Fractional Occupation Numbers Research Articles

Next generation extended Lagrangian first principles molecular dynamics.

Extended Lagrangian Born-Oppenheimer molecular dynamics [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] is formulated for general Hohenberg-Kohn density-functional theory and compared with the extended Lagrangian framework of first principles molecular dynamics by Car and Parrinello [Phys. Rev. Lett. 55, 2471 (1985)]. It is shown how extended Lagrangian Born-Oppenheimer molecular dynamics overcomes several shortcomings of regular, direct Born-Oppenheimer molecular dynamics, while improving or maintaining important features of Car-Parrinello simulations. The accuracy of the electronic degrees of freedom in extended Lagrangian Born-Oppenheimer molecular dynamics, with respect to the exact Born-Oppenheimer solution, is of second-order in the size of the integration time step and of fourth order in the potential energy surface. Improved stability over recent formulations of extended Lagrangian Born-Oppenheimer molecular dynamics is achieved by generalizing the theory to finite temperature ensembles, using fractional occupation numbers in the calculation of the inner-product kernel of the extended harmonic oscillator that appears as a preconditioner in the electronic equations of motion. Material systems that normally exhibit slow self-consistent field convergence can be simulated using integration time steps of the same order as in direct Born-Oppenheimer molecular dynamics, but without the requirement of an iterative, non-linear electronic ground-state optimization prior to the force evaluations and without a systematic drift in the total energy. In combination with proposed low-rank and on the fly updates of the kernel, this formulation provides an efficient and general framework for quantum-based Born-Oppenheimer molecular dynamics simulations.

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.

Calculating the optical properties of defects and surfaces in wide band gap materials

The optical properties of a material critically depend on its defects, and understanding that requires substantial and accurate input from theory. This paper describes recent developments in the electronic structure theory of defects in wide band gap materials, where the standard local or semi-local approximations of density functional theory fail. The success of the HSE06 screened hybrid functional is analyzed in case of Group-IV semiconductors and TiO2, and shown that it is the consequence of error compensation between semi-local and non-local exchange, resulting in a proper derivative discontinuity (reproduction of the band gap) and a total energy which is a linear function of the fractional occupation numbers (removing most of the electron self-interaction). This allows the calculation of electronic transitions with accuracy unseen before, as demonstrated on the single-photon emitter NV(-) center in diamond and on polaronic states in TiO2. Having a reliable tool for electronic structure calculations, theory can contribute to the understanding of complicated cases of light-matter interaction. Two examples are considered here: surface termination effects on the blinking and bleaching of the light-emission of the NV(-) center in diamond, and on the efficiency of photocatalytic water-splitting by TiO2. Finally, an outlook is presented for the application of hybrid functionals in other materials, as, e.g., ZnO, Ga2O3 or CuGaS2.

Analytic hyperpolarizability and polarizability derivative with fractional occupation numbers for large extended systems

The analytic hyperpolarizability and polarizability derivative with fractional occupation numbers are derived using Wigner's 2n + 1 rule. The formulation contains no terms that blow up for quasi-degenerate systems. The density-functional tight-binding method is used for implementation, which makes it possible to compute these third-order derivatives for systems containing up to one thousand atoms within 8 h using 24 CPU cores. A comparison between analytic and numerical non-resonance Raman activity spectra indicates that the numerical differentiation approach can give a significant deviation unless the strength of perturbative electric field is carefully chosen. With extremely high electronic temperatures, the polarizability and hyperpolarizability should converge to zero.

The Fractional Occupation Number Weighted Density as a Versatile Analysis Tool for Molecules with a Complicated Electronic Structure.

The fractional occupation number weighted density (FOD) analysis is explored as a general theoretical diagnostic for complicated electronic structures. Its main feature is to provide robustly and quickly the information on the localization of "hot" (strongly correlated and chemically active) electrons in a molecule. We demonstrate its usage in four different prototypical applications: 1) As a new and fast measure of the biradical character of polycyclic aromatic hydrocarbons, 2) for the selection of active orbital spaces in multiconfigurational or complete active space self consistent field (MCSCF/CASSCF) treatments, 3) as a possibility to describe molecular-energy landscapes consistently in regions with varying biradical character, as exemplified by partial double-bond torsions, and 4) as a powerful visualization method for static electron correlation effects in large biomolecules in connection with an efficient semi-empirical tight-binding molecular orbital scheme. The last application opens a full quantum-mechanical, unbiased route to the automatic detection of errors in experimental protein X-ray structures, such as false protonation states or misplaced atoms. In the first example, the complete (unfragmented) quantum-chemical calculation of the FOD for an entire metalloprotein with more than 7500 atoms is described.

Self-consistent implementation of ensemble density functional theory method for multiple strongly correlated electron pairs.

The spin-restricted ensemble-referenced Kohn-Sham (REKS) method is based on an ensemble representation of the density and is capable of correctly describing the non-dynamic electron correlation stemming from (near-)degeneracy of several electronic configurations. The existing REKS methodology describes systems with two electrons in two fractionally occupied orbitals. In this work, the REKS methodology is extended to treat systems with four fractionally occupied orbitals accommodating four electrons and self-consistent implementation of the REKS(4,4) method with simultaneous optimization of the orbitals and their fractional occupation numbers is reported. The new method is applied to a number of molecular systems where simultaneous dissociation of several chemical bonds takes place, as well as to the singlet ground states of organic tetraradicals 2,4-didehydrometaxylylene and 1,4,6,9-spiro[4.4]nonatetrayl.

Quantum chemical prediction of vibrational spectra of large molecular systems with radical or metallic electronic structure

Quantum chemical simulation of infrared (IR) and Raman spectra for molecules with open-shell, radical, or multiradical electronic structure represents a major challenge. We report analytic second-order geometrical derivatives of the Mermin free energy for the second-order self-consistent-charge density-functional tight-binding (DFTB2) method with fractional occupation numbers (FONs). This new method is applied to the evaluation of NO radical stretching modes in various open-shell molecules and to the prediction of the evolution of IR and Raman spectra of graphene nanoribbons with increasing molecular size.

Degenerate density perturbation theory

Fractional occupation numbers can be used in density functional theory to create a symmetric Kohn-Sham potential, resulting in orbitals with degenerate eigenvalues. We develop the corresponding perturbation theory and apply it to a system of $N_d$ degenerate electrons in a harmonic oscillator potential. The order-by-order expansions of both the fractional occupation numbers and unitary transformations within the degenerate subspace are determined by the requirement that a differentiable map exists connecting the initial and perturbed states. Using the X$\alpha$ exchange-correlation (XC) functional, we find an analytic solution for the first-order density and first through third-order energies as a function of $\alpha$, with and without a self-interaction correction. The fact that the XC Hessian is not positive definite plays an important role in the behavior of the occupation numbers.

Generalization of the Kohn-Sham system that can represent arbitrary one-electron density matrices

Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.

Second-Order Perturbation Theory for Fractional Occupation Systems: Applications to Ionization Potential and Electron Affinity Calculations.

Recently, we have developed an integration approach for the calculations of ionization potentials (IPs) and electron affinities (EAs) of molecular systems at the level of second-order Møller-Plesset (MP2) (Su, N. Q.; Xu, X. J. Chem. Theory Comput. 11, 4677, 2015), where the full MP2 energy gradient with respect to the orbital occupation numbers was derived but only at integer occupations. The theory is completed here to cover the fractional occupation systems, such that Slater's transition state concept can be used to have accurate predictions of IPs and EAs. Antisymmetrized Goldstone diagrams have been employed for interpretations and better understanding of the derived equations, where two additional rules were introduced in the present work specifically for hole or particle lines with fractional occupation numbers.

Using the GVB Ansatz to develop ensemble DFT method for describing multiple strongly correlated electron pairs.

Ensemble density functional theory (DFT) furnishes a rigorous theoretical framework for describing the non-dynamic electron correlation arising from (near) degeneracy of several electronic configurations. Ensemble DFT naturally leads to fractional occupation numbers (FONs) for several Kohn-Sham (KS) orbitals, which thereby become variational parameters of the methodology. The currently available implementation of ensemble DFT in the form of the spin-restricted ensemble-referenced KS (REKS) method was originally designed for systems with only two fractionally occupied KS orbitals, which was sufficient to accurately describe dissociation of a single chemical bond or the singlet ground state of biradicaloid species. To extend applicability of the method to systems with several dissociating bonds or to polyradical species, more fractionally occupied orbitals must be included in the ensemble description. Here we investigate a possibility of developing the extended REKS methodology with the help of the generalized valence bond (GVB) wavefunction theory. The use of GVB enables one to derive a simple and physically transparent energy expression depending explicitly on the FONs of several KS orbitals. In this way, a version of the REKS method with four electrons in four fractionally occupied orbitals is derived and its accuracy in the calculation of various types of strongly correlated molecules is investigated. We propose a possible scheme to ameliorate the partial size-inconsistency that results from perfect spin-pairing. We conjecture that perfect pairing natural orbital (NO) functionals of reduced density matrix functional theory (RDMFT) should also display partial size-inconsistency.

The electronic configurations of LnX (Ln = La–Eu, X = O, S, Se, Te): A FON–DFT investigation

Relativistic density functional theory calculation has been carried out for lanthanide monochalcogenide molecules LnX (Ln=La–Eu, X=O, S, Se, Te). Fractional occupation number approach was applied to investigate the electronic structure. LnX are mainly ionic molecules with 5d–np (n=2–5 for O–Te) covalent contributions. The ground state formal configuration is 4fkσ1 (k=0–6 for La–Eu) when k<3, and 4fk+1σ0 when k>3. NdX (k=3) molecules mostly behave as a mixture between them. Molecular orbital analysis and Mulliken population indicate orbital mixing of Ln (4f, 5d, 6s) in most molecules. Spin–orbit coupling has an increasing effect from La to Eu on bond lengths and dipole moments.

A practicable real-space measure and visualization of static electron-correlation effects.

The inclusion of dynamical and static electron correlation (SEC) is mandatory for accurate quantum chemistry (QC). SEC is particularly difficult to calculate and hence a qualitative understanding is important to judge the applicability of approximate QC methods. Existing scalar SEC diagnostics, however, lack the important information where the SEC effects occur in a molecule. We introduce an analysis tool based on a fractional occupation number weighted electron density (ρ(FOD) ) that is plotted in 3D for a pre-defined contour surface value. The scalar field is obtained by finite-temperature DFT calculations with pre-defined electronic temperature (e.g. TPSS at 5000 K). FOD plots only show the contribution of the "hot" (strongly correlated) electrons. We discuss illustrative plots for a broad range of chemical systems from small molecules to large conjugated molecules with polyradicaloid character. Spatial integration yields a single number which can be used to globally quantify SEC.

Local reduced-density-matrix-functional theory: Incorporating static correlation effects in Kohn-Sham equations

We propose a novel scheme to bring reduced density matrix functional theory (RDMFT) into the realm of density functional theory (DFT) that preserves the accurate density functional description at equilibrium, while incorporating accurately static and left-right correlation effects in molecules and keeping the good computational performance of DFT-based schemes. The key ingredient is to relax the requirement that the local potential is the functional derivative of the energy with respect to the density. Instead, we propose to restrict the search for the approximate natural orbitals within a domain where these orbitals are eigenfunctions of a single-particle hamiltonian with a local effective potential. In this way, fractional natural occupation numbers are accommodated into Kohn-Sham equations allowing for the description of molecular dissociation without breaking spin symmetry. Additionally, our scheme provides a natural way to connect an energy eigenvalue spectrum to the approximate natural orbitals and this spectrum is found to represent accurately the ionization potentials of atoms and small molecules.

A mathematical perspective on density functional perturbation theory

This article is concerned with the mathematical analysis of the perturbation method for extended Kohn–Sham models, in which fractional occupation numbers are allowed. All our results are established in the framework of the reduced Hartree–Fock (rHF) model, but our approach can be used to study other kinds of Kohn–Sham models, under some assumptions on the mathematical structure of the exchange–correlation functional. The classical results of density functional perturbation theory in the non-degenerate case (that is when the Fermi level is not a degenerate eigenvalue of the mean-field Hamiltonian) are formalized, and a proof of Wigner's (2n + 1) rule is provided. We then focus on the situation when the Fermi level is a degenerate eigenvalue of the rHF Hamiltonian, which has not been considered so far.

Investigation on spin-flip reaction of Re + CH3CN by relativistic density functional theory.

To explore the integrated reaction mechanisms for Re atom with acetonitrile theoretically, density functional theory with zero-order regular approximation (ZORA) relativistic corrections has been employed at the BP86/TZ2P level. There have been three adiabatic potential energy surfaces in the study along sextet, quartet and doublet spin states. However, the detailed minimum energy reaction pathway altogether contains six stationary states () to (), five transition states (), and two intersystem crossings with spin inversion (marked by ⇒): (6)Re + CH3CN → η(1)-ReNCCH3 () → ⇒ η(2)-Re(NC)CH3 () → → η(3)-HRe(NCCH2) () → → CH3-ReNC () → → CH2[double bond, length as m-dash]Re(H)NC () ⇒ → CH[triple bond, length as m-dash]Re(H)2NC (). Thereinto, the lowest energy crossing points (LECP) have been determined by the DFT fractional-occupation-number (FON) approach. The first spin inversion has transferred the potential energy surfaces from high-spin sextet to the quartet intermediate () with the subsequent C-C bond breakage. The second one from the quartet to the low-spin doublet state accompanies the C-H activation, decreasing the transition barrier by 157 kJ mol(-1). The overall reaction could be exothermic by about 210 kJ mol(-1). Harmonic vibration frequencies and NBO, WBO analysis are also applied to verified the experimental observed information.

Calculation of thermodynamic potentials with the inclusion of fractional occupation numbers and investigation of FCC-BCC structural phase transitions in alkaline-earth metals

The smearing near the Fermi level has been taken into account in the calculations of the thermodynamic characteristics of metals in order to improve the convergence of the performed calculations and to increase the quality of the obtained results. The choice of the smearing parameter usually has not been explained, although the results of the calculations differ significantly for different values of this parameter. Possible schemes for calculating the thermodynamic potentials with the inclusion of the smearing parameter and additional parameters of the volume and energy shifts have been considered. The influence of these parameters on the calculations of the thermodynamic properties of alkaline-earth metals under pressure and on the description of the structural phase transition has been analyzed.

Spin-flip reactions of Zr + C2H6 researched by relativistic density functional theory

Density functional theory (DFT) with relativistic corrections of zero-order regular approximation (ZORA) has been applied to explore the reaction mechanisms of ethane dehydrogenation by Zr atom with triplet and singlet spin-states. Among the complicated minimum energy reaction path, the available states involves three transition states (TS), and four stationary states (1) to (4) and one intersystem crossing with spin-flip (marked by -->): (3) Zr + C 2 H 6 → (3) Zr-CH 3 -CH 3 ((3)1) → (3)TS 1/2 → (3) ZrH-CH 2 -CH 3 ((3)2) → (3) TS 2/3 --> (1) ZrH2-CH2 = CH2 ((1) 3) → (1) TS 3/4 → (1) ZrH 3 -CH = CH 2 ((1)4). The minimum energy crossing point is determined with the help of the DFT fractional-occupation-number (FON) approach. The spin inversion leads the reaction pathway transferring from the triplet potential energy surface (PES) to the singlet's accompanying with the activation of the second C-H bond. The overall reaction is calculated to be exothermic by about 231 kJ mol(-1). Frequency and NBO analysis are also applied to confirm with the experimental observed data.

Direct ΔMBPT(2) method for ionization potentials, electron affinities, and excitation energies using fractional occupation numbers

A direct method (D-ΔMBPT(2)) to calculate second-order ionization potentials (IPs), electron affinities (EAs), and excitation energies is developed. The ΔMBPT(2) method is defined as the correlated extension of the ΔHF method. Energy differences are obtained by integrating the energy derivative with respect to occupation numbers over the appropriate parameter range. This is made possible by writing the second-order energy as a function of the occupation numbers. Relaxation effects are fully included at the SCF level. This is in contrast to linear response theory, which makes the D-ΔMBPT(2) applicable not only to single excited but also higher excited states. We show the relationship of the D-ΔMBPT(2) method for IPs and EAs to a second-order approximation of the effective Fock-space coupled-cluster Hamiltonian and a second-order electron propagator method. We also discuss the connection between the D-ΔMBPT(2) method for excitation energies and the CIS-MP2 method. Finally, as a proof of principle, we apply our method to calculate ionization potentials and excitation energies of some small molecules. For IPs, the ΔMBPT(2) results compare well to the second-order solution of the Dyson equation. For excitation energies, the deviation from equation of motion coupled cluster singles and doubles increases when correlation becomes more important. When using the numerical integration technique, we encounter difficulties that prevented us from reaching the ΔMBPT(2) values. Most importantly, relaxation beyond the Hartree-Fock level is significant and needs to be included in future research.

Polaronic effects in TiO2calculated by the HSE06 hybrid functional: Dopant passivation by carrier self-trapping

Metal oxides are extremely challenging for defect calculations within density-functional theory because of the underestimation of the band gap and of the polaronic effects due to spurious electron self-interaction in the standard (semi)local implementations. We show here that---similarly to Group-IV semiconductors---the HSE06 screened hybrid functional provides not only a good description of the ground-state properties and an accurate gap of ${\mathrm{TiO}}_{2}$ modifications, but fulfills the generalized Koopmans' theorem for host-related defect states---independent of localization. This means that the total energy has the correct dependence on the fractional occupation number, lending credibility to the calculated optical and thermodynamic charge transition levels for acceptors (Al, Ga, In, Sc, and Y on Ti site) and donors (Nb on Ti site, and interstitial H). We find deep, localized hole polarons in anatase and electron polarons in rutile. Therefore, $p$-type doping in anatase and $n$-type doping in rutile are counteracted by carrier self-trapping. Donors in anatase and the acceptors in rutile give rise to effective masslike states, but the latter are deeper. In fact, the polaron bound at In in anatase is about as deep as the effective masslike state induced by In in rutile. The shallow state in anatase and the deep one in rutile for the Nb donor, as well as the deep state for Al in anatase, agree well with experimental observations.