Direct approximation of the long- and short-range components of the exchange-correlation Kohn-Sham potential

An approximation scheme was developed for the Kohn-Sham exchange-correlation potential v_xcσ, making use of a partitioning of v_xcσ into a long-range screening v_scrσ and a short-range response v_resp component. For the response part, a model v^mod_respσ was used, which represents v_resp as weighted orbital density contributions, the weights being determined by the orbital energies. v^mod_respσ possesses the proper short-range behavior and the atomic-shell stepped structure characteristic for v_resp. For the screening part, two model potentials v^mod_scrσ were used, one with the accurate Slater potential; the other one with the generalized gradient approximation (GGA) for the exchange part. Both use the GGA for the Coulomb correlation contribution to v_scrσ. The scheme provides an adequate approximation to v_xcσ in the outer-valence region with both the proper asymptotics and a rather accurate estimate of the ionization potential from the highest one-electron energy and a reasonable estimate of atomic E_xc and total energies E_tot. © 1997 John Wiley & Sons, Inc.     

Accurate local spin-polarized exchange potential: Reconciliation of generalized Slater and Kohn–Sham methods

Using simple physical arguments, a local spin-polarized exchange potential, V_xσ, is constructed from the single-particle Hartree–Fock (HF ) potentials (generalized Slater method) that reduces to the usual Kohn–Sham (KS ) result in the uniform gas limit. Numerical results for 10 closed subshell atoms demonstrate that the total energy calculated employing this V_xσ is closer to the exact KS results than those of other standard exchange approximations with electron densities and highest occupied orbital eigenvalues that closely approximate the HF results.     

Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal–vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn–Sham density functional theory (KS‐DFT) exchange‐correlation potential energy v_xc(r) in terms of the irreducible electron self‐energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange‐correlation potential energy is derived to be v_xc(z → ∞) = ?α_KS,xc/z, and its exchange and correlation components to be v_x(z → ∞) = ?α_KS,x/z and v_c(z → ∞) = ?α_KS,c/z, respectively. The analytical expressions for the coefficients α_KS,xc and α_KS,x show them to be dependent on the bulk‐metal Wigner–Seitz radius and the barrier height at the surface. The coefficient α_KS,c = 1/4 is determined in the plasmon‐pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of v_xc(z) in the vacuum region is image‐potential‐like but not the commonly accepted one of ?1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q‐DFT) shows that both the Pauli W_x(z → ∞) and lowest‐order correlation‐kinetic W(z → ∞) components of the exchange potential energy v_x(z → ∞), and the Coulomb W_c(z → ∞) and higher‐order correlation‐kinetic components of the correlation potential energy v_c(z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation‐kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS‐DFT is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

The Flipping of CO Ligand Groups in Metal Carbonyl Compounds and its Frequency in Fe(CO)5

It has been proven qualitatively by a number of authors using variable temperature NMR experiments that most metal carbonyl complexes are nonrigid. A quantitative determination of the ligand exchange frequency v_e is often achieved by a line shape analysis or by measurement of the transverse relaxation time T₂ using the Carr-Purcell method. In the case of a “very fast” exchange, however, both methods prove unsuccessful. It is shown in this study that a simultaneous fit of IR or Raman spectra on the one hand and NMR spectra on the other can make possible the determination of v_e for the “very fast” exchange and can also facilitate the determination of v_e in “slow” and “medium” exchange cases considerably. The ligand exchange frequency thus found for Fe(CO)₅, 1.1 × 10¹⁰s^?1, is unexpectedly high; comparison with variable temperature measurements on solid Fe(CO)₅, yields similar energy barriers. A mechanism of exchange closely related to the “Berry mechanism” is proposed. Finally the consequences of this surprisingly large ligand exchange rate are discussed with respect to IR band assignments for molecular “fragments” M(CO)^x (where x=coordination number, and M is a transition metal, typically lanthanoid or actinoid).     

Exchange energy for two closed shells generated by a bare Coulomb potential energy −<Emphasis Type="Italic">Ze</Emphasis><Superscript>2</Superscript>/<Emphasis Type="Italic">r</Emphasis> in the limit of large <Emphasis Type="Italic">Z</Emphasis>, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze ²/r and (ii) the exchange energy density e_x(r){\varepsilon_x({\bf r})} . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential 2e_x(r)/n(r){2\varepsilon_x(r)/n(r)} , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.     

