José L. Gázquez

Piecewise polynomial electronic wavefunctions

Piecewise polynomials are examined as basis functions for electronic wavefunctions. The spline function method is a special case, which is shown to be less accurate, for a fixed set of mesh points, than a method based directly on Hermite’s interpolation formula. The determination of a suitable mesh is discussed both inductively and deductively, and a logarithmic formula for the 1s orbital of helium is ’’derived.’’ The accuracy is shown to depend on the number of points N+1 and on the polynomial order 2s+1, approximately according to the formula, δE∼N−4s−2, for appropriate meshes. A striking result is the possibility for systematically increasing the accuracy of the energy by systematically increasing the number of points, without encountering linear dependence problems, is demonstrated by calculations on the helium atom. With a 16‐point theoretically derived mesh, and with seventh order polynomials, we obtain a Hartree–Fock energy for helium of −2.8616799956122 a.u.

Electronegativities and hardnesses of open shell atoms

A Taylor series expansion of the energy of an atomic system around the neutral atom value, which introduces the first and second derivatives of the energy with respect to the number of electrons (electronegativity χ, and hardness η, respectively) is proposed. The relaxed first derivative and the unrelaxed second derivative of the Xα and hyper‐Hartree–Fock methods are used to relate χ and η with the Lagrange multiplier ei, and the self‐repulsion integral J(i) of the highest occupied atomic orbital for the case of an open shell. A simple model, based on screening effects, is developed to get a better representation of a relaxed second derivative. This model replaces J(i) by 1/2 〈r−1〉i and leads to η= 1/4 〈r−1〉i. The use of this relation, together with the Xα expression for electronegativity, χ=−ei, and a simple charge transfer model for electronegativity equalization leads to values of molecular electronegativities which are in very good agreement with the values obtained through the use of atomic or molecu...

Non-empirical improvement of PBE and its hybrid PBE0 for general description of molecular properties

Imposition of the constraint that, for the hydrogen atom, the exchange energy cancels the Coulomb repulsion energy yields a non-empirical re-parameterization of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-correlation energy functional, and of the related PBE hybrid (PBE0). The re-parameterization, which leads to an increase of the gradient contribution to the exchange energy with respect to the original PBE functional, is tested through the calculation of heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies, for some well known test sets designed to validate energy functionals. The results for the re-parameterized PBE GGA, called PBEmol, give substantial improvement over the original PBE in the prediction of the heats of formation, while retaining the quality of the original PBE functional for description of all the other properties considered. The results for the hybrids indicate that, although the PBE0 functional provides a rather good description of these properties, the predictions of the re-parameterized functional, called PBEmolβ0, are, except in the case of the ionization potentials, modestly better. Also, the results for PBEmolβ0 are comparable to those of B3LYP. In particular, the mean absolute error for the bond distance test set is 17% lower than the corresponding error for B3LYP. The re-parameterization for the pure GGA (PBEmol) differs from that for the hybrid (PBEmolβ0), illustrating that improvement at the GGA level of complexity does not necessarily provide the best GGA for use in a hybrid.

Local softness and chemical reactivity of maleimide: nucleophilic addition

Abstract The local softness and the Fukui function of maleimide are determined through a finite differences scheme in order to show the usefulness of these concepts for rationalizing the inherent chemical reactivity of a given molecule. The local softness surface diagram of maleimide indicates that soft nucleophiles interact with the a carbon atoms, whereas hard nucleophiles interact with the carbonyl carbon atoms, in agreement with the experimental evidence.

Density Functional Theory

The constrained search approach, mappings to external potentials, and virial-like theorems for electron-density and one-matrix energy-functional theories.- Density matrices, reduced density matrices, a geometric investigation of their properties, and applications to density functional theory.- Properties of one-matrix energy functionals.- Self-interaction correction.- Some recent developments in density functional theory.- Electron gas models and density functional theory.- Electron structure calculations for heavy atoms: A local density approach.- Density functionals obtained from models of the electron first and second order density matrices.- Free energy density functionals for non-uniform classical fluids.

Revisiting the definition of the electronic chemical potential, chemical hardness, and softness at finite temperatures

We extend the definition of the electronic chemical potential (μe) and chemical hardness (ηe) to finite temperatures by considering a reactive chemical species as a true open system to the exchange of electrons, working exclusively within the framework of the grand canonical ensemble. As in the zero temperature derivation of these descriptors, the response of a chemical reagent to electron-transfer is determined by the response of the (average) electronic energy of the system, and not by intrinsic thermodynamic properties like the chemical potential of the electron-reservoir which is, in general, different from the electronic chemical potential, μe. Although the dependence of the electronic energy on electron number qualitatively resembles the piecewise-continuous straight-line profile for low electronic temperatures (up to ca. 5000 K), the introduction of the temperature as a free variable smoothens this profile, so that derivatives (of all orders) of the average electronic energy with respect to the average electron number exist and can be evaluated analytically. Assuming a three-state ensemble, well-known results for the electronic chemical potential at negative (-I), positive (-A), and zero values of the fractional charge (-(I + A)/2) are recovered. Similarly, in the zero temperature limit, the chemical hardness is formally expressed as a Dirac delta function in the particle number and satisfies the well-known reciprocity relation with the global softness.

Local and linear chemical reactivity response functions at finite temperature in density functional theory

We explore the local and nonlocal response functions of the grand canonical potential density functional at nonzero temperature. In analogy to the zero-temperature treatment, local (e.g., the average electron density and the local softness) and nonlocal (e.g., the softness kernel) intrinsic response functions are defined as partial derivatives of the grand canonical potential with respect to its thermodynamic variables (i.e., the chemical potential of the electron reservoir and the external potential generated by the atomic nuclei). To define the local and nonlocal response functions of the electron density (e.g., the Fukui function, the linear density response function, and the dual descriptor), we differentiate with respect to the average electron number and the external potential. The well-known mathematical relationships between the intrinsic response functions and the electron-density responses are generalized to nonzero temperature, and we prove that in the zero-temperature limit, our results recover well-known identities from the density functional theory of chemical reactivity. Specific working equations and numerical results are provided for the 3-state ensemble model.

Improved constraint satisfaction in a simple generalized gradient approximation exchange functional

Though there is fevered effort on orbital-dependent approximate exchange-correlation functionals, generalized gradient approximations, especially the Perdew-Burke-Ernzerhof (PBE) form, remain the overwhelming choice in calculations. A simple generalized gradient approximation (GGA) exchange functional [A. Vela, V. Medel, and S. B. Trickey, J. Chem. Phys. 130, 244103 (2009)] was developed that improves substantially over PBE in energetics (on a typical test set) while being almost as simple in form. The improvement came from constraining the exchange enhancement factor to be below the Lieb-Oxford bound for all but one value of the exchange dimensionless gradient, s, and to go to the uniform electron gas limit at both s = 0 and s → ∞. Here we discuss the issue of asymptotic constraints for GGAs and show that imposition of the large s constraint, lim(s→∞)s(1/2)F(xc)(n,s)<∞, where F(xc)(n, s) is the enhancement factor and n is the electron density, upon the Vela-Medel-Trickey (VMT) exchange functional yields modest further improvement. The resulting exchange functional, denoted VT{8,4}, is only slightly more complicated than VMT and easy to program. Additional improvement is obtained by combining VT{8,4} or VMT exchange with the Lee-Yang-Parr correlation functional. Extensive computational results on several datasets are provided as verification of the overall performance gains of both versions.

Generalized gradient approximation exchange energy functional with correct asymptotic behavior of the corresponding potential

A new non-empirical exchange energy functional of the generalized gradient approximation (GGA) type, which gives an exchange potential with the correct asymptotic behavior, is developed and explored. In combination with the Perdew-Burke-Ernzerhof (PBE) correlation energy functional, the new CAP-PBE (CAP stands for correct asymptotic potential) exchange-correlation functional gives heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies on standard test sets that are fully competitive with those obtained from other GGA-type functionals that do not have the correct asymptotic exchange potential behavior. Distinct from them, the new functional provides important improvements in quantities dependent upon response functions, e.g., static and dynamic polarizabilities and hyperpolarizabilities. CAP combined with the Lee-Yang-Parr correlation functional gives roughly equivalent results. Consideration of the computed dynamical polarizabilities in the context of the broad spectrum of other properties considered tips the balance to the non-empirical CAP-PBE combination. Intriguingly, these improvements arise primarily from improvements in the highest occupied and lowest unoccupied molecular orbitals, and not from shifts in the associated eigenvalues. Those eigenvalues do not change dramatically with respect to eigenvalues from other GGA-type functionals that do not provide the correct asymptotic behavior of the potential. Unexpected behavior of the potential at intermediate distances from the nucleus explains this unexpected result and indicates a clear route for improvement.

Two‐parameter statistical model for atoms

A simple two‐parameter charge density of the form ρ (r) =C/(1+βr)n with β=α/n, where C is determined by normalization and α and n are determined by minimization of the total energy, is examined in connection with an energy density functional that contains kinetic energy terms up to fourth order in the gradient expansion and the classical Dirac exchange term. This charge density is finite at the nucleus, is monotonically decreasing, becomes pure exponential when n→∞, is extremely accurate for determining two‐electron Hartree–Fock densities, and gives good energy values for first‐row atoms. For higher atomic number, the energy values are much superior to the traditional Thomas–Fermi–Dirac values. Also presented, following Fermi [Z. Phys. 48, 73 (1928)] is a prediction with this charge density of the number of electrons with a given quantum number l in an atom with atomic number Z. The agreement with the known values is excellent: whereas Fermi found that the d shell begins to fill at Z=19.33 and the f shell...