Errors In Atomization Energies Research Articles

Segmented Contracted Error-Consistent Basis Sets of Quadruple-ζ Valence Quality for One- and Two-Component Relativistic All-Electron Calculations.

Segmented contracted basis sets of quadruple-ζ quality for exact two-component (X2C) calculations are presented for the elements H-Rn. These sets are the all-electron relativistic counterparts of the Karlsruhe "def2" and "dhf" systems of bases, which were designed for Hartree-Fock and density functional treatments and-with a somewhat extended set-also for correlated treatments. The bases were optimized with analytical basis set gradients and the finite nucleus model based on a Gaussian charge distribution at the scalar-relativistic X2C level. Extensions are provided for self-consistent two-component treatments to describe spin-orbit coupling, polarization effects, and nuclear magnetic resonance (NMR) shielding constants. The basis sets were designed to yield comparable errors in atomization energies, orbital energies, dipole moments, and NMR shielding constants all across the periodic table of elements. A test set of more than 360 molecules representing (nearly) all elements in their common oxidation states was utilized for the valence properties, and a test set of more than 250 closed-shell molecules was employed for the NMR shielding constants. The quality of the developed basis sets is compared to other frequently used relativistic all-electron bases.

The Elephant in the Room of Density Functional Theory Calculations.

Using multiwavelets, we have obtained total energies and corresponding atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a test set of 211 molecules with an unprecedented and guaranteed μHartree accuracy. These quasi-exact references allow us to quantify the accuracy of standard all-electron basis sets that are believed to be highly accurate for molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric atom-centered orbitals (NAOs), and full-potential augmented plane wave (APW) methods. We show that NAOs are able to achieve the so-called chemical accuracy (1 kcal/mol) for the typical basis set sizes used in applications, for both total and atomization energies. For GTOs, a triple-ζ quality basis has mean errors of ∼10 kcal/mol in total energies, while chemical accuracy is almost reached for a quintuple-ζ basis. Due to systematic error cancellations, atomization energy errors are reduced by almost an order of magnitude, placing chemical accuracy within reach also for medium to large GTO bases, albeit with significant outliers. In order to check the accuracy of the computed densities, we have also investigated the dipole moments, where in general only the largest NAO and GTO bases are able to yield errors below 0.01 D. The observed errors are similar across the different functionals considered here.

Intensive Atomization Energy: Re-Thinking a Metric for Electronic Structure Theory Methods

Abstract The errors in atomization energies (AE) of molecules have long been used to measure the errors of wavefunction or density functional methods for electronic structure calculations. In particular, the G3 set of Pople and collaborators (for sp-bonded molecules from the first rows of the periodic table) has become a standard benchmark for such methods. But the mean absolute error of AE tends to increase with increasing number N at of atoms in a molecule. In fact, AE is an extensive variable, which diverges as N at →∞. Here, as did Savin and Johnson 2015, we define an intensive atomization energy, IAE = AE/N at or atomization energy per atom, which tends to the finite cohesive energy (per atom) of a large cluster or solid (N at →∞). We find that the mean absolute error of the G3 molecular IAE from accurate density functionals remains close to 1 kcal/mol as the average molecular size increases. This makes it possible to estimate in advance the magnitude of the error in AE for a molecule similar to most of those in the G3 set. It also allows us identify the G3 “outlying molecules”, and to more directly compare the accuracy of a given functional for different kinds of molecules (such as those containing transition-metal atoms) to that for G3-type molecules, by removing the otherwise-uncontrolled size factor. Finally, we point out that the familiar concept of “chemical accuracy” needs to be qualified.

Parameterization of DFTB3/3OB for Sulfur and Phosphorus for Chemical and Biological Applications.

We report the parametrization of the approximate density functional tight binding method, DFTB3, for sulfur and phosphorus. The parametrization is done in a framework consistent with our previous 3OB set established for O, N, C, and H, thus the resulting parameters can be used to describe a broad set of organic and biologically relevant molecules. The 3d orbitals are included in the parametrization, and the electronic parameters are chosen to minimize errors in the atomization energies. The parameters are tested using a fairly diverse set of molecules of biological relevance, focusing on the geometries, reaction energies, proton affinities, and hydrogen bonding interactions of these molecules; vibrational frequencies are also examined, although less systematically. The results of DFTB3/3OB are compared to those from DFT (B3LYP and PBE), ab initio (MP2, G3B3), and several popular semiempirical methods (PM6 and PDDG), as well as predictions of DFTB3 with the older parametrization (the MIO set). In general, DFTB3/3OB is a major improvement over the previous parametrization (DFTB3/MIO), and for the majority cases tested here, it also outperforms PM6 and PDDG, especially for structural properties, vibrational frequencies, hydrogen bonding interactions, and proton affinities. For reaction energies, DFTB3/3OB exhibits major improvement over DFTB3/MIO, due mainly to significant reduction of errors in atomization energies; compared to PM6 and PDDG, DFTB3/3OB also generally performs better, although the magnitude of improvement is more modest. Compared to high-level calculations, DFTB3/3OB is most successful at predicting geometries; larger errors are found in the energies, although the results can be greatly improved by computing single point energies at a high level with DFTB3 geometries. There are several remaining issues with the DFTB3/3OB approach, most notably its difficulty in describing phosphate hydrolysis reactions involving a change in the coordination number of the phosphorus, for which a specific parametrization (3OB/OPhyd) is developed as a temporary solution; this suggests that the current DFTB3 methodology has limited transferability for complex phosphorus chemistry at the level of accuracy required for detailed mechanistic investigations. Therefore, fundamental improvements in the DFTB3 methodology are needed for a reliable method that describes phosphorus chemistry without ad hoc parameters. Nevertheless, DFTB3/3OB is expected to be a competitive QM method in QM/MM calculations for studying phosphorus/sulfur chemistry in condensed phase systems, especially as a low-level method that drives the sampling in a dual-level QM/MM framework.

