One-electron Basis Set Research Articles

Quantum-Mechanical Energy Calculations in Chemistry

Truncation of the one-electron basis set is, in general, the main source of error in nonempirical quantum-chemical energy calculations. Total-energy estimates for infinite basis sets are needed. We use an efficient extrapolation to zero in linear dependences between total energies of atoms (H, He, Ne, Ar) and molecules (H, H2, HF, HCl, H2O, CO, (Ne)2, (Ar)2, (HF)2, (H2O)2) and a reciprocal total number of basis-set functions; this is equivalent to the extrapolation to the infinite basis set. The procedure works satisfactorily when the coupled-cluster method [CCSD(T)] is employed with correlation-consistent polarized-valence (cc-PVXZ) basis sets. Energy changes rather than absolute energies play a role in molecular sciences. When calculating ΔE, attention has to be paid to associations of the chemical and van der Waals type, where basis set superposition error should be considered, even with an extensive basis set. The original function counterpoise method by Boys and Bernardi is applied. In chemistry and in the whole area of biodisciplines, it is Gibbs energy rather than energy which plays a key role. When statistical mechanics for obtaining Gibbs energy changes and related characteristics are used, it is always desirable to investigate the propagation of error and to establish the critical, i.e., the error-determining quantity.

Fluorofluoroxydioxirane and Other CF2O3 Isomers

Seven structural isomers, and relevant transition states, on the lowest-energy spin-singlet potential energy surface of CF2O3 are characterized using correlated ab initio electronic structure methods. On the basis of preliminary comparative calculations, second-order Møller−Plesset perturbation theory with a correlation-consistent polarized valence double-ζ one-electron basis set (MP2/cc-pVDZ) was chosen with which to describe equilibrium structures and harmonic vibrational frequencies of all relevant rotamers. Accurate energy differences were determined using the coupled cluster method, with perturbative inclusion of triple excitations, and a valence triple-ζ basis (CCSD(T)/cc-pVTZ). The lowest energy isomer is predicted to be the molozonide of difluorocarbene, whose adiabatic decay is prevented by a sizable barrier. Plausible mechanisms for the isomerization of the titled compound to the low-energy fluoroformyl peroxyhypofluorite and nearly isenergetic carbonyl hypofluorite are suggested. An analogous reaction path may be of importance to the isomerization of difluorodioxirane (CF2O2) and is included in the study.

Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave function/basis set calculations

The accuracy of standard ab initio wave-function calculations of atomization energies and reaction enthalpies has been assessed by comparing with experimental data for 16 small closed-shell molecules and 13 isogyric reactions. The investigated wave-function models are Hartree–Fock (HF), Møller–Plesset second-order perturbation theory (MP2), coupled-cluster theory with singles and doubles excitations (CCSD) and CCSD with perturbative triple-excitation corrections [CCSD(T)]; the one-electron basis sets used are the correlation-consistent cc-pVxZ and cc-pCVxZ basis sets with cardinal numbers x=D, T, Q, 5, and 6. Results close to the basis-set limit have been obtained by using two-point extrapolations. In agreement with previous studies, it is found that the intrinsic error of the CCSD(T) method is less than chemical accuracy (≈4 kJ/mol) for both atomization energies and reaction enthalpies. The mean and maximum absolute errors of the best CCSD(T) calculations are 0.8 and 2.3 kJ/mol for the atomization energies and 1.0 and 2.3 kJ/mol for the reaction enthalpies. Chemical accuracy is obtained already from the extrapolations based on the cc-pCVTZ and cc-pCVQZ basis sets—with mean and maximum absolute errors of 1.7 and 4.0 kJ/mol for atomization energies and 1.3 and 3.1 kJ/mol for reaction enthalpies. The intrinsic errors of the Hartree–Fock, MP2, and CCSD wave-function models are significantly larger than for CCSD(T). For CCSD and MP2, the mean absolute errors in the basis set limit are about 32 kJ/mol for the atomization energies and about 10 and 15 kJ/mol, respectively, for the reaction enthalpies. For the Hartree–Fock model, the mean absolute errors are 405 and 29 kJ/mol for atomization energies and reaction enthalpies, respectively. Correlation of the core electrons is important in order to obtain accurate results with CCSD(T). Without compromising the accuracy, the core contribution may be calculated with a basis set that has one cardinal number lower than that used for the valence correlation contribution. Basis-set extrapolation should be used for both the core and the valence contributions.

