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     

Development and testing of a compact basis set for use in effective core potential calculations on rhodium complexes

We present a set of effective core potential (ECP) basis sets for rhodium atoms which are of reasonable size for use in electronic structure calculations. In these ECP basis sets, the Los Alamos ECP is used to simulate the effect of the core electrons while an optimized set of Gaussian functions, which includes polarization and diffuse functions, is used to describe the valence electrons. These basis sets were optimized to reproduce the ionization energy and electron affinity of atomic rhodium. They were also tested by computing the electronic ground state geometry and harmonic frequencies of [Rh(CO)₂μ‐Cl]₂, Rh(CO)₂ClPy, and RhCO (neutral and its positive, and negative ions) as well as the enthalpy of the reaction of [Rh(CO)₂μ‐Cl]₂ with pyridine (Py) to give Rh(CO)₂ClPy, at different levels of theory. Good agreement with experimental values was obtained. Although the number of basis functions used in our ECP basis sets is smaller than those of other ECP basis sets of comparable quality, we show that the newly developed ECP basis sets provide the flexibility and precision required to reproduce a wide range of chemical and physical properties of rhodium compounds. Therefore, we recommend the use of these compact yet accurate ECP basis sets for electronic structure calculations on molecules involving rhodium atoms. © 2012 Wiley Periodicals, Inc.     

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     

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.     

Quality of correlating functions generated from commonly used basis sets

Tests have been performed on the quality of correlating functions generated from commonly used Gaussian basis sets, such as the 4-31G and MIDI-4 sets. The atoms tested were carbon, nitrogen, and oxygen. Self-consistent field and configuration interaction (CI) calculations were performed for the ground and lower excited states of neutral atoms as well as for positive and negative ions, using the original sets. Next, after adding (1) one d, and (2) two d and one f primitive Gaussian-type functions (GTFs) to the original sets, the CI calculations were repeated. In order to investigate the quality of the correlating orbitals generated from the GTF sets, parallel calculations to those for the GTF sets were carried out with an extended set of Slater-type functions. It was found that the excitation energies change in a stepwise manner as the basis sets changed from the original sets to the original set + 1d and the original set +2d1f. The improvements in excitation energies and ionization energies were almost independent of the original sets and were found to be strongly dependent on the augmented correlation functions. © 1996 by John Wiley & Sons, Inc.     

Contracted Gaussian-type basis functions revisited. III. Atoms K through Kr

Six minimal basis sets of contracted Gaussian-type functions (GTFs) are developed for the third-row atoms K through Kr. The smallest and largest sets for transition metal atoms are (3333/33/3) and (8433/84/8), respectively, where a slash distinguishes the s, p, and d symmetries and single-digit figures in the parentheses denote the numbers of primitive GTFs. The two largest sets, (7433/74/7) and (8433/84/8), surpass the (62111111/33111/311) set of Schaefer et al. in the associated total energies. Our (8433/84/8) set is also superior to their (842111/631/411) set. The quality of the present basis sets is tested by self-consistent field (SCF) and configuration interaction (CI) calculations on the Cu₂ molecule. As the accuracy of the basis set increases, SCF calculations show a decrease in the dissociation energy and an increase in the equilibrium internuclear distance. The same tendencies are found in the results of CI calculations with and without a Davidson correction. All the present basis sets are freely available at the internet address: http://202.35.198.41/∼htatewak/. Received: 17 June 1998 / Accepted: 4 August 1998 / Published online: 23 November 1998     

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.     

Medium-size polarized basis sets for high-level-correlated calculations of molecular electric properties

Summary The first-order polarized basis sets for the use in high-level-correlated investigations of molecular electric properties have been generated for Pb, Bi, Po, and At. The performance of the standard [10.17.14.5/13.11.8.2] and extended [20.17.14.9/13.11.8.4] basis sets has been examined in nonrelativistic and quasirelativistic calculations for atoms and simple closed-shell hydrides. The relativistic contributions to electric dipole properties of those systems have been evaluated by using the recently developed quasirelativistic scheme. The predicted dipole polarizability of At is in good agreement with the results of other relativistic calculations. The calculated quasirelativistic dipole moments of BiH₃ (–0.499 a.u.), PoH₂ (–0.207 a.u.), and AtH (+0.036 a.u.) involve a significant relativistic contribution which amounts to —0.230 a.u., –0.177 a.u., and –0.097 a.u., respectively. The basis set details append this paper. They are also available as a part of the basis set library of the MOLCAS system.     

General variation method for the relativistic calculations of atoms and molecules

Use of the general variation method of Weinstein and MacDonald for the relativistic calculation of atoms and molecules is proposed. It is shown from the numerical calculations for hydrogenlike atomic systems that this method is useful in judging an accuracy of energies and wave functions obtained with a relativistic Hamiltonian whose spectra are not bounded. It is also shown that this method can be used to find spurious solutions such as 1p_½ or 2d_3/2 appearing in atomic systems. Problems in extending the method to many-electron atoms and molecules are discussed.     

Segmented contracted basis sets for atoms H through Xe: Sapporo-(DK)-<Emphasis Type="Italic">n</Emphasis>ZP sets (<Emphasis Type="Italic">n</Emphasis> = D,T, Q)

For the 54 atoms from H to Xe, compact yet accurate segmented Gaussian-type basis sets have been constructed for all electron calculations. Non-relativistic nZP (Sapporo-nZP) sets for Li–Xe and relativistic nZP (Sapporo-DK-nZP) sets for K–Xe are developed (n = D, T, Q), which efficiently incorporate valence and core electron correlations. Test calculations at the coupled-cluster level of theory are performed for spectroscopic constants of 12 hydrides of s- and d-block atoms and 12 diatomics of p-block atoms in their ground states. For all molecules, the calculated spectroscopic constants approach to the experimental values smoothly as the basis set quality increases.     

The error due to neglecting the core spin–orbit splitting in valence ZORA calculations with frozen core approximation and its elimination

In valence zeroth-order regular approximation (ZORA) calculations with frozen core approximation, when the basis set optimized to the related scalar relativistic ZORA calculations is used, neglecting the core spin–orbit splitting may result in additional basis set truncation errors. It is found that the error is negligible for most elements except the 6p-block elements. When the basis set is extended by a p-type STO function put on the 6p element atoms with the ζ value proper to 5p_1/2 orbitals, the error can be reduced to be negligible. The calculated atomic properties related to valence orbitals can be improved greatly by use of this extended basis set. The frozen core approximation calculations of some molecules containing Tl, Pb and Bi with closed shells show that neglecting the core spin–orbit splitting only slightly affects the calculated bond lengths and bond energies, and the calculated molecular property can also be improved slightly by use of the extended basis sets.     

Electrically polarized valence basis sets for the SBKJC effective core potential developed for calculations of dynamic polarizabilities and Raman intensities

Sadlej’s electric polarization method of Gaussian basis functions was applied to the double-zeta effective core potential basis sets of Stevens, Basch, Krauss, Jasien and Cundari to generate a new augmented polarized valence double-zeta set, named as pSBKJC, which is appropriate for the calculation of dynamic polarizabilities and Raman intensities. The pSBKJC basis set was developed for the atoms of families 14–17 (from C to F, Si to Cl, Ge to Br and Sn to I). In order to assess the performance of this new basis set, these properties were compared to those evaluated using Sadlej’s set, available in the EMSL online library under the name of Sadlej-pVTZ. In these tests, Hartree-Fock/pSBKJC calculations have proved to be less demanding of the computer than the Hartree-Fock/Sadlej-pVTZ ones but give results in excellent agreement with those from the Sadlej-pVTZ basis set. Since the Stevens et al. pseudopotential can represent the scalar relativistic effects, the results obtained at the Hartree-Fock/pSBKJC level show a better agreement with the results of Dirac-Hartree-Fock/Sadlej-pVTZ relativistic calculations using Dyall’s spin-free Hamiltonian. When comparing Hartree-Fock/pSBKJC data of Raman scattering activities, at the excitation wavelength of 488 nm, with those of spin-free Dirac-Hartree-Fock/Sadlej-pVTZ calculations, a very good agreement is observed, where the RMS error is 8.5 Å⁴a.m.u.^?1 and the averaged percentage error is 3.4%. In terms of computer savings in calculations of dynamic Raman intensities, a 20% reduction in the CPU time in the coupled cluster singles and doubles intensities of C₆H₆ and about 40% reduction in the time-dependent Hartree-Fock intensities for C₆F₆ molecules were attained.     

Ab initio calculations of relativistic and electron correlation effects in polyatomics using the universal Gaussian basis set: XeF2

Ab initio accurate all-electron relativistic molecular orbital Dirac–Fock self-consistent field calculations are reported for the linear symmetric XeF₂ molecule at various internuclear distances with our recently developed relativistic universal Gaussian basis set. The nonrelativistic limit Hartree–Fock calculations were also performed for XeF₂ at various internuclear distances. The relativistic correction to the electronic energy of XeF₂ was calculated as ～ ?215 hartrees (?5850 eV) by using the Dirac–Fock method. The dominant magnetic part of the Breit interaction correction to the nonrelativistic interelectron Coulomb repulsion was included in our calculations by both the Dirac–Fock–Breit self-consistent field and perturbation methods. The calculated Breit correction is ～6.5 hartrees (177 eV) for XeF₂. The relativistic Dirac–Fock as well as the nonrelativistic HF wave functions predict XeF₂ to be unbound, due to neglect of electron correlation effects. These effects were incorporated for XeF₂ by using various ab initio post Hartree–Fock methods. The calculated dissociation energy obtained using the MP 2(full) method with our extensive basis set of 313 primitive Gaussians that included d and f polarization functions on Xe and F is 2.77 eV, whereas the experimental dissociation energy is 2.78 eV. The calculated correlation energy is ～ ?2 hartrees (?54 eV) at the predicted internuclear distance of 1.986 Å, which is in excellent agreement with the experimental Xe—F distance of 1.979 Å in XeF₂. In summary, electron correlation effects must be included in accurate ab initio calculations since it has been shown here that their inclusion is crucial for obtaining theoretical dissociation energy (D_e) close to experimental value for XeF₂. Furthermore, relativistic effects have been shown to make an extremely significant contribution to the total energy and orbital binding energies of XeF₂. © 1995 John Wiley & Sons, Inc.     