Inverting the external-potential-to-electron density map in systems of electrons by minimization of a least-squares fit functional for analytic potentials

In a system of electrons, there is a map connecting any external potential v with its electron density ρ _v. In this work, we describe a procedure for inverting this potential-to-density map, so that potentials (if any) corresponding to a target density ρ_t can be obtained. We give the trial external potential v _α , an analytic expression depending on a number of parameters α = (α₁, …) and then minimize the least-squares integral ∫(ρ _α − ρ_t)² d r by the conjugate gradient method, where ρ _α is the density corresponding to v _α . The implementation takes advantage of the analytic nature of v _α . The procedure can be applied to any system and quantum chemistry model, and works both for ground and excited states, as well as for ensembles of states. The method is tested with some excited states of the particle-in-a-box model, confirming the lack of a Hohenberg–Kohn theorem for excited states. It is also applied to the first singlet excited state of the helium atom, where, apart from the nucleus–electron attraction potential, some generalized external potentials are found.     

Step structure in the atomic Kohn-Sham potential

In this work we analyze the exchange-correlation potentialv _xc within the Kohn-Sham approach to density functional theory for the case of atomic systems. The exchange-correlation potential is written as the sum of two potentials. One of these potentialsv _xc,scr is the long-range. Coulombic potential of the coupling constant integrated exchange-correlation hole which represents the screening of the two-particle interactions due to exchange-correlation effects. The other potentialv _xc,scr ^resp contains the functional derivative with respect to the electron density of the coupling constant integrated pair-correlation function representing the sensitivity of this exchange-correlation screening to density variations. As explicit expression of the exchange-part of this functional derivative is derived using an approximation for the Greens function of the Kohn-Sham system and is shown to display a distinct atomic shell structure. The corresponding potentialv _xc,scr ^resp has a clear step structure and is constant within the atomic shells and changes rapidly at the atomic shell boundaries. Numerical examples are presented for the Be and Kr atoms using the Optimized Potential Model (OPM).     

Density-gradient analysis for density functional theory: Application to atoms

We present an analysis of local or semilocal density functionals for the exchange-correlation energy by decomposing them into their gradients r_s (local Seitz radius), ζ (relative spin polarization), and s (reduced density gradient). We explain the numerical method pertaining to this kind of analysis and present results for a few atoms and ions. The atomic shell structure is prominent, and only the ranges 0 < r_s < 10 and 0 < s < 3 are important. The low-density and large-gradient domains, where the approximations for the exchange-correlation energy are least trustworthy, have very little weight. © 1997 John Wiley & Sons, Inc.     

A simple model for the Slater exchange potential and its performance for solids

A simple local model for the Slater exchange potential is determined by least square fit procedure from Hartree–Fock (HF) atomic data. Since the Slater potential is the exact exchange potential yielding HF electron density from Levy‐Perdew‐Sahni density functional formalism (Levy et al., Phys. Rev. A 1984, 30, 2745), the derived local potential is significantly more negative than the conventional local density approximation. On the set of 22 ionic, covalent and van der Waals solids including strongly correlated transition metal oxides, it has been demonstrated, that this simple model potential is capable of reproducing the band gaps nearly as good as popular meta GGA potentials in close agreement with experimental values.     

Cluster calculations of the H2O/Pt(111) system

The electronic interaction between water and a Pt(111) surface as evaluated for different Pt_x(H₂O)_y clusters is discussed. Hartree–Fock–Slater (HFS ) one-electron ground state energies, ionization potentials, partial densities of states, and Mulliken occupation numbers are related to bonding shifts, as well as initial and final state screening for different orientations of the molecule. The formation of Pt? H₂O bonds are sensitive to the orientation since surface oriented H atoms bridge the spatial separation between O 2p and Pt 5d orbitals and thus increase the intermixing of metal and adsorbate orbitals. The dipole moment and the net charge of the H₂O molecule is also discussed. Finally, approximations of the metal–H₂O potential for use in statistical models of the liquid–metal interface are suggested.     

