Fractional Number Of Electrons Research Articles

Potential-functional embedding theory for molecules and materials

We introduce a potential-functional embedding theory by reformulating a recently proposed density-based embedding theory in terms of functionals of the embedding potential. This potential-functional based theory completes the dual problem in the context of embedding theory for which density-functional embedding theory has existed for two decades. With this potential-functional formalism, it is straightforward to solve for the unique embedding potential shared by all subsystems. We consider charge transfer between subsystems and discuss how to treat fractional numbers of electrons in subsystems. We show that one is able to employ different energy functionals for different subsystems in order to treat different regions with theories of different levels of accuracy, if desired. The embedding potential is solved for by directly minimizing the total energy functional, and we discuss how to efficiently calculate the gradient of the total energy functional with respect to the embedding potential. Forces are also derived, thereby making it possible to optimize structures and account for nuclear dynamics. We also extend the theory to spin-polarized cases. Numerical examples of the theory are given for some homo- and hetero-nuclear diatomic molecules and a more complicated test of a six-hydrogen-atom chain. We also test our theory in a periodic bulk environment with calculations of basic properties of bulk NaCl, by treating each atom as a subsystem. Finally, we demonstrate the theory for water adsorption on the MgO(001)surface.

Improving Band Gap Prediction in Density Functional Theory from Molecules to Solids

A novel nonempirical scaling correction method is developed to tackle the challenge of band gap prediction in density functional theory. For finite systems the scaling correction largely restores the straight-line behavior of electronic energy at fractional electron numbers. The scaling correction can be generally applied to a variety of mainstream density functional approximations, leading to significant improvement in the band gap prediction. In particular, the scaled version of a modified local density approximation predicts band gaps with an accuracy consistent for systems of all sizes, ranging from atoms and molecules to solids. The scaled modified local density approximation thus provides a useful tool to quantitatively characterize the size-dependent effect on the energy gaps of nanostructures.

Density-orbital embedding theory

In the article density-orbital embedding (DOE) theory is proposed. DOE is based on the concept of density orbital (DO), which is a generalization of the square root of the density for real functions and fractional electron numbers. The basic feature of DOE is the representation of the total supermolecular density ${\ensuremath{\rho}}_{s}$ as the square of the sum of the DO ${\ensuremath{\varphi}}_{a}$, which represents the active subsystem $A$ and the square root of the frozen density ${\ensuremath{\rho}}_{f}$ of the environment $F$. The correct ${\ensuremath{\rho}}_{s}$ is obtained with ${\ensuremath{\varphi}}_{a}$ being negative in the regions in which ${\ensuremath{\rho}}_{f}$ might exceed ${\ensuremath{\rho}}_{s}$. This makes it possible to obtain the correct ${\ensuremath{\rho}}_{s}$ with a broad range of the input frozen densities ${\ensuremath{\rho}}_{f}$ so that DOE resolves the problem of the frozen-density admissibility of the current frozen-density embedding theory. The DOE Euler equation for the DO ${\ensuremath{\varphi}}_{a}$ is derived with the characteristic embedding potential representing the effect of the environment. The DO square ${\ensuremath{\varphi}}_{a}^{2}$ is determined from the orbitals of the effective Kohn-Sham (KS) system. Self-consistent solution of the corresponding one-electron KS equations yields not only ${\ensuremath{\varphi}}_{a}^{2}$, but also the DO ${\ensuremath{\varphi}}_{a}$ itself.

Predicting polaronic defect states by means of generalized Koopmans density functional calculations

AbstractLattice defects in semiconductors and wide‐gap materials which create deep levels in an open‐shell electronic configuration can give rise to so‐called defect bound small polarons. This type of defects present a challenge for electronic structure methods because the localization of the defect state and the associated energy levels depend sensitively on the ability of the total‐energy functional to satisfy the physical condition that the energy E(N) must be a piecewise linear function of the fractional electron number N. For practical applications the requirement of a linear E(N) is re‐cast as a generalized Koopmans condition. Since most functionals do not fulfill this condition accurately, we use parameterized perturbations that cancel the non‐linearity of E(N) and recover the correct Koopmans behavior. Starting from standard density functionals, we compare two types of parameterized perturbations, i.e., the addition of on‐site potentials and the mixing of non‐local Fock exchange in hybrid‐functionals. Surveying a range of acceptor‐type defects in II–VI and III–V semiconductors, we present a classification scheme that describes the relation between hole localization and the lattice relaxation of the polaronic state.