Theoretical resolution of theH−resonance spectrum up to then=4threshold. I. States of1Po,1Do,and1Fosymmetries

We report on a theoretical approach to the calculation of wave functions, energies $E,$ and widths $\ensuremath{\Gamma}$ of high-lying resonances of ${\mathrm{H}}^{\ensuremath{-}},$ with application to the identification of 76 states of ${}^{1}{P}^{o},$ ${}^{1}{D}^{o},$ and ${}^{1}{F}^{o}$ symmetries up to the $n=4$ threshold, with widths down to about $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$--$1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}\mathrm{a}.\mathrm{u}.,$ depending on symmetry and threshold. The overwhelming majority of these resonances have not been detected experimentally. Previous calculations by different methods allowed the identification of 35 of these states, with only very few cases having a level of accuracy comparable to the one of the present work. We suggest that the measurement of these resonances might become possible via two-step excitation mechanisms using ultrasensitive techniques capable of dealing with the problems of very small widths and preparation cross-sections. In this work, the ${}^{1}D$ state at $10.872\mathrm{eV}$ above the ${\mathrm{H}}^{\ensuremath{-}}{1s}^{2}{}^{1}S$ ground state, already prepared and measured by electron scattering as well as by two-photon absorption, is considered as the stepping stone for the possible probing of resonances of ${}^{1}{P}^{o},$ ${}^{1}{D}^{o},$ and ${}^{1}{F}^{o}$ symmetries via absorption of tunable radiation of high resolution. By classifying the results according to the Gailitis-Damburg model of dipole resonances (a product of a ${1/r}^{2}$-like potential) we find that there are unperturbed as well as perturbed series, in analogy with the Rydberg spectra of neutrals and positive ions (a product of a $1/r$-like potential). For the former, the agreement with the Gailitis-Damburg predictions as to the relationship of the extent of the outer orbital and of the energies and widths of states is excellent. The perturbed series result from interchannel coupling and the remaining electron correlation. One of the effects is the existence of overlapping resonances. For example, for two ${}^{1}{P}^{o}$ states below the $n=3$ threshold there is degeneracy on the energy axis ${(E}_{1}=\ensuremath{-}0.0555763612\mathrm{a}.\mathrm{u}.$ and ${E}_{2}=\ensuremath{-}0.0555763099\mathrm{a}.\mathrm{u}.)$ but the widths differ $({\ensuremath{\Gamma}}_{1}=1.14\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\mathrm{eV}$ and ${\ensuremath{\Gamma}}_{2}=5.45\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\mathrm{eV}).$ We also comment on whether consideration of the relativistic Lamb shift splitting of the hydrogen thresholds is sufficient for deciding the truncation of the resonance series. Our calculations were carried out by implementing previously published theories, whereby the resonance $E\mathrm{'}\mathrm{s}$ and \ensuremath{\Gamma}'s are determined from properly selected complex eigenvalues of non-Hermitian Hamiltonian matrices constructed in terms of physically relevant square integrable real and complex function spaces representing the localized and asymptotic parts of the resonance eigenfunctions. For the ${\mathrm{H}}^{\ensuremath{-}}$ series of resonances, the physical relevance of the real functions implies the systematic construction of basis sets with average $〈r〉$ extending to thousands of atomic units, in order to account for the extreme diffuseness of the outer orbital as each threshold is approached. The complex one-electron basis sets are Slater-type orbitals of a complex coordinate ${\mathrm{re}}^{\ensuremath{-}i\ensuremath{\theta}}.$ Their inclusion into the overall calculation and their optimization via the variation of nonlinear parameters (including \ensuremath{\theta}) accounts for the contribution of the asymptotic part of the resonance, and for the energy width and shift beyond the real energy ${E}_{o}$ of the localized part.

Nuclear quadrupole coupling constant of 21Ne in the neon dimer and its influence on the T1 NMR relaxation time in fluid neon

The coupled cluster singles and doubles (CCSD) and CCSD combined with a perturbative correction for triple excitations (CCSD(T)) wave-function models are applied to study the effect of interatomic interactions in the Ne dimer on the electric field gradient (EFG) at the Ne atoms. Large one-electron basis sets are used to describe simultaneously the correlation corrections and the effect of the interatomic interaction on the EFG. The resulting potential and EFG curves are used in molecular dynamics simulations to determine the T 1 NMR relaxation time in supercritical and liquid 21Ne. The role of various approximations and the accuracy of the obtained results can now be systematically analyzed, and the calculations therefore make a reliable prediction of the relaxation times possible.

Comparison of relativistic effective core potential and all-electron Dirac-Coulomb calculations of mercury transition energies by the relativistic coupled-cluster method

Relativistic effective core potential (RECP) and all-electron Dirac-Coulomb calculations of transition energies for low-lying states of the mercury atom and its ions are carried out with equivalent basis sets by the relativistic coupled-cluster method. RECP results are compared with corresponding all-electron values to estimate the accuracy of different RECP versions. Contributions from correlations of different electron shells to the calculated transition energies are studied. Effects of different nuclear models and of basis set truncation at different orbital angular momenta as well as errors of the Gaussian approximation of the generalized RECP components are reported. We show that at least 34 external electrons of the mercury atom should be correlated and the one-electron basis set should contain up to h-type functions in order to attain consistent agreement within 200 cm-1 with experimental data.

How to choose a one-electron basis set to reliably describe a dipole-bound anion

The problem of choosing appropriate atomic orbital basis sets for ab initio calculations on dipole-bound anions has been examined. Such basis sets are usually constructed as combination of a standard valence-type basis, designed to describe the neutral molecular core, and an extra diffuse set designed to describe the charge distribution of the extra electron. As part of the present work, it has been found that the most commonly used valence-type basis sets (e.g., 6-31CCG or 6-311CG), when so augmented, are subject to unpredictable under- or overestimating electron binding energies for dipole-bound anions. Whereas, when the aug-cc-pVDZ, aug-cc-pVTZ (or other medium-size polarized (MSP) basis sets) are so augmented, more reliable binding energies are obtained especially when the electron binding energy is calculated at the CCSD(T) level of theory. The issue of designing and centering the extra diffuse basis functions for the excess electron has also been studied, and our findings are discussed here. c 2000 John Wiley & Sons, Inc. Int J Quantum Chem 80: 1024-1038, 2000

Dynamics of the O(3P)+HCl reaction on the 3A″ electronic state: A new ab initio potential energy surface, quasi-classical trajectory study, and comparison to experiment

A new potential energy surface for the lowest 3A″ electronic state of the O(3P)+HCl system is presented. This surface is based on electronic energies calculated at the multireference configuration interaction level of theory with the Davidson correction (MR-CI+Q) using the Dunning cc-pVTZ one-electron basis sets. The ab initio energies thus obtained are scaled using the scaled external correlation (SEC) method of Brown and Truhlar. The SEC-scaled energies are fitted to a simple analytical expression to yield a potential energy surface which correlates the reactants O(3P)+HCl(1Σ+) to the products OH(2Π)+Cl(2P). The reaction barrier on this surface lies at an O–H–Cl angle of 131.4° at an energy of 9.78 kcal/mol above the asymptotic O+HCl minimum. This barrier is 1.3 kcal/mol higher than that on the potential energy surface obtained by Koizumi, Schatz, and Gordon (KSG) [J. Chem. Phys. 95, 6421 (1991)] and 1.1 kcal/mol lower than the S2 surface of Ramachandran, Senekowitsch, and Wyatt (RSW) [J. Mol. Struct. (Theochem) 454, 307 (1998)]. The dynamics of the reaction O(3P)+HCl(v=2; j=1,6,9)→OH(v′,j′)+Cl on this potential surface is studied using quasi-classical trajectory (QCT) propagation and the results are compared to the experimental observations of Zhang et al. [R. Zhang, W. J. van der Zande, M. J. Bronikowski, and R. N. Zare, J. Chem. Phys. 94, 2704 (1991)]. The broad distribution of collision energies in the experiment is modeled by computing weighted averages of the quantities of interest with the weighting factor at each collision energy determined by the collision energy distribution.

Simple Approximation of Core-Correlation Effects on Binding Energies

The majority of the ab initio electronic structure calculations that are performed today include the correlation energy of only the valence electrons. This is sometimes called the frozen-core (FC) approximation. Even when correlated calculations are carried out without the FC approximation, one cannot make a realistic estimate of core correlation energy unless one adds extra basis functions in the core region, and standard basis sets 1 do not include these. The idea that only the valence electrons contribute to binding has a long and venerable history, dating to the early days of quantum chemistry, and it is essentially correct, rightfully constituting one of the key elements for understanding why the periodic table explains so much of chemistry. 2 However recent years have seen great progress in the calculation of molecular binding energies, 3 and in some cases errors due to neglecting the contribution of core correlation energy may exceed all other errors combined. In fact, over the past few years a number of researchers 4-17 have shown that the change in correlation energy of core electrons must be accounted for quantitatively to predict accurate molecular structures or energetics. Accounting for the correlation energy of core electrons by explicitly including them in correlated electronic structure calculations increases the computational cost by a very significant amount, due both to the need for larger one-electron basis sets and also to the larger number of manyelectron configurations that need be considered. This is prohibitively expensive for larger molecules. The purpose of the present paper is to point out that one can formulate a useful approximation to core correlation effects on binding energies by counting the heavy-atom bonds in a molecule.

