797 publications from this institution
While most molecules and solids are spin-unpolarized, most chemically-active atoms are partly spin-polarized. As a result, the errors of the spin-dependence of a density functional are much more troublesome for atomization energies than they are for typical reaction or formation energies. This observation explains why the atomization energy errors of approximate functionals do not correlate with their other errors, and why the errors of atomization energies for a given functional can be radically reduced by fitting the energies of the atoms. We present an illustrative example from the recent nonempirical construction of the SCAN meta-generalized gradient approximation.
Lieb and Oxford (1981) derived rigorous lower bounds, in the form of local functionals of the electron density, on the indirect part of the Coulomb repulsion energy. The greatest lower bound for a given electron number N depends monotonically upon N, and the N-> infinity limit is a bound for all N. These bounds have been shown to apply to the exact density functionals for the exchange- and exchange-correlation energies that must be approximated for an accurate and computationally efficient description of atoms, molecules, and solids. A tight bound on the exact exchange energy has been derived therefrom for two-electron ground states, and is conjectured to apply to all spin-unpolarized electronic ground states. Some of these and other exact constraints have been used to construct two generations of non-empirical density functionals beyond the local density approximation: the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the strongly constrained and appropriately normed (SCAN) meta-GGA.
The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue'. (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.
Solid, liquid and alloyed phases of gallium play a role in a variety of important technological applications. While many of the gallium phases involved in these applications are metallic, some have been proposed or are known to contain covalently bound Ga dimers. Thus, understanding the nature of bonding in Ga is crucial to the development of Ga-based materials. The solid phase of gallium at ambient conditions, <i>α</i>-Ga, is metallic and composed of molecular dimers, and can serve as a testing ground for studying gallium bonding with electronic structure calculations. We use density functional theory-based molecular dynamics simulations in conjunction with maximally localised Wannier functions to examine the nature of chemical bonding in <i>α</i>-Ga. We propose a geometric criterion for defining various bonding environments, which enables the quantification of covalent and weak bonds in solid gallium. We additionally connect the bonding structure of <i>α</i>-Ga to its phonon density of states and discuss similarities and differences with diatomic halogen crystals.
The van der Waals interaction is a weak, long-range correlation, arising from quantum electronic charge fluctuations. This interaction affects many properties of materials. A simple and yet accurate estimate of this effect will facilitate computer simulation of complex molecular materials and drug design. Here we develop a fast approach for accurate evaluation of dynamic multipole polarizabilities and van der Waals (vdW) coefficients of all orders from the electron density and static multipole polarizabilities of each atom or other spherical object, without empirical fitting. Our dynamic polarizabilities (dipole, quadrupole, octupole, etc.) are exact in the zero- and high-frequency limits, and exact at all frequencies for a metallic sphere of uniform density. Our theory predicts dynamic multipole polarizabilities in excellent agreement with more expensive many-body methods, and yields therefrom vdW coefficients C(6), C(8), C(10) for atom pairs with a mean absolute relative error of only 3%.
The ionization energy for a metallic cluster of radius R is I(R)=W+\ensuremath{\alpha}${\mathit{e}}^{2}$/R+O(${\mathit{R}}^{\mathrm{\ensuremath{-}}2}$), where W is the work function. Two different classical values for \ensuremath{\alpha} have been derived: 1/2 from the spherical-capacitor approach, and 3/8 from the image-potential approach. We present a ``classical jellium'' or Wigner-crystal model in which \ensuremath{\alpha}=1/2 and W=9/10${\mathit{e}}^{2}$/${\mathit{r}}_{\mathit{s}}$. In the image-potential approach to the ``perfect-conductor'' model, we discover an additional work 1/8${\mathit{e}}^{2}$/R, resolving the contradiction in favor of \ensuremath{\alpha}=1/2. The experimental deviation from \ensuremath{\alpha}=1/2 is a quantum effect.
Triangulene and its analogue metal-free magnetic systems have garnered increasing attention since their discovery. Predicting the magnetic couplings and spin polarization energy with quantitative accuracy is beyond the predictive power of today’s density-functional theory (DFT) due to their intrinsic multi-reference character. Herein, we create a benchmark dataset of 25 magnetic systems with non-local spin densities, including the triangulene monomer, dimer, and their analogues. We calculate the magnetic coupling (J) and spin-polarization energy (ΔEspin) of these systems using complete active space self-consistent field (CASSCF) and coupled cluster methods as high-quality reference values. This reference data is then used to benchmark 22 DFT functionals commonly used in material science. Our results show that, while some functionals consistently correctly predict the qualitative character of the ground state, achieving quantitative accuracy with small relative errors is currently not feasible. PBE0, M06-2X, and MN15 are predicting the correct electronic ground state for all systems investigated here, and also have the lowest mean absolute error for predicting both ΔEspin (0.34 eV, 0.32 eV and 0.31 eV) and J (11.74 meV, 12.66 meV and 10.64 meV). They may therefore also serve as starting points for higher-level methods such as the GW or the random phase approximation. As other functionals fail for the prediction of the ground state, they cannot be recommended for metal-free magnetic systems.
We show that known or knowable information about the high-$({r}_{s}\ensuremath{\rightarrow}0)$ and low-density $({r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})$ asymptotes can be used to predict the correlation energy per electron, ${e}_{c}({r}_{s},\ensuremath{\zeta})$, of the three-dimensional uniform gas over the whole range of the density parameter $(0\ensuremath{\le}{r}_{s}<\ensuremath{\infty})$ and relative spin polarization $(0\ensuremath{\le}|\ensuremath{\zeta}|\ensuremath{\le}1)$, without quantum Monte Carlo or other input. For $\ensuremath{\zeta}=0$, the high-density limit through order ${r}_{s}$ is known exactly from many-body perturbation theory, and for all $\ensuremath{\zeta}$, the low-density limit through order $1/{r}_{s}^{2}$ is known accurately from a simple, intuitive, and accurate model. We propose a single interpolation formula with the expected analytic structure to all orders in both limits, and use it to predict ${e}_{c}({r}_{s},0)$ in excellent agreement with quantum Monte Carlo data. For $|\ensuremath{\zeta}|>0$, we derive the $\ensuremath{\zeta}$ dependence of the coefficient ${a}_{1}(\ensuremath{\zeta})$ of the ${r}_{s}\text{ }\text{ln}\text{ }{r}_{s}$ term, previously known only for $|\ensuremath{\zeta}|=0$ and 1. For ${b}_{1}(\ensuremath{\zeta})$, the coefficient of the ${r}_{s}$ term (not yet derived for $\ensuremath{\zeta}\ensuremath{\ne}0$), we approximately extend the known ${b}_{1}(0)$ by using a simplification of the available quantum Monte Carlo information that replaces the second-order transition over $50<{r}_{s}<100$ by a sudden transition to full spin polarization at ${r}_{s}=75$.
A simple analytic model is proposed for the angle- and system-averaged exchange hole of a many-electron system. The model hole depends on the local density and density gradient. It recovers a nonoscillatory local-spin density (LSD) approximation to the exchange hole for a vanishing density gradient. The model hole reproduces the exchange energy density of the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) for exchange, and facilitates a detailed understanding of the PBE GGA. The hole model is applied to atoms and molecules, and a comparison is made to exact and LSD angle- and system-averaged exchange holes. We find that the GGA hole model significantly improves upon the LSD model. Furthermore, the GGA hole model accurately describes the change in the exchange hole upon the formation of single bonds, but is less accurate for the formation of multiple bonds, where it misses the appearance of a long-range tail.
