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.     

Performance of numerical basis set DFT for aluminum clusters

We have investigated and compared the ability of numerical and Gaussian-type basis sets combined with density functional theory (DFT) to accurately describe the geometries, binding energies, and electronic properties of aluminum clusters, Al12XHn (X = Al, Si; n = 0, 1, 2). DFT results are compared against high-level benchmark calculations and experimental data where available. Properties compared include geometries, binding energies, ionization potentials, electron affinities, and HOMO-LUMO gaps. Generally, the PBE functional with the double numerical basis set with polarization (DNP) performs very well against experiment and the analytical basis sets for considerably less computational expense.     

All-electron scalar relativistic basis sets for the 6p elements

New segmented all-electron relativistically contracted (SARC) basis sets have been developed for the elements ₈₁Tl–₈₆Rn, thus extending the SARC family of all-electron basis sets to include the 6p block. The SARC basis sets are separately contracted for the second-order Douglas–Kroll–Hess and the zeroth-order regular approximation scalar relativistic Hamiltonians. Their compact size and segmented construction are best suited to the requirements of routine density functional theory (DFT) applications. Evaluation of the basis sets is performed in terms of incompleteness and contraction errors, orbital properties, ionization energies, electron affinities, and atomic polarizabilities. From these atomic metrics and from computed basis set superposition errors for a series of homonuclear dimers, it is shown that the SARC basis sets achieve a good balance between accuracy and size for efficient all-electron scalar relativistic DFT applications.     

Substituent effects in second-row molecules: Basis set performance in calculations of normal valency phosphorus and sulfur compounds

Ab inito molecular orbital calculations of the phosphorus- and sulfur-containing series PH₂X, PH₃X⁺, SHX, and SH₂X⁺ (X = H, CH₃, NH₂, OH, F) have been carried out over a range of Gaussian basis sets and the results (optimized geometrical structures, relative energies, and electron distributions) critically compared. As in first-row molecules there are large discrepancies between substituent interaction energies at different basis set levels, particularly in electron-rich molecules; use of basis sets lower than the supplemented 6-31G basis incurs the risk of obtaining substituent stabilizations with large errors, including the wrong sign. Only a small part of the discrepancies is accounted for by structural differences between the optimized geometries. Supplementation of low level basis sets by d functions frequently leads to exaggerated stabilization energies for π-donor substituents. Poor performance also results from the use of split valence basis sets in which the valence shell electron density is too heavily concentrated in diffuse component of the valence shell functions, again likely to occur in electron-rich molecules. Isodesmic reaction energies are much less sensitive to basis set variation, but d function supplementation is necessary to achieve reliable results, suggesting a marginal valence role for d functions, not merely polarization of the bonding density. Optimized molecular geometries are relatively insensitive to basis set and electron population analysis data, for better-than-minimal bases, are uniform to an unexpected degree.     

Compact basis sets for LCAO-LSD calculations. Part II: Tests for Cr2 and Ni4

The compact orbital and auxiliary basis sets for LCAO-LSD calculations introduced in Part I are tested in molecular calculations on Cr₂ and Ni₄. The present results for spectroscopic constants and valence orbital energies obtained using medium size orbital expansions with a double-zeta representation for valence orbitals are in very good agreement with those previously calculated with very extended sets. Since the computational time of the present calculations is reduced severalfold compared with the extended basis set calculations, the present basis sets allow increased efficiency of the LCAO-LSD calculations and allow the method to be extended to larger systems.     

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.     

The effect of conjugated spacer on novel carbazole derivatives for dye‐sensitized solar cells: Density functional theory/time‐dependent density functional theory study

The ground‐state structure and frontier molecular orbital of D‐π‐A organic dyes, CFT1A, CFT2A, and CFT1PA were theoretically investigated using density functional theory (DFT) on B3LYP functional with 6‐31G(d,p) basis set. The vertical excitation energies and absorption spectra were obtained using time‐dependent DFT (TD‐DFT). The adsorptions of these dyes on TiO₂ anatase (101) were carried out by using a 38[TiO₂] cluster model using Perdew–Burke–Ernzerhof functional with the double numerical basis set with polarization (DNP). The results showed that the introduction of thiophene–thiophene unit (T–T) as conjugated spacer in CFT2A could affect the performance of intramolecular charge transfer significantly due to the inter‐ring torsion of T–T being decreased compared with phenylene–phenylene (P–P) spacer of CFP2A in the researhcers' previous report. It was also found that increasing the number of π‐conjugated unit gradually enhanced charge separation between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of these dyes, leading to a high‐efficiency photocurrent generation. The HOMO–LUMO energy gaps were calculated to be 2.51, 2.37, and 2.50 eV for CFT1A, CFT2A, and CFT1PA respectively. Moreover, the calculated adsorption energies of these dyes on TiO₂ cluster were ～14 kcal/mol, implying that these dyes strongly bind to TiO₂ surface. Furthermore, the electronic HOMO and LUMO shapes of all dye–TiO₂ complexes exhibited injection mechanism of electron via intermolecular charge‐transfer transition. © 2012 Wiley Periodicals, Inc.     

