Efficiency of numerical basis sets for predicting the binding energies of hydrogen bonded complexes: evidence of small basis set superposition error compared to Gaussian basis sets

Binding energies of selected hydrogen bonded complexes have been calculated within the framework of density functional theory (DFT) method to discuss the efficiency of numerical basis sets implemented in the DFT code DMol3 in comparison with Gaussian basis sets. The corrections of basis set superposition error (BSSE) are evaluated by means of counterpoise method. Two kinds of different numerical basis sets in size are examined; the size of the one is comparable to Gaussian double zeta plus polarization function basis set (DNP), and that of the other is comparable to triple zeta plus double polarization functions basis set (TNDP). We have confirmed that the magnitudes of BSSE in these numerical basis sets are comparative to or smaller than those in Gaussian basis sets whose sizes are much larger than the corresponding numerical basis sets; the BSSE corrections in DNP are less than those in the Gaussian 6-311+G(3df,2pd) basis set, and those in TNDP are comparable to those in the substantially large scale Gaussian basis set aug-cc-pVTZ. The differences in counterpoise corrected binding energies between calculated using DNP and calculated using aug-cc-pVTZ are less than 9 kJ/mol for all of the complexes studied in the present work. The present results have shown that the cost effectiveness in the numerical basis sets in DMol3 is superior to that in Gaussian basis sets in terms of accuracy per computational cost.     

Hartree-Fock calculation of the harmonic force constants and equilibrium geometry of formaldehyde

Using the force method, complete sets of harmonic force constants have been obtained for formaldehyde from Hartree-Fock wavefunctions. The agreement with experiment is considered particularly satisfying for the off-diagonal constants. This holds not only for a near-Hartree-Fock Gaussian basis set but also for a small but polarized 7, 3/3/1 basis set. The value even of such a small calculation is underlined by frequencies calculated from force constants corrected for almost systematic errors in the diagonal constants. Experimental force fields are critically examined, and an explanation for the surprisingly large coupling between CO and CH stretching is indicated.     

Cubic-grid Gaussian basis sets for electron scattering calculations. II. Matrix elements

In this article, we report an efficient computational procedure for electron scattering matrix elements in the previously developed cubic-grid Gaussian basis sets. The Green function matrix elements derived for the cubic-grid basis set are simpler and easier to calculate than are those available in the literature for conventional Gaussian basis sets. Special features of the cubic-grid basis sets may also be exploited for a very efficient computation of Coulomb and exchange integrals. Inelastic scattering amplitudes for vibrational excitations may be efficiently calcualted in the harmonic approximation by numerical differention of the T-matrix elements. © 1995 John Wiley & Sons, Inc.     

Numerically stable optimized effective potential method with balanced Gaussian basis sets

A solution to the long-standing problem of developing numerically stable optimized effective potential (OEP) methods based on Gaussian basis sets is presented by introducing an approach consisting of an exact exchange OEP method with an accompanying construction and balancing scheme for the involved auxiliary and orbital Gaussian basis sets that is numerically stable and that properly represents an exact exchange Kohn-Sham method. The method is a purely analytical method that does not require any numerical grid, scales like Hartree-Fock or B3LYP procedures, is straightforward to implement, and is easily generalized to take into account orbital-dependent density functionals other than the exact exchange considered in this work. Thus, the presented OEP approach opens the way to the development and application of novel orbital-dependent exchange-correlation functionals. It is shown that adequately taking into account the continuum part of the Kohn-Sham orbital spectrum is crucial for numerically stable Gaussian basis set OEP methods. Moreover, it is mandatory to employ orbital basis sets that are converged with respect to the used auxiliary basis representing the exchange potential. OEP calculations in the past often did not meet the latter requirement and therefore may have led to erroneously low total energies.     

Benchmarking NMR indirect nuclear spin-spin coupling constants: SOPPA, SOPPA(CC2), and SOPPA(CCSD) versus CCSD

Accurate calculations of NMR indirect nuclear spin-spin coupling constants require especially optimized basis sets and correlated wave function methods such as CCSD or SOPPA(CCSD). Both methods scale as N(6), where N is the number of orbitals, which prevents routine applications to molecules with more than 10-15 nonhydrogen atoms. We have therefore developed a modification of the SOPPA(CCSD) method in which the CCSD singles and doubles amplitudes are replaced by CC2 singles and doubles amplitudes. This new method, called SOPPA(CC2), scales only as N(5), like the original SOPPA-method. The performance of the SOPPA(CC2) method for the calculation of indirect nuclear spin-spin coupling constants is compared to SOPPA and SOPPA(CCSD) employing a set of benchmark molecules. We also investigate the basis set dependence by employing three different basis sets optimized for spin-spin coupling constants, namely the HuzIV-su4, ccJ-pVTZ, and ccJ-pVQZ basis sets. The results of the corresponding CCSD calculations are used as a theoretical reference.     

Theoretical determination of multiple exchange couplings and magnetic susceptibility data in inorganic solids: the prototypical case of Cu2(OH)3NO3

Density functional theory based on hybrid functionals and localized atomic type basis sets is employed to calculate the exchange couplings in the layered three-dimensional compound Cu2(OH)3NO3. We assign accurate values to the six different in-plane exchange couplings. Interlayer exchange interactions through hydrogen bonds are also quantified. The calculated exchange coupling constants are then employed to perform quantum Monte Carlo simulations to yield magnetic susceptibility data, which compare successfully with experiments. Our approach sets the foundations of a viable methodology to extract reliable magnetic susceptibilities from density functional data.     

Proton affinities of molecules containing nitrogen and oxygen: Comparing density functional results to experiment

Summary Proton affinities were calculated using density functional theory for 11 small molecules whose primary protonation site is on nitrogen, and eight small molecules that protonate on oxygen. Calculations were performed using both the local spin density approximation and nonlocal gradient corrections to the exchange correlation functional. The results were not sensitive to whether the nonlocal gradient correction was implemented on the final local spin density optimized geometry or whether the correction was included in the self-consistent calculation of the energy at each optimization step. Although negligible basis set dependence was found using the analytic Gaussian basis sets, numerical basis sets required augmentation by a double set of polarization functions to achieve reasonable agreement with experiment. All calculations systematically underestimated oxygen proton affinities.     

A study of the performance of numerical basis sets in DFT calculations on sulfur-containing molecules

The performance of numerical basis sets in relation to Gaussian basis sets is examined, by studying 20 small sulfur-containing molecules. The results of geometry optimization calculations are reported for each molecule using both density functional and Hartee-Fock methods. In comparison with experimental data, it is shown that the use of numerical bases tend to overestimate structural parameters, particularly bond lengths, and, in most cases, more than Gaussian basis sets. It is also shown that the use of a larger Gaussian basis set in DFT calculations has the effect of reducing bond lengths. © 1996 John Wiley & Sons, Inc.     

Geometric approximation to nuclear spin–spin coupling constants in the water molecule

The geometric aproximation is used within the framework of triple perturbation theory to evaluate the contributions to nuclear spin–spin coupling constants in the water molecule provided by the Fermi contact, the spin–orbit, and the spin–dipolar interactions. The results, obtained with SCF wave functions expanded over Gaussian basis sets of increasing quality, are compared with corresponding coupled Hartree–Fock estimates. The limits of the geometric approximation to coupling constants are discussed.     

