Bonds between, and to, transition-metal atoms often involve strong electron correlation which cannot be handled by conventional density-functional-theory approximations. The recent “B13” functional of Becke [J. Chem. Phys. 138, 074109 (2013) and J. Chem. Phys. 138, 161101 (2013)] models dynamic, static, and strong correlation in an exact-exchange-based framework. We test B13 on bond energies of transition-metal diatomics in this work, with promising results.
A completely numerical method for the computation of Hartree–Fock–Slater wave functions in diatomic systems has been previously reported and applied to the calculation of dissociation energies for selected first-row molecules. The previous results were obtained using spin-restricted orbitals. In this note, the results of new, fully spin-unrestricted calculations are presented.
Since its formal inception in 1964–1965, Kohn-Sham density-functional theory (KS-DFT) has become the most popular electronic structure method in computational physics and chemistry. Its popularity stems from its beautifully simple conceptual framework and computational elegance. The rise of KS-DFT in chemical physics began in earnest in the mid 1980s, when crucial developments in its exchange-correlation term gave the theory predictive power competitive with well-developed wave-function methods. Today KS-DFT finds itself under increasing pressure to deliver higher and higher accuracy and to adapt to ever more challenging problems. If we are not mindful, however, these pressures may submerge the theory in the wave-function sea. KS-DFT might be lost. I am hopeful the Kohn-Sham philosophical, theoretical, and computational framework can be preserved. This Perspective outlines the history, basic concepts, and present status of KS-DFT in chemical physics, and offers suggestions for its future development.
We have compiled a benchmark set of mean ligand-removal enthalpies for 32 transition-metal complexes of relevance in organometallic and catalysis chemistry. Our recent exact-exchange-based density-functional model, DF07 ( J. Chem. Phys. 2007, 127 (12), 124108 ), is assessed on this benchmark set along with other representative GGA, meta-GGA, and hybrid functionals. DF07 performs remarkably well, despite its exact-exchange foundation, indicating that it properly describes nondynamical correlation in transition-metal–ligand bonds.
We have developed a fully numerical, basis-set-free algorithm for solution of the Schrödinger single-particle equation in polyatomic molecules. As a test of the algorithm, the Hartree–Fock energy of H+3 is computed and compared with previous momentum-space benchmarks. The present calculations are the first successful basis-set-free calculations in coordinate space on a polyatomic molecular system.
We describe a new algorithm for the generation of 3D grids for the numerical evaluation of multicenter molecular integrals in density functional theory. First, we use the nuclear weight functions method of Becke [A. D. Becke, J. Chem. Phys. 88, 2547 (1988)] to decompose a multicenter integral ∫F(r) dr into a sum of atomic-like single-center integrals. Then, we apply automatic numerical integration techniques to evaluate each of these atomic-like integrals, so that the total integral is approximated as ∫F(r) dr≊∑iωiF(ri). The set of abscissas ri and weights ωi constitutes the 3D grid. The 3D atomic-like integrals are arranged as three successive monodimensional integrals, each of which is computed according to a recently proposed monodimensional automatic numerical integration scheme which is able to determine how many points are needed to achieve a given accuracy. When this monodimensional algorithm is applied to 3D integration, the 3D grids obtained adapt themselves to the shape of the integrand F(r), and have more points in more difficult regions. The function F(r), which, upon numerical integration, yields the 3D grid, is called the generating function of the grid. We have used promolecule densities as generating functions, and have checked that grids generated from promolecule densities are also accurate for other integrands. Our scheme is very reliable in the sense that, given a relative tolerance ε, it generates 3D grids which are able to approximate multicenter integrals with relative errors smaller than ε for all the molecules tested in this work. Coarser or finer grids can be obtained using greater or smaller tolerances. For a series of 21 molecules, the average number of points per atom for ε=2.0⋅10−3, ε=2.0⋅10−4, ε=2.0⋅10−5, ε=2.0⋅10−6, and ε=2.0⋅10−7 is respectively 3141 (2.9⋅10−4), 10271 (2.4⋅10−5), 27184 (3.1⋅10−6), 72266 (1.9⋅10−7), and 164944 (5.2⋅10−9) (in parentheses are the maximum errors obtained when integrating the density). It is possible to reduce the number of points in the grid by taking advantage of molecular symmetry. It seems that our method achieves a given accuracy with fewer points than other recently proposed methods.
In this work, our exact-exchange-based static + dynamical correlation density functional [A. D. Becke, J. Chem. Phys. 122, 064101 (2005)]10.1063/1.1844493 is generalized to include “strong” correlation, i.e., accurate computations on dissociating chemical systems without breaking space or spin symmetries and without using multi-determinantal reference states. Also, we introduce a strong-correlation benchmark set composed of space- and spin-symmetrized open-shell atoms on which the generalized functional is tested. Initial results are very promising.
In a recent paper [H. L. Schmider and A. D. Becke, J. Chem. Phys. 108, 9624 (1998)], we applied a systematic method for the determination of exchange-correlation functionals within the generalized gradient approximation (GGA) to the extended G2 test set of standard heats of formation of Curtiss et al. [J. Chem. Phys. 106, 1063 (1997)]. In the present work, we apply a similar methodology that goes beyond the GGA by taking second-order gradients and the (noninteracting) kinetic-energy density into account. The resulting improvement in the reproduction of thermochemical data brings us very close to the quality of G2 theory itself. Our lowest mean absolute error for standard heats of formation, 1.60 kcal/mol, is only marginally greater than the G2 value (1.58 kcal/mol). The corresponding largest deviation is 9.97 kcal/mol, as compared to 8.2 kcal/mol for G2 theory.
We present a new coordinate-space model of spherically averaged exchange-hole functions in inhomogeneous systems that depends on local values of the density and its gradient and Laplacian, and also the kinetic energy density. Our model is completely nonempirical, incorporates the uniform-density electron gas and hydrogenic atom limits, and yields the proper 1/r asymptotic exchange potential in finite systems. Comparisons of model exchange energies, holes, and potentials with exact Hartree-Fock results in selected atoms are very encouraging.
We introduce in this work a new approach to the identification of localized electronic groups in atomic and molecular systems. Our approach is based on local behavior of the Hartree–Fock parallel-spin pair probability and is completely independent of unitary orbital transformations. We derive a simple ‘‘electron localization function’’ (ELF) which easily reveals atomic shell structure and core, binding, and lone electron pairs in simple molecular systems as well.
Real-space models of nondynamical correlation between electrons of opposite spin and of parallel spin in multicenter (molecular) systems are discussed. These models are designed to be partnered with Hartree–Fock or exact Kohn–Sham exchange. Thus the numerous and well-known problems of local density-functional exchange-correlation approximations, especially in stretched odd-electron systems, are circumvented.
Despite the remarkable thermochemical accuracy of Kohn–Sham density-functional theories with gradient corrections for exchange-correlation [see, for example, A. D. Becke, J. Chem. Phys. 96, 2155 (1992)], we believe that further improvements are unlikely unless exact-exchange information is considered. Arguments to support this view are presented, and a semiempirical exchange-correlation functional containing local-spin-density, gradient, and exact-exchange terms is tested on 56 atomization energies, 42 ionization potentials, 8 proton affinities, and 10 total atomic energies of first- and second-row systems. This functional performs significantly better than previous functionals with gradient corrections only, and fits experimental atomization energies with an impressively small average absolute deviation of 2.4 kcal/mol.
First singlet (S1) excitations are of primary importance in the photoluminescence spectra of organic chromophores. However, due to the multi-determinantal nature of the singlet excited states, standard Kohn-Sham density-functional theory (DFT) is not applicable. While linear-response time-dependent DFT is the method of choice for the computation of excitation energies, it fails severely for excitations with charge-transfer character. Becke’s recent virial exciton model [A. D. Becke, J. Chem. Phys. 148, 044112 (2018)] offers a promising solution to employ standard DFT for calculation of the S1 excitation energy in molecular systems. Here, it is shown that the virial exciton model is free of charge-transfer error. It is equally reliable for S1 excitations with significant charge-transfer character as for other classes of transitions.
A simple formalism for the evaluation of 〈S2〉 in terms of the two-particle density matrix is presented. The implementation of the formalism in the restricted open-shell Hartree–Fock (ROHF), unrestricted HF (UHF) and density functional (DFT) based theories is discussed. Rules governing the nonzero S2 matrix elements in the UHF based methods are presented. Further examples are given of 〈S2〉 in several atomic and radical systems from very simple density functional models.