Simultaneous benchmarking of ground- and excited-state properties with long-range-corrected density functional theory

We present benchmark calculations using several long-range-corrected (LRC) density functionals, in which Hartree-Fock exchange is incorporated asymptotically using a range-separated Coulomb operator, while local exchange is attenuated using an ansatz introduced by Iikura et al. [J. Chem. Phys. 115, 3540 (2001)]. We calculate ground-state atomization energies, reaction barriers, ionization energies, and electron affinities, each as a function of the range-separation parameter mu. In addition, we calculate excitation energies of small- and medium-sized molecules, again as a function of mu, by applying the LRC to time-dependent density functional theory. Representative examples of both pure and hybrid density functionals are tested. On the basis of these results, there does not appear to be a single range-separation parameter that is reasonable for both ground-state properties and vertical excitation energies. Reasonable errors in atomization energies and barrier heights are achieved only at the expense of excessively high excitation energies, at least for the medium-sized molecules, whereas values of mu that afford reasonable excitation energies yield some of the largest errors for ground-state atomization energies and barrier heights in small molecules. Notably, this conclusion is obscured if the database of excitation energies includes only small molecules, as has been the case in previous benchmark studies of LRC functionals.

Dual-basis self-consistent field methods: 6-31G* calculations with a minimal 6-4G primary basis

The feasibility of the strongest possible dual-basis approximation to polarized valence double zeta Hartree–Fock and density functional theory calculations is explored. Specifically, to approximate 6-31G* and 6-31 + G* calculations, a minimal 6-4G basis is constructed from 6-31G exponents. The self-consistent field calculation is performed in this minimal basis, followed by a single diagonalization step in the target basis. Statistical errors in atomization energies, molecular structures, and harmonic frequencies approach those of the target basis for HF, B3LYP and EDF1, and are far superior to calculations that do not include polarization functions. Speedups of approximately 5–10 are obtained for the energy, with speedups of 2–3 for the gradient, using existing algorithms, with significant potential for future improvements.

Truncation of the correlation consistent basis sets: Application to extended systems

AbstractThe systematic reduction of the correlation consistent basis sets [cc‐pVnZ, where n = D (2), T (3), and Q (4)] for the hydrogen atom as a means to reduce the computational cost of ab initio calculations has been extended to methane through decane and the hydrogen‐containing molecules of the G3/99 test suite. Møller–Plesset second‐order perturbation theory single‐point computations were performed with the full and truncated cc‐pVnZ basis sets utilizing the B3LYP/6‐31G(d) density functional optimized geometry. Complete basis set (CBS) limits were determined for atomization energies utilizing the full cc‐pVnZ basis set series and several truncated basis set series. The difference between the CBS limit for atomization energies determined with a series of truncated basis sets as compared to that of the full basis set series is relatively constant from molecule to molecule. We introduce a correction factor per hydrogen, which when paired with CBS limits determined with a series of truncated basis sets results in errors in atomization energies of less than 0.5 kcal/mol from CBS limits determined with a series of full basis sets. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Basis-set extensions for two-component spin–orbit treatments of heavy elements

The accuracy of standard basis sets of quadruple-zeta and lower quality for the use in two-component self-consistent field procedures including spin-orbit coupling is investigated for the elements In-I and Au-At. Spin-orbit coupling leads to energetic and spatial splittings of inner shells, which are not described accurately with standard basis sets optimized for scalar relativistic calculations. This results in large errors in total atomic energies and significant errors in atomization energies of compounds containing these atoms. We show how these errors can be corrected by adding just a few steep sets of basis functions and demonstrate the quality of the resulting extended basis sets.

The magnitude of pseudo-potential errors for density functional calculations

It is shown that the use of pseudo-potentials for replacing the core electrons leads to errors in relative energies for density functional calculations which are comparable to the errors from using double or triple zeta type basis sets for systems composed of first- and second-row elements, respectively. The pseudo-potential error is ∼10 kJ/mol for relative energies, although errors in atomization energies are significantly larger.