Exchange energy density and exchange potential via a hartree–fock plus mp2 study of the electron liquid in the ground-state conformer of glycine

Very recent criticisms of existing exchange-correlation functionals by Wanko et al. applied to systems of biological interest have led us to reopen the question of the ground-state conformer of glycine: the simplest amino acid. We immediately show that the global minimum of the Hartree–Fock (HF) ground-state leads to a planar structure of the five non-hydrogenic nuclei, in the non-ionized form NH₂–CH₂–COOH. This is shown to lie lower in energy than the zwitterion structure NHB₃ ⁺–CH₂–COO^?, as required by experiment. Refinement of the nuclear geometry using second-order Møller–Plesset perturbation theory (MP2) is also carried out, and bond lengths are found to accord satisfactorily with experimentally determined values. The ground-state electron density for the MP2 geometry is then redetermined by HF theory and equidensity contours are displayed. The HF first-order density matrix γ( r , r ′) is then used to obtain similar exchange-energy density (ε _x ( r )) contours for the lowest conformer of glycine. At first sight, their shape looks almost the same as for the density ρ( r ), which seems to vindicate the LDA proportional to ρ( r )^3/4. However, by way of an analytically soluble model for an atomic ion, it is shown that this has to be corrected to obtain an accurate HF exchange energy E_x as the volume integral of ε _x ( r ). Finally, recognizing that for larger amino acids, the use of HF plus MP2 perturbation corrections will become prohibitive, we have used the HF information for ε _x ( r ) and ρ( r ) to plot the truly non-local exchange potential proposed by Slater, from the density matrix γ( r , r ′). This latter calculation should be practicable for large amino acids, but there adopting Becke's one-parameter form of ε _x ( r ) correcting LDA exchange. Some future directions are suggested.     

Doublet instability and the molecular structure of AlO2

The possible C_s, C_2v, and C_∞v structures of AlO₂ corresponding to the two lowest electronic states which dissociate into the neutral Al(²P) and O₂(³Σ_g^?) fragments have been investigated at the ab initio self-consistent field (SCF) and CI levels using nonempirical pseudopotentials. The most stable structure corresponds to a C_2v symmetry in the ²A₂ electronic state. However, this structure presents the three-center three-electron Hartree-Fock instability and CASSCF calculations were necessary to unequivocally characterize it as true minimum. Moreover, only another stable structure, of C_2v geometry, was found to be a minimum, corresponding to a low-lying excited state of ²A₁ symmetry. The optimized C_∞v structures were not minima on the corresponding potential energy surfaces and no evidence of any stable C_s structure was found. Calculating values are compared with the different experimental data obtained from the reaction of Al and O₂ in frozen gas inert matrices.     

Tuning the Electrochemical Performance of Titanium Carbide MXene by Controllable In Situ Anodic Oxidation

MXenes are a class of two‐dimensional (2D) transition metal carbides, nitrides and carbonitrides that have shown promise for high‐rate pseudocapacitive energy storage. However, the effects that irreversible oxidation have on the surface chemistry and electrochemical properties of MXenes are still not understood. Here we report on a controlled anodic oxidation method which improves the rate performance of titanium carbide MXene (Ti₃C₂T_x, T_x refers to ‐F, =O, ‐Cl and ‐OH) electrodes in acidic electrolytes. The capacitance retention at 2000 mV s^?1 (with respect to the lowest scan rate of 5 mV s^?1) increases gradually from 38 % to 66 % by tuning the degree of anodic oxidation. At the same time, a loss in the redox behavior of Ti₃C₂T_x is evident at high anodic potentials after oxidation. Several analysis methods are employed to reveal changes in the structure and surface chemistry while simultaneously introducing defects, without compromising electrochemically active sites, are key factors for improving the rate performance of Ti₃C₂T_x. This study demonstrates improvement of the electrochemical performance of MXene electrodes by performing a controlled anodic oxidation.     

Electronic structure of UF5

Non-relativistic and relativistic self-consistent Hartree—Fock—Slater and Dirac—Slater models have been used to calculate one-electron energy levels and ionization energies for UF₅. The calculations were performed in an assumed structure of C_4v symmetry with the uranium atom at the center of mass of the molecule. The spacing and level ordering are compared with earlier results obtained with the MS Xα method using the muffin-tin approximation. Connections with the multi-photon isotope separation scheme of UF₆ are discussed.     

Slater sum for an inhomogeneous liquid of Fermions generated by a sech2 x potential in one dimension

In one dimension, the Slater sum S(x, β), which is the diagonal element of the canonical density matrix, satisfies a known partial differential equation characterised by a one-body potential V(x). Here, for the case of a sech² x potential in one dimension, it is stressed that S(x, β) is explicitly related to the limit S ₀(β) as V(x) → 0 and to V(x) itself. This is the same input information as in the Thomas–Fermi result. The relevance to density functional theory is emphasised.     

