Federico Moscardó

A simple, reliable and efficient scheme for automatic numerical integration

A scheme for automatic numerical integration is presented. It uses the change of variable x=1+(2π){[1+23(1-z2)]z√1-z2 -arccosz} to transform the integral to be computed, ∫1-1f(x) dx, into (163π)∫1-1f(x)(1-z2)√1-z2 dz, which is approximated by successive n-points Gauss-Chebyshev quadrature formulas of the second kind (In). Due to the special nature of their abscissas and weights, a sequence of formulas I1, I3, I7,…, I(n-1)2, In, I2n+1 may be generated, such that I2n+1 may be computed with only n + 1 new integrand evaluations, using the previous value of In. An error estimation is proposed for I2n+1, which only needs two previous values of the sequence (In and I(n+1)2). The algorithm may be implemented by a very short program (a FORTRAN 77 version is included) that spends practically all its running time in integrand evaluations. It is compared with other methods for automatic numerical integration (trapezoidal rule, Simpsons rule, Rombergs method, an adaptive Gauss-Kronrod rule and Clenshaw-Curtis method) over a broad set of 20 functions. We conclude that the present method is very simple and reliable and is the most efficient among the methods tested here. Possible applications in density functional theory are explored.

A comparison between DFT and other ab initio schemes on the activation energy in the automerization of cyclobutadiene

Abstract The influence of the multireference character in the transition state for the automerization reaction of cyclobutadiene is considered. We have analyzed two forms of taking into account this effect: either by the use of two-body density functionals or traditional density functional theory (DFT) correlation functionals conveniently modified. Comparison has also been made with conventional density functional theory Kohn–Sham (DFT–KS) [15] approaches. It is shown that only when the aforementioned multideterminantal character is included in the computational scheme is the activation energy in accord with accurate benchmark calculations.

Usefulness of the Colle–Salvetti model for the treatment of the nondynamic correlation

In this work, the behavior of the Colle–Salvetti correlation functional is examined for strongly correlated systems with non-negligible nondynamic effects. Used with an appropriate multideterminantal wave function, it is able to reproduce accurately previous multireference coupled-cluster results for the problem of the automerization of cyclobutadiene, as well as to provide the correct energetical profiles for diatomic molecules under dissociation. The results confirm the current quality of the functional for complicated chemical problems, in spite of the fact that the functional does not satisfy some known exact properties.

New approach to the design of density functionals

The prevailing approach in density-functional theory makes use of universal functionals, valid for any number of electrons (N). In this article and following the work of Lieb [E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983)], we argue that the use of N-dependent functionals could be an equally valid approach. Size consistency (dissociation into proper fragments) puts restrictions on the form that N-dependent functionals may adopt. We propose a simple procedure for correcting the size-consistency problem of existing N-dependent functionals that also provides an original way of designing new ones.

V-N vertical transition of planar ethylene

The V-N vertical transition of planar ethylene has been investigated by means of a large Gaussian basis set, taking into account both the SCF and correlation energies. It is shown that in this case the SCF function, which would lead to a valuation of the Rydberg character at around 80%, does not provide an adequate description of the V state. The correlative effects bring about a lowering of the percentage of the Rydberg character to 38%. The calculated value for the transition energy is in excellent accord with the experimental data.

High-level ab initio calculations of the torsional potential of glyoxal

Abstract The torsional barrier of glyoxal has been extensively studied using state-of-the-art ab initio methodologies ranging from Hartree–Fock (HF) and Moller–Plesset theories up to several developments of the coupled-cluster (CC) formalism, namely: CCD, CCSD, CCSD(T), BD and BD(T). We present a thorough discussion about the influence of the basis set size on the results, as well as a comparison between all the theories employed. The values obtained from the CCSD or CCSD(T) treatment achieve the so-called `calibration accuracy (±1 kJ / mol ) from which an extremely accurate torsional potential has been derived.

On the application of the Kohn–Sham theory to the calculation of potential energy curves

Abstract The application of single-determinant Kohn–Sham theory to the calculation of potential energy curves has been analyzed on the basis of the calculations for three diatomic molecules. Both the non-polarized and the polarized results show that the one-determinant approach does not provide exact potential energy curves for the ground state. These deficiencies cannot be corrected by the correlation potential without explicitly including a dependence on more Slater determinants.

Electron correlation in the Coulomb hole model. Comparison of methods

Methods, based both on the density functional and on the correlation factor, are analyzed by applying them to the correlation energy calculation of two‐ and four‐electron closed shell atoms; so that only the short range electron correlation is considered. The method based on the correlation factor seems to be more promising than those of the density functional. However, in the present state of development, there are some serious deficiencies that limit their application to atomic and molecular calculations.

Results of the correlation energy functionals versus the wavefunction type

Abstract The results of applying several correlation energy functionals (Gombas, Vosko-Wilk-Nussair, Perdew, Becke, Colle-Salvetti, Moscardo-San-Fabian and Lee-Yang-Parr) to different wave-function types (RHF, GVB and CI) were analyzed. The test was carried out on atomic systems (He, Li, Be, F and Ne) and on H2 and LiH molecules at several internuclear distances. The potential energy curve of the hydrogen molecule was calculated; its accuracy was analyzed by checking the results for (i) some molecular properties such as Re and De and (ii) the Born-Oppenheimer vibrational quanta, as obtained from the Kolos and Wolnievicz potential energy curve.

Comparison between ab initio and density functional calculations in the reaction O+(4S) + H2(1∑g+) → OH+(3∑−) + H(2S)

Abstract The reaction between O+ and H2 has been studied by means of density functional methods. We adopt a combined procedure in which a Hartree-Fock, or GVB-PP calculation, is complemented by the calculation of the correlation energy using a density functional. Four different correlation functionals are used: Perdews (PF), Beckes (BF), and the more recent ones of Moscardo and San-Fabian (MSF), and Moscardo and Perez-Jimenez (MP). We have also carried out Kohn-Sham calculations combining different exchange-correlation functionals. The functionals MSF and MP, together with the GVB-PP wavefunction, provide the best numerical results. Bond lengths, dissociation energies, and the more difficult to obtain category the reaction energy, compare favorably with the result of very extensive MRCI-MP2 calculations.