Machine-learning creates a density functional that accounts for fractional charge and spin.
In a standard Kohn-Sham density functional calculation, the total energy of a crystal at zero temperature is evaluated for a perfect static lattice of nuclei and minimized with respect to the lattice constant. Sometimes a zero-point vibrational energy, whose anharmonicity expands the minimizing or equilibrium lattice constant, is included in the calculation or (as here) is used to correct the experimental reference value for the lattice constant to that for a static lattice. A simple model for this correction, based on the Debye and Dugdale-MacDonald approximations, requires as input only readily available parameters of the equation of state, plus the experimental Debye temperature. However, particularly because of the rough Dugdale-MacDonald estimation of Gr\"uneisen parameters for diatomic solids, this simple model is found to overestimate the correction by about a factor of two for some solids in diamond and zinc-blende structures. Using the quasiharmonic phonon frequencies calculated from density functional perturbation theory gives a more accurate zero-point anharmonic expansion (ZPAE) correction. However, the error statistics for the lattice constants of various semilocal density functionals for the exchange--correlation energy are little changed by improving the ZPAE correction. The Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) for solids and the revised Tao-Perdew-Staroverov-Scuseria (revTPSS) meta-GGA, the latter of which is implemented self-consistently here in the band-structure program BAND and applied to a test set of 58 solids, remain the most accurate of the functionals tested, with MAREs below 0.7$%$ for the lattice constants. The most positive and most negative revTPSS relative errors tend to occur for solids for which full nonlocality (missing from revTPSS) may be important.
Equation (B3) has a transcription error. In this equation, tan -1 (u) should be replaced by tan -1 (1/u). This correction does not affect the rest of the paper. A detailed derivation is presented in Appendix A of Ref.
From a global perspective, the density of an atom is strongly inhomogeneous and not at all like the density of a uniform or nearly-uniform electron gas. But, from the semi-local or myopic perspective of standard density functional approximations to the exchange-correlation energy,it is not so easy to tell an atom from an electron gas. We address the following problem: Given the ground-state electron density n and orbital kinetic energy density in the neighborhood of a point r, can we construct an "inhomogeneity index" w(r) which approaches zero for weakly-inhomogeneous densities and unity for strongly-inhomogeneous ones? The solution requires not only the usual local ingredients of a meta-generalized gradient approximation (n,rn,r2n, ),but also r and r2 . The inhomogeneity index is displayed for atoms, and for model densities of metal surfaces and bulk metals. Scaling behavior and a possible application to functional interpolation are discussed.
The role that nonlocal short-range correlation plays at metal surfaces is investigated by analyzing the correlation surface-energy into contributions from dynamical density fluctuations of various two-dimensional wave vectors. Although short-range correlation is known to yield considerable correction to the ground-state energy of both uniform and nonuniform systems, short-range correlation effects on intermediate and short-wavelength contributions to the surface formation energy are found to compensate one another. As a result, our calculated surface energies, which are based on a nonlocal exchange-correlation kernel that provides accurate total energies of a uniform electron gas, are found to be very close to those obtained in the random-phase approximation, and support the conclusion that the error introduced by the local-density approximation is small.
The large relative error of the local density approximation for the transverse-photon-electron or Breit energy in atoms and solids is shown to arise from spurious self-interaction of the 1s electrons. Orbital self-interaction correction dramatically reduces this error. On the other hand, the large relative error of the local density approximation for the Breit contribution to the 2s electron removal energy is almost unchanged by self-interaction correction.
The random phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of iso-electronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spin-polarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for H2 and N2, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched H2 or N2. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched H2+), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all one-electron densities, in the same simple semilocal way that the correlation energy densities of many meta-generalized gradent approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.
Local effective electron-ion potentials for metals are constructed from a priori calculations of the electron density around an ion embedded in an electron gas. These effective potentials fold certain higher order contributions to the cohesive energy into a lower order calculation. The dynamical matrix is evaluated to second and third order in the effective potential. When band structure effects are included in the energy denominators of the response functions, the lithium phonon frequencies calculated with no empirical input are in excellent agreement with experiment.
The new variational-principle, density-functional theory of the spin susceptibility $\ensuremath{\chi}$ is used to make a priori calculations of $\ensuremath{\chi}$ for the alkalis. Crystalline effects are calculated by the spherical-cell method and the local spin-density approximation is used for the exchange-correlation functionals. The excellent agreement between the results and recent experiments establishes the validity of this new theory and the correctness of the theoretical values for the exchange-correlation enhancement of $\ensuremath{\chi}$ for a homogeneous electron gas for ${\mathcal{r}}_{s}\ensuremath{\lesssim}5$.