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.     

An efficient methodology to study cyclodextrin clusters: application to α-CD hydrated monomer, dimer, trimer and tetramer

The hydrated α-cyclodextrin (α-CD) clusters resulting from the following process: nα-CD + n(H₂O)₆ → α-CD_n · 6nH₂O, with n = 1, 2, 3, 4, have been investigated using semiempirical (PM3), ab initio Hartree-Fock and Density Functional Theory (BLYP functional) levels of theory. The largest structure containing 576 atoms and 5,760 contracted basis functions (6-31G(d,p) basis set) poses as a considerable hard task for quantum chemical calculations. As the number of basis function increases rapidly with the cluster size, an alternative procedure to make the calculations feasible is certainly welcome, in order to perform BLYP calculations with an adequate basis set. Through the aid of a computer program that we developed, it became of practical use the selection of atom by atom basis sets, using the common chemical sense, enabling quantum mechanical calculations to be performed for very large molecular interacting systems (inclusion complexes), at an affordable computational cost. In this article we show how an appropriate selection of basis functions, leaving the CH_n groups with a minimal basis set and the oxygen atoms (and OH groups) with a better quality basis set, lower considerably the computational cost with no significant loss in the calculated interaction energies. A regular pattern is observed for α-CD hydrated monomer, dimer, trimer and tetramer, therefore adding support to the use of this procedure when studying larger hydrogen bonded clusters where electron correlation effects are important. We show that the procedure reported here enables DFT calculations for hydrated cyclodextrin using basis set up to the 6-311++G(3df,3pd) triple zeta quality .     

Ab initio relativistic effective potentials with spin-orbit operators. VI. Fr through Pu

Ab initio averaged relativistic effective core potentials (AREP ), spin-orbit (SO ) operators, and valence basis sets are reported for the elements Fr through Pu in the form of expansions in Gaussian-type functions. Gaussian basis sets with expansion coefficients for the low-energy states of each atom are given. Atomic orbital energies calculated under the j–j coupling scheme within the self-consistent field approximation and employing the AREP 'S in their unaveraged form (REP 'S) agree to within 10% of orbital energies due to numerical all-electron Dirac–Fock calculations. The accuracy of the AREP 'S and so operators is also shown to be good through comparisons of calculated so splitting energies with all-electron Dirac–Fock results.     

Basis sets for geometry optimizations of second-row transition metal inorganic and organometallic complexes

Modest-sized basis sets for the second-row transition metal atoms are developed for use in geometry optimization calculations. Our method is patterned after previous work on basis sets for first-row transition metal atoms. The basis sets are constructed from the minimal basis sets of Huzinaga and are augmented with a set of diffuse p and d functions. The exponents of these diffuse functions are chosen to minimize both the difference between the calculated and experimental equilibrium geometries and the total molecular energies for several second-row transition metal inorganic and organon etallic complexes. Slightly smaller basis sets, based on the same Huzinaga minimal sets but augmented with a set of diffuse s and p functions rather than diffuse p and d functions, are also presented. The performance of these basis sets is tested on a wide variety of second-row transition metal inorganic and organometallic complexes and is compared to pseudopotential basis sets incorporating effective core potentials.     

Ab initio study of hydrogen bonding in the phenol–water system

Three hydrogen-bonded minima on the phenol-water, C₆H₅OH—H₂O, potential energy surface were located with 3–21G and 6–31G** basis sets at both Hartree–Fock and MP2 levels of theory. MP2 binding energies were computed using large “correlation consistent” basis sets that included extra diffuse functions on all atoms. An estimate of the effect of expanding the basis set to the triple-zeta level (multiple f functions on carbon and oxygen and multiple d functions on hydrogen) was derived from calculations on a related prototype system. The best estimates of the electronic binding energies for the three minima are –7.8, –5.0, and –2.0 kcal/mol. The consequences of uncertainties in the geometries and limitations in the level of correlation recovery are analyzed. It is suggested that our best estimates will likely underestimate the complete basis set, full CI values by 0.1–0.3 kcal/mol. Vibrational normal modes were determined for all three minima, including an MP2/6–31G** analysis for the most strongly bound complex. Computational strategies for larger phenol–water complexes are discussed. © John Wiley & Sons, Inc.