Estimating the Hartree—Fock limit from finite basis set calculations

It is demonstrated that the polarization-consistent basis sets, which are optimized for density functional methods, are also suitable for Hartree–Fock calculations, and can be used for estimating the Hartree–Fock basis set limit to within a few micro-hartree accuracy. Various two- and three-point extrapolation schemes are tested and exponential functions are found to be superior compared to functions depending on the inverse power of the highest angular momentum function in the basis set. Total energies can be improved by roughly an order of magnitude, but atomization energies are only marginally improved by extrapolation.     

A detailed test study of barrier heights for the HO2 + H2O + O3 reaction with various forms of multireference perturbation theory

We report an ab initio multireference perturbation theory investigation of the HO(2) + H(2)O + O(3) reaction, with particular emphasis on the barrier heights for two possible reaction mechanisms: oxygen abstraction and hydrogen abstraction, which are identified by two distinct saddle points. These saddle points and the corresponding pre-reactive complexes were optimized at the CASSCF(11,11) level while the single point energies were calculated with three different MRPT2 theories: MRMP, CASPT2, and SC-NEVPT2. Special attention has been drawn on the "intruder state" problem and the effect of its corrections on the relative energies. The results were then compared with single reference coupled-cluster methods and also with our recently obtained Kohn-Sham density functional theory (KS-DFT) calculations [L. P. Viegas and A. J. C. Varandas, Chem. Phys., (2011)]. It is found that the relative energies of the pre-reactive complexes have a very good agreement while the MRPT2 classical barrier heights are considerably higher than the KS-DFT ones, with the SC-NEVPT2 calculations having the highest energies between the MRPT2 methods. Possible explanations have been given to account for these differences.     

Computational methods in organic thermochemistry. 1. Hydrocarbon enthalpies and free energies of formation

Standard state enthalpies and free energies of formation can be computed with reasonable accuracy (usually within 4 and often 2 kJ/mol) using high level model chemistries. A comparison set of nearly 300 organic compounds ranging from 1 to 10 carbon atoms having a variety of functional groups for which enthalpy and free energy literature values are available has been examined using G2, G2MP2, G3, G3MP2, G3B3, G3MP2B3, CBS-QB3, and density functional (B3LYP/6-311+G(3df,2p)) model chemistries. G3 gives an average mean absolute deviation of 3.0 and 13.4 kJ/mol for the enthalpies and free energies, respectively, using the atomization method and 3.1 and 3.7 kJ/mol when bond separation reactions are employed. G3 and G3B3 are the most accurate overall; the related G3MP2 and G3MP2B3 are nearly as accurate and can compute larger molecules. CBS-QB3 was also found to be accurate but is more limited in the size of molecules that can be computed. The density functional energies were found to have large deviations from the literature values using either the atomization or the bond separation method. Regardless of the model employed, the free energies are increasingly underestimated by computation as the size of the molecule increases. A series of corrections applied to the aliphatic hydrocarbons is presented, which usually reduces the deviations to less than 4 kJ/mol regardless of the size of the molecule.     

Energy correctors for accurate prediction of molecular energies

Energy correctors are introduced for the calculation of molecular energies of compounds containing first row atoms (Li-F) to modify ab initio molecular orbital calculations of energies to better reproduce experimental results. Four additive correctors are introduced to compensate for the differences in the treatment of molecules with different spin multiplicities and multiplicative correctors are also calculated for the electronic and zero-point vibrational energies. These correctors, individually and collectively yield striking improvements in the atomization energies for several ab initio methods. We use as training set the first row subset of molecules from the G1 basis of molecules; when the correctors are applied to other molecules not included in the training set, selected from the G3 basis, similar improvements in the atomization energies are obtained. The special case of the B3PW91/cc-pVTZ yields an average error of 1.2 kcal/mol, which is already within a chemical accuracy and comparable to the Gaussian-n theories accuracy. The very inexpensive B3PW91/6-31G** yields an average error of 2.1 kcal/mol using the correctors. Methods considered unsuitable for energetics such as HF and LSDA yield corrected energies comparable to those obtained with the best highly correlated methods.     

Databases for transition element bonding: metal-metal bond energies and bond lengths and their use to test hybrid, hybrid meta, and meta density functionals and generalized gradient approximations

We propose a data set of bond lengths for 8 selected transition metal dimers (Ag(2), Cr(2), Cu(2), CuAg, Mo(2), Ni(2), V(2), and Zr(2)) and another data set containing their atomization energies and the atomization energy of ZrV, and we use these for testing density functional theory. The molecules chosen for the test sets were selected on the basis of the expected reliability of the data and their ability to constitute a diverse and representative set of transition element bond types while the data sets are kept small enough to allow for efficient testing of a large number of computational methods against a very reliable subset of experimental data. In this paper we test 42 different functionals: 2 local spin density approximation (LSDA) functionals, 12 generalized gradient approximation (GGA) methods, 13 hybrid GGAs, 7 meta GGA methods, and 8 hybrid meta GGAs. We find that GGA density functionals are more accurate for the atomization energies of pure transition metal systems than are their meta, hybrid, or hybrid meta analogues. We find that the errors for atomization energies and bond lengths are not as large if we limit ourselves to dimers with small amounts of multireference character. We also demonstrate the effects of increasing the fraction of Hartree-Fock exchange in multireference systems by computing the potential energy curve for Cr(2) and Mo(2) with several functionals. We also find that BLYP is the most accurate functional for bond energies and is reasonably accurate for bond lengths. The methods that work well for transition metal bonds are found to be quite different from those that work well for organic and other main group chemistry.     

Effect of the calculation method and the basis set on the structure and electrical properties of (4,4) carbon nanotubes with different lengths and open ends

The structure and electrical properties of open carbon nanotube with chirality (4,4), consisting of 5-15 segments, are calculated within four quantum chemical models: AM1, PM3, LSDA/3-21G*, and B3LYP/6-31G. Size effects and the effect of the model choice on the geometry, energy, enthalpy and Gibbs energy of the formation (atomization), Mulliken atomic charges, polarizability, and predicted adsorption properties of nanotubes are discussed.     

Simple energy corrections for precise atomization energies of CHON molecules

We test a few ways to improve atomization energies of CHON molecules and found that the best way to do it is simply by correcting the atom energies of the participating atoms. Extraordinary improvement on the average errors is obtained. For the HF/6-31G** level of theory an average error of 271.2 kcal/mol on 115 molecules is improved to 6.7 kcal/mol simply by correcting the atomic energy of the four chon atoms. The corrections to density functional methods allow us to reach a chemical accuracy of 2 kcal/mol.     

Density functionals for inorganometallic and organometallic chemistry

We present a database of 21 bond dissociation energies for breaking metal-ligand bonds. The molecules in the metal-ligand bond energy database are AgH, CoH, CoO+, CoOH+, CrCH3+, CuOH2+, FeH, Fe(CO)5, FeO, FeS, LiCl, LiO, MgO, MnCH3NiCH2+, Ni(CO)4, RhC, VCO+, VO, and VS. We have also created databases of metal-ligand bond lengths and atomic ionization potentials. The molecules used for bond lengths are AgH, BeO, CoH, CoO+, FeH, FeO, FeS, LiCl, LiO, MgO, RhC, VO, and VS and the ionization potentials are for the following atoms: C, Co, Cr, Cu, Ni, O, and V. The data were chosen based on their diversity and expected reliability, and they are used along with three previously developed databases (transition metal dimer bond energies and bond lengths and main-group molecular atomization energies) for assessing the accuracy of several kinds of density functionals. In particular, we report tests for 42 previously defined functionals: 2 local spin density approximation (LSDA) functionals, 14 generalized gradient approximation (GGA) methods, 13 hybrid GGA methods, 7 meta GGA methods, and 8 hybrid meta GGA methods. In addition to these functionals, we also examine the effectiveness of scaling the correlation energy by testing 13 functionals with scaled or no gradient-corrected correlation energy, and we find that functionals of this kind are more accurate for metal-metal and metal-ligand bonds than any of the functionals already in the literature. We also present a readjusted GGA and a hybrid GGA with parameters adjusted for metals. When we consider these 57 functionals for metal-ligand and metal-metal bond energies simultaneously with main-group atomization energies, atomic ionization potentials, and bond lengths we find that the most accurate functional is G96LYP, followed closely by MPWLYP1M (new in this article), XLYP, BLYP, and MOHLYP (also new in this article). Four of these five functionals have no Hartree-Fock exchange, and the other has only 5%. As a byproduct of this work we introduce a convenient diagnostic, called the B1 diagnostic, for ascertaining the multireference character in a bond.     

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.     

Theoretical studies of molecular structures and properties of platinum (II) antitumor drugs

The molecular structure and vibration spectra of carboplatin were investigated by the different density functional models (mPW1PW, BPV86, HCTH, PBEPBE, LSDA and PBE1PBE) using several basis sets including LANL2DZ, SDD, LANL2MB, CEP-4G, CEP-31G and CEP-121G. The results indicate that LSDA/SDD and LSDA/LANL2DZ levels are clearly superior to all the remaining density functional methods in predicting the structures of carboplatin. Mean absolute deviation between the calculated harmonic and observed fundamental vibration frequencies for each method indicates that PBE1PBE/CEP-121G and PBE1PBE/SDD methods are sufficient to predict vibration spectrum of carboplatin comparing with other DFT methods.     

Approaching chemical accuracy with quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.     

