Quadratic in Fractional Occupation Number.

Approximate functionals used in practical density functional theory (DFT) deviate from the piecewise linear behavior of the exact functional for fractional charges. This deviation causes excess charge delocalization, which leads to incorrect densities, molecular properties, barrier heights, band gaps, and excitation energies. We present a simple delocalization function for characterizing this error and find it to be almost perfectly linear vs the fractional electron number for systems spanning in size from the H atom to the C12H14 polyene. This causes the delocalization energy error to be a quadratic polynomial in the fractional electron number, which permits us to assess the comparative performance of 47 popular and recent functionals through the curvature. The quadratic form further suggests that information about a single fractional charge is sufficient to eliminate the principal source of delocalization error. Generalizing traditional two-point information like ionization potentials or electron affinities to account for a third, fractional charge-based data point could therefore permit fitting/tuning of functionals with lower delocalization error.

On the many-electron self-interaction error of the semilocal exchange hole based meta-GGA level range-separated hybrid with the B88 hybrids

Delocalization error or many electron self-interaction error related to the fractional occupation number is studied within the framework of the density matrix expansion based meta-GGA level range-separated hybrid functional. The present study mainly focuses on the performance of that functional with the BECKE88 exchange based local, global and range-separated hybrids. It is observed that using the Kohn–Sham kinetic energy dependent exchange hole, several properties relating to the fractional occupation number are significantly improved without hindering the atomization energy accuracy. The present systematic assessment and functional performance for G2/148 test set and (non-) hydrogen barrier heights indicate that beyond the BECKE88 based range-separated functional, present meta-GGA level range-separated functional can be used for further molecular simulations.

Fractional-occupation-number based divide-and-conquer coupled-cluster theory

We have extended the divide-and-conquer (DC) coupled-cluster with singles and doubles (CCSD) to a fractional occupation number (FON) formalism, denoted as FON-DC-CCSD, using the thermal Wick theorem. The motivation is to address the inconsistency in the treatment of orbital occupations between the DC-based Hartree–Fock and the DC-CCSD methods, which adopt the Fermi distribution function and the step function for orbital occupation, respectively. Numerical applications involving polyene chains and single-walled carbon nanotubes confirm that the proposed FON-DC-CCSD method reduces both energy errors and computational costs compared with the conventional DC-CCSD method.

Communication: Coupled cluster and many-body perturbation theory for fractional charges and spins.

The study of systems with fractional charges and spins has become an extremely important tool to understand errors in approximate electronic structure methods, particularly in the context of density functional theory. Meanwhile, similar studies with wavefunction (WF)-based methods beyond second-order perturbation theory have been lacking. In this contribution, we study the performance of different coupled cluster (CC) and many-body perturbation theory (MBPT)-based methods for fractional charges. The use of the conventional and renormalized formulations of fractional-charge MBPT is discussed. The fractional spin behavior of the coupled cluster doubles (CCD) method is also investigated. Overall, all tested WF methods show very promising performance for the fractional charge problem. CCD is also quite accurate for the fractional spin problem in He+ across most of the range, although it breaks down to near Hartree-Fock quality in the strongly correlated limit. Beyond the study of fractional charge and spin curves, the implementation of CC methods with fractional occupation numbers offers a promising route to treating problems with multi-reference character in a single-reference framework.

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.

The role of topology in organic molecules: origin and comparison of the radical character in linear and cyclic oligoacenes and related oligomers.

We discuss the nature of electron-correlation effects in carbon nanorings and nanobelts using an analysis tool known as fractional occupation number weighted electron density (ρFOD) and the RAS-SF method, revealing for the first time significant differences in static correlation effects depending on how the rings (i.e. chemical units) are fused and/or connected until closing the loop. We choose to study in detail linear and cyclic oligoacene molecules of increasing size, and relate the emerging differences with the difficulties for the synthesis of the latter due to their radicaloid character. We finally explore how minor structural modifications of the cyclic forms can alter these results, showing the potential use of these systems as molecular templates for the growth of well-shaped carbon nanotubes as well as the usefulness of theoretical tools for molecular design.

Directly patching high-level exchange-correlation potential based on fully determined optimized effective potentials.

The key element in Kohn-Sham (KS) density functional theory is the exchange-correlation (XC) potential. We recently proposed the exchange-correlation potential patching (XCPP) method with the aim of directly constructing high-level XC potential in a large system by patching the locally computed, high-level XC potentials throughout the system. In this work, we investigate the patching of the exact exchange (EXX) and the random phase approximation (RPA) correlation potentials. A major challenge of XCPP is that a cluster's XC potential, obtained by solving the optimized effective potential equation, is only determined up to an unknown constant. Without fully determining the clusters' XC potentials, the patched system's XC potential is "uneven" in the real space and may cause non-physical results. Here, we developed a simple method to determine this unknown constant. The performance of XCPP-RPA is investigated on three one-dimensional systems: H20, H10Li8, and the stretching of the H19-H bond. We investigated two definitions of EXX: (i) the definition based on the adiabatic connection and fluctuation dissipation theorem (ACFDT) and (ii) the Hartree-Fock (HF) definition. With ACFDT-type EXX, effective error cancellations were observed between the patched EXX and the patched RPA correlation potentials. Such error cancellations were absent for the HF-type EXX, which was attributed to the fact that for systems with fractional occupation numbers, the integral of the HF-type EXX hole is not -1. The KS spectra and band gaps from XCPP agree reasonably well with the benchmarks as we make the clusters large.