Fractional Occupation Numbers Research Articles

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.

Photochemistry of DNA Fragments via Semiclassical Nonadiabatic Dynamics

Forming upon absorption of a UV photon, excited states of DNA are subject to nonadiabatic evolution, via either internal conversion (IC) back to the ground state or mutagenesis. Nonadiabatic processes following the formation of the first singlet excited states, S1, in 10 different small DNA fragments--4 single 4'H-nucleosides, 2 Watson-Crick base pairs, and 4 nucleotide quartets--have been investigated. Simulations were done via the nonadiabatic direct trajectory surface hopping semiclassical dynamics. The electronic wave function was obtained with configuration interaction, based on the semiempirical AM1 and PM3 Hamiltonians with fractional orbital occupation numbers. The evolution of the electronic wave function was governed by the time-dependent Schrödinger equation with a locally diabatic representation, intrinsically stable near surface crossings. The nuclei evolved on adiabatic potential energy surfaces, as prescribed by classical Newtonian dynamics, with sudden hops between potential energy surfaces to account for nonadiabatic transitions. The "fewest switches" surface hopping algorithm coupled the quantum and classical parts of the system. The dynamics simulations revealed several routes of nonadiabatic relaxation in these systems, which were not reported previously, and also recovered known routes of IC.

Generalized Koopmans density functional calculations reveal the deep acceptor state ofNOin ZnO

Applying a generalized Koopmans condition to recover the linear behavior of the energy with respect to the fractional occupation number, we find that substitutional nitrogen NO in ZnO is a deep acceptor with an ionization energy of 1.6 eV, which is prohibitively large for p-type conductivity. Testing the generalized Koopmans condition in computationally more demanding hybrid-functional calculations, we obtain a very similar result for NO, but find that the simultaneous correction of defect acceptor-level and host band-gap properties remains challenging in hybrid methods. The deep character of anion-site acceptors in ZnO has important consequences for the concept of codoping, as we show that nominally charge-compensated impurity pairs such as NO-GaZn or CO-TiZn have positively charged states in the gap that act as hole traps.

Physical meaning of the natural orbitals: Analysis of exactly solvable models

We investigate the suitability of natural orbitals as a basis for describing many-body excitations. We analyze to which extent the natural orbitals describe both bound as well as ionized excited states and show that depending on the specifics of the excited state the ground-state natural orbitals may yield a good approximation. We show that the success of reduced density-matrix functional theory in describing molecular dissociation lies in the flexibility provided by fractional occupation numbers while the role of the natural orbitals is minor.

Photochemistry of DNA Fragments Via Semiclassical Nonadiabatic Dynamics

Forming upon absorption of a UV photon, excited states of DNA are subject to nonadiabatic evolution. Though photo-excited DNA mostly undergoes internal conversion back to the ground state, various routes of mutagenesis are also possible. Ultimately, the accumulation of errors in the genome can result in cancer. Here, nonadiabatic processes following the formation of the first singlet excited states, S1, in ten different small DNA fragments have been investigated: four single 4′H-nucleosides, two Watson-Crick base pairs, and four nucleotide quartets. Simulations were done via the nonadiabatic direct trajectory surface hopping semiclassical dynamics. The electronic wavefunction was obtained with configuration interaction, based on the semiempirical PM3 Hamiltonian with fractional orbital occupation numbers. The evolution of the electronic wavefunction was governed by the time-dependent Schrodinger's equation with a locally-diabatic representation, intrinsically stable near surface crossings. The nuclei evolved on adiabatic potential energy surfaces, as prescribed by classical Newtonian dynamics. The “fewest switches” surface hopping algorithm coupled the quantum and classical parts of the system. The dynamics simulations revealed several routes of nonadiabatic relaxation in these systems, which were not reported previously.View Large Image | View Hi-Res Image | Download PowerPoint Slide

Koopmans’ springs to life

The meaning of orbital energies (OOEs) in Kohn-Sham (KS) density functional theory (DFT) is subject to a longstanding controversy. In local, semilocal, and hybrid density functionals (DFs) a Koopmans' approach, where OOEs approximate negative ionization potentials (IPs), is unreliable. We discuss a methodology based on the Baer-Neuhauser-Livshits range-separated hybrid DFs for which Koopmans' approach "springs to life." The OOEs are remarkably close to the negative IPs with typical deviances of +/-0.3 eV down to IPs of 30 eV, as demonstrated on several molecules. An essential component is the ab initio motivated range-parameter tuning procedure, forcing the highest OOE to be exactly equal to the negative first IP. We develop a theory for the curvature of the energy as a function of fractional occupation numbers to explain some of the results.