An improved long-range corrected hybrid functional with vanishing Hartree–Fock exchange at zero interelectronic distance (LC2gau-BOP)

We present a new long-range corrected (LC) density functional theory (DFT) scheme, named "LC2gau," which combines the best features of our two recently developed hybrid functionals, "LCgau" [J.-W. Song et al., J. Chem. Phys. 127, 154109 (2007)] and "LCgau-core" [J.-W. Song et al., J. Chem. Phys. 129, 184113 (2008)]. By introducing a flexible mixing of Hartree-Fock and DFT exchange in the LCgau scheme, we showed that a DFT functional could simultaneously achieve high accuracy in the reproduction of thermochemical properties, molecular geometries, as well as charge transfer and valence-Rydberg excitation energies. With an alternative mixing of short-range exchange, LCgau-core can reproduce core excitations with high accuracy, especially in the C, N, and O atoms, but at the expense of slightly higher atomization energy errors. We now show that LC2gau can simultaneously perform well for all types of excitations, as well as thermochemistry. In contrast to the previously proposed LC functionals, a notable feature is the inclusion of 100% DFT exchange as the interelectronic distance vanishes, showing that pure DFT approximations can be successfully used at short range, and the importance of including an appropriate correction in the midrange. This is achieved using two Gaussian functions in combination with the error function to describe the exchange partitioning. We rationalize the success of LC2gau by demonstrating a near-linear behavior of the total energies of the C atom as a function of the fractional number of electrons, both in the valence and core regions, which indicates an alleviation of significant self-interaction errors observed with other functionals.

Core-excitation energy calculations with a long-range corrected hybrid exchange-correlation functional including a short-range Gaussian attenuation (LCgau-BOP)

We report the calculations of core-excitation energies of first-row atoms using the time-dependent density functional theory (DFT) and the long-range correction (LC) scheme for exchange-correlation functionals, including LC-BOP, Coulomb-attenuated method BLYP, and our recently developed LCgau-BOP method, which includes a flexible portion of short-range Hartree-Fock (HF) exchange through the inclusion of a Gaussian function in the LC scheme. We show that the LC scheme completely fails to improve the poor accuracy of conventional generalized gradient approximation functionals, while the LCgau scheme gives an accuracy which is an order of magnitude better than BLYP and significantly better than B3LYP. A reoptimization of the two parameters controlling the inclusion of short-range HF exchange in the LCgau method enables the errors to be reduced to the order of 0.1 eV which is competitive with the best DFT methods we are aware of. This reparametrization does not affect the LC scheme and therefore maintains the high accuracy of predicted reaction barrier heights. Moreover, while there is some loss in accuracy in thermochemical predictions compared to the previously optimized LCgau-BOP, rms errors in the atomization energies over the G2 test set are found to be comparable to B3LYP. Finally, we attempt to rationalize the success of the LC and LCgau schemes in terms of the well-known self-interaction error (SIE) of conventional functionals. To estimate the role of the SIE, we examine the total energy calculations for systems with a fractional number of electrons, not only in the highest occupied molecular orbital but also in the 1s-characterized core orbital. Our conclusion is that the inclusion of short-range HF exchange in LC-type functionals can significantly alleviate the problems of the SIE in the core region. In particular, we confirm that the absence of the SIE diagnostics in the core orbital energies correlates with the accurate prediction of core-excitation energies using the newly optimized LCgau approach.

Degeneracy and size consistency in electronic density functional theory

The electronic structure calculations based upon energy density functionals are highly successful and widely used both in solid state physics and quantum chemistry. Moreover, the Hohenberg-Kohn theorems and the Kohn-Sham method provide them with a firm basis. However, several basic issues are not solved, and hamper the progress to achieve high accuracy. In this paper we focus on the conceptual problem of size consistency, basing our analysis on the non-intensive character of the (spin) electronic density in the presence of degeneracy. We also briefly discuss some of the issues concerning fractional electron numbers from the same point of view, analyzing the behavior of the exact functionals for the He and the Hooke's atom series when the number of electrons fluctuates between one and two.

Ab initio quantum mechanical/molecular mechanical simulation of electron transfer process: fractional electron approach.

Electron transfer (ET) reactions are one of the most important processes in chemistry and biology. Because of the quantum nature of the processes and the complicated roles of the solvent, theoretical study of ET processes is challenging. To simulate ET processes at the electronic level, we have developed an efficient density functional theory (DFT) quantum mechanical (QM)/molecular mechanical (MM) approach that uses the fractional number of electrons as the order parameter to calculate the redox free energy of ET reactions in solution. We applied this method to study the ET reactions of the aqueous metal complexes Fe(H(2)O)(6)(2+/3+) and Ru(H(2)O)(6)(2+/3+). The calculated oxidation potentials, 5.82 eV for Fe(II/III) and 5.14 eV for Ru(II/III), agree well with the experimental data, 5.50 and 4.96 eV, for iron and ruthenium, respectively. Furthermore, we have constructed the diabatic free energy surfaces from histogram analysis based on the molecular dynamics trajectories. The resulting reorganization energy and the diabatic activation energy also show good agreement with experimental data. Our calculations show that using the fractional number of electrons (FNE) as the order parameter in the thermodynamic integration process leads to efficient sampling and validate the ab initio QM/MM approach in the calculation of redox free energies.

Acidity of meta- and para-substituted aromatic acids: a conceptual DFT study

Quantum chemical computations with full geometry optimization at the B3LYP/6-31G(d) level in aqueous phase at 298 K are performed to calculate a set of molecular properties for 45 aromatic organic acids such as benzoic acid, phenol, cinnamic acid, benzohydroxamic acid, anilinium ion and their meta- and para-substituted derivatives. In a separate set of calculations the 6-31+G(d) basis set is employed for substituted benzoic and cinnamic acids as representative test cases, to test the effect of diffuse functions. Molecular descriptors such as ionization potential, electron affinity, hardness, chemical potential, global electrophilicity and free energy of deprotonation of these acids are computed. The local descriptors such as Fukui functions and group philicity (ω+g) are calculated with Mulliken Population Analysis (MPA) and Hirschfeld Population Analysis (HPA) schemes. These aromatic acids are reacted with a strong base, OH−. The computed Gibbs free energy of deprotonation, the fractional number of electrons transferred, ΔN and electrophilicity based charge transfer index (ECT) in acid–base reaction with a strong base, OH−, ω+g and group charges are correlated with the experimental pKa values of these acids. It is found that this approach is particularly effective in discussing trends of changes in acidity of intimately related molecules. The effect of substituents on these descriptors is also studied. These parameters are correlated with experimental Hammett substituent constants (σ). The ECT, fractional number of electron transfer (ΔN) and group charge correlate strongly with pKa and σ in separate groups of aromatic acids. Minimum energy, maximum hardness and minimum electrophilicity principles are tested for the acid–base reactions.

Development of exchange-correlation functionals with minimal many-electron self-interaction error

New exchange-correlation functionals that address the important issue of many-electron self-interaction are developed. This is carried out by considering the performance of the functional on systems with fractional numbers of electrons at the same time as more standard thermochemical tests. The inclusion of Coulomb-attenuated exchange in the functional is facilitated by use of the adiabatic connection coupled with a short-range and long-range splittings. The new functionals have a good performance on thermochemistry and a much improved description of the total energy versus number of electrons and henceforth a much smaller many-electron self-interaction error.

Tests of functionals for systems with fractional electron number

In the exact theory, the ground state energy of an open system varies linearly when the electron number is changed between two adjacent integers. This linear dependence is not reproduced by common approximate density functionals. Deviation from linearity in this dependence has been suggested as a basis for the concept of many-electron self-interaction error (SIE). In this paper, we quantify many-electron SIE of a number of approximations by performing calculations on fractionally charged atoms. We demonstrate the direct relevance of these studies to such problems of common approximate functionals as instabilities of anions, spurious fractional charges on dissociated atoms, and poor description of charge transfer. Semilocal approximations have the largest many-electron SIE, which is only slightly reduced in typical global hybrids. In these approximations the energy versus fractional electron number curves upward, while in Hartree-Fock theory the energy curves downward. Perdew-Zunger self-interaction correction [Phys. Rev. B 23, 5048 (1981)] significantly reduces the many-electron SIE of semilocal functionals but impairs their accuracy for equilibrium properties. In contrast, a long-range corrected hybrid functional can be nearly many-electron SIE-free in many cases (for reasons we discuss) and at the same time performs remarkably well for many molecular properties.

Fractional numbers of electrons in Kohn–Sham theory

A semi-empirical self-interaction correction, based on the fractional number of electrons q A surrounding each atom A, is investigated in generalised gradient approximation density functional theory. The correction, which involves locally scaling the exchange functional (Becke 1988) by q A 0.371, leads to a significantly improved potential energy curve for H 2 +. It also shifts the H 2 potential energy curve towards that of Hartree–Fock, and increases the barrier for the reaction H + H 2→H 2 + H. An alternative correction q A −0.174 leads to a much improved dissociation of H 2 in a restricted formalism.

Degenerate ground states and a fractional number of electrons in density and reduced density matrix functional theory

For a linear combination of electron densities of degenerate ground states, it is shown that the value of any energy functional is the ground state energy, if the energy functional is exact for ground state densities, size consistent, and translational invariant. The corresponding functional of kinetic and interaction energy is the linear combination of the functionals of the degenerate densities. Without invoking ensembles, it is shown that the energy functional of fractional number electrons is a series of straight lines interpolating its values at integers. These results underscore the importance of grand canonical ensemble formulation in density functional theory.

Ensuring proper short-range and asymptotic behavior of the exchange-correlation Kohn-Sham potential by modeling with a statistical average of different orbital model potentials

The long-range asymptotic behavior of the exchange-correlation Kohn-Sham (KS) potential {nu}{sub xc} and its relation to the exchange-correlation energy E{sub xc} are considered using various approaches. The line integral of {nu}{sub xc}([{rho}];r) yielding the exchange-correlation part {Delta}E{sub xc} of a relative energy {Delta}E of a finite system, shows that a uniform constant shift of {nu}{sub xc} never shows up in any physically meaningful energy difference {Delta}E. {nu}{sub xv} may thus be freely chosen to tend asymptotically to zero or to some nonzero constant. Possible choices of the asymptotics of the potential are discussed with reference to the theory of open systems with a fractional number of electrons. The authors adhere to the conventional choice {nu}{sub xc}({infinity}) = 0 for the asymptotics of the potential leading to {epsilon}{sub N} = {minus}I{sub p} for the energy {epsilon}{sub N} of the highest occupied orbital. A statistical average of orbital dependent model potentials is proposed as a way to model {nu}{sub xc}. An approximate potential {nu}{sub xco}{sup SAOP} with exact {minus}1/r asymptotics is developed using the statistical average of, on the one hand, a model potential {nu}{sub xc{sigma}}{sup Ei} for the highest occupied KS orbital {psi}{sub N{sigma}} and, on the other hand, a modelmore » potential {nu}{sub xc}{sup GLB} for other occupied orbitals. It is demonstrated for the well-studied case of the Ne atom, that calculations with the new model potential can, in principle, reproduce perfectly all energy characteristics.« less

Differential virial theorem for the fractional electron number: Derivative discontinuity of the Kohn-Sham exchange-correlation potential

We extend the differential virial theorem put forth by Holas and March [Phys. Rev. A 51, 2040 (1995)] to a statistical mixture of densities corresponding to different numbers of electrons. This allows us to derive an expression for the gradient of the exchange-correlation potential for densities that lead to noninteger numbers of electrons. We apply this to study the jump in the exchange-correlation potential as the electron number increases through an integer. We show that the difference between the gradient of the exchange-correlation potential for $N$ electrons, where $N$ is an integer, and that for $N+f$ electrons, where $f$ is a small positive number, is similar to the electric field in and around a charged metal sphere: It is zero inside, peaks at the surface, and then decays. Consequently, the corresponding potential is constant in the interior of the system and decays sharply at the surface.

Aromaticity, electronic structure and molecular dimension in the adsorption of organic compounds on mercury—I. The metal/adsorbate interaction

The adsorbability of a wide series of aromatic derivatives (38) on mercury from aqueous solution at constant potential ( E pzc or close to it) cannot be simply accounted for by the “aromaticity” of the compounds, at any rate modulated by substituent groups on the aromatic moiety, even if the adsorbates show a common planar disposition on the surface in the adsorbed state. The interfacial activity can be successfully related to the process of formation of a “soft-soft” couple (in the Pearson's concept) between mercury and the organic, in competition with the “hard-soft” couple between the metal and water. Desorption is then explained with a higher “hard” character of mercury at increasing negative potential. In this view, the feasibility of a partial charge transfer, as expressed by λ, calculated for neutral organic adsorbates, corresponding to a positive Δ N value (Δ N, fractional number of electrons transferred from the adsorbate to the mercury) can be confirmed. A simple relation between adsorbability and “bulkyness” of the compounds, in relation to the dislodgement of water molecules from the surface occurring in the adsorption process is proposed and discussed.

Extensions of the density-functional theory (DFT) to systems with a fractional number of electrons: Consequences for the meaning of the DFT energy levels.

The relation between the ionization potential I(N) and the energy ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{N}}$(N), relative to the value of the external potential at infinity, of the highest occupied orbital level in the density-functional theory (DFT) for finite systems is examined on the basis of the extensions of the energy functional to the case where the number of electrons of the system is not an integer. The difference between I(N) and -${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{N}}$(N) is analyzed, as in other works, as a relaxation of the system following the removal of an electron. A problem closely related to the value of this difference is that of the constant value of the exchange and correlation contribution to the effective single-electron potential entering in the DFT, far away from the system. To illustrate this, we propose a sum rule that can be considered as a constraint for the exact energy functional.

Electrosorption of ‘hard’ and ‘soft’ aromatic derivatives on mercury

An ‘intrinsic’ value of the adsorption Gibbs energy (ΔG°int)∥, not directly proportional to the measured ΔG∥ads, has been evaluated for a set of 28 aromatic, heterocyclic and naphthalenic derivatives. The structural feature of (ΔG°int)∥ is rationalised in terms of a ‘soft’ interaction with mercury at the point of zero charge, on the basis of the parr and Pearson concept, accounting on a quantitative basis for the different adsorbability of the compounds. Moreover, the partial charge-transfer coefficient, λ, has been related in a satisfactory way to ΔN(the fractional number of electrons transferred) calculated taking into account the absolute electronegativitiy, χ, and the hardness, η, both of the adsorbate and of mercury.

Energy formulas for atoms and molecules

Exact expressions are derived for the nonrelativistic energy of an atom in terms of its chemical potential and the electrostatic potential at the nucleus. The formulas do not require the consideration of fractional numbers of electrons. Generalizations to multinuclear systems are included.

Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy

The Hohenberg-Kohn theorem is extended to fractional electron number $N$, for an isolated open system described by a statistical mixture. The curve of lowest average energy ${E}_{N}$ versus $N$ is found to be a series of straight line segments with slope discontinuities at integral $N$. As $N$ increases through an integer $M$, the chemical potential and the highest occupied Kohn-Sham orbital energy both jump from ${E}_{M}\ensuremath{-}{E}_{M\ensuremath{-}1}$ to ${E}_{M+1}\ensuremath{-}{E}_{M}$. The exchange-correlation potential $\frac{\ensuremath{\delta}{E}_{\mathrm{xc}}}{\ensuremath{\delta}n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}$ jumps by the same constant, and $\frac{{\mathrm{lim}}_{r\ensuremath{\rightarrow}\ensuremath{\infty}}\ensuremath{\delta}{E}_{\mathrm{xc}}}{\ensuremath{\delta}n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}>~0$.