Basis set effects on relative energies and HOMO–LUMO energy gaps of fullerene C36

Fifteen C₃₆ isomers were examined to determine the influence that the quality of basis sets has on the geometry parameters, the relative stability and HOMO–LUMO energy gaps of fullerene isomers calculated with density functional theory. It is worthwhile to note that the geometry parameters of all C₃₆ isomers are insensitive to basis sets. On the other hand, one set of d-type polarization functions plays an important role in evaluating relative stability and HOMO–LUMO energy gaps, while diffuse functions are not effective. To obtain reliable energies, at least a double-zeta plus polarization basis set is required, and a triple-zeta plus polarization basis set is suggested to lead to accurate energies at a reasonable computational cost.     

High resolution electron momentum spectroscopy of the valence orbitals of water

The development of a third-generation electron momentum spectrometer with significantly improved energy and momentum resolutions at Tsinghua University (ΔE = 0.45–0.68 eV, Δθ = ±0.53° and Δ? = ±0.84°) has enabled a reinvestigation of the valence orbital electron momentum distributions of H₂O with improved statistical accuracy. The measurements have been conducted at impact energies of 1200 eV and 2400 eV in order to check the validity of the plane wave impulse approximation. The obtained ionization spectra and electron momentum distributions have been compared with the results of computations carried out with Hartree Fock [HF] theory, density functional theory in conjunction with the standard B3LYP functional, one-particle Green’s function [1p-GF] theory along with the third-order algebraic diagrammatic construction scheme [ADC(3)], symmetry adapted cluster configuration interaction [SAC-CI] theory, and a variety of multi-reference [MR-SDCI, MR-RSPT2, MR-RSPT3] theories. The influence of the basis set on the computed momentum distributions has been investigated further, using a variety of basis sets ranging from 6-31G to the almost complete d-aug-cc-pV6Z basis set. A main issue in the present work pertains to a shake-up band of very weak intensity at 27.1 eV, of which the related momentum distribution was analyzed for the first time. The experimental evidences and the most thorough theoretical calculations demonstrate that this band borrows its ionization intensity from the 2a₁ orbital.     

Study on the maximum accuracy of the pseudopotential density functional method with localized atomic orbitals versus plane-wave basis sets

A detailed study on the accuracy attainable with numerical atomic orbitals in the context of pseudopotential first-principles density functional theory is presented. Dimers of first- and second-row elements are analyzed: bond lengths, atomization energies, and Kohn-Sham eigenvalue spectra obtained with localized orbitals and with plane-wave basis sets are compared. For each dimer, the cutoff radius, the shape, and the number of the atomic basis orbitals are varied in order to maximize the accuracy of the calculations. Optimized atomic orbitals are obtained following two routes: (i) maximization of the projection of plane wave results into atomic orbital basis sets and (ii) minimization of the total energy with respect to a set of primitive atomic orbitals as implemented in the OPENMX software package. It is found that by optimizing the numerical basis, chemical accuracy can be obtained even with a small set of orbitals.     

A computational study on CunN0,±1 (n=1–4) clusters by density functional methods

Clusters of the type Cu_nN^0,±1 (n=1–4) are investigated computationally using density functional theory methods. Equilibrium geometries are optimized under the constraint of well-defined point-group symmetries at the B3LYP level employing a pseudo-potential method in conjunction with double-zeta basis sets. In this article, different molecular properties such as total energies, electron affinities, ionization potentials, fragmentation energies and equilibrium geometries of the Cu_nN^0,±1 (n=1–4) clusters are systematically calculated and discussed. In particular, the photoelectron spectra of the anionic Cu_nN⁻¹ (n=2–4) clusters are calculated showing a good agreement with the available experimental results. In addition, Mulliken and natural orbital population analyses, and natural orbital configurations are calculated in order to elucidate the charge distributions in the clusters.     

A Systemic DFT Study on Several 5<Emphasis Type="Italic">d</Emphasis>-Electron Element Dimers: Hf<Subscript>2</Subscript>, Ta<Subscript>2</Subscript>, Re<Subscript>2</Subscript>, W<Subscript>2</Subscript>, and Hg<Subscript>2</Subscript>

Eleven kinds of density functionals in conjunction with three different basis sets are employed to investigate the homonuclear 5d-electron dimers: Hf₂, Ta₂, Re₂, W₂ and Hg₂. The computed bond lengths, vibrational frequencies and dissociation energies of these molecules are used to compare with available experimental data to find the appropriate combination of functional and basis set. The different functionals and basis sets favor different ground electronic state for Hf₂ and Re₂ molecules, indicating that these two dimers are sensitive to the functionals used. The molecular properties of Hg₂ dimer depend strongly on both functionals and basis sets used. It is found that the BP86 and PBEPBE functionals are generally successful in describing the 5d-electron dimers. For the ground states of these dimers, the bonding patterns are determined by natural bond orbital (NBO) analysis. Natural electron configurations show that the 6s and 5d orbitals in the bonding atoms hybrid with each other for the studied dimers except for Hg₂.     

Study on the behavior of energy convergence in ab initio crystal orbital calculations

Summary This article studies the dependence on the cutoff scheme of ab initio crystal orbital calculations with no long-range correction. We have thoroughly studied the Namur cutoff and cell-wise cutoff schemes through calculations of polyethylene and LiH chains. The Namur cutoff gives the fastest energy convergence with respect to the number of neighbors (N ₀). The energy convergence behavior with respect to N ₀ depends on the basis set. The Namur cutoff shows the fastest convergence with the STO-3G basis set, intermediate convergence with the MINI basis set, and the slowest convergence with the (7s4p/3s) basis set. The cell-wise cutoff shows exactly the reverse order of the Namur cutoff. The Namur cutoff destroys the translational symmetry. Both the Namur cutoff and cell-wise cutoff schemes introduce slight asymmetry on the two equivalent C-C bonds of polyethylene when calculating with a C₂H₄ unit cell. The asymmetry with the Namur cutoff can be made to disappear by increasing N ₀ a little. The calculations on two different unit-cell structures of trans-polyacetylene show the effect of the cutoff scheme on the total energy. Only the symmetric cutoff energies are the same. Disagreement related to the Namur cutoff disappears at N ₀ = 20, however, that related to the cell-wise and modified symmetric cutoff schemes remains at N ₀ 20. The optimized geometry and vibrational frequency are not as sensitive to the cutoff method except with the symmetric cutoff. A compilation of all results shows that the Namur cutoff is the superior cutoff scheme when calculating the insulator using the minimal basis set, especially the STO-3G basis set.     

An improved generator coordinate Hartree–Fock method applied to the choice of contracted Gaussian basis sets for first‐row diatomic molecules

Accurate Gaussian basis sets (18s for Li and Be and 20s11p for the atoms from B to Ne) for the first‐row atoms, generated with an improved generator coordinate Hartree–Fock method, were contracted and enriched with polarization functions. These basis sets were tested for B₂, C₂, BeO, CN⁻, LiF, N₂, CO, BF, NO⁺, O₂, and F₂. At the Hartree–Fock (HP), second‐order Møller–Plesset (MP2), fourth‐order Møller–Plesset (MP4), and density functional theory (DFT) levels, the dipole moments, bond lengths, and harmonic vibrational frequencies were studied, and at the MP2, MP4, and DFT levels, the dissociation energies were evaluated and compared with the corresponding experimental values and with values obtained using other contracted Gaussian basis sets and numerical HF calculations. For all diatomic molecules studied, the differences between our total energies, obtained with the largest contracted basis set [6s5p3d1f], and those calculated with the numerical HF methods were always less than 3.2 mhartree. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 15–23, 2000     

Thermochemical data from ab initio calculations. I. Estimation of SCF energies for augmented basis sets and Hartree–Fock limit energies

A procedure is outlined which allows an estimation of molecular energies both for a finite basis set including polarization functions and for the Hartree–Fock limit. It is shown that the orbital error of a given minimal basis is covered to a certain relatively constant percentage by an augmented basis set calculation. Thus an improvement factor Q_av can be determined by analyzing the corresponding results of small molecules where reasonable estimates of HF limit energies can be taken from the literature. For a combination of Pople's STO -3G and 6-31G* basis sets Q_av turns out to be 0.955.     

Ab initio and density functional theory study of the enthalpies of formation of F2SOx and FClSOx (x=1, 2)

Enthalpies of formation of F₂SO, F₂SO₂, FClSO and FClSO₂ molecules have been determined using ab initio molecular orbital theory and density functional theory (DFT) calculations. Different DFT approaches and levels of the Gaussian-3 and the complete basis set (CBS) ab initio model chemistries have been employed to calculate enthalpies of formation from both total atomization energies and isodesmic reaction schemes. The best values at 298 K for F₂SO, F₂SO₂, FClSO and FClSO₂ as derived from an average of G3, G3B3, CBS-Q and CBS-QB3 isodesmic energies are −140.6, −181.1, −92.6 and −132.3 kcal mol⁻¹, respectively. The results obtained suggest that the accumulated small component errors found in the DFT-based methods are significantly reduced at the ab initio levels employed. Structural properties, harmonic vibrational frequencies, mode assignations and infrared intensities derived from B3LYP and mPW1PW91 functional with the 6-311+G(3df) basis set are presented.     

Performance of TDDFT Vertical Excitation Energies of Core-Substituted Naphthalene Diimides

We have evaluated the performance of various density functionals, covering generalized gradient approximation (GGA), global hybrid (GH) and range-separated hybrid (RSH), using time dependent density functional theory (TDDFT) for computing vertical excitation energies against experimental absorption maximum (λ_max) for a set of 10 different core-substituted naphthalene diimides (cNDI) recorded in dichloromethane. The computed excitation in case of GH PBE0 is most accurate while the trend is most systematic with RSH LCY-BLYP compared to λ_max. We highlight the importance of including solvent effects for optimal agreement with the λ_max. Increasing the basis set size from TZ2P to QZ4P has a negligible influence on the computed excitation energies. Notably, RSH CAMY-B3LYP gave the least error for charge-transfer excitation. The poorest agreement with λ_max is obtained with semi-local GGA functionals. Use of the optimally-tuned RSH LCY-BLYP* is not recommended because of the high computational cost and marginal improvement in results.     

A note on the importance of d orbitals in the calculation of effective interactions and the excitation spectra of atoms and molecules

The focal point of our discussion is the examination of truncated basis sets used in obtaining an accurate first principles clculation of the effective valence shell Hamiltonian by the canonical transformation-cluster expansion approasch. Subsequent diagonalization of this effecitve valence shell hamiltonian yields the valence shell transition energies. A detailed analysis of numerical results obtained using a number of different basis sets of hydrogen-like orbitals together with rigorous symmetry arguments celarly demonstrates the special role played by d orbitals in computing the ³P → ¹D transition energy in carbon. The failure of early attempts to calculate the effective Hamiltonian for ethylene from first principles is examined in the light of recent ab initio calculations on ethylene involving d orbitals and the computations reported in this paper. We conclude that accurate calculations of the effective valence shell Hamiltonian for molecules must consider d orbitals in the excited orbital basis set.     

s-四嗪-水簇复合物的理论研究

用量子化学B3LYP方法和6-31＋＋G**基函数研究了s-四嗪-水簇复合物基态分子间相互作用, 并进行了构型优化和频率计算, 分别得到无虚频稳定的s-四嗪-(水)₂复合物、s-四嗪-(水)₃复合物和s-四嗪-(水)₄复合物6个、9个和12个. 复合物存在较强的氢键作用, 复合物结构中形成一个N…H—O氢键并终止于O…H—C氢键的氢键水链构型最稳定. 经基组重叠误差和零点振动能校正后, 最稳定的1∶2, 1∶3和1∶4(摩尔比)复合物的结合能分别是41.35, 70.9和 94.61 kJ/mol. 振动分析显示氢键的形成使复合物中水分子H—O键对称伸缩振动频率减小(红移). 研究表明N…H键越短, N…H—O键角越接近直线, 稳定化能越大, 氢键作用越强. 同时, 用含时密度泛函理论方法在TD-B3LYP/6-31＋＋G**水平计算了s-四嗪单体及其氢键复合物的第一¹(n, p*)激发态的垂直激发能.