Nonempirical hyper-generalized-gradient functionals constructed from the Lieb-Oxford bound

The information energy is proposed as a possible measure for electron correlation energy in atoms and molecules. For neutral atoms and their cations, we proposed an expression for correlation energy with explicit dependence on the information energy and atomic number. The obtained correlation energies have been used to compute the first ionization potentials of the ground state of the main group elements from hydrogen through krypton.It is shown that the information energy of a molecule can be well-estimated by use of the information energies of its constituent atoms. On the basis of the molecular information energy and number of electrons in the molecule, a simple model has been proposed for evaluation of molecular correlation energy.

From a knowledge of the Hartree-Fock and exact non-relativistic energies of atoms, the correlation energy Ec, as defined by Lowdin, may be calculated. For atoms this correlation is defined as dynamic correlation. The separate like-spin and unlike-spin contributions, Ecσσ, Ecαβ, may be calculated as a sum of pair energies from quantum chemistry; we have used the unrestricted Møller-Plesset second-order algorithm, and then scaled them to give Ec. These three values may also be computed using dynamic correlation functionals, with the Stoll partitioning. The VWN, LYP and P91 functionals were studied for the atoms from H to Ar. Although the total correlation energies of LYP and P91 are similar, only P91 gives a semi-sensible breakdown into the Ecσσ, and Ecαβ components. It is immediately apparent that a new functional, OPTC, derived from the P91 components as 0.6625 x Ecσσ, + 1. 1015 x Ecαβ is an improvement (its mean absolute error is only 0.006 Eh). Using the recently introduced improved exchange functional OPTX (obtained through a fit to the HF energies of atoms), Kohn-Sham calculations were performed on the atoms using the OPT(=OPTX + OPTC) functional. The total energies have a mean absolute error of 0.006 Eh. The study then moves to molecules. First it is shown that the dynamic correlation energy contribution to the dissociation energies is very similar (within 2kcalmol−1 in most cases), whether it is calculated with LYP, P91 or OPTC. A calculation is then made of the HF contribution, the dynamic contribution through OPTC and the left-right contribution through OPTX, to molecular binding. In many cases the sum agrees with the observed value, but in some cases the prediction is significantly in error, e.g. O2 is overbound by 10 kcal mol−1. Thus either OPTX or OPTC or both are inadequate. An attempt was made to determine improved local exchange and correlation functionals by fitting to both atomic and molecular data, but this was unsuccessful. The conclusion is that the method is close to the limit of accuracy achievable from separately optimized local exchange and correlation functionals. Finally, a new hybrid functional O3LYP, which is a substantial improvement on B3LYP, is presented.

The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total and exchange-correlation energy functionals to the extent needed in applications of DFT to atomic, molecular and solid-state physics and quantum chemistry. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.

The realization of fractional quantum chemistry is presented. Adopting the integro‐differential operators of the calculus of arbitrary‐order, we develop a general framework for the description of quantum nonlocal effects in the complex electronic environments. After a brief overview of the historical and fundamental aspects of the calculus of arbitrary‐order, various classes of fractional Schrödinger equations are discussed and pertinent controversies and open problems around their applications to model systems are detailed. We provide a unified approach toward fractional generalization of the quantum chemical models such as Hartree–Fock and Kohn–Sham density functional theory and develop fractional variants of the fundamental molecular integrals and correlation energy. Furthermore, we offer various strategies for modeling static‐ and dynamic‐order quantum nonlocal effects through constant‐ and variable‐order fractional operators, respectively. Possible directions for future developments of fractional quantum chemistry are also outlined.

In this article we report a series of benchmark calculations on a series of small- and medium-scale computers of interest to solid-state theorists and quantum chemists. These computers include Digital Equipment Corpation VAX 11–730, 11–750, 11–780, 2040, 2060, and 11/23 computers. In addition, some test results on an IBM 3033N and a CDC Cyber 175 are included by means of a control. Finally, a series of studies on the new Floating Point Systems FPS-164 attached processor under the SJE operating system are also included. It is concluded from these studies that for an environment in which one is running many lengthy batch programs of a number-crunching-type, that a small computer such as a VAX augmented by an FPS-164 attached processor (or its equivalent) is a cost effective and convenient way to achieve high performance computing power.

Progress in Numerical Fluid Dynamics. Edited by H. J. Witz. (Springer-Verlag, 1975.) [Pp. vi+471.] DM37; $15.20. Scope: Survey. Level: Specialist/postgraduate. Physics. Third edition. By K. R. Atkins. (John Wiley, 1076.) [Pp. xii + 818.] £10.00. Scope: Textbook; Library. Level: Undergraduate. Research Fields in Physics at United Kingdom Universities and Polytechnics. Fourth edition. (The Institute of Physics, 1975.) [Pp. xvii + 340.] £11.00 (limp cover), $25.00. Annuaire Informatique Physique, 1975–76. (C.N.R.S., Paris, 1975.) [Sectionalized.] F80.00. Diatomic Molecules. A Critical Bibliography of Spectroscopic Data. Established by R. F. Barrow. Volume II, 1976. A volume of International Tables of Constants published by CNRS, Paris. The Radio Universe. Second Edition. By T. S. Hey. (Pergamon, 1975.) [Pp. viii+264.] Flexi £3.95; hardcover £6.25. Scope: Survey. Level: Undergraduate? School; General reader. Polymeric Carbons. By G. M. Jenkins and K. Kawamura. (Cambridge University Preas, 1976.) [Pp. vi+178.] £8.50. Scope: Undergraduate/Postgraduate materials science, solid state physics and chemistry. Laser Doppler Measurements. B. M. Watrasiewicz and W. J. Rudd. (Butterworths. 1975.) [Pp. viii + 160.] £6.90. Scope: Survey; Library(reference). Level: For specialists only. Reviews of Plasma Physics. Volume 6. Edited by M. A. Leontovich. (Plenum Press (Consultants Bureau), 1975.) [Pp. xiv + 331.] $33.50. Scope: Specialists; Graduates. The Oklo Phenomenon. Proceedings of a Symposium at Libreville, June 1975. Organized by the International Atomic Energy Agency in cooperation with the French Atomic Energy Commission and the Government of the Republic of Gabon. (International Atomic Agency, Vienna; Her Majesty's Stationery Office, London, 1975.) [Pp. 650.] £17.50, $36.09.

The electronic structure calculations based upon energy density functionals are highly successful and widely used both in solid state physics and quantum chemistry. Moreover, the Hohenberg-Kohn theorems and the Kohn-Sham method provide them with a firm basis. However, several basic issues are not solved, and hamper the progress to achieve high accuracy. In this paper we focus on the conceptual problem of size consistency, basing our analysis on the non-intensive character of the (spin) electronic density in the presence of degeneracy. We also briefly discuss some of the issues concerning fractional electron numbers from the same point of view, analyzing the behavior of the exact functionals for the He and the Hooke's atom series when the number of electrons fluctuates between one and two.

The Jahn-Teller effect is one of the most fascinating phenomena in modern physics and chemistry, providing a general approach to understanding the properties of molecules and crystals and their origins. The effect inspired one of the most important recent scientific discoveries, the concept of high-temperature superconductivity. This comprehensive volume presents the background of the theory and its main applications in physics and chemistry, along with more recent achievements. Full descriptions are presented alongside thorough references to original material. The 2006 book contains over 200 figures to aid visual explanation and avoids bulky mathematical deductions and overly technical language. It is intended for graduate students and academic researchers working in solid state physics, quantum chemistry, crystallography, spectroscopy, and materials science.

Atomic valence correlation energies from ab initio model potential calculations

Detonation characteristics of hydrocarbon fuels are important indexes for their safety evaluation and protection. To investigate the detonation characteristic of alkane homologs series and the influence of different molecular energies, alkane homologs series’ critical energy to direct detonation (hereinafter related to as critical energy) and detonation cellular width were measured by a self-designed shock tube. Then, the molecular total energy (E) and bond dissociation energy (BDE) of fuel molecules were calculated by the quantum chemistry method. After the comparison between detonation characteristics and molecular energies, it is found that for alkane homologs series, molecular total energy was found to have a good exponent relation between critical energy and detonation cellular width, which indicated that the detonation sensitivity of alkane homologs series can be predicted based on total energy of fuel molecules with preferable accuracy. Extra experiment and calculation was also carried out on gaseous hydrocarbon fuels (ethane, ethylene and acetylene) with same carbon atom number. Results indicated that the bond dissociation energy may have the impact on the detonation cellular width for these three gaseous fuels. The bigger bond dissociation energy leads to smaller cellular width.

We construct a reference benchmark set for atomic and molecular random phase approximation (RPA) correlation energies in a density functional theory framework at the complete basis-set limit. This set is used to evaluate the accuracy of some popular extrapolation schemes for RPA all-electron molecular calculations. The results indicate that for absolute energies, accurate results, clearly outperforming raw data, are achievable with two-point extrapolation schemes based on quintuple- and sextuple-zeta basis sets. Moreover, we show that results in good agreement with the benchmark can also be obtained by using a semiempirical extrapolation procedure based on quadruple- and quintuple-zeta basis sets. Finally, we analyze the performance of different extrapolation schemes for atomization energies.

Heats of atomization Ha (per atom) are compared for for 28 diatomic oxides, 14 diatomic fluorides, 26 solid and 2 polyatomic gaseous oxides, 9 solid and 5 polyatomic gaseous fluorides. There is no universal trend toward lower Ha with increasing Z in a column of the Periodic Table; d-group, post-d-group, and partly filled 4f and 5f shells encourage smaller Ha but barium, lanthanum, lutetium, hafnium, tantalum, (to the largest extent thorium), and uranium show unexpected high Ha. This 'Barium-Thorium Effect' can be rationalized by known facts of correlation energy in atoms and (light) compounds, possibly including indirect relativistic effects due to distorted AO in LCAO.

The availability of quantitatively accurate total energies $({E}_{\mathrm{tot}})$ of atoms, molecules, and solids, enabled by the development of density functional theory (DFT), has transformed solid state physics, quantum chemistry, and materials science by allowing direct calculations of measureable quantities, such as enthalpies of formation $(\mathrm{\ensuremath{\Delta}}{H}_{f})$. Still, the ability to compute ${E}_{\mathrm{tot}}$ and $\mathrm{\ensuremath{\Delta}}{H}_{f}$ values does not, necessarily, provide insights into the physical mechanisms behind their magnitudes or chemical trends. Here, we examine a large set of calculated ${E}_{\mathrm{tot}}$ and $\mathrm{\ensuremath{\Delta}}{H}_{f}$ values obtained from the DFT+$U$-based fitted elemental-phase reference energies (FERE) approach [V. Stevanovi\ifmmode \acute{c}\else \'{c}\fi{}, S. Lany, X. Zhang, and A. Zunger, Phys. Rev. B 85, 115104 (2012)] to probe relationships between the ${E}_{\mathrm{tot}}/\mathrm{\ensuremath{\Delta}}{H}_{f}$ of metal-nonmetal compounds in their ground-state crystal structures and properties describing the compound compositions and their elemental constituents. From a stepwise linear regression, we develop a linear model for ${E}_{\mathrm{tot}}$, and consequently $\mathrm{\ensuremath{\Delta}}{H}_{f}$, that reproduces calculated FERE values with a mean absolute error of \ensuremath{\sim}80 meV/atom. The most significant contributions to the model include calculated total energies of the constituent elements in their reference phases (e.g., metallic iron or gas phase ${\mathrm{O}}_{2}$), atomic ionization energies and electron affinities, Pauling electronegativity differences, and atomic electric polarizabilities. These contributions are discussed in the context of their connection to the underlying physics. We also demonstrate that our ${E}_{\mathrm{tot}}/\mathrm{\ensuremath{\Delta}}{H}_{f}$ model can be directly extended to predict the ${E}_{\mathrm{tot}}$ and $\mathrm{\ensuremath{\Delta}}{H}_{f}$ of compounds outside the set used to develop the model.

For selected first- and second-row atoms, correlation-optimized Gaussian k functions have been determined and used in the construction of septuple-ζ basis sets for the correlation-consistent cc-pVXZ and aug-cc-pVXZ series. Restricted Hartree–Fock (RHF) and second-order Møller–Plesset (MP2) total and pair energies were computed for H, N, O, F, S, H2, N2, HF, H2O, and (H2O)2 to demonstrate the consistency of the new septuple-ζ basis sets as extensions of the established (aug)-cc-pVXZ series. The pV7Z and aug-pV7Z sets were then employed in numerous extrapolation schemes on the test species to probe the accuracy limits of the conventional MP2 method vis-à-vis explicitly correlated (MP2-R12/A) benchmarks. For (singlet, triplet) pairs, (X+12)−n functional forms with n=(3, 5) proved best for extrapolations. The (mean abs. relative error, std. dev.) among the 73 singlet pair energies in the dataset is (1.96%, 0.54%) and (1.72%, 0.51%) for explicit computations with the pV7Z and aug-pV7Z basis sets, respectively, but only (0.07%, 0.09%) after two-point, 6Z/7Z extrapolations with the (X+12)−3 form. The effects of k functions on molecular relative energies were examined by application of the septuple-ζ basis sets to the barrier to linearity and the dimerization energy of water. In the former case, an inherent uncertainty in basis set extrapolations persists which is comparable in size to the error (≈20 cm−1) in explicit aug-pV7Z computations, revealing fundamental limits of orbital expansion methods in the domain of subchemical accuracy (0.1 kcal mol−1).

Hierarchic computation of atomic correlation energies using a p-version finite element method