Basis Set Convergence Study Research Articles

The eight-valence-electron systems re-examined: convergence of the coupled-cluster series and performance of quasiperturbative methods for quadruple excitations

We present the results of a coupled-cluster basis-set convergence study on 12 eight-valence-electron systems. These small systems span a wide range of correlation character (from purely dynamical correlation to pathological static correlation) as well as of chemical bonding (from ionic with some backdonation to quadruple covalent bonding). Indicators such as the correlation entropy and the CASSCF 1-C02, which primarily sample type A static correlation, are good predictors for the energetic importance of connected quintuple excitations, T5, and to a lesser extent of higher order triple excitations, T3–(T). The percentage of the total atomisation energy recovered by connected triple excitations, %TAE[(T)], is however a fair predictor for the importance of connected quadruples, T4. Of the various ansätze for quasiperturbative connected quadruples, CCSDT(Q)/B appears to be the most robust, even as it is still outperformed by the much more expensive CCSDT(Q)Λ.

Ab Initio Ground- and Excited-State Intermolecular Potential Energy Surfaces for the NO–Ne and NO–Ar van der Waals Complexes

The ground- [NO(X(2)Π)] and excited-state [NO(A(2)Σ(+))] intermolecular potential energy surfaces (IPESs) of the NO-Ne and NO-Ar van der Waals complexes are evaluated using the RCCSD(T) spin-restricted coupled cluster method and d-aug-cc-pVQZ basis set extended with a set of 3s3p2d1f1g midbond functions. These bases are selected from the results of a systematic basis-set convergence study carried out for the NO(A(2)Σ(+))-Ar state. We fit the interaction energies to analytic functions and compare the results to those previously available. The NO-Ar (NO-Ne) IPESs are characterized by absolute minima of -120 and -75 cm(-1) (-58 and -5 cm(-1)) at the ground and first excited state, respectively, located close to the T-shaped geometries for the ground states and at linear dispositions in the case of the excited states. The potentials are further used in the evaluation of the rovibrational spectra of the complexes, and the results are compared to those available in the literature.

Performance of Effective Core Potentials for Density Functional Calculations on 3d Transition Metals.

The performance of popular Hartree-Fock-based effective core potentials in Hartree-Fock and density functional calculations of 3d transition metals has been evaluated by basis-set convergence studies for ten cases: the equilibrium bond dissociation energy (De) for dissociation of ground-state Ti2 to ground and excited atoms, the ground-state dissociation energies of FeO, Cu2, ScH, TiH, Sc2, Fe2, and TiV(+), and the first excitation energy (Ex) of Ti atom. Each case is studied with 11 or 13 density functionals. For comparison, the accuracy of the all-electron def2-TZVP basis set is tested with both relativistic and nonrelativistic treatments. Convergence and accuracy are assessed by comparing to relativistic all-electron calculations with a nearly complete relativistic basis set (NCBS-DK, which denotes the cc-pV5Z-DK basis set for 3d metals and hydrogen and the ma-cc-pV5Z-DK basis set for oxygen) and to nonrelativistic all-electron calculations with a nearly complete nonrelativistic basis set (NCBS-NR, which denotes the cc-pV5Z basis set for 3d metals and hydrogen and the ma-cc-pV5Z basis set for oxygen). As compared to NCBS-DK results, all ECP calculations perform worse than def2-TZVP all-electron relativistic calculations when averaged over all 130 data (13 functionals and ten test cases). The compact effective potential (CEP) relativistic effective core potential (RECP) combined with a valence basis set developed for the many-electron Dirac-Fock (MDF10) RECP performs best in effective core potential calculations and has an average basis-set incompleteness error of 3.7 kcal/mol, which is much larger than that (0.9 kcal/mol) of def2-TZVP relativistic all-electron results. Hence, the def2-TZVP relativistic all-electron calculations are recommended for accurate DFT calculations on 3d transition metals. In addition to our general findings, we observed that all kinds of density functionals do not show the same trends. For example, when ECPs are used with hybrid functionals, which sometimes are not recommended for calculations of transition metal systems, they are found to perform better at achieving the basis-set limit than when used with local functionals and meta-GGA functionals. The most successful combination of RECP and basis set has a basis-set incompleteness error of 1.7-2.4 kcal/mol for hybrid generalized gradient approximations, which is smaller than that of nonrelativistic NCBS calculations (whose average basis-set incompleteness error for hybrid functionals is 2.7-2.9 kcal/mol). The average basis-set incompleteness error in Hartree-Fock calculations is 1.0-4.4 kcal/mol for five of the ECP basis sets but is 5.8-10.8 kcal/mol for six others.

