Generator Coordinate Dirac-Fock Method Research Articles

Relativistic adapted Gaussian basis sets free of variational prolapse of small and medium size for cesium through radon.

Relativistic adapted Gaussian basis sets of small and medium sizes are presented in this study for all elements from cesium to radon, including some alternative electron configurations. Both basis sets are made free of variational prolapse, being developed by means of a polynomial version of the generator coordinate Dirac-Fock method. In addition, these sets were designed to be promptly used with two popular finite nuclear models, uniform sphere and Gaussian nuclei. The largest basis set errors found with the uniform sphere nucleus are 27.3 and 10.6 mHartree, respectively, for the small- and medium-size sets. The largest basis set errors obtained with the Gaussian nuclear model are smaller, reaching 23.2 and 7.1 mHartree for the small- and medium-size sets, respectively. Soon, these basis sets will be augmented with polarization functions to be properly used in molecular calculations.

Relativistic adapted Gaussian basis sets free of variational prolapse of small and medium size for hydrogen through xenon.

Two new relativistic adapted Gaussian basis sets of small and medium size are presented for all elements from Hydrogen through Xenon. These sets are free of variational prolapse and were developed with the polynomial generator coordinate Dirac-Fock method to be used with two finite nuclear models, uniform sphere and Gaussian. The largest basis set errors for electronic configurations from the Aufbau principle are around 10.0 and 4.7 mHartree for the small- and medium-size sets, respectively, which is in accordance with the accuracy level expected in each case. Hence, to our knowledge, these are the smallest prolapse free basis sets developed for these elements.

Relativistic Prolapse-Free Gaussian Basis Set of Quadruple-ζ Quality: (aug-)RPF-4Z. I. The s- and p-Block Elements

This study reports a new relativistic prolapse-free Gaussian basis set series of quadruple-ζ quality, RPF-4Z, and an augmented version that includes extra diffuse functions, aug-RPF-4Z, for all the s- and p-block elements. The relativistic adapted Gaussian basis sets (RAGBSs), which are free of variational prolapse, were used as the starting primitive sets. Exponents of correlating/polarization functions were taken from a polynomial version of the generator coordinate Dirac-Fock (p-GCDF) method, in which the previously optimized RAGBS parameters are applied. By using such procedure we aimed to reduce the computational demand of these sets in comparison with fully optimized ones. The effect of these basis set increments on the correlation energy was evaluated by atomic multireference configuration interaction calculations with single and double excitations out of the valence shell. Finally, atomic and molecular calculations of fundamental properties (bond lengths, vibrational frequencies, dipole moments and electron affinities) corroborate the quadruple-ζ quality of these new sets that are also about half-time-consuming than the correspondent Dyall's v4z sets. The read-to use format of these (aug-)RPF-4Fz sets are available as Supporting Information files and can also be found at http://basis-sets.iqsc.usp.br/ .

Accurate relativistic adapted gaussian basis sets for francium through ununoctium without variational prolapse and to be used with both uniform sphere and gaussian nucleus models

Accurate relativistic adapted Gaussian basis sets (RAGBSs) for 87 Fr up to 118 Uuo atoms without variational prolapse were developed here with the use of a polynomial version of the Generator Coordinate Dirac-Fock method. Two finite nuclear models have been used, the Gaussian and uniform sphere models. The largest RAGBS error, with respect to numerical Dirac-Fock results, is 15.4 miliHartree for Ununoctium with a basis set size of 33s30p19d14f functions.

Accurate relativistic adapted Gaussian basis sets for Cesium through Radon without variational prolapse and to be used with both uniform sphere and Gaussian nucleus models

Accurate relativistic adapted Gaussian basis sets (RAGBSs) from Cs (Z = 55) through Rn (Z = 86) without variational prolapse were developed by using the polynomial version of the Generator Coordinate Dirac-Fock method. The RAGBSs presented here can be used with any of two popular finite nucleus models, the uniform sphere and the Gaussian models. The largest RAGBS error is 4.5 mHartree for Radon with a size of 30s27p17d11f.

Accurate relativistic adapted Gaussian basis sets for hydrogen through xenon without variational prolapse and to be used with both uniform sphere and Gaussian nucleus models

Accurate relativistic adapted Gaussian basis sets (RAGBSs) from H (Z = 1) through Xe (Z = 54) without variational prolapse have been developed by employing a polynomial version of the Generator Coordinate Dirac-Fock (p-GCDF) method. Two nuclear models have been used in this work: (1) the finite nucleus of uniform proton-charge distribution, and (2) the finite nucleus with a Gaussian proton–charge distribution. The largest errors observed are only 1.5 mHartree (silver and cadmium) and the RAGBS sizes are much smaller than previous accurate relativistic Gaussian basis sets that were shown to be free of variational prolapse. © 2005 Wiley Periodicals, Inc. J Comput Chem 27: 61–71, 2006

Relativistic Gaussian basis sets obtained with a polynomial version of the generator coordinate Dirac–Fock method: Ionization energies of some closed‐shell atomic systems

AbstractThe purpose of this work is to analyze the relativistic Gaussian basis sets obtained with a recently developed polynomial version of the generator coordinate Dirac–Fock method. The atomic systems chosen to evaluate the basis sets are some alkaline earth and noble gas atoms: He, Be, Ne, Mg, Ar, and Ca. Hartree–Fock, relativistic, and electronic correlation contributions to the atomic total energy of the neutral and ionized atomic species were estimated using these relativistic Gaussian basis sets. The basis sets developed were able to attain energies that deviate at most by 0.47 and 0.65 mHartree, respectively, from numerical Hartree–Fock and Dirac–Fock–Coulomb results and are also competitive with fully optimized relativistic Gaussian basis sets of similar sizes. The electronic correlation energy contributions were obtained after two nonrelativistic calculations: a Hartree–Fock and a full quadratic configuration interaction, including single and double substitutions. The uncontracted basis sets obtained here along with polarization functions chosen to have equal exponents to selected functions already present in the original set were adequate in describing not only the neutral but also the highly ionized atomic species. The comparison of some calculated shell ionization energies with experimental results presented a maximum deviation of 3% (for a total of 15 ionization values ranging from 18 to 10,599 eV), reinforcing the quality of the basis sets developed in this work. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

A polynomial version of the generator coordinate Dirac-Fock method.

A polynomial version of the Generator Coordinate Dirac-Fock (p-GCDF) method is introduced and applied to develop Adapted Gaussian Basis Sets (AGBS) for helium- and beryllium-like atomic species (He, Ne +8, Ar +16, Sn +48, Be, Ne +6, Ar +14, and Sn +46) and for Kr and Xe atoms. The Dirac-Fock-Coulomb and Dirac-Fock-Breit energies obtained with these basis sets are in excellent agreement with numerical finite-difference calculations. Moreover, the sizes of the AGBS generated here with the p-GCDF method are significantly smaller than the size of previous relativistic Gaussian basis sets.

Highly accurate relativistic gaussian basis sets for closed-shell atoms from He through to No

We have used the generator coordinate Dirac–Fock method for closed shell atoms, to generate accurate adapted Gaussian basis sets for ab initio relativistic self-consistent field calculations of some atoms from He ( Z=2) through to No ( Z=102). For all atoms studied, our Dirac–Fock–Coulomb and Dirac–Fock–Breit total energies are better than the corresponding ones obtained with previous larger Gaussian basis sets. Except for the Hg atom, our Dirac–Fock–Coulomb total energies are always equal to or lower than those calculated with the numerical finite-difference method (GRASP2 package). For No, we compare our Dirac–Fock–Coulomb and Dirac–Fock–Breit orbital energies with the corresponding values obtained with other approaches.

Accurate adapted Gaussian basis sets for helium- and beryllium-like atomic species to be used in Dirac–Fock calculations

The closed-shell Generator Coordinate Dirac–Fock method is applied to perform Dirac–Fock–Coulomb and Dirac–Fock–Breit calculations for He- and Be-like atomic species. With the Generator Coordinate Dirac–Fock method, the integral Dirac–Fock equations are integrated numerically in order to generate accurate adapted Gaussian basis sets for the atomic species under study. The Dirac–Fock–Coulomb and Dirac–Fock–Breit energies obtained here for He- and Be-like atomic species with the Generator Coordinate Dirac–Fock formalism are in general better than the corresponding energies obtained with previous Gaussian-type functions. The Dirac–Fock–Coulomb energies obtained for the atomic species here studied are in excellent agreement with numerical-finite-difference calculations, and for several atomic species the Dirac–Fock–Coulomb energy results obtained with our adapted Gaussian basis sets are lower than the corresponding energies obtained from numerical-finite-difference calculations.

The generator coordinate Dirac–Fock method for open-shell atomic systems

Recently we developed generator coordinate Dirac–Fock and Dirac–Fock–Breit methods for closed-shell systems assuming finite nucleus and have reported Dirac–Fock and Dirac–Fock–Breit energies for the atoms He through Nobelium (Z=102) [see Refs. Reference 10Reference 11Reference 12Reference 13]. In this paper, we generalize our earlier work on closed-shell systems and develop a generator coordinate Dirac–Fock method for open-shell systems. We present results for a number of representative open-shell heavy atoms (with nuclear charge Z>80) including the actinide and superheavy transactinide (with Z>103) atomic systems: Fr (Z=87), Ac (Z=89), and Lr (Z=103) to E113 (eka-thallium, Z=113). The high accuracy obtained in our open-shell Dirac–Fock calculations is similar to that of our closed-shell calculations, and we attribute it to the fact that the representation of the relativistic dynamics of an electron in a spherical ball finite nucleus near the origin in terms of our universal Gaussian basis set is as accurate as that provided by the numerical finite difference method. The DF SCF energies calculated by Desclaux [At. Data. Nucl. Data Tables 12, 311 (1973)] (apart from a typographic error for Fr pointed out here) are higher than those reported here for atoms of some of the superheavy transactinide elements by as much as 5 hartrees (136 eV). We believe that this is due to the use by Desclaux of much larger atomic masses than the currently accepted values for these elements.

The generator coordinate Dirac-Fock method applied to helium-like atomic species

The closed-shell Generator Coordinate Dirac-Fock method is applied to perform Dirac-Fock-Coulomb and Dirac-Fock-Breit calculations for the He atom and the helium-like ionic species Ne+8, Ar+16, and Sn+48. With the Generator Coordinate Dirac-Fock method, the integral Dirac-Fock equations are integrated numerically so as to generate Gaussian basis sets for the atomic species under study. The Dirac-Fock-Coulomb and Dirac-Fock-Breit energies obtained here for He and He-like ionic species with the Generator Coordinate Dirac-Fock formalism are better than the corresponding energies obtained with variational calculations using even-tempered Gaussian-type functions. The Dirac-Fock-Coulomb energies obtained for Ne+8 and Ar+16 with the Generator Coordinate Dirac-Fock method are in excellent agreement with Desclaux’s numerical calculations, and for He and Sn+48 our Dirac-Fock-coulomb energy results are lower than the corresponding energies obtained with Desclaux’s numerical-finite-difference program.

Adapted Gaussian basis sets for closed-shell atoms from samarium to nobelium generated with the generator coordinate Dirac-Fock method

The Generator Dirac-Fock method for closed-shell atoms is applied to generate adapted Gaussian basis sets for the heavy atoms from Sm ( Z = 62) to No ( Z = 102) to be used in the calculations of electronic properties of atomic and molecular systems. Our Dirac-Fock-Coulomb and Dirac-Fock-Breit energies, for all atoms studied here are better than the corresponding Dirac-Fock-Coulomb and Dirac-Fock-Breit energies attained with previous Gaussian basis sets. Besides, our Dirac-Fock-Coulomb energies (except for Yb that is equal) are lower than the corresponding ones obtained by numerical-finite-difference calculations. A discussion between the generation of universal Gaussian basis sets and adapted Gaussian basis sets is presented.

Accurate universal gaussian basis set for helium through calcium generated with the generator coordinate Dirac-Fock method

The recently developed generator coordinate Dirac-Fock method for closed-shell atoms is applied to generate an accurate universal gaussian basis set for the atoms from He ( Z = 2) through Ca ( Z = 20). The Dirac-Fock-Coulomb and Dirac-Fock-Breit energy results obtained with our universal gaussian basis set are compared with previous universal gaussian basis set Dirac-Fock-Coulomb and Dirac-Fock-Breit results, and with corresponding results obtained by numerical-finite-difference calculations. In general, our Dirac-Fock-Coulomb energies are better than the corresponding Dirac-Fock-Coulomb energies attained with the previous universal gaussian basis set, and are in excellent agreement with those obtained by the numerical-finite-difference method. For He and Be atoms, the Dirac-Fock-Coulomb energies obtained with our universal gaussian basis set are lower than the corresponding Dirac-Fock-Coulomb energies obtained by both numerical-finite-difference calculations and a previous universal gaussian basis set.

Adapted Gaussian basis sets for the relativistic closed-shell atoms from helium to barium generated with the generator coordinate Dirac-Fock method

The generator coordinate Dirac-Fock method for closed-shell atoms is applied to generate adapted Gaussian basis sets for the relativistic closed-shell atoms from He ( Z = 2) to Ba ( Z = 56). Our Dirac-Fock-Coulomb and Dirac-Fock-Breit energies, for all the atoms studied here (except for He and Mg), are better than the corresponding Dirac-Fock-Coulomb and Dirac-Fock-Breit energies attained with previous Gaussian basis sets. Except for the Mg, Si, Ar and Ca atoms our Dirac-Fock-Coulomb energies are lower than the corresponding ones obtained by numerical finite-difference calculations. A discussion on the generation of universal and adapted Gaussian basis sets is presented.

On the inclusion of the Breit interaction term in the closed-shell generator coordinate Dirac–Fock formalism

The generator coordinate Dirac–Fock formalism with the inclusion of the frequency-independent Breit interaction term is presented. The integration of the Dirac–Fock equations is performed through an integral discretization technique in order to preserve the continuous character of the original generator coordinate formalism. The inclusion of the Breit interaction term in the generator coordinate Dirac–Fock formalism is assessed through Dirac–Fock–Breit calculations for the relativistic closed-shell atoms with d and f orbitals, i.e., from zinc (Z=30) up to nobelium (Z=102), by using a universal Gaussian basis set generated previously by employing the generator coordinate Dirac–Fock method. The Dirac–Fock–Breit energy results presented in this work are the best, so far, obtained for the relativistic closed-shell atoms from Zn (Z=30) through No (Z=102) by using Gaussian-type functions, and show the efficacy of the generator coordinate Dirac–Fock method in performing accurate Dirac–Fock–Breit calculations.

The generator coordinate Dirac–Fock method applied to beryllium-like atomic species

The recently formulated generator coordinate Dirac–Fock method for relativistic closed-shell atoms is applied to the Be atom and Be-like ions Ne6+, Ar14+, and Sn46+ in order to assess its efficacy for light atomic systems. The Dirac–Fock equations are integrated numerically in the generator coordinate Dirac–Fock method so as to generate Gaussian basis sets for the atomic species under study. The results obtained with the application of the generator coordinate Dirac–Fock method in this work for Dirac–Fock–Coulomb and Dirac–Fock–Breit energies for Be-like ions are in excellent agreement with Declaux's benchmark numerical calculations, and are better than the Dirac–Fock–Coulomb and Dirac–Fock–Breit energies obtained with even-tempered Gaussian-type function calculations. For the Be atom, the Dirac–Fock–Coulomb energy result obtained with the generator coordinate Dirac–Fock method is lower than the corresponding value obtained with the Declaux's numerical finite-difference program. Key words: Dirac–Fock–Coulomb energy, Dirac–Fock–Breit energy, Gaussian basis sets, generator coordinate Dirac–Fock method.

A generator coordinate version of the closed-shell Dirac–Fock equations

A generator coordinate version of the Dirac–Fock equations for relativistic closed-shell atoms is presented. The integration of the Dirac–Fock equations is performed through the integral discretization technique so as to preserve the continuous character of the generator coordinate formalism. With the new formalism we generate a universal Gaussian basis set for relativistic closed-shell atoms with d and f orbitals (zinc up to nobelium). The results obtained with the universal Gaussian basis set for Dirac–Fock–Coulomb self-consistent-field energies are compared with numerical-finite-difference results and Dirac–Fock–Coulomb energies obtained by using other Gaussian basis sets. The excellent performance of our universal Gaussian basis set is attributed to the integral discretization technique of the generator coordinate Dirac–Fock method, as with it we are capable of generating Gaussian-type function exponents that are able to represent properly the relativistic kinematics of an electron inside the nucleus.