Improving the calculation of Electron Paramagnetic Resonance hyperfine coupling tensors for d-block metals

Calculation of hyperfine coupling constants (HFCs) of Electron Paramagnetic Resonance from first principles can be a beneficial complement to experimental data in cases where the molecular structure is unknown. We have recently investigated basis set convergence of HFCs in d-block complexes and obtained a set of basis functions for the elements Sc-Zn, which were saturated with respect to both the Fermi contact and spin-dipolar components of the hyperfine coupling tensor [Hedeg?rd et al., J. Chem. Theory Comput., 2011, 7, 4077-4087]. Furthermore, a contraction scheme was proposed leading to very accurate, yet efficient basis sets for the elements Sc-Zn. Here this scheme is tested against a larger test set of molecules and a wider range of DFT functionals. We further investigate the regular aug-cc-pVTZ and core-valence correlation aug-cc-pCVTZ basis sets as well as another core-property basis set, CP(PPP). While aug-cc-pVTZ-J provides hyperfine coupling constants that are almost identical to the converged series (aug-cc-pVTZ-Juc), we observe that not only the regular but also the core-valence correlation basis sets provide results far from the converged results. The usage of specialized core-basis sets leads to a large and highly significant improvement of the calculated hyperfine couplings in comparison with experimental data.     

Magnitude of Finite‐Nucleus‐Size Effects in Relativistic Density Functional Computations of Indirect NMR Nuclear Spin–Spin Coupling Constants

A spherical Gaussian nuclear charge distribution model has been implemented for spin‐free (scalar) and two‐component (spin–orbit) relativistic density functional calculations of indirect NMR nuclear spin–spin coupling (J‐coupling) constants. The finite nuclear volume effects on the hyperfine integrals are quite pronounced and as a consequence they noticeably alter coupling constants involving heavy NMR nuclei such as W, Pt, Hg, Tl, and Pb. Typically, the isotropic J‐couplings are reduced in magnitude by about 10 to 15 % for couplings between one of the heaviest NMR nuclei and a light atomic ligand, and even more so for couplings between two heavy atoms. For a subset of the systems studied, viz. the Hg atom, Hg₂²⁺, and Tl? X where X=Br, I, the basis set convergence of the hyperfine integrals and the coupling constants was monitored. For the Hg atom, numerical and basis set calculations of the electron density and the 1s and 6s orbital hyperfine integrals are directly compared. The coupling anisotropies of TlBr and TlI increase by about 2 % due to finite‐nucleus effects.     

Interaction energies and NMR indirect nuclear spin-spin coupling constants in linear HCN and HNC complexes

The cooperativity effects on both the electronic energy and NMR indirect nuclear spin-spin coupling constants J of the linear complexes (HCN)n and (HNC)n (n = 1-6) are discussed. The geometries of the complexes were optimized at the MP2 level by using the cc-pVTZ basis sets. The spin-spin coupling constants were calculated at the level of the second-order polarization propagator approximation with use of the local dense basis set scheme based on the cc-pVTZ-J basis sets. We find strong correlations in the patterns of different properties such as interaction energy, hydrogen bond distances, and spin-spin coupling constants for both series of compounds. The intramolecular spin-spin couplings are with two exceptions dominated by the Fermi contact (FC) mechanism, while the FC term is the only nonvanishing contribution for the intermolecular couplings. The latter do not follow the Dirac vector model and are important only between nearest neighbors.     

Optimization of augmentation functions for correlated calculations of spin-spin coupling constants and related properties

A new hierarchy of augmented basis sets optimized for the calculation of molecular properties such as indirect spin-spin coupling constants is presented. Based on the Dunning hierarchy of cc-pVXZ (X = D, T, Q, and 5) basis sets augmentation functions with tight exponents have been optimized for coupled-cluster calculations of indirect spin-spin coupling constants. The optimal exponents for these tight functions have been obtained by optimizing the sum of the absolute values of all contributions to the coupling constant. On the basis of a series of test cases (CO, HF, N(2), F(2), H(2)O, NH(3), and CH(4)) we propose a set of tight s, p, and d functions to be added to the uncontracted Dunning basis sets, and, subsequently, to recontract. The resulting ccJ-pVXZ (X = D, T, Q, and 5) basis sets demonstrate excellent cost efficiency in benchmark calculations. These new basis sets should generally be applicable for the calculation of spin-spin coupling constants and other properties that have a strong dependence on powers of 1r or even contain a delta distribution for correlated ab initio methods.     

About the calculation of exchange coupling constants using density-functional theory: the role of the self-interaction error

The effect of the correction of the self-interaction error on the calculation of exchange coupling constants with methods based on density-functional theory has been tested in simple model systems. The inclusion of the self-interaction correction cancels the nondynamical correlation energy contributions simulated by the commonly used functionals. Hence, such correction should be important in the accurate determination of exchange coupling constants. We have also tested several recent functionals to calculate exchange coupling constants in transition-metal complexes, such as meta-GGA functionals or new formulations of hybrid functionals. The influence of the basis set and of the use of pseudopotentials on the calculated J values has also been evaluated for a Fe(III) dinuclear complex in which the paramagnetic centers bear several unpaired electrons.     

The optimum contraction of basis sets for calculating spin–spin coupling constants

The previously proposed pcJ-n basis sets, optimized for calculating indirect nuclear spin–spin coupling constants using density functional methods, are re-evaluated for finding the optimum contraction scheme as a compromise between computational efficiency and minimizing contraction errors. An exhaustive search is performed for the H₂, F₂ and P₂ molecules, and candidates for optimum contraction schemes are evaluated for a larger test set of 21 molecules. Using the criterion that the contraction error should not exceed the basis set error relative to the basis set limit, the optimum contraction is defined for each basis set. The results show that it is difficult to contract basis sets for calculating spin–spin coupling constants to any significant degree without losing the inherent accuracy. The work provides guidelines for searching for optimum contraction schemes for other properties and/or at theoretical levels where a systematic search is impractical.     

Ab initio molecular orbital calculations of nuclear spin-spin coupling constants in PH2, PH3, PH4 and P2H4

Ab initio molecular orbital calculations of nuclear spin-spin coupling constants in PH⁻₂, PH₃, PH⁺₄ and P₂H₄ have been carried out employing SCF perturbation theory. Basis set dependence of all the four contributing terms has been studied in order to find the criterion for the selection of basis sets to be employed for computing this property. The dependence of the coupling constants of PH⁻₂ on its geometry has also been found. This study also discusses the requirement for satisfactory computation of couplings in cases where none of the coupling nuclei is a proton. It is found that bond-centred functions along with at least double zeta basis sets reproduce coupling constants quite satisfactorily. In all the cases studied, uncontracted core basis functions yield couplings which are in better agreement with experimental couplings than those obtained with contracted core functions.     

Basis-set quality and basis-set bias in molecular property calculations

Quality measures for Gaussian basis sets are proposed that are based on principal angles between the basis set and reference molecular orbitals. The principal angles are obtained from the cosine-sine (CS) decomposition of orthogonal matrices and yield detailed information about basis-set convergence with respect to different regions of space. Principal angles for occupied orbitals show excellent correlation with basis-set errors in ground-state energies. Furthermore, ground-state bias in finite basis sets can be estimated from the relation between principal angles for occupied and Rydberg orbitals. Ground-state bias is observed in basis sets including extensive diffuse augmentation and affects the quality of computed molecular response properties. Principal angles and ground-state bias are investigated for the H-Ne atoms and a series of diatomics using numerical Hartree-Fock calculations as a reference. Convergence of ground-state energies and static polarizabilities is studied for the hierarchies of correlation-consistent and Karlsruhe segmented def2 basis sets including different levels of diffuse augmentation.     

An ab initio investigation of the geometry,bonding and coupling constants of BF2

Comparative calculations, using five different basis sets of contracted Gaussian functions, of the geometry, bonding and hyperfine coupling constants of BF₂ are reported. The best calculation, using a near Hartree-Fock atomic basis, predicts a bond angle of 120° and a bond length of 2.50 a.u. (=1.32 Å) for the X ² A ₁ ground state. The geometries of three low-lying excited states are also presented.