Theoretical spectroscopy of organic systems

The complete active space (CAS) SCF method in conjunction with the multiconfigurational second-order perturbation theory (CASPT2) has been applied to study the electronically excited states of basic organic compounds. As shown in the lecture with a number of examples, the CASPT2 method is capable of yielding accurate results for relative energies and other properties of excited states, provided that flexible one-electron basis sets are employed. The applications comprise an ample range of systems and problems, including polyenes, conjugated and unconjugated dienes, alternant and nonalternant hydrocarbons, polyenals, etc. As a whole these studies enable both qualitative and quantitative understanding of electronic spectra of organic molecules of fundamental importance.

Adaptation of one-electron basis sets to spatial confinements

A method that generates out of a given orbital basis set its analog appropriate for describing the effects of a spatial confinement is presented. The method is based on the requirement that the one-particle model spaces for the confined and for the unconfined systems are equivalent in the sense of a criterion derived from the basis-set-generating eigenvalue equations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 341–347, 1999

Dispersion coefficients for first hyperpolarizabilities using coupled cluster quadratic response theory

The frequency dependence of third-order properties can in the normal dispersion region be expanded in a Taylor series in the frequency arguments. The dispersion coefficients thus obtained provide an efficient way of expressing the dispersion of frequency-dependent properties and are transferable between different optical processes. We derive analytic expressions for the dispersion coefficients of third-order properties in coupled cluster quadratic response theory and report an implementation for the three coupled cluster models CCS, CC2, and CCSD. Calculations are performed for the first hyperpolarizability of the NH3 molecule. The convergence of the dispersion expansion with the order of the coefficients is examined and we find good convergence up to about half the frequency at which the first pole in the hyperpolarizability occurs. Pade approximants improve the convergence dramatically and extend the application range of the dispersion expansion to frequencies close to the first pole. The sensitivity of the dispersion coefficients on the dynamic correlation treatment and on the choice of the one-electron basis set is investigated. The results demonstrate that, contrary to presumptions in the literature, the dispersion coefficients are sensitive to basis set effects and correlation treatment similar to the static hyperpolarizabilities.

Comment on “Assessment of complete basis set methods for calculation of enthalpies of formation” [J. Chem. Phys. 108, 692 (1998)

Large errors recently reported in the calculated CBS-Q, CBS-q, and CBS-4 heats of formation of two-butyne and naphthalene are shown to be an artifact of near linear dependence in the one-electron basis set. The corrected results are in much better agreement with experiment.

Quasi-ab initio dynamics: a test trajectory study of the H+H 2 reaction using energies and gradients based on scaling of the external correlation

We have carried out a test dynamics study of the prototype H+H 2 exchange reaction by running quasi-classical trajectories `on-the-fly' using ab initio correlated energies which have been previously corrected semiempirically by the double many-body expansion plus scaling of the external correlation method to account for size-limitations of the one-electron basis set and configuration interaction expansion. The method is general and gives results in agreement with conventional trajectory calculations carried out on the accurate double many-body expansion potential energy surface for H 3.

Electron affinities, excitation energies and ionization potentials of the transition metals (I) Sc and Ti

The electron affinities of the Sc and Ti atoms have been obtained by configuration interaction calculations. Energy convergence with respect to the systematic expansion of both the one-electron basis set and the configuration space was investigated for valence electrons, and the inclusion of correlation contributions from core electrons and relativistic effects gave the electron affinities of 0.181 eV and 0.163 eV for Sc and Ti, respectively. These are in excellent agreement with the observed values of 0.189 ± 0.020 eV and 0.080 eV. The same approach was applied for the first excited states and positive ions of both atoms. Excellent agreement with the experimental results was also obtained for these states.

Oscillator strength sum rules with an external electromagnetic field

We demonstrate that the Bethe, and therefore the Thomas-Reiche-Kuhn, sum rule is unaffected by the presence of an applied external electromagnetic field in the exact case. We use the consequence that the first-order perturbation contribution must also vanish to derive a necessary condition for the completeness of computational one-electron basis sets.

Electronic Structure of Dipole-Bound Anions

Dipole-bound anionic states of HCN, (HF)2, CH3CN, C3H2, C4H2, C5H2, and stretched CH3F are studied using extended one-electron basis sets at the coupled cluster level of theory with single, double, and noniterative triple excitations (CCSD(T)). Orbital relaxation and electron correlation corrections to the Koopmans' theorem prediction of electron binding energy are analyzed, and a physical interpretation of low-order corrections is proposed. It is demonstrated that the second-order dispersion interaction between the loosely bound electron and the electrons of the neutral host should be included into physical models of dipole-bound anions. Higher-order electron correlation corrections are also found to be important, and a slow convergence of the Moller−Plesset series for electron binding energies is documented. Modifications of the potential energy surfaces of the above polar molecules upon electron attachment are studied at the second-order Moller−Plesset level, and Franck−Condon factors for the anion/neu...

Atomic configuration interaction and studies of He, Li, Be, and Ne ground states

The atomic configuration interaction (CI) is reconsidered. We compare the algebraic and geometric approaches to the construction of the CI matrix and point out advantages of the latter. One-electron basis sets of quality comparable to numerical multiconfigurational Hartree-Fock are readily obtained. The generation of large CI lists of symmetry eigenfunctions is monitored by a prescription establishing an {ital a priori} identification of relevant contributions to the wave function. Systematic and well-defined truncations to multireference CI{close_quote}s are examined. A formula for energy contributions of six-excited unlinked clusters is derived and shown to give reasonable estimates. Ensuing nonrelativistic CI calculations on He, Li, Be, and Ne ground states yield the lowest upper bounds in the literature, capturing 99.978, 99.923, 99.919, and 99.59{percent} of the accepted correlation energies, respectively. {copyright} {ital 1997} {ital The American Physical Society}

Efficient Calculation of Isotropic Hyperfine Constants of Phosphorus Radicals Using Density Functional Theory

Calculations of the isotropic hyperfine coupling constants of phosphorus nuclei in different environments have been carried out using density functional theory with both B3LYP and B3PW91 functionals and a variety of one-electron basis sets. A set of 35 radicals, radical cations, and triplet species containing P have been analyzed, including the set recently examined by Cramer and Lim (J. Phys. Chem. 1994, 98, 5024) using the UMP2 method. The dependency of the calculated spin densities with respect to the methods, basis sets, and geometries have been investigated. Overall, the B3LYP method, in conjunction with a TZVP basis optimized for DFT calculations and further augmented by tight 1s-functions on all heavy atoms, appears to be the most efficient treatment, presumably owing to better cancelations of intrinsic errors. Depending on the size of the species examined and/or the spin contamination of UHF references, use of UMP2 geometries is preferred, otherwise B3LYP/6-311G(d,p) geometries are a reasonable choice. In both cases, linear correlation between computed and observed values have been found with slopes close to unity and small intercepts ≤10 G.

Energies of dipole-bound anionic states

Dipole-bound anionic states of CH3CN, C3H2, and (HF)2 were studied using highly correlated electronic structure methods and extended one-electron basis sets. The electron detachment energies were calculated using the coupled cluster method with single, double, and noniterative triple excitations. Geometrical relaxation of the molecular framework upon electron attachment and the difference in the harmonic zero-point vibrational energies between the neutral and the dipole-bound anionic species were calculated at the MP2 level of theory. We demonstrate that the dispersion interaction between the loosely bound electron and the electrons of the neutral molecule is an important component of the electron binding energy, comparable in magnitude to the electrostatic electron–dipole stabilization. The geometrical relaxation upon electron attachment and the change in the zero-point vibrational energy is important for the weakly bound HF dimer. The predicted values of the vertical electron detachment energies for the dipole bound states of CH3CN and C3H2 of 112 and 188 cm−1, respectively, are in excellent agreement with the recent experimental results of 93 and 171±50 cm−1, respectively. For (HF)2−, the predicted value of adiabatic electron detachment energy is 396 cm−1, whereas the experimental vertical detachment energy is 508±24 cm−1. The possibility of formation of the neutral dimer in an excited vibrational state is considered. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64: 183–191, 1997