Fractional Number Of Electrons Research Articles

Ensemble Ground State of a Many-Electron System with Fractional Electron Number and Spin: Piecewise-Linearity and Flat-Plane Condition Generalized.

Description of many-electron systems with a fractional electron number (Ntot) and fractional spin (Mtot) is of great importance in physical chemistry, solid-state physics, and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system and characterize the dependence of the energy and the spin-densities on both Ntot and Mtot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at a constant Ntot, as a jump in the Kohn-Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.

Quantum states of physical domains in molecular systems: A three-state model approach

The physical regions (domains or basins) within the molecular structure are open systems that exchange charge between them and, consequently, house a fractional number of electrons (net charge). The natural framework describing the quantum states for these domains is the density matrix (DM) in its grand-canonical version, which corresponds to a convex expansion into a set of basis states of an integer number of electrons. In this report, it is shown that the solution for these quantities is supported by the DM expansion into three states of different numbers of particles: the neutral and two (edge) ionic states. The states and the average number of particles in the domains (fractional occupation population) are determined by the coefficients of the expansion in terms of the fundamental transference magnitudes, revealing the donor/acceptor character of the domains by which the quantum accessible states are discussed.

Perturbed reactivity descriptors in the two parabolas model of fractional electron number

Perturbed reactivity descriptors in the two parabolas model of fractional electron number

Understanding Density-Driven Errors for Reaction Barrier Heights.

Delocalization errors, such as charge-transfer and some self-interaction errors, plague computationally efficient and otherwise accurate density functional approximations (DFAs). Evaluating a semilocal DFA non-self-consistently on the Hartree-Fock (HF) density is often recommended as a computationally inexpensive remedy for delocalization errors. For sophisticated meta-GGAs like SCAN, this approach can achieve remarkable accuracy. This HF-DFT (also known as DFA@HF) is often presumed to work, when it significantly improves over the DFA, because the HF density is more accurate than the self-consistent DFA density in those cases. By applying the metrics of density-corrected density functional theory (DFT), we show that HF-DFT works for barrier heights by making a localizing charge-transfer error or density overcorrection, thereby producing a somewhat reliable cancellation of density- and functional-driven errors for the energy. A quantitative analysis of the charge-transfer errors in a few randomly selected transition states confirms this trend. We do not have the exact functional and electron densities that would be needed to evaluate the exact density- and functional-driven errors for the large BH76 database of barrier heights. Instead, we have identified and employed three fully nonlocal proxy functionals (SCAN 50% global hybrid, range-separated hybrid LC-ωPBE, and SCAN-FLOSIC) and their self-consistent proxy densities. These functionals are chosen because they yield reasonably accurate self-consistent barrier heights and because their self-consistent total energies are nearly piecewise linear in fractional electron number─two important points of similarity to the exact functional. We argue that density-driven errors of the energy in a self-consistent density functional calculation are second order in the density error and that large density-driven errors arise primarily from incorrect electron transfers over length scales larger than the diameter of an atom.

Assessment of drug loading and release efficiencies of zigzag (8, 0) single-walled carbon nanotube as a Bendamustine hydrochloride drug delivery system in silico: DFT approach

ABSTRACT In view of non-immunogenic containers of biomolecules, we investigated zigzag (8, 0) single-walled carbon nanotube (SWCNT) as a vehicle for the selectivity delivery of Bendamustine hydrochloride (Benda-HCl) drug to the cell target in silico using the density functional theory (DFT) approach. During drug interaction with the carbon nanotube surface, at about 56 kJ/mol of energy is liberated in the most stable structure of the Benda-HCl@SWCNT complex. Therefore, there is an attractive van der Waals (vdW) interaction between the drug molecule and SWCNT during the exothermic adsorption. The kinetic and thermodynamic stability of the Benda-HCl@SWCNT complex is evaluated with the reactivity descriptors such as individual energy change of donor (ΔE B(A)), the fractional number of electrons (ΔN), the overall stabilisation energy (ΔE SE) and the individual energy change of acceptor (ΔE A(B)). The analysis of the electronic properties shows that no sensible charge transfer occurs between the nanotube and the drug molecule due to the physisorption process. Furthermore, the binding affinity of the drug molecule to SWCNT is decreased in an acidic environment. The findings of the present research work suggest that carbon nanotube can be effectively utilised as a Bendamustine hydrochloride drug delivery vehicle. Abbreviations: Benda-HCl: bendamustine hydrochloride; BSSE: basis set superposition error; DFT: density functional theory; E inter: interaction energy; HOMO: the highest occupied molecular orbital; LUMO: the lowest unoccupied molecular orbital; NBO: natural bond orbital analysis; PDOS: projected density of state; TDOS: total density of states

Dissociating limit in density functional theory with Coulomb optimal transport cost

In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multi-marginals optimal transport cost with Coulombian electron-electron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of N=2 electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak* convergence of measures.