Ab initio HF-SCF study of naphthazarin: Geometries,isomerism, hydrogen bonding,and vibrational spectrum

The structural properties and intramolecular hydrogen bonding of a series of structures of naphthazarin molecule were investigated by ab initio HF-SCF methods. The geometries of theC _2v ,C _2h ,D _2h , andC _s symmetry structures were optimized using split-valence basis sets. MP2/6-31G^*// HF/6-31G single-point energy calculations indicate that theC _2v isomer (5,8-dihydroxy-1,4-naphthoquinone) is the lowest energy structure of the molecule and that theC _2h symmetry one (4,8-dihydroxy-1,5-naphthoquinone), lying 37 kJ/mol above theC _2v form, is the other stable isomer of naphthazarin. At the HF/6-31G level, the intramolecular proton exchange between two equivalentC _2v structures is a two-step process where each proton can be independently transferred through an unsymmetrical potential having a 1,5-quinone intermediate, theC _2h symmetry structure, and two equivalent transition states ofC _s symmetry, with a barrier height equal to 38 kJ/ mol (MP2/6-31G^*//HF/6-31G). The study of naphthazarin molecule is flanked by a theoretical investigation on theC _2v andC _2h isomers of the parent naphthoquinone and dihydroxynaphthalene molecules. The SCF vibrational spectrum of the ground state of naphthazarin, harmonic frequencies, and infrared and Raman band intensities were computed at the HF/6-31G level. The results of the calculations are compared with the matrix isolation FT-IR spectroscopy measurements and with the infrared and Raman spectra of the crystal molecule.     

Ionic bonding of lanthanides,as influenced by d‐ and f‐atomic orbitals,by core–shells and by relativity

Lanthanide trihalide molecules LnX₃ (X = F, Cl, Br, I) were quantum chemically investigated, in particular detail for Ln = Lu (lutetium). We applied density functional theory (DFT) at the nonrelativistic and scalar and SO‐coupled relativistic levels, and also the ab initio coupled cluster approach. The chemically active electron shells of the lanthanide atoms comprise the 5d and 6s (and 6p) valence atomic orbitals (AO) and also the filled inner 4f semivalence and outer 5p semicore shells. Four different frozen‐core approximations for Lu were compared: the (1s²–4d¹⁰) [Pd] medium core, the [Pd+5s²5p⁶ = Xe] and [Pd+4f¹⁴] large cores, and the [Pd+4f¹⁴+5s²5p⁶] very large core. The errors of Lu? X bonding are more serious on freezing the 5p⁶ shell than the 4f¹⁴ shell, more serious upon core‐freezing than on the effective‐core‐potential approximation. The Ln? X distances correlate linearly with the AO radii of the ionic outer shells, Ln³⁺‐5p⁶ and X^?‐np⁶, characteristic for dominantly ionic Ln³⁺‐X^? binding. The heavier halogen atoms also bind covalently with the Ln‐5d shell. Scalar relativistic effects contract and destabilize the Lu? X bonds, spin orbit coupling hardly affects the geometries but the bond energies, owing to SO effects in the free atoms. The relativistic changes of bond energy BE, bond length R_e, bond force k, and bond stretching frequency v_s do not follow the simple rules of Badger and Gordy (R_e～BE～k～v_s). The so‐called degeneracy‐driven covalence, meaning strong mixing of accidentally near‐degenerate, nearly nonoverlapping AOs without BE contribution is critically discussed. © 2015 Wiley Periodicals, Inc.     

Identification of unknown diffusion coefficient in pure diffusive linear model of chronoamperometry. I. The theory

Coefficient identification problem for diffusion equation u _t (x, t) = (D(x)u _x (x, t)) _x arising in chronoamperometry is studied. The adjoint problem approach is developed for the case when the output measured data is given in the form of left/right flux. Analytical formulas for determination of the values D(0), D(L) at the endpoints x = 0; L, of the unknown coefficient D(x), via the solution v(x, t) of the constant coefficient equation v _t (x, t) = D v _xx (x, t) is obtained. The integral identity relating solutions of the forward and corresponding adjoint problems is derived. This integral identity permits one to prove the monotonicity and invertibility of input-output map, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems.