On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies

The leading cause of error in standard coupled cluster theory calculations of thermodynamic properties such as atomization energies and heats of formation originates with the truncation of the one-particle basis set expansion. Unfortunately, the use of finite basis sets is currently a computational necessity. Even with basis sets of quadruple zeta quality, errors can easily exceed 8 kcal/mol in small molecules, rendering the results of little practical use. Attempts to address this serious problem have led to a wide variety of proposals for simple complete basis set extrapolation formulas that exploit the regularity in the correlation consistent sequence of basis sets. This study explores the effectiveness of six formulas for reproducing the complete basis set limit. The W4 approach was also examined, although in lesser detail. Reference atomization energies were obtained from standard coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) calculations involving basis sets of 6ζ or better quality for a collection of 141 molecules. In addition, a subset of 51 atomization energies was treated with explicitly correlated CCSD(T)-F12b calculations and very large basis sets. Of the formulas considered, all proved reliable at reducing the one-particle expansion error. Even the least effective formulas cut the error in the raw values by more than half, a feat requiring a much larger basis set without the aid of extrapolation. The most effective formulas cut the mean absolute deviation by a further factor of two. Careful examination of the complete body of statistics failed to reveal a single choice that out performed the others for all basis set combinations and all classes of molecules.     

Local orbital eigenvalues and basis set balance

Consideration of the chemical potential of an electron in a wavefunction suggests that a quantity called the local orbital eigenvalue and its variation in space provides a method of testing the balance of a basis set as a function of spatial position. The Hartree-Fock method as applied to the helium and neon atoms is used as an example.     

Basis set convergence of post-CCSD contributions to molecular atomization energies

Basis set convergence of correlation effects on molecular atomization energies beyond the coupled cluster with singles and doubles (CCSD) approximation has been studied near the one-particle basis set limit. Quasiperturbative connected triple excitations, (T), converge more rapidly than L(-3) (where L is the highest angular momentum represented in the basis set), while higher-order connected triples, T3-(T), converge more slowly--empirically, proportional to L(-5/2). Quasiperturbative connected quadruple excitations, (Q), converge smoothly as proportional to L(-3) starting with the cc-pVTZ basis set, while the cc-pVDZ basis set causes overshooting of the contribution in highly polar systems. Higher-order connected quadruples display only weak, but somewhat erratic, basis set dependence. Connected quintuple excitations converge very rapidly with the basis set, to the point where even an unpolarized double-zeta basis set yields useful numbers. In cases where fully iterative coupled cluster up to connected quintuples (CCSDTQ5) calculations are not an option, CCSDTQ(5) (i.e., coupled cluster up to connected quadruples plus a quasiperturbative connected quintuples correction) cannot be relied upon in the presence of significant nondynamical correlation, whereas CCSDTQ(5)(Lambda) represents a viable alternative. Connected quadruples corrections to the core-valence contribution are thermochemically significant in some systems. We propose an additional variant of W4 theory [A. Karton et al., J. Chem. Phys. 125, 144108 (2006)], denoted W4.4 theory, which is shown to yield a rms deviation from experimental atomization energies (active thermochemical tables, ATcT) of only 0.05 kcal/mol for systems for which ATcT values are available. We conclude that "3sigma 相似文献   

The sensitivity of B3LYP atomization energies to the basis set and a comparison of basis set requirements for CCSD(T) and B3LYP

The atomization energies of the 55 G2 molecules are computed using the B3LYP approach with a variety of basis sets. The 6–311 + G(3df) basis set is found to yield superior results to those obtained using the augumented-correlation-consistent valence-polarized triple-zeta set. The atomization energy of SO₂ is found to be the most sensitive to basis set and is studied in detail. Including tight d functions is found to be important for obtaining good atomization energies. The results for SO₂ are compared with those obtained using the coupled-cluster singles and doubles approach including a perturbational estimate of the triple excitations.     

Counterpoise orbital basis in SCF computations of conformational energies of molecules

The functional counterpoise idea of Boys and Bernardi is extended to SCF computations, to improve the accuracy of computed values of conformational energies of molecules. It consists of using the same orbital basis to obtain the SCF energies for different molecular conformations. The procedure is found to be effective for two examples discussed in the paper.     

Effects of contraction and reduction of basis set size on the He2 interaction energy components

The effect of basis set contraction and elimination of primitive Gaussian orbitals on the He₂ interaction energy components have been studied within the SCF counterpoise corrected approach supplemented by a dispersion term calculated within the variation-perturbation scheme. Despite elimination of almost half of the primitive Gaussian functions from the saturated sp basis set and complete contraction of the remaining ones, the components of interaction energy in He₂ suffer a remarkably small loss of accuracy except for the short range charge transfer contribution.     

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.     

Extrapolation of electron correlation energies to finite and complete basis set targets

The electron correlation energy of two-electron atoms is known to converge asymptotically as approximately (L+1)(-3) to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence approximately X(-3) with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X(-beta) that for an effective exponent beta=beta(eff)(X,X+1,X+N) provides the correct energy E(X+N), when extrapolating from results for two smaller basis sets, E(X) and E(X+1). This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations. beta(eff) is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent beta(opt)(X,X+1,infinity) for the entire set of molecules, and compared to similar approaches reported in the literature.     

The effect of basis set superposition error on the convergence of interaction energies

 For the intermolecular interaction energies of ion-water clusters [OH⁻(H₂O) _n (n=1,2), F⁻(H₂O), Cl⁻(H₂O), H₃O⁺(H₂O) _n (n=1,2), and NH₄ ⁺(H₂O) _n (n=1,2)] calculated with correlation-consistent basis sets at MP2, MP4, QCISD(T), and CCSD(T) levels, the basis set superposition error is nearly zero in the complete basis set (CBS) limit. That is, the counterpoise-uncorrected intermolecular interaction energies are nearly equal to the counterpoise-corrected intermolecular interaction energies in the CBS limit. When the basis set is smaller, the counterpoise-uncorrected intermolecular interaction energies are more reliable than the counterpoise-corrected intermolecular interaction energies. The counterpoise-uncorrected intermolecular interaction energies evaluated using the MP2/aug-cc-pVDZ level is reliable. Received: 14 March 2001 / Accepted: 25 April 2001 / Published online: 9 August 2001     

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.     

Revisiting the extrapolation of correlation energies to complete basis set limit

The extrapolation scheme of correlation energy is revisited to evaluate the complete basis set limit from double‐zeta (DZ) and triple‐zeta levels of calculations. The DZ level results are adjusted to the standard asymptotic behavior with respect to the cardinal number, observed at the higher levels of basis sets. Two types of adjusting schemes with effective scaling factors, which recover errors in extrapolations with the DZ level basis set, are examined. The first scheme scales the cardinal number for the DZ level energy, while the second scheme scales the prefactor of the extrapolation function. Systematic assessments on the Gaussian‐3X and Gaussian‐2 test sets reveal that these calibration schemes successfully and drastically reduce errors without additional computational efforts. © 2015 Wiley Periodicals, Inc.     

A theoretical study of the photoelectron spectra of dichloroketene with accurate computation of ionization energies via complete basis set limit extrapolation

We calculated the equilibrium geometries and harmonic vibrational frequencies of the ground state and five cationic states of dichloroketene using (TD-)B3LYP, PBE0, and M06/M06-2X approaches. The photoelectron spectra of dichloroketene were simulated by computing Franck-Condon factors. The ionization energies were computed using the CCSD(T) approach with extrapolation to the complete basis set (CBS) limit. We propose two new CBS energy formulas (E = E_CBS + Aexp(-x) + B/(x−1) ⁿ, n = 2 or 3) and compare the performance of different CBS approaches. A new ionic state of dichloroketene belonging to the C_s point group is reported. This state is identified as the first excited state of Cl₂CCO⁺ having a double-well potential-energy curve along the CCO bending mode with a barrier height of 1.335 eV. The simulated photoelectron spectra are in agreement with the experiment. The vertical ionization energies calculated via spectral simulation are more accurate compared with those obtained at the ground-state structure. Among the CBS formulas used, the proposed ansatz with n = 2 performs best, with a mean absolute error of 0.021 and 0.012 eV for the adiabatic and vertical ionization energies, respectively.     

Basis set effects on frontier molecular orbital energies and energy gaps: a comparative study between plane waves and localized basis functions in molecular systems

In order to study the Kohn-Sham frontier molecular orbital energies in the complete basis limit, a comparative study between localized functions and plane waves, obtained with the local density approximation exchange-correlation functional is made. The analyzed systems are ethylene and butadiene, since they are theoretical and experimentally well characterized. The localized basis sets used are those developed by Dunning. For the plane-waves method, the pseudopotential approximation is employed. The results obtained by the localized basis sets suggest that it is possible to get an estimation of the orbital energies in the limit of the complete basis set, when the basis set size is large. It is shown that the frontier molecular orbital energies and the energy gaps obtained with plane waves are similar to those obtained with a large localized basis set, when the size of the supercell and the plane-wave expansion have been appropriately calibrated.     

The basis set dependence of structures and energies of various states of cyclodisiloxane

Several common basis sets, ranging from minimal to double-zeta, are applied to study the neutral singlet and triplet as well as positive- and negative-ion doublet states of cyclodisiloxane. The effect of d-polarization function exponents on the equilibrium geometries and energies is analyzed. The d-type functions seem to be essential in the basis set of silicon, whereas their presence on oxygen is less critical. The optimum exponents (with respect to SCF energy) are determined to be 0.45 for Si and 0.60 for O, very close to those recommended for the 6–31G^** basis set. The best structural predictions are obtained with the 6–31G(2d, p) basis set, which contains two sets of d functions on the heavy atoms. The predicted Si? O bond length is 166 pm; the Si? Si and O? O distances are 237 and 232 pm, respectively, which correspond to an O—Si? O angle of 88.6°. The ground state is found to be a singlet. All higher states have longer Si? O bonds and Si—Si distances, whereas O—O distances are shorter. The energy separation between the singlet and other states is modified by electron correlation (MP treatment) by only a few kcal/mol.     

A contracted bromine basis set for use in calculation of molecular energies

A contracted [9s6p2d] basis set derived from Dunning's (14s11p5d) primitive Gaussian set for bromine has been used in ab initio molecular orbital calculations of the dissociation energies of HBr, CH₃Br, and Br₂, the ionization potentials of Br and HBr, and the electron affinity of Br. The calculated energies are within 0.1 eV of the experimental values. This is similar to the accuracy obtained in a previous study, also using a contracted [9s6p2d] basis set, of the dissociation and ionization energies of the GeH_n, AsH_n, and SeH_n hydrides.     

Comparative theoretical investigation of the vertical excitation energies and the electronic structure of [MoVOCl4]-: influence of basis set and geometry

The electronic structures, geometries, and vibration frequencies of the open-shell molybdenum(V) ion, [MoOCl(4)](-), have been calculated at the extended Hückel, semiempirical ZINDO/1, ZINDO/S, and PM3(tm), as well as ab initio and DFT theoretical levels. Electronic structure calculations suggest that the expected metal-fold orbital order can be satisfied only at the DFT level. The time-dependent density functional theory (TDDFT) approach has been used for the calculation of the vertical excitation energies in the UV-vis region with different basis sets, starting geometries, and exchange-correlation functionals. A good agreement between the predicted and the experimental electronic absorption and MCD spectra of the complex, [MoOCl(4)](-), was observed when the B3LYP and B3P86 exchange-correlation functionals were used with a full electron valence double-zeta with polarization basis set for the molybdenum and 6-311G(d) for all other atoms. Similar results were obtained when the LANL2DZ effective core potential for molybdenum atom and 6-31G(d) for all other atoms were used. The best absolute deviation of 0.13 and mean deviation of 0.01 eV were calculated for the bands in the UV-vis region by B3P86, while the results for the B3LYP exchange-correlation functional were less satisfactory. Compared to polarization functions, the inclusion of diffuse functions resulted in little improvement. The calculated excitations energies and charge-transfer band intensities are found to be sensitive to the Mo=O distance and O-Mo-Cl angle.     

π和σ体系彻底分离的高度定域的键轨道基组的建立

高度定域的、对称的、键轨道基组的建立是一个多步的计算程序：(1)以定域片断轨道[φk,φi,φj]为基,对分子作有条件的RHF运算,算得FUL和DSI°态的片断分子轨道[Φ°l',Φ°n,Φ°m]和[Φl,Φn,Φm]。在基组[φk,φi,φj]中,φi∈双占据和空σ片断分子轨道(FMOs)组,φj∈πFMO组,φk∈单占据σFMO组,它们都精确地定域在各自的片断内;(2)利用Φ°l'与Φ°l间的重叠积分值(Sl'l>0.5),可以从DSI°态中,自动地选出Ns个对称的、由单占据轨道线性组合而成的分子轨道Φ°l'=Σakl'φk(k=1,2,…,Ns),接着,用Φ°l'取代FUL态中同类的、非对称轨道组Φl=Σaklφk(k=1,2,…,Ns);(3)以上述新的轨道组[Φ°l',Φn,Φm]为基(其中,Φ°l'∈DSI°态,它们离域于整个分子;双占据及空σFMO组Φm和πFMO组Φm属于FUL态),按FUL态的条件,再次对分子作有条件的RHF运算,从中得到一组对称的、闭壳层正则FMOs,而且每一个FMO均有正确的电子占据数;(4)利用Perkin原理,将第3步所得的正则FMO组定域成一个对称的键轨道基组[Φl',Φn',Φm']。在这个基组中,π体系Φm'与σ构架Φn'是彻底分离的,而且这两个轨道组始终精确地定域在各自的片断内。     

Vibrational energies of PH3 calculated variationally at the complete basis set limit

The potential energy surface for the electronic ground state of PH(3) was calculated at the CCSD(T) level using aug-cc-pV(Q+d)Z and aug-cc-pVQZ basis sets for P and H, respectively, with scalar relativistic corrections included. A parametrized function was fitted through these ab initio points, and one parameter of this function was empirically adjusted. This analytical PES was employed in variational calculations of vibrational energies with the newly developed program TROVE. The convergence of the calculated vibrational energies with increasing vibrational basis set size was improved by means of an extrapolation scheme analogous to the complete basis set limit schemes used in ab initio electronic structure calculations. The resulting theoretical energy values are in excellent agreement with the available experimentally derived values.     

Ab initio molecular orbital calculation of fe-porphine with a double zeta basis set

An ab initio LCAO SCF MO calculation was performed on planar Fe-porphine with a double zeta basis set consisting of 300 CGTO 's. SCF wave functions of several states of Fe-porphine and its cation were obtained. The net charge of Fe is in the range of 1.39 to 1.53. The highest occupied orbital is ascertained to be a pure porphine π-MO , 1a_1u. The calculated ionization potentials of the two highest occupied orbitals, 1a_1u and 5a_2u are 5.98 and 6.43 eV, respectively. They are in good agreement with experiments. The role of the porphine macrocycle on the oxidation of Fe is discussed in terms of gross atomic populations and with contour maps of the density difference.