Polaronic hole localization and multiple hole binding of acceptors in oxide wide-gap semiconductors

Acceptor-bound holes in oxides often localize asymmetrically at one out of several equivalent oxygen ligands. Whereas Hartree-Fock (HF) theory overly favors such symmetry-broken polaronic hole localization in oxides, standard local-density (LD) calculations suffer from spurious delocalization among several oxygen sites. These opposite biases originate from the opposite curvatures of the energy as a function of the fractional occupation number $n$, i.e., ${d}^{2}E/d{n}^{2}<0$ in HF and ${d}^{2}E/d{n}^{2}>0$ in LD. We recover the correct linear behavior, ${d}^{2}E/d{n}^{2}=0$, that removes the (de)localization bias by formulating a generalized Koopmans condition. The correct description of oxygen hole localization reveals that the cation-site nominal single acceptors in ZnO, ${\text{In}}_{2}{\text{O}}_{3}$, and ${\text{SnO}}_{2}$ can bind multiple holes.

Spin-Flip Reaction of Re + CH4—A Relativistic Density Functional Theory Investigation

To explore the reaction mechanisms of methane dehydrogenation by gas-phase Re atom, the sextet, quartet, and doublet potential energy surfaces have been performed using density functional theory (DFT) and zero-order regular approximation relativistic corrections at the PW91/TZ2P level. The minimum energy reaction path is found to proceed through the following steps: (6)Re + CH(4) --> ReCH(4) ((6)1) --> H(3)CReH ((4)2) --> (4)TS2/3 --> H(2)CReH(2) ((4)3) --> (2)TS3/4 --> HCReH(3) ((2)4). Also, the reaction path involves the spin inversion twice in the different reaction steps. To better understand the spin inversion processes, the low energy crossing point is determined with the help of the density functional fractional occupation number approach. The first spin inversion, from the sextet state to the quartet state, makes the activation of the C-H bond energetically spontaneous. The second transition from the quartet state to the doublet state facilitates the cleavage of the second C-H bond, lowering the barrier from 186.1 to 24.2 kJ/mol. The overall reaction is calculated to be exothermic by 149.8 kJ/mol, and the final products in three spin states are investigated by NBO analysis, to compare the Re-C bonds and the C-H bonds.

A MinMax self-consistent-field approach for auxiliary density functional theory

A MinMax self-consistent-field (SCF) approach is derived in the framework of auxiliary density functional theory. It is shown that the SCF convergence can be guided by the fitting coefficients that arise from the variational fitting of the Coulomb potential. An in-core direct inversion of the iterative subspace (DIIS) algorithm is presented. Due to its reduced memory demand this new in-core DIIS method can be applied without overhead to very large systems with tens of thousands of basis and auxiliary functions. Due to the new DIIS error definition systems with fractional occupation numbers can be treated, too.

Evaluation of Electronic Coupling in Transition-Metal Systems Using DFT: Application to the Hexa-Aquo Ferric-Ferrous Redox Couple.

We present a density-functional theory (DFT) approach, with fractionally occupied orbitals, for studying the prototypical ferric-ferrous electron-transfer (ET) process in liquid water. We use a recently developed ab initio method to calculate the transfer integral (also named electronic-coupling or ET matrix element) between the solvated ions. The computed transfer integral is combined with previous ab initio values of the reorganization energy, within the framework of Marcus' theory, to estimate the rate of the electron self-exchange reaction. The self-interaction correction incorporated (through an appropriate treatment of the electronic correlation effects) into a Hubbard U extension to the DFT scheme leads to a theoretical value of the ET rate relatively close to an experimental estimate from kinetic measurements. The use of fractional occupation numbers (FON) turned out to be crucial for achieving convergence in most self-consistent calculations because of the open-shell d-multiplet electronic structure of each iron ion and the near degeneracy of the redox groups involved. We provide a theoretical justification for the FON approach, which allows a description of the chemical potential and orbital relaxation, and possible extension to other transition-metal redox systems. Accordingly, the methodology developed in this paper, which rests on a suitable combination of Hubbard U correction and a FON approach to DFT, seems to offer a fruitful approach for the quantitative description of ET reactions in biochemical systems.

