Improving numerical integration through basis set expansion

Calculations are presented to assess a theorem presented by S.F. Boys [(1969) Proc. R. Soc. A. 309:195], regarding the accuracy of numerical integration in quantum chemical calculations. The theorem states that the error due to numerical integration can be made proportional to the error due to basis set truncation, and thus goes to zero in the limit of a complete basis. We test this theorem on the hydrogen atom, showing that with a solution-spanning basis, the numerically exact orbital energy can indeed be calculated with a small number of integration points. Moreover, tests for H and H₂⁺ demonstrate that even when only a near-complete basis is employed, Boys Theorem can significantly reduce integration error. However, for other systems, like the oxygen atom and the CO₂ molecule, the theorem yields no advantage for some occupied orbitals. It is concluded that the theorem would be most useful for calculations that demand large basis sets.     

On basis set superposition error corrected stabilization energies for large n-body clusters

In this contribution, we propose an approximate basis set superposition error (BSSE) correction scheme for the site-site function counterpoise and for the Valiron-Mayer function counterpoise correction of second order to account for the basis set superposition error in clusters with a large number of subunits. The accuracy of the proposed scheme has been investigated for a water cluster series at the CCSD(T), CCSD, MP2, and self-consistent field levels of theory using Dunning's correlation consistent basis sets. The BSSE corrected stabilization energies for a series of water clusters are presented. A study regarding the possible savings with respect to computational resources has been carried out as well as a monitoring of the basis set dependence of the approximate BSSE corrections.     

Comparison of basis set effects and the performance of ab initio and DFT methods for probing equilibrium fluctuations

The electronic absorption and emission spectra of large molecules reflect the extent and timescale of electron-vibration coupling and therefore the extent and timescale of relaxation/reorganization in response to a perturbation. In this paper, we present a comparison of the calculated absorption and emission spectra of NADH in liver alcohol dehydrogenase (LADH), using quantum mechanical/molecular mechanical methods, in which we vary the QM component. Specifically, we have looked at the influence of basis set (STO-3G, 3-21G*, 6-31G*, CC-pVDZ, and 6-311G**), as well as the influence of applying the DFT TD-B3LYP and ab initio TD-HF and CIS methods to the calculation of absorption/emission spectra and the reorganization energy (Stokes shift). The ab initio TD-HF and CIS methods reproduce the experimentally determined Stokes shift and spectral profiles to a high level of agreement, while the TD-B3LYP method significantly underestimates the Stokes shift, by 45%. We comment on the origin of this problem and suggest that ab initio methods may be naturally more suited to predicting molecular behavior away from equilibrium geometries.     

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.     

Franck-Condon matrix elements for bound-continuum vibrational transitions calculated by numerical integration and basis set expansion techniques

Franck-Condon factor distributions for bound-to-continuum transitions of one-dimensional vibrational states are calculated by a) using numerical integration, b) employing a finite number of square integrable harmonic oscillator functions. The methods are generally applicable to any kind of bound or repulsive potential involved. Results are presented and compared to model potential calculations previously reported by Krüger [1].     

Complete basis set ab initio computational study of three-dimensional aromaticity in highly symmetric hydrogen clusters

The aromaticity of planar and highly symmetric three-dimensional hydrogen clusters were evaluated with the complete basis set ab initio computational method. The energy of formation of the hydrogen clusters from the hydrogen molecule and hydrogen molecular ions were used in comparison to their relative stabilities. The aromaticity of planar hydrogen clusters, as well as hydrogen clusters in the three dimensions, arranged as highly symmetric regular polyhedra (Plato's polyhedras), were discussed with respect to the Hückel and Möbius aromatic rules.     

Geometrical basis set for molecular computations

To simplify the computation of many center two-electron integrals in large molecules a new type of basis set - called geometrical - is proposed. Its flexibility is tested for atoms from Z = 1 to 38 and for positive and negative ions. This basis is designed mainly for improving large-molecule computations but we have tested it with an accuracte computation for H₂O.     

A correlation-consistent basis set for Fe

 A series of correlation-consistent basis sets are developed for Fe. Our best computed ⁵F–⁵D separation in the Fe atom is in excellent agreement with experiment. Our best estimate for the FeCO D ₀ value is in good agreement with experiment. The ⁵Σ⁻–³Σ⁻ separation in FeCO has an error of 3.6 kcal/mol; while the origin of this error is not clear, it is probably not due to the basis set. Received: 5 March 2001 / Accepted: 2 May 2001 / Published online: 9 August 2001     

Gaussian basis set of double zeta quality for atoms K through Kr: application in DFT calculations of molecular properties

Contracted basis sets of double zeta (DZ) quality for the atoms from K to Kr are presented. They were determined from fully optimized basis sets of primitive Gaussian-type functions generated in atomic Hartree-Fock calculations. Sets of Gaussian polarization functions optimized at the Möller-Plesset second-order level were added to the DZ basis set. This extends earlier work on segmented contracted DZ basis set for atoms H-Ar. From this set, using the BP86 nonhybrid and B3LYP hybrid functionals, dissociation energy, geometric parameters, harmonic vibrational frequency, and electric dipole moment of a set of molecules were calculated and compared with results obtained with other basis sets and with experimental data reported in the literature. In addition, ⁵⁷Fe and ⁷⁷Se nuclear magnetic resonance chemical shifts in Fe(C₅H₅)₂, H₂Se, and CSe₂ were calculated using density functional theory and gauge-including atomic orbitals and, then, compared with theoretical and experimental values previously published in the literature. Except for chemical shift, one verifies that our results give the best agreement with experimental and benchmark values. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2008     

Comparative analysis of a full-electron basis set and pseudopotential for the iodine atom in DFT quantum-chemical calculations of iodine-containing compounds

A systematic study was performed to examine the possibilities of the B3LYP DFT method in a dgdzvp full-electron basis and of the method including a pseudopotential for iodine compounds. The full-electron basis generally gives better agreement for X-I bond lengths and reaction enthalpies of iodination of organic compounds and equally good agreement in calculations of the IR vibrations of the X-I bond length compared with the studies using the pseudopotential. The full-electron basis also allows adequate calculations of the quadrupole coupling constants of iodine atoms and is generally characterized by smaller computing times.     

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.     

The 6-31G++ basis set: An economical basis set for correlated wavefunctions

The 6-31G ⁺⁺ basis set is described. This basis set is very similar to the existing 6-31G ^** set but is somewhat smaller through the use of five (rather than six) second-order Gaussians (d functions) and has polarization function exponents optimized for correlated rather than Hartree–Fock wavefunctions. The performance of 6-31G ⁺⁺ is compared with that of the 6-31G ^** and 6-31G ^** basis sets through calculation of the geometries and atomization energies for the set of molecules LiH, FH, H₂O, NH₃, CH₄, N₂, CO, HCN, and HCCH.     

Reaction of aluminum clusters with water

The atomistic mechanism of rapid hydrogen production from water by an aluminum cluster is investigated by ab initio molecular dynamics simulations on a parallel computer. A low activation-barrier mechanism of hydrogen production is found, in which a pair of Lewis acid and base sites on the cluster surface plays a crucial role. Hydrogen production is assisted by rapid proton transport in water via a chain of hydrogen-bond switching events similar to the Grotthuss mechanism, where hydroxide ions are converted to water molecules at the Lewis-acid sites and hydrogen atoms are supplied at the Lewis-base sites. The activation free energy is estimated along various reaction paths associated with hydrogen production, and the corresponding reaction rates are discussed based on the transition state theory.     

Adsorption of small aromatic molecules on gold: a DFT localized basis set study including van der Waals effects

Schrödinger equation for harmonium and related models may be transformed to the biconfluent Heun equation. The solubility of this equation and its applications in quantum chemistry are briefly discussed.     

Effects of global orbital cutoff value and numerical basis set size on accuracies of theoretical atomization energies

Numerical basis sets are known for their rapid convergence in density functional theory calculations. The selections of global orbital cutoff values and numerical basis set sizes are important to the computational accuracies and efficiencies. In this study, the effects of global orbital cutoff values and numerical basis set sizes on the theoretical atomization energies (D ₀) were investigated using density functional theory with the generalized gradient approximation. Our results on the total energies of seven atoms and D ₀ of a set of 44 molecules demonstrate that the numerical orbital cutoff value should be larger than 6.5 Å to get the converged energetic properties. Through comparing the D ₀ of these 44 molecules obtained by using four kinds of different numerical basis sets, DN, DND, DNP, and TNP, it demonstrates that the DNP basis set is good enough to predict accurate D ₀ with affordable computational cost.     

Intramolecular basis set superposition errors

The effects of intramolecular basis set superposition errors are less well documented than the corresponding intermolecular effects. The intramolecular basis set superposition errors are examined, using the approach of Jensen, for several basis sets developed by Pople and his co‐workers, which are widely used in studies of larger molecules. Prototype calculations are reported for the ground state of the water molecule using both the matrix Hartree–Fock method and the many‐body perturbation expansion for the correlation energy taken through second order. A similar investigation is carried out for some of the correlation consistent basis sets published by Dunning and his collaborators. Specifically, the following aspects are investigated: (i) the magnitude of the intramolecular basis set superposition error, (ii) the nonadditivity of intramolecular counterpoise corrections when applied in a pairwise fashion, and (iii) the use of multiple “ghost” centers. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 282–292, 2001     

Hydrogen dissociation on small aluminum clusters

Transition states and reaction paths for a hydrogen molecule dissociating on small aluminum clusters have been calculated using density functional theory. The two lowest spin states have been taken into account for all the Al(n) clusters considered, with n=2-6. The aluminum dimer, which shows a (3)Π(u) electronic ground state, has also been studied at the coupled cluster and configuration interaction level for comparison and to check the accuracy of single determinant calculations in this special case, where two degenerate configurations should be taken into account. The calculated reaction barriers give an explanation of the experimentally observed reactivity of hydrogen on Al clusters of different size [Cox et al., J. Chem. Phys. 84, 4651 (1986)] and reproduce the high observed reactivity of the Al(6) cluster. The electronic structure of the Al(n)-H(2) systems was also systematically investigated in order to determine the role played by interactions of specific molecular orbitals for different nuclear arrangements. Singlet Al(n) clusters (with n even) exhibit the lowest barriers to H(2) dissociation because their highest doubly occupied molecular orbitals allow for a more favorable interaction with the antibonding σ(u) molecular orbital of H(2).     

孪函数N电子基组的组态

分析了各类孪函数Ｎ电子基组态展开式的特点以及它们对体系相关能的贡献，提出了一种在享函数Ｎ电子基矢下进行多组态自洽场计算时的组态选取方法，并依此方法在ＳＴＯ－６－３１Ｇ基组下对ＬｉＨ分子的基态能量做了计算，结果表明，用该组态选取方法只需选取少量的组态波函数便可得到相当精确的计算结果。