Accuracy of Effective Core Potentials and Basis Sets for Density Functional Calculations, Including Relativistic Effects, As Illustrated by Calculations on Arsenic Compounds.

For molecules containing the fourth-period element arsenic, we test (i, ii) the accuracy of all-electron (AE) basis sets from the def2-xZVP and ma-xZVP series (where xZ is S, TZ, or QZ), (iii) the accuracy of the 6-311G series of AE basis sets with additional polarization and diffuse functions, and (iv) the performance of effective core potentials (ECPs). The first set of tests involves basis-set convergence studies with eleven density functionals for five cases: equilibrium dissociation energy (De) of As2, vertical ionization potential (VIP) of As2, IP of As, acid dissociation of H3AsO4, and De of FeAs. A second set of tests involves the same kinds of basis-set convergence studies for the VIP and De values of As3 and As4 clusters. Both relativistic and nonrelativistic calculations are considered, including in each case both AE calculations and calculations with ECPs. Convergence and accuracy are assessed by comparing to relativistic AE calculations with the cc-pV5Z-DK or ma-cc-pV5Z-DK basis and to nonrelativistic AE calculations with the cc-pV5Z or ma-cc-pV5Z basis. The primary objective of this study is to evaluate the abilities of ECPs with both their recommended basis sets and other basis sets to reproduce the results of all-electron relativistic calculations. The performance of the def2 and ma series basis sets is consistent with their sizes, and quadruple-ζ basis sets are the best. The def2-TZVP basis set performs better than most of the 6-311G series basis sets, which are the most commonly used basis sets in the previous studies of arsenic compounds. However, relativistic def2-TZVP calculations are not recommended. The large-core ECPs, which are the only available ECPs for arsenic in the popular Gaussian program, have average errors of 9-12 kcal/mol for the arsenic systems studied; therefore, these ECPs are not recommended. The triple-ζ small-core relativistic ECP (RECP) basis set cc-pVTZ-PP is found to have performance better than that of the def2-TZVP basis set, and it is highly recommended for arsenic-containing systems. The double-ζ RECP basis set ma-sc-SVP is recommended for large arsenic systems for which the def2-TZVP and cc-pVTZ-PP basis sets are unaffordable, if a basis-set error of ∼2 kcal/mol can be tolerated.

Basis set convergence studies of Hartree–Fock calculations of molecular properties within the resolution of the identity approximation

Within the resolution of the identity (RI) method, the convergence of the Hartree-Fock (HF) total molecular energy and the multipole moments in the course of the combined regular expansion of the molecular and auxiliary (RI) basis sets is studied. Dunning's cc-pVXZ series is used for both the molecular and the RI basis sets. The results show the calculated quantities converge to the HF limit when both the molecular and the RI basis sets are expanded from correlation-consistent polarized valence double zeta to correlation-consistent polarized valence sextuple zeta. Combinations of molecular/RI basis sets sufficient for convergence of the total energy and of the multipole moments at various accuracy levels have been determined. A measure of the RI basis set incompleteness is suggested and discussed. As it is significantly faster than the standard HF algorithm for small and midsize molecules, the RI-HF method, together with appropriate expanding series of both molecular and RI basis sets, provide an efficient tool to estimate and control the error of the Hartree-Fock calculations due to the finite basis set.

Accurate intermolecular ground state potential of the Ar-N2 van der Waals complex

After carrying out a systematic basis set convergence study, we evaluate several ground state potential energy surfaces of the Ar-N(2) van der Waals complex at the coupled cluster singles and doubles model including connected triples corrections. We use the aug-cc-pVXZ (X=5,Q,D) and the daug-cc-pVQZ basis sets augmented with a set of 3s3p2d1f1g (denoted 33211) and 3s3p2d2f1g (denoted 33221) midbond functions, respectively. aug-cc-pVTZ-33211 results were available in the literature. The aug-cc-pV5Z-33211 (daug-cc-pVQZ-33221) surface is characterized by a T-shaped minimum at R(e)=3.709 (3.701) A and of 99.01 (102.50) cm(-1), and a linear saddle point at 4.260 (4.257) A and D(e)=75.28 (79.73) cm(-1). These results are compared with the values provided by the semiempirical potentials available, and those of previous theoretical studies. The basis set convergence of the intermolecular potentials is also analyzed. From the potentials the rovibronic spectroscopic properties are determined. We study the basis set convergence of the rotational frequencies. The binding parameters that characterized the aug-cc-pVTZ-33211 surface are reasonable, but the surface is not good enough to evaluate the microwave spectra. The aug-cc-pVQZ-33211 basis set results considerably improve the triple zeta and are close to the aug-cc-pV5Z-33211. Considering the small differences between the quadruple and the quintuple zeta surfaces, the latter results can be expected to be close to convergence. At this level the differences with respect to the accurate experimental frequencies are in the order of 0.7%. In the case of the daug-cc-pVXZ-33211,33221 (X=5,Q,T,D) series, the convergence of the interaction energies with respect to basis set improvement is not so smooth. The errors in the frequencies obtained with the daug-cc-pVQZ-33221 basis set with respect to experiment are in the order of 0.4%.

Polarizability and optical rotation calculated from the approximate coupled cluster singles and doubles CC2 linear response theory using Cholesky decompositions.

A new implementation of the approximate coupled cluster singles and doubles CC2 linear response model using Cholesky decomposition of the two-electron integrals is presented. Significantly reducing storage demands and computational effort without sacrificing accuracy compared to the conventional model, the algorithm is well suited for large-scale applications. Extensive basis set convergence studies are presented for the static and frequency-dependent electric dipole polarizability of benzene and C60, and for the optical rotation of CNOFH2 and (-)-trans-cyclooctene (TCO). The origin-dependence of the optical rotation is calculated and shown to persist for CC2 even at basis set convergence.

An improved method for density functional calculations of the frequency-dependent optical rotation

Density functional theory calculations of the frequency-dependent optical rotation (OR) provide a simple and effective way to determine the absolute configuration of chiral molecules. We report two major improvements of the method here. Firstly, the dipole velocity form of the OR is introduced. In contrast to the commonly employed dipole length form, the dipole velocity form is gauge-origin invariant in any basis set. We present a basis set convergence study which shows that basis sets of augmented double zeta quality yield sufficiently accurate results in most cases. The deviation between the length and velocity forms provides a convenient measure of the basis set error. Secondly, we extend the resolution of the identity (RI) approximation to frequency-dependent OR calculations. Using the Coulomb norm and auxiliary basis sets optimized for ground-state calculations, we obtain RI errors that are negligible compared to errors due to the basis set and the density functional. Total computation times are reduced by a up to a factor of 10, as is demonstrated for a benchmark set of eight different molecules.

A coupled cluster response study of the electric dipole polarizability, first and second hyperpolarizabilities of HCl

The electric dipole polarizability and first and second hyperpolarizabilities of HCl are computed within coupled cluster response theory. A systematic basis set convergence study with Dunning's correlation consistent basis sets is first performed. Correlation effects are studied using the coupled cluster singles and doubles and CC3 models. The latter takes iteratively into account the effect of connected triple excitations. Emphasis is given to the accurate prediction of the frequency dependence and the appropriate inclusion of the effect of molecular vibrations. The results should provide the most accurate theoretical estimates to-date for the electric dipole (hyper)polarizabilities of HCl. The agreement with experiment is excellent.

Linear response coupled cluster calculation of Raman scattering cross sections

The linear response coupled cluster singles and doubles (CCSD) method has been used to evaluate electric dipole polarizabilities, their geometric derivatives and Q-branch Raman scattering cross sections for five diatomic molecules: N2, CO, HF, HCl, and Cl2. A systematic study of basis set convergence has been carried out. Geometric derivatives of the frequency dispersion coefficients have been computed. This allowed to evaluate Raman intensities for arbitrary laser frequencies outside the resonance region. The geometric derivatives of polarizabilities, and therefore Raman intensities, exhibit stronger basis set dependence and dispersion effects than the polarizabilities themselves. Whereas the calculated polarizabilities are very close to experiment, the agreement of their geometric derivatives and of the Raman scattering cross sections with experiment is less excellent. We attribute this mainly to the limitations of Placzek’s theory (for Raman intensities) and to the low precision of the reference experimental data.

Basis set convergence study of the atomization energy, geometry, and anharmonic force field of SO2: The importance of inner polarization functions

The total atomization energies, geometries, and anharmonic force fields of the SO and SO2 molecules have been studied at the augmented coupled cluster [CCSD(T)] level near the one-particle basis set limit. The effect of core correlation has been accounted for. The addition of high-exponent d and f “inner polarization functions” to the sulfur basis set was found to be essential for obtaining reliable molecular geometries. The differential effect of core correlation on computed properties is in fact much less important. The CCSD(T) one-particle basis set limit for the symmetric stretching frequencies appears to be higher than the exact value, while the antisymmetric stretching frequency benefits from an error cancellation. The basis set extension effects of diffuse functions and inner polarization functions appear to be nearly perfectly additive. Our best computed atomization energies and geometries agree to within 0.1 kcal/mol, 0.0004 Å, and 0.03 degrees with experiment: The errors in the fundamentals of SO2 are +3.9, −0.4, and +0.4 cm−1. The best computed quartic force field for SO2 should provide a good starting point for a new experimental refinement.

Basis-set convergence of highly defected sites in amorphous carbon.

A systematic first-principles pseudopotential plane-wave and linear combination of atomic orbitals basis-set convergence study has been performed for a 64-atom amorphous ``tetrahedral'' carbon network (Drabold et al.). Converged results of the two methods are consistent with each other and indicate that when large, site-dependent bond-angle and bond-length distortions are present, variational freedom within a basis-set representation is crucial for an accurate representation of atomic forces. Minimal basis-set/non-self-consistent methods are not adequate for these systems, but are more applicable to systems with minimal site-dependent, electron-density distortions.

Model potential in spectral representation form. Its fidelity to the frozen-core operator and comparison with other model potentials

The model potential, in which the Coulomb and exchange core operators are converted to the matrix elements by use of the spectral representation technique, should be equivalent to the frozen-core operator, if the basis set of the spectral representation is complete. Its fidelity to the frozen-core operator is examined in this paper through a thorough study of basis set convergence in both the spectral space and the atomic orbital space, along with comparison with the Sakai–Huzinaga and the Huzinaga–Seijo–Barandiaran–Klobukowski model potentials. The RHF molecular calculations on N2, P2, As2, Sb2, PN, and AsN show that the present model potential can approximate, with reliable accuracy, the results obtained from the frozen-core calculations. Keywords: model potential, spectral representation, molecular orbital calculation, electronic structure.

Accurate calculation of core-electron binding energies by the density-functional method

Using a basis set of atomic natural orbitals of valence triple-zeta plus double polarization quality, the core-electron binding energies (CEBEs) of HF, H 2O, N 2 and CO were computed by the deMon density-functional program with a combined functional of Becke's exchange (B88) with Perdew's correlation (P86). Of four different versions of the transition-state method tested, the unrestricted generalized transition-state (uGTS) model showed most promise. Two other functionals were tested but found to be inferior to the B88/P86 combination. The molecules HCN, CO 2 and H 2CO were added in the comparative study of basis set convergence. The best procedure found (the uGTS model using B88/P86 functional with Dunning's cc-pV5Z basis set) was applied to other molecules. The average deviation from experiment for 20 computed CEBEs is 0.23 eV.

A basis set convergence study of conventional and HSF electron densities in the Li 2 molecule

Calculations of nonnuclear, Hartree-Fock HSF and CHSF electron densities are reported for the first time. The positions of critical points in the conventional, HSF, and CHSF electron densities of the Li 2 molecule (including the nonnuclear maximum) and corresponding values of the electron density are computed for a sequence of systematically improved basis sets. The basis set convergence of these topological properties, as well as that of the densities at nuclei, are examined. Quantities derived from HSF and CHSF electron densities are found to converge more rapidly than their conventional counterparts.