The Heyd-Scuseria-Ernzerhof (HSE) density functionals are popular for their ability to improve the accuracy of standard semilocal functionals such as Perdew-Burke-Ernzerhof (PBE), particularly for semiconductor band gaps. They also have a reduced computational cost compared to hybrid functionals, which results from the restriction of Fock exchange calculations to small inter-electron separations. These functionals are defined by an overall fraction of Fock exchange and a length scale for exchange screening. We systematically examine this two-parameter space to assess the performance of hybrid screened exchange (sX) functionals and to determine a balance between improving accuracy and reducing the screening length, which can further reduce computational costs. Three parameter choices emerge as useful: "sX-PBE" is an approximation to the sX-LDA screened exchange density functionals based on the local density approximation (LDA); "HSE12" minimizes the overall error over all tests performed; and "HSE12s" is a range-minimized functional that matches the overall accuracy of the existing HSE06 parameterization but reduces the Fock exchange length scale by half. Analysis of the error trends over parameter space produces useful guidance for future improvement of density functionals.

Last year, Salfi et al. made the first direct measurements of a donor wave function and found extremely good theoretical agreement with atomistic tight-binding [Salfi et al., Nat. Mater. 13, 605 (2014)]. Here, we show that multi-valley effective mass theory, applied properly, does achieve close agreement with tight-binding and hence gives reliable predictions. To demonstrate this, we variationally solve the coupled six-valley Shindo-Nara equations, including silicon's full Bloch functions. Surprisingly, we find that including the full Bloch functions necessitates a tetrahedral, rather than spherical, donor central cell correction to accurately reproduce the experimental energy spectrum of a phosphorous impurity in silicon. We cross-validate this method against atomistic tight-binding calculations, showing that the two theories agree well for the calculation of donor-donor tunnel coupling. Further, we benchmark our results by performing a statistical uncertainty analysis, confirming that derived quantities such as the wave function profile and tunnel couplings are robust with respect to variational energy fluctuations. Finally, we apply this method to exhaustively enumerate the tunnel coupling for all donor-donor configurations within a large search volume, demonstrating conclusively that the tunnel coupling has no spatially stable regions. Though this instability is problematic for reliably coupling donor pairs for two-qubit operations, we identify specific target locations where donor qubits can be placed with scanning tunneling microscopy technology to achieve reliably large tunnel couplings.

We present all-electron G 0 W 0 calculations for the electronic structure of the organic semiconductor copper phthalocyanine, based on semi-local and hybrid density functional theory (DFT) starting points. We show that G 0 W 0 calculations improve the quantitative agreement with high resolution photoemission and inverse photoemission experiments. However, the extent of the improvement provided by G 0 W 0 depends significantly on the choice of the underlying DFT functional, with the hybrid functional serving as a much better starting point than the semi-local one. In particular, strong starting point dependence is observed in the energy positions of highly localized molecular orbitals. This is attributed to self-interaction errors, due to which the orbitals obtained from semi-local DFT do not approximate the quasi-particle orbitals as well as those obtained from hybrid DFT. Our findings establish the viability of the G 0 W 0 approach for describing the electronic structure of metal-organic systems, given a judiciously chosen DFT-based starting point.

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In a recent Letter [1], the authors construct a machine learning (ML) model of molecular atomization energies, which they compare to bond counting (BC) and the PM6 semiempirical method [2]. However, their ML model was trained and tested on density functional theory (DFT) energies while BC and PM6 are fit to standard enthalpies. For fair comparison, bond energies are refit to DFT data and PM6 is converted to an electronic energy using peratom corrections [3]. BC and PM6 both perform better than the ML model and are free of large outliers in their error distributions as shown in Fig. 1.As noted in [25] of [1], some ML model error may originate from the coordinate system choice. The n eigenvalues of the Coulomb matrix correspond to an equienergy 2n-dimensional space of n-atom molecules rather than one molecule. For n ¼ 3, this corresponds to the 3 translations and 3 rotations that naturally preserve the energy of an isolated molecule. For n > 3, the space includes unphysical molecular deformations that destroy structural rigidity. Figure 2 shows this with a distortion of acetylene (C 2 H 2 ) that preserves its ML energy and coordinate, (53.058, 21.149, 0.290, 0.219).It is suggested in [25] of [1] that the n 2 sorted entries of a Coulomb matrix might be utilized instead of its n eigenvalues as a ML coordinate system. This eliminates the dimensional deficiency, but produces identical coordinates for homometric molecules [5] that do not necessarily have equal energies. A computationally expensive alternative is the equivalence class of permuted Coulomb matrices with distance metricfor Coulomb matrices M and M 0 , permutation matrices P, and the Frobenius matrix norm. Another possible source of ML model error is its lack of size-consistency. Even if the energy of two molecules A and B are accurately modeled in isolation, there are no guarantees that the well-separated pair of molecules A þ B will be similarly accurate. This requires explicitly filling the chemical compound space with a sufficiently dense set of training molecules, which likely leads to an Oð n Þ computational complexity for n atoms ( > 1). While benchmarks are favorable for n 7, the ML model cannot scale favorably compared to OðnÞ classical force fields or Oðn 3 Þ DFT or semiempirical methods. Alternative ML methods [6] enforce size consistency by modeling an intensive quantity, per-atom energy, rather than directly modeling the extensive total energy and control costs by exploiting nearsightedness [7].Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a