Piecewise nonlinearity and capacitance in the joint density functional theory of extended interfaces

The ab initio simulation of charged interfaces in the framework of density functional theory (DFT) is heavily employed for the study of electrochemical energy conversion processes. The capacitance is the primary descriptor for the response of the electrochemical interface. It is essentially equal to the inverse of the energy curvature as a function of electron number, and as such there appears a conflict with the fundamental principle of piecewise linearity in DFT that requires the energy curvature to be zero at fractional electron numbers, i.e. almost everywhere. To resolve this conflict, we derive an exact expression between the energy curvature and the Kohn-Sham density of states, the local density of states, and the Fukui potential. We find that the piecewise linearity requirement does not hold for the volume- or area-specific energy of extended systems and surfaces. Applied to the joint density functional theory of an electrode-electrolyte interface, including the ionic and dielectric response of the electrolyte, the same expression represents a rigorous basis for the partitioning of the total interfacial capacitance into contributions of the quantum capacitance, space-charge capacitance, and electrochemical double-layer capacitance. It provides insight into the influence of the electrode material, thickness, and temperature on the charging characteristics, as demonstrated by results for a bulk gold electrode, a single-layer gold electrode, and a single-layer graphene electrode.

Theoretical study on carbonaceous materials as high efficient carriers for crizotinib drug in liquid water by density functional theory approach

In order to get a precise insight into the nature of the intermolecular interaction of crizotinib drug with carbon nanomaterials, i.e., fullerene C70 and graphene nanosheet as high efficient carriers, the density functional theory calculations are applied on the binding strength, the molecular structures, electronic properties, and charge transfer in the water phase. Based on the calculated adsorption energy values, the physical nature for the interaction of the adsorbed drug with carbon nanostructures is approved. The obtained results suggest that crizotinib drug exhibits the strongest affinity for the adsorption on graphene nanosheet in comparison to fullerene C70. By considering the optimized geometries of the considered complexes, the π–π stacking interactions between π orbitals of the carbon nanostructures and the six-membered aromatic ring of the adsorbed molecule are observed. As a result of the physisorption process, no structural or electronic change of the adsorbed drug is found. Analysis of the energetic parameter of atoms-in-molecule method suggests medium strength intermolecular interaction of crizotinib drug with carbon nanostructures with partially covalent nature. The interactions of crizotinib drug and carbon nanostructures on the basis of two reactivity descriptors such as the overall stabilization energy and the fractional number of electrons transferred have been explained. The theoretical results show that the interaction of graphene nanosheet with crizotinib agent is more favorable than fullerene C70. The analysis of the natural bond orbital shows the capability of the carbon nanostructures to accept precisely the electron from crizotinib drug molecule during the adsorption process.

N‐centered ensemble density‐functional theory for open systems

AbstractTwo (so‐called left and right) variants of N‐centered ensemble density‐functional theory (DFT) are presented. Unlike the original formulation of the theory, these variants allow for the description of systems with a fractional electron number. While conventional DFT for open systems uses only the true electron density as basic variable, left/right N‐centered ensemble DFT relies instead on (a) a fictitious ensemble density that integrates to a central (integral) number N of electrons, and (b) a grand canonical ensemble weight α which is equal to the deviation of the true electron number from N. Within such a formalism, the infamous derivative discontinuity that appears when crossing an integral number of electrons is described exactly through the dependence in α of the left and right N‐centered ensemble Hartree‐exchange‐correlation density functionals. Incorporating N‐centered ensembles into existing density‐functional embedding theories is expected to pave the way toward the in‐principle‐exact description of an open fragment by means of a pure‐state N‐electron many‐body wavefunction. Work is currently in progress in this direction.

Fractional Charge by Fixed-Node Diffusion MonteCarlo Method.