The limitations of Slater's element-dependent exchange functional from analytic density-functional theory

Our recent formulation of the analytic and variational Slater-Roothaan (SR) method, which uses Gaussian basis sets to variationally express the molecular orbitals, electron density, and the one-body effective potential of density-functional theory, is reviewed. Variational fitting can be extended to the resolution of identity method, where variationality then refers to the error in each two-electron integral and not to the total energy. However, a Taylor-series analysis shows that all analytic ab initio energies calculated with variational fits to two-electron integrals are stationary. It is proposed that the appropriate fitting functions be charge neutral and that all ab initio energies be evaluated using two-center fits of the two-electron integrals. The SR method has its root in Slater's Xalpha method and permits an arbitrary scaling of the Slater-Gàspàr-Kohn-Sham exchange-correlation potential around each atom in the system. The scaling factors are Slater's exchange parameters alpha. Of several ways of choosing these parameters, two most obvious are the Hartree-Fock (HF) alpha(HF) values and the exact atomic alpha(EA) values. The former are obtained by equating the self-consistent Xalpha energy and the HF energies, while the latter set reproduces exact atomic energies. In this work, we examine the performance of the SR method for predicting atomization energies, bond distances, and ionization potentials using the two sets of alpha parameters. The atomization energies are calculated for the extended G2 set of 148 molecules for different basis-set combinations. The mean error (ME) and mean absolute error (MAE) in atomization energies are about 25 and 33 kcal/mol, respectively, for the exact atomic alpha(EA) values. The HF values of exchange parameters alpha(HF) give somewhat better performance for the atomization energies with ME and MAE being about 15 and 26 kcal/mol, respectively. While both sets give performance better than the local-density approximation or the HF theory, the errors in atomization energy are larger than the target chemical accuracy. To further improve the performance of the SR method for atomization energies, a new set of alpha values is determined by minimizing the MAE in atomization energies of 148 molecules. This new set gives atomization energies half as large (MAE approximately 14.5 kcal/mol) and that are slightly better than those obtained by one of the most widely used generalized-gradient approximations. Further improvements in atomization energies require going beyond Slater's functional form for exchange employed in this work to allow exchange-correlation interactions between electrons of different spins. The MAE in ionization potentials of 49 atoms and molecules is about 0.5 eV and that in bond distances of 27 molecules is about 0.02 A. The overall good performance of the computationally efficient SR method using any reasonable set of alpha values makes it a promising method for study of large systems.     

Hydrogen-bonding interaction of 5,6-dihydrouracil with RNA bases: DFT and MP2 studies

5,6-Dihydrouracil (DHU) is a rare pyrimidine base naturally occurring in tRNAs, it differs from the base uracil due to the saturation of the C₅–C₆ bond. This work presents the interaction energies of complexes formation involving DHU bound to the natural RNA bases adenine (A), uracil (U), guanine (G), and cytosine (C). Full geometry optimization has been performed for the studied complexes by B3LYP/6-31+G(d,p) and MP2/6-31+G(d,p) calculations. The interaction energies were corrected for the basis-set superposition error (BSSE), using the full Boys–Bernardi counterpoise correction scheme. We find that the stability order is DHU:G > DHU:A > DHU:C  DHU:U.     

Density functional study on the structure and stability of positive iron rare-gas complexes, FeXn (X=Ar, Xe; n=1–6)

The structures and atomization energies of positively charged complexes of iron with argon and xenon, Fe⁺X_n (X=Ar, Xe; n=1–6) are investigated by density functional theory calculations. We explain the special stability of some of these complexes (“magic numbers”) – that has been observed in previous laser ablation and multi-photon ionization experiments – and predict their geometries.     

Diminished gradient dependence of density functionals: constraint satisfaction and self-interaction correction

The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation for the exchange-correlation energy functional has two nonempirical constructions, based on satisfaction of universal exact constraints on the hole density or on the energy. We show here that, by identifying one possible free parameter in exchange and a second in correlation, we can continue to satisfy these constraints while diminishing the gradient dependence almost to zero (i.e., almost recovering the local spin density approximation or LSDA). This points out the important role played by the Perdew-Wang 1991 nonempirical hole construction in shaping PBE and later constructions. Only the undiminished PBE is good for atoms and molecules, for reasons we present, but a somewhat diminished PBE could be useful for solids; in particular, the surface energies of solids could be improved. Even for atoms and molecules, a strongly diminished PBE works well when combined with a scaled-down self-interaction correction (although perhaps not significantly better than LSDA). This shows that the undiminished gradient dependence of PBE and related functionals works somewhat like a scaled-down self-interaction correction to LSDA.     

High levelab initio stabilization energies of benzene

Summary G2 theory is shown to be reliable for calculating isodesmic and homodesmotic stabilization energies (ISE and HSE, respectively) of benzene. G2 calculations give HSE and ISE values of 92.5 and 269.1 kJ mol^–1 (298 K), respectively. These agree well with the experimental HSE and ISE values of 90.5±7.2 and 268.7±6.3 kJ mol^–1, respectively. We conclude that basis set superposition error corrections to the enthalpies of the homodesmotic or isodesmic reactions are not necessary in calculations of the stabilization energies of benzene using G2 theory. The calculated values of the enthalpies of formation of such molecules containing multiple bonds such as benzene ands-trans 1,3-butadiene, which are found from the enthalpies of isodesmic and homodesmotic reactions rather than of atomization reactions, demonstrate good performance of G2 theory. Estimates of theH _f ^o value for benzene from the G2 calculated enthalpies of homodesmotic reaction (2) and isodesmic reaction (3) are 80.9 and 82.5 kJ mol^–1 (298 K), respectively. These are very close to the experimentalH _f ^o value of 82.9±0.3 kJ mol^–1. TheH _f ^o value ofs-trans 1,3-butadiene calculated using the G2 enthalpy of isodesmic reaction (4) is 110.5 kJ mol^–1 and is in excellent agreement with the experimentalH _f ^o value of 110.0±1.1 kJ mol^–1.     

Nonempirical calculations of potential-energy surfaces for nucleophilic addition of H− and F− to an acetylene molecule

The minimal energy paths for the nucleophilic addition of a hydride ion (H^–) and a fluoride ion (F^–) to a molecule of acetylene (A) have been calculated with the use of 3–21++G and 3–21+G double basis sets in the framework of the Hartree-Fock-Roothaan method. The values of the total energies of the reactants, transition states, and products have been refined by means of calculations with more complete basis sets [6–31++G// 3–21++G and 6–31++G*//3–21++G for reaction (1); 6–31+G*//3–21+G and 6–31++G**//3–21+G for reaction (2)] and by taking into account the correlation energy for reaction (1) in the framework of the SCEP/6–31++*//3–21++G method. It has been established that the activation energy of reaction (2) is 15.94 kJ/mole lower than that for reaction (1), that reaction (1) is exothermic, and that the enthalpy change accompanying reaction (2) is close to zero. The character of the distribution of the electron density along the minimal energy paths of both reactions has been analyzed, and the differences appearing as a result of the replacement of the soft nucleophile H^– by the harder nucleophile F^– have been ascertained. The results of the calculations have been compared with the results available in the literature for reaction (1).Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 2, pp. 149–155, March–April, 1989.     

Assessment of a composite method based on selected density functional theory methods and complete basis set extrapolation formulas

Extrapolation formulas based on power and exponential formulas, as well as alternatives from a Taylor series, were tested and used with density functional theory (DFT) for the calculation of enthalpies of formation. The following four functionals were analyzed: B3LYP, BMK, M06-2X, and B2PLYP. Preliminary tests pointed to B2PLYP and B3LYP as the best and worst functionals, respectively. Taylor series expressions were as accurate as the power formulas and presented better performance than the exponential equation. The power formula (Equation (2)) and one of the simplest Taylor expressions (Equation (13)) were selected for the calculations with B3LYP and B2PLYP, with further empirical adjustments based on the higher level correction (HLC) and scaling of the experimental atomization energies used to calculate enthalpies of formation. HLC improved the B3LYP mean absolute error (MAE) from approximately 4.3 to 3.5 kcal mol⁻¹ using both extrapolation alternatives. For B2PLY, the MAEs were improved from 2.7 to 2.6 kcal mol⁻¹. Regarding the G3/05 test set, a significant improvement in the MAEs around 2.5 and 1.5 kcal mol⁻¹ were achieved using B3LYP and B2PLYP, respectively. The accuracy obtained from these empirical corrections was equivalent to other composite methods. The MAEs from B3LYP and B2PLYP may be suggested as ranges for the possible accuracy to be achieved by some DFT methods. The empirical corrections suggested in this work are improvements that may be considered to provide acceptable accuracy for enthalpies of formation and possibly other properties.     

Design of low band gap polymers employing density functional theory—hybrid functionals ameliorate band gap problem

Band gaps in solids and excitation energies in finite systems are underestimated significantly if estimated from differences between eigenvalues obtained within the local spin density approximation (LSDA). In this article we present results on 20 small- and medium-sized π-systems which show that HOMO–LUMO energy differences obtained with the B3LYP, B3P86, and B3PW91 functionals are in good agreement with vertical excitation energies from UV-absorption spectra. The improvement is a result of the use of the exact Hartree–Fock exchange with hybrid methods. Negative HOMO energies and negative LUMO energies do not provide good estimates for IPs and EAs. In contrast to Hartree–Fock theory, where IPs are approximated well and EAs are given poorly, DFT hybrid methods underestimate IPs and EAs by about the same amount. LSDA yields reasonable EAs but poor IPs. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1943–1953, 1997