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.     

Systematic Sequences of Geometric Relativistic Basis Sets. I:s- and p-Block Elements up to Xe

In this work a scheme for constructing systematic sequences of relativistic SCF basis sets at a reasonable computational cost is presented and applied to atoms of the s- and p-block up to Xe. This scheme, which couples simplex optimization and the use of geometric series given by four-term polynomial expressions for the logarithm of the exponents, allows for the construction of basis sets that exhibit very regular patterns of convergence to the numerical reference values of atomic total energies, spinor energies and radial expectation values. This regularity, together with the broad range of basis set sizes presented, enables these sets to be used as building blocks for basis sets applicable in both routine and benchmark relativistic calculations on atomic and molecular systems.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Optimized Slater-type basis sets for the elements 1-118

Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.     

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     

All-electron scalar relativistic basis sets for the elements Rb–Xe

Segmented all-electron relativistically contracted (SARC) basis sets are presented for the elements ₃₇Rb–₅₄Xe, for use with the second-order Douglas–Kroll–Hess approach and the zeroth-order regular approximation. The basis sets have a common set of exponents produced with established heuristic procedures, but have contractions optimized individually for each scalar relativistic Hamiltonian. Their compact size and loose segmented contraction, which is in line with the construction of SARC basis sets for heavier elements, makes them suitable for routine calculations on large systems and when core spectroscopic properties are of interest. The basis sets are of triple-zeta quality and come in singly or doubly polarized versions, which are appropriate for both density functional theory and correlated wave function theory calculations. The quality of the basis sets is assessed against large decontracted reference basis sets for a number of atomic and ionic properties, while their general applicability is demonstrated with selected molecular examples.     

Reduced-size polarized basis sets for calculations of molecular electric properties. IV. First-row transition metals

Recent studies of the perturbation-dependent basis sets have indicated the possibility of a significant reduction of the size of the usual CGTO sets without considerable loss of accuracy in calculations of molecular electric properties. The resulting (ZPolX) basis sets have been developed for several atoms of the first and second row of the Periodic Table. The same method of the ZPolX basis set generation is extended for the first-row transition metals and the corresponding contracted ZPolX basis sets of the size [6s5p3d1f] are determined for both nonrelativistic and scalar relativistic calculations. The performance of the ZPolX basis sets is verified in calculations on the first-row transition metal oxides at the level of the ROHF, ROHF/CASPT2, and ROHF/CCSD(T) approximations. Also the study of the dipole polarizability of TiCl₄ confirms the excellent features of these very compact basis sets. The ZPolX basis sets for nonrelativistic and relativistic calculations of molecular electric properties are available on the web page http://www.chem.uni.torun.pl/zchk/basis-sets.html.     

Performance of numerical atom‐centered basis sets in the ground‐state correlated calculations of noncovalent interactions: Water and methane dimer cases

Numerical atom‐centered basis sets (orbitals) (NAO) are known for their compactness and rapid convergence in the Hartree–Fock and density‐functional theory (DFT) molecular electronic‐structure calculations. To date, not much is known about the performance of the numerical sets against the well‐studied Gaussian‐type bases in correlated calculations. In this study, one instance of NAO [Blum et al., The Fritz Haber Institute ab initio Molecular Simulations Package (FHI‐aims), 2009] was thoroughly examined in comparison to the correlation‐consistent basis sets in the ground‐state correlated calculations on the hydrogen‐bonded water and dispersion‐dominated methane dimers. It was shown that these NAO demonstrate improved, comparing to the unaugmented correlation‐consistent based, convergence of interaction energies in correlated calculations. However, the present version of NAO constructed in the DFT calculations on covalently‐bound diatomics exhibits enormous basis‐set superposition error (BSSE)—even with the largest bases. Moreover, these basis sets are essentially unable to capture diffuse character of the wave function, necessary for example, for the complete convergence of correlated interaction energies of the weakly‐bound complexes. The problem is usually treated by addition of the external Gaussian diffuse functions to the NAO part, what indeed allows to obtain accurate results. However, the operation increases BSSE with the resulting hybrid basis sets even further and breaks down the initial concept of NAO (i.e., improved compactness) due to the significant increase in their size. These findings clearly point at the need in the alternative strategies for the construction of sufficiently‐delocalized and BSSE‐balanced purely‐numerical bases adapted for correlated calculations, possible ones were outlined here. For comparison with the considered NAOs, a complementary study on the convergence properties of the correlation‐consistent basis sets, with a special emphasis on BSSE, was also performed. Some of its conclusions may represent independent interest. © 2013 Wiley Periodicals, Inc.     

Reduced-size polarized basis sets for calculations of molecular electric properties. I. The basis set generation

Following the recent studies of basis sets explicitly dependent on oscillatory external electric field we have investigated the possibility of some further truncation of the so-called polarized basis sets without any major deterioration of the computed data for molecular dipole moments, dipole polarizabilities, and related electric properties of molecules. It has been found that basis sets of contracted Gaussian functions of the form [3s1p] for H and [4s3p1d] for the first-row atoms can satisfy this requirement with particular choice of contractions in their polarization part. With m denoting the number of primitive GTOs in the contracted polarization function, the basis sets devised in this article will be referred to as the ZmPol sets. In comparison with earlier, medium-size polarized basis sets (PolX), these new ZmPol basis sets are reduced by 2/3 in their size and lead to the order of magnitude computing time savings for large molecules. Simultaneously, the dipole moment and polarizability data remain at almost the same level of accuracy as in the case of the PolX sets. Among a variety of possible applications in computational chemistry, the ZmPolX are also to be used for calculations of frequencies and intensities in the Raman spectra of large organic molecules (see Part II, this issue).     

