Under a general coordinate transformation, the exchange hole can be made localized, as shown by Springborg [Chem. Phys. Lett. 308, 83 (1999)]. While the conventional or untransformed hole is referenced to the position of an electron, the maximally localized one is referenced to the center of mass of an electron pair. The benefit of a more localized hole is that semilocal density functionals model it and the associated energy density more easily than they model the conventional ones. We show that, out of the class of general coordinate transformations, one can identify a subset (including the maximally localized case) which we call appropriate. Under an appropriate coordinate transformation, while the exchange hole is no longer always normalized, it retains other familiar and useful features such as the conventional on-top value and uniform-density limit. In particular, its system average remains invariant, retaining the normalization sum rule and the negativity property. Therefore, unlike the exchange energy density ex(r), the real-space analysis ẽx(u) of the exchange energy [into contributions from different electron–electron separations (u)] is uniquely defined. Thus the real-space analysis provides an alternative way to make simple and fair but detailed comparisons of approximate and exact exchange. As a byproduct, we show how to improve the accuracy of the Negele–Vautherin model for the density matrix expansion of the exchange energy by imposing negativity and sum rule constraints on the system average of its maximally localized hole.
Gallium phosphide (GaP) nanowire photocathodes synthesized using a surfactant-free solution-liquid-solid (SLS) method were investigated for their photoelectrochemical evolution of hydrogen. Zinc as a p-type dopant was introduced into the nanowires during synthesis to optimize the photocathode's response. Investigation of the electrical properties of Zn-doped GaP nanowires confirmed their p-type conductivity. After optimization of the nanowire diameter and Zn doping concentration, higher absorbed photon-to-current efficiency (APCE) over the spectrum was achieved. The versatility of the SLS synthesis and the capability to control the electrical properties suggest that our approach could be generalized to other III-V and II-VI semiconductors.
We investigate the performance of contemporary semilocal and hybrid density functionals for bond energetics, structures, dipole moments, and harmonic frequencies of 3d transition-metal (TM) compounds by comparison with gas-phase experiments. Special attention is given to the nonempirical metageneralized gradient approximation (meta-GGA) of Tao, Perdew, Staroverov, and Scuseria (TPSS) [Phys. Rev. Lett. 91, 146401 (2003)], which has been implemented in TURBOMOLE for the present work. Trends and error patterns for classes of homologous compounds are analyzed, including dimers, monohydrides, mononitrides, monoxides, monofluorides, polyatomic oxides and halogenides, carbonyls, and complexes with organic pi ligands such as benzene and cyclopentadienyl. Weakly bound systems such as Ca(2), Mn(2), and Zn(2) are discussed. We propose a reference set of reaction energies for benchmark purposes. Our all-electron results with quadruple zeta valence basis sets validate semilocal density-functional theory as the workhorse of computational TM chemistry. Typical errors in bond energies are substantially larger than in (organic) main group chemistry, however. The Becke-Perdew'86 [Phys. Rev. A 38, 3098 (1988); Phys. Rev. B 33, 8822 (1986)] GGA and the TPSS meta-GGA have the best price/performance ratio, while the TPSS hybrid functional achieves a slightly lower mean absolute error in bond energies. The popular Becke three-parameter hybrid B3LYP underbinds significantly and tends to overestimate bond distances; we give a possible explanation for this. We further show that hybrid mixing does not reduce the width of the error distribution on our reference set. The error of a functional for the s-d transfer energy of a TM atom does not predict its error for TM bond energies and bond lengths. For semilocal functionals, self-interaction error in one- and three-electron bonds appears to be a major source of error in TM reaction energies. Nevertheless, TPSS predicts the correct ground-state symmetry in the vast majority of cases and rarely fails qualitatively. This further confirms TPSS as a general purpose functional that works throughout the periodic table. We also give workstation timing comparisons for the 645-atom protein crambin.
It is well known that the density-gradient expansion of the Hartree-Fock exchange energy for the bare-Coulomb interaction contains divergent terms of order ${e}^{4}$ where $e$ is the electronic charge. We argue that the exchange energy evaluated with Kohn-Sham orbitals (i.e., those derived from a local effective potential) is purely of order ${e}^{2}$ and therefore its gradient expansion is well defined. This density-gradient expansion, with the a priori coefficient of Sham, is shown to converge by comparison with numerically refined values for the exact exchange energy of a metal surface in the linear-potential model. As the electron density profile becomes more slowly varying, the relative error of the zeroth-order (local-density) term tends to zero. We present here the first demonstration that, in addition, the absolute error is increasingly canceled by the second-order (gradient) term. Like the gradient expansion for the kinetic energy but unlike the one for correlation, the gradient expansion for exchange gives useful results even for "physical" surface profiles. One- and many-electron atoms are also discussed. It is observed that, as the atomic number increases, the relative errors of the local-density and gradient-expansion approximations decrease in magnitude, but the gradient term corrects only a small fraction of the error of the local-density approximation. This is a consequence of the fact that the convergence condition $\frac{|\ensuremath{\nabla}n|}{2}{k}_{F}n\ensuremath{\lesssim}1$ is increasingly satisfied as the atomic number increases but the second convergence condition $\frac{|{\ensuremath{\nabla}}^{2}n|}{2{k}_{F}|\ensuremath{\nabla}n|}\ensuremath{\ll}1$ is not so well satisfied.
In many situations, the most long-ranged parts of the exchange and correlation holes surrounding an electron cancel one another. Apparently for this reason, local spin density and generalized gradient approximations are more accurate for exchange and correlation together than for either alone. A study is made of the ability of these density functionals, and also the unmodified second-order gradient expansion, to describe various short-range effects in atoms: the correlation contribution to the interacting kinetic energy, the antiparallel-spin correlation energy, and the correction to the random phase approximation. Generalized gradient approximations, constructed with no adjustable parameter from the electron gas of slowly varying density, are found to give results of useful accuracy for real atoms. Prospects are discussed for use of the new functionals to improve the accuracy of electronic-structure calculations. © 1993 John Wiley & Sons, Inc.
