The properties of the simple metals are controlled largely by three density parameters: the equilibrium average valence electron density 3/4\ensuremath{\pi}${\mathit{r}}_{\mathit{s}}^{3}$, the valence z, and the density on the surface of the Wigner-Seitz cell, represented here by the equilibrium number ${\mathit{N}}_{\mathrm{int}}$ of valence electrons in the interstitial region. To demonstrate this fact, and as a refinement of the ``stabilized jellium'' or ``structureless pseudopotential'' model, we propose a structured local electron-ion pseudopotential w(r) which depends upon either ${\mathit{r}}_{\mathit{s}}$ and z (``universal'' choice for ${\mathit{N}}_{\mathrm{i}\mathrm{n}\mathrm{t})}$, or ${\mathit{r}}_{\mathit{s}}$, z, and ${\mathit{N}}_{\mathrm{int}}$ for each metal (``individual'' potential). Calculated binding energies, bulk moduli, and pressure derivatives of bulk moduli, evaluated in second-order perturbation theory, are in good agreement with experiment for 16 simple metals, and the bulk moduli are somewhat better than those calculated from first-principles nonlocal norm-conserving pseudopotentials. Structural energy differences agree with those from a nonlocal pseudopotential calculation for Na, Mg, and Al, but not for Ca and Sr. Our local pseudopotential w(r) is analytic for all r, and displays an exponential decay of the core repulsion as r\ensuremath{\rightarrow}\ensuremath{\infty}. The decay length agrees with that of the highest atomic core orbital of s or p symmetry, corroborating the physical picture behind this ``evanescent core'' form. The Fourier transform or form factor w(Q) is also analytic, and decays rapidly as Q\ensuremath{\rightarrow}\ensuremath{\infty}; its first and only zero ${\mathit{Q}}_{0}$ is close to conventional or empirical values. In comparison with nonlocal pseudopotentials, local ones have the advantages of computational simplicity, physical transparency, and suitability for tests of density functional approximations against more-exact many-body methods.
Some fundamental issues in ground-state density functional theory are discussed without equations: (1) The standard Hohenberg-Kohn and Kohn-Sham theorems were proven for a Hamiltonian that is not quite exact for real atoms, molecules, and solids. (2) The density functional for the exchange-correlation energy, which must be approximated, arises from the tendency of electrons to avoid one another as they move through the electron density. (3) In the absence of a magnetic field, either spin densities or total electron density can be used, although the former choice is better for approximations. (4) "Spin contamination" of the determinant of Kohn-Sham orbitals for an open-shell system is not wrong but right. (5) Only to the extent that symmetries of the interacting wave function are reflected in the spin densities should those symmetries be respected by the Kohn-Sham noninteracting or determinantal wave function. Functionals below the highest level of approximations should however sometimes break even those symmetries, for good physical reasons. (6) Simple and commonly used semilocal (lower-level) approximations for the exchange-correlation energy as a functional of the density can be accurate for closed systems near equilibrium and yet fail for open systems of fluctuating electron number. (7) The exact Kohn-Sham noninteracting state need not be a single determinant, but common approximations can fail when it is not. (8) Over an open system of fluctuating electron number, connected to another such system by stretched bonds, semilocal approximations make the exchange-correlation energy and hole-density sum rule too negative. (9) The gap in the exact Kohn-Sham band structure of a crystal underestimates the real fundamental gap but may approximate the first exciton energy in the large-gap limit. (10) Density functional theory is not really a mean-field theory, although it looks like one. The exact functional includes strong correlation, and semilocal approximations often overestimate the strength of static correlation through their semilocal exchange contributions. (11) Only under rare conditions can excited states arise directly from a ground-state theory.
We report the first reliable theoretical calculation of the quantum size correction c which yields the asymptotic ionization energy I(R)=W+(12+c)/R+O(R−2) of a simple-metal cluster of radius R. Restricted-variational electronic density profiles are used to evaluate two sets of expressions for the bulk work function W and quantum size correction c: the Koopmans expressions, and the more accurate and profile-insensitive ΔSCF expressions. We find c≈−0.08 for stabilized (as for ordinary) jellium, and thus for real simple metals. We present parameters from which the density profiles may be reconstructed for a wide range of cluster sizes, including the planar surface. We also discuss how many excess electrons can be bound by a neutral cluster of given size. Within a continuum picture, the criterion for total-energy stability of a negatively charged cluster is less stringent than that for existence of a self-consistent solution.
2-Aminoethoxydiphenylborate (2-APB) is a broad-spectrum modulator of various membrane proteins. Specifically, it exhibits concentration dependent modulation of calcium signaling through store-operated calcium (SOC) channels: low micromolar concentration of 2-APB stimulates SOC entry while a higher concentration induces complete inhibition. Ab initio quantum chemical calculations show that the relative stability of the two major isomers of 2-APB (cyclic and extended) is about 8 kcal/mol. The dual functionality of 2-APB for SOC channels is thus likely associated with its ability to switch among isomeric forms, suited to different binding sites in the SOC channels with distinct binding affinities. Importantly, the moderate relative stability of different isomers results from a delicate balance between the intramolecular boron–nitrogen coordinate bond with strength about −45 kcal/mol and ring strain engendered by cyclic oligomerization. The synergistic effect of these two factors likely makes 2-APB an ideal dual effect drug.
The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not. We present two related methods for the self-interaction correction (SIC) of any density functional for the energy; correction of the self-consistent one-electron potenial follows naturally from the variational principle. Both methods are sanctioned by the Hohenberg-Kohn theorem. Although the first method introduces an orbital-dependent single-particle potential, the second involves a local potential as in the Kohn-Sham scheme. We apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole, while substantially improving the description of its shape. We apply this method to a number of physical problems, where the uncorrected LSD approach produces systematic errors. We find systematic improvements, qualitative as well as quantitative, from this simple correction. Benefits of SIC in atomic calculations include (i) improved values for the total energy and for the separate exchange and correlation pieces of it, (ii) accurate binding energies of negative ions, which are wrongly unstable in LSD, (iii) more accurate electron densities, (iv) orbital eigenvalues that closely approximate physical removal energies, including relaxation, and (v) correct longrange behavior of the potential and density. It appears that SIC can also remedy the LSD underestimate of the band gaps in insulators (as shown by numerical calculations for the rare-gas solids and CuCl), and the LSD overestimate of the cohesive energies of transition metals. The LSD spin splitting in atomic Ni and $s\ensuremath{-}d$ interconfigurational energies of transition elements are almost unchanged by SIC. We also discuss the admissibility of fractional occupation numbers, and present a parametrization of the electron-gas correlation energy at any density, based on the recent results of Ceperley and Alder.
The atomization energies of molecules from first-principles density functional approximations improve from the local spin-density approximation (LSDA) to the Perdew-Burke-Ernzerhof (PBE)) generalized gradient approximation (GGA) to the strongly constrained and appropriately normed (SCAN) meta-GGA, and their sensitivities to non-spherical components of the density increase in the same order. Thus, these functional advances increase density sensitivity and imitate the exact constrained search over correlated wavefunctions better than that over ensembles. The diatomic molecules studied here, singlet C2 and F2 plus triplet B2 and O2, have cylindrically symmetric densities. Because the densities of the corresponding atoms are non-spherical, the approximate Kohn-Sham potentials for the atoms have a lower symmetry than that of the external (nuclear) potential, so that the non-interacting wavefunctions are not eigenstates of the square of total orbital angular momentum, breaking a symmetry that yields a feature of the exact ground-state density. That spatial symmetry can be preserved by a non-self-consistent approach in which a self-consistent equilibrium-ensemble calculation is followed by integer re-occupation of the Kohn-Sham orbitals, as the first of several steps. The symmetry-preserving approach is different from symmetry restoration based upon projection. First-step space- (and space-spin-) symmetry preservation in atoms is shown to have a small effect on the atomization energies of molecules, quantifying earlier observations by Fertig and Kohn. Thus, the standard Kohn-Sham way of calculating atomization energies, with self-consistent symmetry breaking to minimize the energy, is justified, at least for the common cases where the molecules cannot break symmetry.
tion, which describes the situation with the crystal in its ground state and no core hole, is also the finalstate Hamiltonian for emission.Initial-and final-state pictures refer to the ways in which transition ma- trix elements are calculated.In the initial-state picture both states entering the matrix element are ei-
Atoms cohere to form solids largely due to exchange and correlation. The volume is set by a balance between the expansive electronic kinetic energy and the compressive exchange-correlation energy. These effects are simply illustrated by the jellium model, in which the valence electrons neutralize a positive background charge that is rigidly uniform. But the formation of free atoms under extreme expansion is found only in the deformable-jellium model. Deformable jellium is condensed matter in miniature, displaying not only bulk cohesion with a realistic equation of state and surface effects, but also phonons and plasmons and their soft mode instabilities. By drawing an analogy with the motion of shoppers in a mall, we also discuss an intuitive picture of exchange and correlation (the tendency of electrons not to bump into other electrons or into themselves).
In the liquid-drop model, the total energy of a system is expanded as a sum of volume, surface, and curvature terms. We derive an expression for the curvature energy of a metal in terms of the electron-density profile for a planar surface, and show that the resulting values agree with the fits of calculated or measured total energies to the liquid-drop expansion. In particular, this expansion accurately describes the formation energies of microscopic voids (including monovacancies) in metals. In our calculations, the curvature energy is determined by the bulk density. It is nearly the same for restricted trial density profiles as for self-consistent Kohn-Sham profiles, for the fourth-order gradient expansion as for the exact kinetic energy, and for jellium as for stabilized jellium. We also report Kohn-Sham results for the surface energy and work function. The stabilized-jellium model, while retaining the simplicity and nonempirical character of jellium, gives a significantly more realistic description of the simple metals, especially those with high bulk densities.
A complete wave-vector analysis has been made of the gradient coefficient for the exchange-correlation energy of a nonuniform electronic system. It is shown that the majority of the contribution comes from a very small but universal region of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space near the origin. From this it can be concluded that random-phase-approximation-like calculations, like the present one or that of Rasolt and Geldart, which treat this region correctly, are likely to provide accurate results for the gradient coefficient and hence for the energy and structure of a system whose density is truly slowly varying. However, it also shows that the criterion for the validity of the gradient approximation itself is much more severe than previously supposed, so that the usual type of application, to say a surface or bulk material, is incorrect. For the surface case this is verified in unequivocal detail. On the other hand, a generalization of the gradient scheme based on an average slope instead of a local slope is proposed. This gives good agreement with limiting cases where they exist, and rough agreement with the interpolation scheme proposed previously by the authors.