Improved efficiency of focal point conformational analysis with truncated correlation consistent basis sets

It has been suggested that the computational cost of correlated ab initio calculations could be reduced efficiently by using truncated basis sets on hydrogen atoms (Mintz et al., J Chem Phys 2004, 121, 5629). We now explore this proposal in the context of conformational analysis of small molecules, such as hydrogen peroxide, dimethyl ether, ethyl methyl ether, formic acid, methyl formate, and several small alcohols. It is found that truncated correlation consistent basis sets that lack certain higher angular momentum functions on hydrogen atoms offer accuracy similar to traditional Dunning's basis sets for conformational analysis. Combination of such basis sets with the basis set extrapolation technique to estimate Hartree-Fock and M?ller-Plesset second order energies provides composite extrapolation model chemistries that are significantly more accurate and faster than analogous single point calculations with traditional correlation consistent basis sets. Root mean square errors of best composite extrapolation model chemistries on the used set of molecules are within 0.03 kcal/mol of traditional focal point conformational energies. The applicability of composite extrapolation methods is illustrated by performing conformational analysis of tert-butanol and cyclohexanol. For comparison, conformational energies calculated with popular molecular mechanics force fields are also given.     

Reduced-size polarized basis sets for calculations of molecular electric properties. II. Simulation of the Raman spectra

The accuracies of the calculated vibrational frequencies and Raman intensities given by two new, highly compact Pol-type basis sets, Z2PolX and Z3PolX, have been determined and compared to the 6-31G(d), PolX, and aug-cc-pVTZ basis sets. Calculation of accurate Raman intensities has previously required large basis sets, but the ZmPolX basis sets are smaller even than PolX, which are the most compact basis sets able to calculate accurate Raman intensities. For the largest compound studied, C5H10O2, Z3PolX required more than an order of magnitude less CPU time than PolX, which has been shown to be 10 times faster than aug-cc-pVTZ. Two sets of test molecules were studied: one was a series of small molecules for which experimental values for absolute Raman activities were available; the second was a series of medium-sized molecules (mainly common organic solvents) where only relative Raman band intensities were available. The accuracies of the Raman intensities given by both of the ZmPolX basis sets were good compared to those of the PolX and aug-cc-pVTZ sets, and much better than the 6-31G(d) values. The errors in even unscaled frequency values <2000 cm(-1) were also acceptable and were slightly lower for Z3PolX than Z2PolX (30 cm(-1) vs. 48 cm(-1)). The combination of good intensity and frequency data meant that for the medium-sized organic molecules there was a close correspondence between the simulated Raman spectra and experimental data, and that the observed bands could easily be assigned on the basis of these calculations. Achieving this level of accuracy in the simulations at modest computational cost should now allow computational methods to be combined with experimental Raman studies much more widely than is currently the case.     

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.     

Basis set selections for relativistic self-consistent field calculations

Practical methods of generating reliable and economic basis sets for relativistic self-consistent fields (RSCF) calculations are developed. Large component basis sets are generated from constrained optimizations of exponents in the nonrelativistic atomic calculations for light atoms. For heavy atoms, large component basis sets for inner core orbitals are generated by fitting numerical atomic spinors of Dirac-Hartree-Fock calculations with appropriate number of Slater-type functions. Small component basis sets are obtained by using the kinetic balance condition and other computational criteria. With judicious selections of the basis sets, virtual orbitals in RSCF calculations become very similar to those in nonrelativistic calculations, implying that relativistic virtual orbitals can be used in electron correlation calculations in the same manner as the conventional nonrelativistic virtual orbitals. It is also evident that the Koopmans' theorem is also valid in RSCF results.     

Computational linear dependence in molecular electronic structure calculations using universal basis sets

Distributed universal even‐tempered basis sets have been developed over recent years that are capable of supporting Hartree–Fock energies to an accuracy approaching the sub‐μHartree level. These basis sets have also been exploited in correlation studies, in applications to polyatomic molecules, and in the calculation of electric properties, such as multipole moments, polarizabilities, and hyperpolarizabilities. Jorge and coworkers have also developed universal basis sets and have recently reported applications to diatomic molecular systems. In this article, we compare the molecular calculations reported by Jorge and coworkers with our previous studies. Particular attention is given to the degree of computational linear dependence associated with the various basis sets employed and the consequential effects of the accuracy of the calculated energies. © 2004 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.     

Polarization functions for the modified m6‐31G basis sets for atoms Ga through Kr

The 2df polarization functions for the modified m6‐31G basis sets of the third‐row atoms Ga through Kr (Int J Quantum Chem, 2007, 107, 3028; Int J. Quantum Chem, 2009, 109, 1158) are proposed. The performances of the m6‐31G, m6‐31G(d,p), and m6‐31G(2df,p) basis sets were examined in molecular calculations carried out by the density functional theory (DFT) method with B3LYP hybrid functional, Møller‐Plesset perturbation theory of the second order (MP2), quadratic configuration interaction method with single and double substitutions and were compared with those for the known 6‐31G basis sets as well as with the other similar 641 and 6‐311G basis sets with and without polarization functions. Obtained results have shown that the performances of the m6‐31G, m6‐31G(d,p), and m6‐31G(2df,p) basis sets are better in comparison with the performances of the known 6‐31G, 6‐31G(d,p) and 6‐31G(2df,p) basis sets. These improvements are mainly reached due to better approximations of different electrons belonging to the different atomic shells in the modified basis sets. Applicability of the modified basis sets in thermochemical calculations is also discussed. © 2013 Wiley Periodicals, Inc.