Fixed-node diffusion MonteCarlo (FNDMC) method is a stochastic quantum many-body approach that has a great potential in electronic structure theory. We examine how FNDMC total energy E(N) satisfies exact constraints, linearity and derivative discontinuity, versus fractional electron number N, if combined with mean-field trial wave functions that miss such features. H and Cl atoms with fractional charge reveal that FNDMC method is well able to restore the piecewise linearity of E(N). The method uses ensemble and projector ingredients to achieve the correct charge localization. A water-solvated Cl^{-} complex illustrates superior performance of FNDMC method for charged noncovalent systems.

Description of intermolecular charge transfer with subsystem density-functional theory.

Efficient quantum-chemical methods that are able to describe intermolecular charge transfer are crucial for modeling organic semiconductors. However, the correct description of intermolecular charge transfer with density-functional theory (DFT) is hampered by the fractional charge error of approximate exchange-correlation (xc) functionals. Here, we investigate the charge transfer induced by an external electric field in a tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) complex as a test case. For this seemingly simple model system, a supermolecular DFT treatment fails with most conventional xc functionals. Here, we present an extension of subsystem DFT to subsystems with a fractional number of electrons. We show that within such a framework, it becomes possible to overcome the fractional charge error by enforcing the correct dependence of each subsystem's total energy on the subsystem's fractional charge. Such a subsystem DFT approach allows for a correct description of the intermolecular charge transfer in the TTF-TCNQ model complex. The approach presented here can be generalized to larger molecular aggregates and will thus allow for modeling organic semiconductor materials accurately and efficiently.

First-principles study of self-trapped holes and acceptor impurities in Ga2O3 polymorphs

We investigate the stability of self-trapped holes (STHs) and the acceptor levels of substitutional Mg and N impurities in $\ensuremath{\alpha}$-, $\ensuremath{\beta}$-, $\ensuremath{\delta}$-, and $\ensuremath{\varepsilon}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ using first-principles calculations based on the hybrid functional approach to assess their $p$-type dopability. When Fock-exchange and screening parameter values in the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional are optimized to satisfy the generalized Koopmans' theorem for a STH level in $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$, the band gap is slightly overestimated, while functionals that well reproduce the band gap show slight convex behavior against the fractional electron number. However, the absolute position of the STH level at a fixed geometry is nearly independent of the parameter value, showing that the results are robust as long as the STH geometry and localized electronic nature are appropriately described and the band edges are well reproduced. In all of the polymorphs, holes localize with high self-trapping energies rather than being delocalized. Furthermore, both Mg and N impurities introduce polaronic acceptor states, and their acceptor levels lie far above the valence band maximum in all the polymorphs. Thus, the $p$-type doping of the four ${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ polymorphs seems unfeasible in terms of the STH formation and the related deep, polaronic acceptor nature of the Mg and N impurities.

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.

Mechanism of CO2 Reduction at Copper Surfaces: Pathways to C2 Products

On the basis of constraints from reported experimental observations and density functional theory simulations, we propose a mechanism for the reduction of CO2 to C2 products on copper electrodes. To model the effects of an applied potential bias on the reactions, calculations are carried out with a variable, fractional number of electrons on the unit cell, which is optimized so that the Fermi level matches the actual chemical potential of electrons (i.e., the applied bias); an implicit electrolyte model allows for compensation of the surface charge so that neutrality is maintained in the overall simulation cell. Our mechanism explains the presence of the seven C2 species that have been detected in the reaction, as well as other notable experimental observations. Furthermore, our results shed light on the difference in activities toward C2 products between the (100) and (111) facets of copper. We compare our methodologies and findings with those in other recent mechanistic studies of the copper-catalyzed C...

Thermodynamic hardness and the maximum hardness principle.

An alternative definition of hardness (called the thermodynamic hardness) within the grand canonical ensemble formalism is proposed in terms of the partial derivative of the electronic chemical potential with respect to the thermodynamic chemical potential of the reservoir, keeping the temperature and the external potential constant. This temperature dependent definition may be interpreted as a measure of the propensity of a system to go through a charge transfer process when it interacts with other species, and thus it keeps the philosophy of the original definition. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures and up to temperatures of chemical interest, one finds that for zero fractional charge, the thermodynamic hardness is proportional to T-1(I-A), where I is the first ionization potential, A is the electron affinity, and T is the temperature. However, the thermodynamic hardness is nearly zero when the fractional charge is different from zero. Thus, through the present definition, one avoids the presence of the Dirac delta function. We show that the chemical hardness defined in this way provides meaningful and discernible information about the hardness properties of a chemical species exhibiting integer or a fractional average number of electrons, and this analysis allowed us to establish a link between the maximum possible value of the hardness here defined, with the minimum softness principle, showing that both principles are related to minimum fractional charge and maximum stability conditions.

On extending Kohn-Sham density functionals to systems with fractional number of electrons.

We analyze four ways of formulating the Kohn-Sham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the Kohn-Sham and the generalized Kohn-Sham (GKS) non-interacting v-representable density domain for integer systems to four different sets of densities for fractional systems. In particular, these density sets are (I) ensemble interacting N-representable densities, (II) ensemble non-interacting N-representable densities, (III) non-interacting densities by the Janak construction, and (IV) non-interacting densities whose composing orbitals satisfy the Aufbau occupation principle. By proving the equivalence of the underlying first order reduced density matrices associated with these densities, we show that sets (I), (II), and (III) are equivalent, and all reduce to the Janak construction. Moreover, for functionals with the ensemble v-representable assumption at the minimizer, (III) reduces to (IV) and thus justifies the previous use of the Aufbau protocol within the (G)KS framework in the study of the ground state of fractional electron systems, as defined in the grand canonical ensemble at zero temperature. By further analyzing the Aufbau solution for different density functional approximations (DFAs) in the (G)KS scheme, we rigorously prove that there can be one and only one fractional occupation for the Hartree Fock functional, while there can be multiple fractional occupations for general DFAs in the presence of degeneracy. This has been confirmed by numerical calculations using the local density approximation as a representative of general DFAs. This work thus clarifies important issues on density functional theory calculations for fractional electron systems.

On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals.

We provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. This suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.

Extraction of state-resolved information from systems with a fractional number of electrons within the framework of time-dependent density functional theory

The determination of the state-resolved physical information within the framework of time-dependent density functional theory has remained a widely open question. We demonstrated the ability to extract the state-resolved probability from the knowledge of only the time-dependent density, which has been used as the basic variable within the time-dependent density functional theory, with the help of state-resolved single-electron capture experiments for collisions of protons on helium in the energy range of 2-100 keV/amu. The present theoretical results for capture into states of H(1s), H(2s), and H(2p) are in good agreement with the most sophisticated experimental results of H+ + He(1s2) system, validating our approach and numerical implementation.

Communication: Two types of flat-planes conditions in density functional theory.

Using results from atomic spectroscopy, we show that there are two types of flat-planes conditions. The first type of flat-planes condition occurs when the energy as a function of the number of electrons of each spin, Nα and Nβ, has a derivative discontinuity on a line segment where the number of electrons, Nα + Nβ, is an integer. The second type of flat-planes condition occurs when the energy has a derivative discontinuity on a line segment where the spin polarization, Nα - Nβ, is an integer, but does not have a discontinuity associated with an integer number of electrons. Type 2 flat planes are rare-we observed just 15 type 2 flat-planes conditions out of the 4884 cases we tested-but their mere existence has implications for the design of exchange-correlation energy density functionals. To facilitate the development of functionals that have the correct behavior with respect to both fractional number of electrons and fractional spin polarization, we present a dataset for the chromium atom and its ions that can be used to test new functionals.

Fractional electron number, temperature, and perturbations in chemical reactions.

We provide a perspective on the role of non-integer electron number in the density functional theory approach to chemical reactivity (conceptual DFT), emphasizing that it is important to not only treat reagents as open systems, but also as non-isolated systems, in contact with their surroundings. The special case of well-separated reagents is treated in some detail, as is the case where reagents interact strongly. The resulting expressions for the chemical potential of an acid, μacid = -(αI + A)/(1 + α), and a base, μbase = -(I + αA)/(1 + α), elucidate and generalize the assumptions inherent in the chemical potential models of Mulliken (α = 1) and Gazquez, Cedillo, and Vela (α = 3). In the strongly-interacting limit, it is appropriate to model the effects of the environment as a state-specific effective temperature, thereby providing a rigorous justification for the phenomenological effective-temperature model one of the authors previously proposed. The framework for the strongly interacting limit subsumes our model for weakly-interacting subsystems at nonzero temperature, the case of open but otherwise noninteracting subsystems, and the zero-temperature limit.