Ab initiocalculation of core-valence-valence Auger spectra in closed shell systems

We propose an ab initio method to evaluate the core-valence-valence (CVV) Auger spectrum of systems with filled valence bands. The method is based on the Cini-Sawatzky theory, and aims at estimating the parameters by first-principles calculations in the framework of density-functional theory (DFT). Photoemission energies and the interaction energy for the two holes in the final state are evaluated by performing DFT simulations for the system with varied population of electronic levels. Transition matrix elements are taken from atomic results. The approach takes into account the non-sphericity of the density of states of the emitting atom, spin-orbit interaction in core and valence, and non quadratic terms in the total energy expansion with respect to fractional occupation numbers. It is tested on two benchmark systems, Zn and Cu metals, leading in both cases to L23M45M45 Auger peaks within 2 eV from the experimental ones. Detailed analysis is presented on the relative weight of the various contributions considered in our method, providing the basis for future development. Especially problematic is the evaluation of the hole-hole interaction for systems with broad valence bands: our method underestimates its value in Cu, while we obtain excellent results for this quantity in Zn.

Excitation Energies from Spin-Restricted Ensemble-Referenced Kohn−Sham Method: A State-Average Approach

A time-independent density functional approach to the calculation of excitation energies from the ground states of molecules typified by the strong nondynamic electron correlation is suggested. The new method is based on the use of the spin-restricted ensemble-referenced Kohn-Sham formalism [Filatov, M.; Shaik, S. Chem. Phys. Lett. 1999, 304, 429] for the calculation of the ground state. In the new method, the average energy of the ground state and a state created by a single excitation thereof is minimized with respect to the Kohn-Sham orbitals and their fractional occupation numbers. The lowest singlet excitation energies obtained with the help of the new formalism for a number of model systems, such as the hydrogen molecule with stretched bond, twisted ethylene, and twisted hexa-1,3,5-triene, are compared with the results of the time-dependent density functional theory, with the results of ab initio CASSCF/CASPT2 calculations, and with the experimental data. Applicability of the new method to the description of photochemical reactions is discussed.

Alternative perspective on density-functional perturbation theory

Perturbation theory is examined in analytic density-functional theory (ADFT), for which V representability means slightly more than in conventional density-functional theory because the potential is fitted. There is synergism between variationality and V representability. Together they redirect the object of perturbation theory from the set of occupied virtual orbital rotations to the change in the fit to the Kohn-Sham potential, which is called the Sambe-Felton potential. This reduces the dimensionality of perturbation theory from order N{sup 2} to order N, where N is the number of basis functions. With variational fitting, no fractional or negative powers of the density appear when using the Slater exchange kernel, which is proportional to the cube root of the spin density. Requiring the Fock matrix and density matrix to commute through each order of perturbation theory determines the role of fractional occupation numbers in density-functional perturbation theory, which are treated via the corresponding nonintegral differences between the occupation numbers of orbitals. This theory is tested by removing a tenth or twentieth of an electron from the highest occupied molecular orbital for a standard set of small molecules, in which case the first- and second-order perturbed energies are accurate to 70%, when compared to themore » energy difference of the two corresponding self-consistent-field (SCF) calculations. For an all-electron ADFT calculation on a C{sub 4v}-symmetric Zr{sub 6}O{sub 12} cluster, the timing for all SCF coupled perturbed iterations is not significant compared to the single required N{sup 4} sum over occupied and virtual orbitals.« less

Assessment of transition operator reference states in electron propagator calculations

The transition operator method combined with second-order, self-energy corrections to the electron propagator (TOEP2) may be used to calculate valence and core-electron binding energies. This method is tested on a set of molecules to assess its predictive quality. For valence ionization energies, well known methods that include third-order terms achieve somewhat higher accuracy, but only with much higher demands for memory and arithmetic operations. Therefore, we propose the use of the TOEP2 method for the calculation of valence electron binding energies in large molecules where third-order methods are infeasible. For core-electron binding energies, TOEP2 results exhibit superior accuracy and efficiency and are relatively insensitive to the fractional occupation numbers that are assigned to the transition orbital.

A simple density functional fractional occupation number procedure to determine the low energy transition region of spin-flip reactions

A computationally simple three-step procedure to survey the energy landscape and to determine the molecular transition structure and activation energy at the intersection of two weakly coupled electronic potential energy surfaces of different symmetry is suggested. Only commercial software is needed to obtain the transition states of, for instance, spin-flip reactions. The computational expense is only two to three times larger than that of the standard determination of an adiabatic reaction path. First, the structures of the two electronic initial and final states along a chosen reaction coordinate are individually optimized. At the "projected crossing," the two states have the same energy at the same value of the reaction coordinate, but different state-optimized partial structures. Second, the unique optimized structure of a low energy crossing point between the two states is determined with the help of the density functional fractional occupation number approach. Finally, the respective energy of the two states at the crossing is estimated by a single point calculation. The prescription is successfully applied to some simple topical examples from organic and from inorganic chemistry, respectively, concerning the spin-flip reactions (3)H(3)CS(+)-->(1)H(2)CSH(+) and (7)MoCO(2)-->(5)MoCO(2)-->(3)OMoCO.

Exciton spectrum for two-dimensional electron systems at fractional occupation numbers of the Landau levels

It is shown that q absorption lines arise in two-dimensional crystal lattices at a rational fractional number p/q of magnetic-field flux quanta per electron. The absorption lines correspond to different exciton formation energies and are determined by the electron-hole Coulomb interaction. The number of the absorption lines is calculated for electron densities corresponding to the fractional quantum Hall effect.

Exact study of the one-dimensional boson Hubbard model with a superlattice potential

We use Quantum Monte Carlo simulations and exact diagonalization to explore the phase diagram of the Bose-Hubbard model with an additional superlattice potential. We first analyze the properties of superfluid and insulating phases present in the hard-core limit where an exact analytic treatment is possible via the Jordan-Wigner transformation. The extension to finite on-site interaction is achieved by means of quantum Monte Carlo simulations. We determine insulator/superfluid phase diagrams as functions of the on-site repulsive interaction, superlattice potential strength, and filling, finding that insulators with fractional occupation numbers, which are present in the hard-core case, extend deep into the soft-core region. Furthermore, at integer fillings, we find that the competition between the on-site repulsion and the superlattice potential can produce a phase transition between a Mott insulator and a charge density wave insulator, with an intermediate superfluid phase. Our results are relevant to the behavior of ultracold atoms in optical superlattices which are beginning to be studied experimentally.

Fractional occupation number approaches for CAS (2,2) systems based on second‐order density

AbstractIn previous work, we applied a fractional occupation number (FON) approach within density functional theory (DFT) for twisted methylene and found that the applicability of FON‐DFT is limited for the neighborhood of the completely degenerate state, implying the necessity of the second‐order density (SOD) expression of the energy functional. In the succeeding work, we have derived the exact expression of SOD of CAS (2,2) by the natural orbitals (NOs) and their occupation numbers, which we call “one‐body representation of second‐order density” (ORSOD). We present here the FON scheme based on ORSOD by directly optimizing the NOs and their occupation numbers. In contrast to the previous FON‐DFT, this FON scheme is found to work well, in particular, the dissociation limit of the heteronuclear systems, showing that this scheme can be an alternative for complete active space self‐consistent field (2,2) methods for such cases. The current scheme provides a guideline for extension and approximations of the FON scheme to investigate mesoscopic systems with large active space. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. III. Systems of noninteracting electrons.

We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBCk) and the irreducible contracted Schrödinger equations (ICSEk) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC1 leads to a Liouville equation for the one-particle density matrix gamma1=gamma, consistent with our earlier analysis that the IBC1 holds both for a pure and an ensemble state. The IBC1 or the ICSE1 must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for gamma. For a closed-shell state gamma is idempotent, i.e., all natural spin orbitals (NSO's) have occupation numbers 0 or 1, and all cumulants lambdak with k> or =2 vanish. For open-shell states there are NSO's with fractional occupation number, and at the same time nonvanishing elements of lambda2, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of gamma. There are essentially three possibilities to construct the diagonal elements of gamma: (i) to consider the perturbation expansion of the characteristic polynomial of gamma, especially the idempotency for closed-shell states, (ii) to rely on the ICSE1, which (at variance with the IBC1) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction.