Relativistic and nonrelativistic finite nucleus optimized double zeta basis sets for the 4p, 5p and 6p elements (Theor Chem Acc (1998) 99:366-371): addendum

 The coefficients of the atomic Foldy-Wout-huysen transformed large component, which can be used in scalar relativistic calculations, are provided in an internet archive for the relativistic double zeta basis sets for the 4p, 5p and 6p elements previously published by the author. Published online: 14 November 2002 Correspondence to: K.G. Dyall e-mail: dyall@schrodinger.com     

Relativistic Gaussian basis sets for the atoms hydrogen to neon

Gaussian basis sets for use in relativistic molecular calculations are developed for atoms and ions with one to ten electrons. A relativistic radial wavefunction coupled to an angular function of l-symmetry is expanded into a linear combination of spherical Gaussians of the form r ^l exp (–r ²). One set of basis functions is used for all large and small components of the same angular symmetry. The expansion coefficients and the orbital exponents have been determined by minimizing the integral over the weighted square of the deviation between the Dirac or Dirac-Fock radial wavefunctions and their analytical approximations. The basis sets calculated with a weighting function inversely proportional to the radial distance are found to have numerical constants very similar to those of their energy-optimized non-relativistic counterparts. Atomic sets are formed by combining l-subsets. The results of relativistic and non-relativistic calculations based on these sets are analyzed with respect to different criteria, e.g. their ability to reproduce the relativistic total energy contribution and the spin-orbit splitting. Contraction schemes are proposed.Dedicated to Prof. Dr. A. Neckel on occasion of his 60th birthday     

Double- and triple-zeta Slater-type basis sets with common exponents

Double- and triple-zeta basis sets of Slater-type functions (STFs) are developed for the 17 atoms from He to Ar. For computational economy, the exponents of STFs corresponding to the same atomic subshell are restricted to be common. Instead, the principal quantum numbers of the STFs are thoroughly optimized within the framework of integer values to reduce the energy loss due to the common exponent restriction. Received: 10 November 1999 / Accepted: 25 January 2000 / Published online: 19 April 2000     

Choosing GTO basis sets for periodic HF calculations

We investigate numerical linear dependencies of Gaussian-type orbital basis sets employed in the framework of the Hartree-Fock self-consistent field method for periodic structures, which so far have hampered the use of extended basis sets for non-ionic crystals. These linear dependencies occur when diffuse basis functions are included in a basis set in an uncontrolled manner. We use the condition number of the overlap matrix to lead us in the construction of extended basis sets for periodic structures which avoid numerical linear dependencies. Extended basis sets of high quality are optimized for a number of periodic structures (fcc He, α-Be, α-BN, and B1 NaF) with respect to the energy of the constituent atoms or ions. The results obtained with our basis sets, which do not require reoptimization in the crystal environment, compare favorably with those obtained with other extended basis sets reported in the literature. Received: 20 July 1998/Accepted: 21 August 1998 / Published online: 19 October 1998     

Fully relativistic correlated benchmark results for uranyl and a critical look at relativistic effective core potentials for uranium

 A complete set of fully-relativistic benchmark results for the bond lengths and vibrational frequencies of uranyl at various levels of correlation treatment are presented. It is shown that the relativistic and correlation effects are of the same magnitude and should be treated on an equal footing. Results of uranyl calculations using various relativistic effective core potentials (RECP) are presented. Appropriate choices for RECPs for use in density functional theory (DFT) calculations in the local density approximation (LDA) and with the gradient-corrected B3LYP exchange-correlation functional are discussed. The conclusion is reached that small-core RECPs need to be used and that the best results as compared to the benchmark values are obtained by using a DFT functional that includes some fraction of Hartree-Fock Exchange. Received: 18 May 2001 / Accepted: 25 July 2001 / Published online: 30 October 2001     

Relativistic Gaussian basis sets for the elements K – Uuo

A series of energy-minimized relativistic Gaussian basis sets for the elements with atomic numbers 19–118 is presented. The basis sets have been derived at the self-consistent field level as weighted average energies of the respective electronic configurations. A spherical Gaussian charge distribution has been used to model the nucleus. The basis sets are constructed as interleaving dual family sets with shared exponents within each family. The quality of the basis sets is better than double zeta. Received: 7 July 2000 / Accepted: 21 September 2000 / Published online: 21 December 2000     

Relativistic Gaussian basis sets for radon through plutonium

Summary Relativistic Gaussian basis sets of neutral atoms Rn-Pu and ions Th⁺⁴, U⁺³ and Pu⁺³ in the configurations of average energies are presented. The exponent parameters of the basis sets are determined by least-squares fitting to the numerical Dirac-Fock wave functions. The total energies obtained are within 0.155 a.u. of the Dirac-Fock limits and the qualities of the basis sets are between double-zeta and triple-zeta in the valence parts. Using the exponent parameters the Breit interaction energies have been calculated by perturbation theory and the self-consistent field treatment.     

The effect of basis set superposition error on the convergence of interaction energies

 For the intermolecular interaction energies of ion-water clusters [OH⁻(H₂O) _n (n=1,2), F⁻(H₂O), Cl⁻(H₂O), H₃O⁺(H₂O) _n (n=1,2), and NH₄ ⁺(H₂O) _n (n=1,2)] calculated with correlation-consistent basis sets at MP2, MP4, QCISD(T), and CCSD(T) levels, the basis set superposition error is nearly zero in the complete basis set (CBS) limit. That is, the counterpoise-uncorrected intermolecular interaction energies are nearly equal to the counterpoise-corrected intermolecular interaction energies in the CBS limit. When the basis set is smaller, the counterpoise-uncorrected intermolecular interaction energies are more reliable than the counterpoise-corrected intermolecular interaction energies. The counterpoise-uncorrected intermolecular interaction energies evaluated using the MP2/aug-cc-pVDZ level is reliable. Received: 14 March 2001 / Accepted: 25 April 2001 / Published online: 9 August 2001     

Convergence of multipole expanded intermolecular interaction energies for Gaussian-type-function and Slater-type-function basis sets

 It is shown that the multipole expansion of each order of the polarization series converges for large enough intermolecular distances when finite basis sets of Gaussian or Slater-type functions are used to approximate molecular response properties. Convergence of the multipole expansion for each order of the polarization series does not imply convergence of the polarization series itself. A corresponding convergence condition is extracted from the general perturbation theory in a finite-dimensional space and is applied to the H + H⁺ problem. Received: 29 September 1999 / Accepted: 22 May 2000 / Published online: 18 August 2000     

Optimization of Gaussian basis sets for Dirac-Hartree-Fock calculations

We investigate the optimization of Gaussian basis sets for relativistic calculations within the framework of the restricted Dirac-Hartree-Fock (DHF) method for atoms. We compare results for Rn of nonrelativistic and relativistic basis set optimizations with a finite nuclear-size. Optimization of separate sets for each spin-orbit component shows that the basis set demands for the lower j component are greater than for the higher j component. In particular, the p _1/2 set requires almost as many functions as the s _1/2 set. This implies that for the development of basis sets for heavy atoms, the symmetry type for which a given number of functions is selected should be based on j, not on l, as has been the case in most molecular calculations performed to date.     

Evaluation of the Coulomb energy in relativistic self-consistent-field theory

 Computational schemes are presented with which to evaluate the electrostatic Coulomb energy in relativistic molecular electronic structure calculations using a basis of four-component Dirac spinor amplitudes. We demonstrate that algorithms may be constructed and implemented which differ only in minor details from those in common use in nonrelativistic quantum chemistry, and that the four-component formalism is neither as complicated nor as expensive as has been suggested recently in the literature. Spherically symmetrical atomic basis sets are presented which indicate that accurate representations of the Coulomb energy may be obtained using modest expansions of the electronic density in a scalar auxiliary basis set of spherical harmonic Gaussian-type functions. Received: 15 April 2002 / Accepted: 15 May 2002 / Published online: 29 July 2002     

A singularity excluded approximate expansion scheme in relativistic density functional theory

 A singularity excluded approximate expansion (SEAX) scheme, which can be considered as one between Breit-Pauli expansion and RA expansion schemes, is proposed to expand the total energy of 4-component relativistic density functional theory. The one-electron equation can be derived variationally from the approximate total energy expression. The Hamiltonian of the one-electron equation is bounded from below and can be dealt with variationally, and the gauge dependency error in the ZORA method is essentially eliminated. It is easier to solve the SEAX equation than the IORA equation. The results related to the valence orbitals by solving the scalar SEAX equation agree very well with those by the scalar ZORA ESA method, and the results related to the inner-shell electrons of heavy elements by the two component SEAX calculations agree quite well with those by the 4-component relativistic density functional calculations. Received: 20 February 2002 / Accepted: 29 April 2002 / Published online: 8 July 2002     

Correlating basis sets for the H atom and the alkali-metal atoms from Li to Rb

 Contracted Gaussian-type function sets are developed for correlating p, d, and f functions for a valence electron of the hydrogen atom and alkali-metal atoms from Li to Rb. A segmented contraction scheme is used for its compactness and efficiency. Contraction coefficients and exponents are determined by minimizing the deviation from the K orbitals of the atoms. The present basis sets yield an accuracy comparable to the correlation-consistent basis set for the hydrogen atom and also give a similar high accuracy for the alkali-metal atoms. In the calculations of spectroscopic constants of alkali hydrides, the decontraction of the p function plays an important role, especially for LiH. The contributions of d and f functions are nontrivial for KH and RbH. Received: 6 September 2002 / Accepted: 13 November 2002 / Published online: 19 March 2003 Acknowledgements. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. Correspondence to: T. Noro e-mail: tashi@sci.hokudai.ac.jp     

Contracted Gaussian-type basis functions revisited. IV. Atoms Rb to Xe

We report minimal-type contracted Gaussian-type function (GTF) sets, #n=(n3333/n33/n3) with n=5 and 6, #7= (74333/743/74), and #8= (84333/843/75), for the fourth-row atoms from Rb to Xe. Test calculations are performed on the Ag₂ molecule. Spectroscopic constants given by split valence sets derived from #5 and #6 are a little contaminated by basis set superposition error. However, we find that the fully valence split #8 set, (8433111/84111/711111), yields essentially the same results as a large GTF set, (22s15p12d), with a general contraction, when p-, d-, and f-type polarization functions are augmented. The present #7 and #8 CGTF sets are recommended for ab initio molecular calculations including fourth-row atoms. Received: 15 January 2002 / Accepted: 16 April 2002 / Published online: 24 June 2002     

The employment of relativistic adapted Gaussian basis sets in Douglas–Kroll–Hess scalar calculations with diatomic molecules

The prolapse-free relativistic adapted Gaussian basis sets (RAGBSs), developed by our research group on the basis of the four-component approach, are used for the first time in Douglas–Kroll–Hess 2nd order scalar relativistic calculations (DKH2) of simple diatomic molecules containing Hydrogen and the halogens from Fluorine up to Iodine: HX and X₂, where X = F, Cl, Br, and I. To this end, the RAGBSs were contracted with the general contraction scheme to triple-, quadruple-, and quintuple-zeta sets. Polarization functions were also added to the basis sets by optimization with the configuration interaction method including single and double excitations into the DKH2 environment, DKH2-CISD. The molecular properties were then calculated with the coupled cluster electronic correlation treatment and the DKH2 scalar relativistic method, DKH2-CCSD(T), and indicated that our RAGBSs should be contracted as quadruple-zeta basis sets. The results achieved with the DKH2-CCSD(T) calculations and the selected quadruple-zeta RAGBSs are able to reproduce the experimental data of equilibrium distances, dissociation energies, and harmonic vibrational frequencies with root-mean-square (rms) errors of 0.015 Å, 3.6 kcal mol⁻¹, and 21.7 cm⁻¹, respectively.     

A study on universal Gaussian basis sets for first-row atoms

Analysis of various optimum and non-optimum Gaussian basis sets for firstrow elements have indicated that with a minimum increase of the basis set size and without loss of accuracy of the calculated total energy, a single universal Gaussian basis set may replace individually optimized Gaussian basis sets for a series of atoms. Such a universal Gaussian basis set may substantially reduce the computational work required for the calculation of molecular integrals in ab initio MO calculations.