Core-valence Correlation Energy Research Articles

Core-core and core-valence correlation energy atomic and molecular benchmarks for Li through Ar.

We have established benchmark core-core, core-valence, and valence-valence absolute coupled-cluster single double (triple) correlation energies (±0.1%) for 210 species covering the first- and second-rows of the periodic table. These species provide 194 energy differences (±0.03 mEh) including ionization potentials, electron affinities, and total atomization energies. These results can be used for calibration of less expensive methodologies for practical routine determination of core-core and core-valence correlation energies.

A density functional for core-valence correlation energy.

A density functional, εCV-DFT(ρc, ρv), describing the core-valence correlation energy has been constructed as a linear combination of εLY P (corr)(ρc), εV WN5 (corr)(ρc, ρv), εPBE (corr)(ρc, ρv), εSlater (ex)(ρc, ρv), εHCTH (ex)(ρc, ρv), εHF (ex)(ρc, ρv), and FCV-DFTNi,Zi, a function of the nuclear charges. This functional, with 6 adjustable parameters, reproduces (±0.27 kcal/mol rms error) a benchmark set of 194 chemical energy changes including 9 electron affinities, 18 ionization potentials, and 167 total atomization energies covering the first- and second-rows of the periodic table. This is almost twice the rms error (±0.16 kcal/mol) obtained with CCSD(T)/MTsmall calculations, but less than half the rms error (±0.65 kcal/mol) obtained with MP2/GTlargeXP calculations, and somewhat smaller than the rms error (±0.39 kcal/mol) obtained with CCSD/MTsmall calculations. The largest positive and negative errors from εCV-DFT(ρc, ρv) were 0.88 and -0.75 kcal/mol with the set of 194 core-valence energy changes ranging from +3.76 kcal/mol for the total atomization energy of propyne to -9.05 kcal/mol for the double ionization of Mg. Evaluation of the εCV-DFT(ρc, ρv) functional requires less time than a single SCF iteration, and the accuracy is adequate for any model chemistry based on the CCSD(T) level of theory.

Systematically Convergent Correlation Consistent Basis Sets for Molecular Core−Valence Correlation Effects: The Third-Row Atoms Gallium through Krypton

The family of correlation consistent polarized valence basis sets has been extended in order to account for core-core and core-valence correlation effects within the third-row, main group atoms gallium through krypton. Construction of the basis sets is similar to that of the atoms boron through argon, where either the difference between core-correlated and valence-only correlation energies were calculated via configuration interaction (CISD) computations on the ground electronic states of the atoms (named cc-pCVnZ) or the sets were optimized with respect to the core-valence correlation energy and a small weight of core-core correlation energy (cc-pwCVnZ). Due to the correlation of 3d orbitals, added shells of higher angular momentum exponents compared to the valence sets are necessary to describe the core region. The pattern of added core-correlating functions is (1s1p1d1f) for double-zeta, (2s2p2d2f1g) for triple-zeta, (3s3p3d3f2g1h) for quadruple-zeta, and (4s4p4d4f3g2h1i) for quintuple-zeta. Atomic and molecular results show good convergence to the CBS limit, with the cc-pwCVnZ sets showing improved convergence compared to the cc-pCVnZ ones for molecular core-valence correlation effects. After testing the basis sets on the homonuclear diatomics Ga2-Kr2 with coupled cluster wave functions, it is concluded that a treatment of core-valence correlation effects is essential for high-accuracy ab initio investigations of third-row-containing molecules. Though the basis sets are optimal for 3s3p3d correlation, preliminary atomic and molecular results show the basis sets to be efficient with respect to 3d-only correlation, and these potentially could be used with 3d-only correlation for more qualitative studies on larger species.

W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions

In an attempt to improve on our earlier W3 theory [A. D. Boese et al., J. Chem. Phys. 120, 4129 (2004)] we consider such refinements as more accurate estimates for the contribution of connected quadruple excitations (T4), inclusion of connected quintuple excitations (T5), diagonal Born-Oppenheimer corrections (DBOC), and improved basis set extrapolation procedures. Revised experimental data for validation purposes were obtained from the latest version of the Active Thermochemical Tables thermochemical network. The recent CCSDT(Q) method offers a cost-effective way of estimating T4, but is insufficient by itself if the molecule exhibits some nondynamical correlation. The latter considerably slows down basis set convergence for T4, and anomalous basis set convergence in highly polar systems makes two-point extrapolation procedures unusable. However, we found that the CCSDTQ-CCSDT(Q) difference converges quite rapidly with the basis set, and that the formula 1.10[CCSDT(Q)cc-pVTZ+CCSDTQcc-pVDZ-CCSDT(Q)cc-pVDZ] offers a very reliable as well as fairly cost-effective estimate of the basis set limit T4 contribution. The T5 contribution converges very rapidly with the basis set, and even a simple double-zeta basis set appears to be adequate. The largest T5 contribution found in the present work is on the order of 0.5 kcal/mol (for ozone). DBOCs are significant at the 0.1 kcal/mol level in hydride systems. Post-CCSD(T) contributions to the core-valence correlation energy are only significant at that level in systems with severe nondynamical correlation effects. Based on the accumulated experience, a new computational thermochemistry protocol for first- and second-row main-group systems, to be known as W4 theory, is proposed. Its computational cost is not insurmountably higher than that of the earlier W3 theory, while performance is markedly superior. Our W4 atomization energies for a number of key species are in excellent agreement (better than 0.1 kcal/mol on average, 95% confidence intervals narrower than 1 kJ/mol) with the latest experimental data obtained from Active Thermochemical Tables. Lower-cost variants are proposed: the sequence W1-->W2.2-->W3.2-->W4lite-->W4 is proposed as a converging hierarchy of computational thermochemistry methods. A simple a priori estimate for the importance of post-CCSD(T) correlation contributions (and hence a pessimistic estimate for the error in a W2-type calculation) is proposed.

Accurate theoretical near-equilibrium potential energy and dipole moment surfaces of HgClO and HgBrO.

The complexes HgBrO and HgClO have been previously determined by ab initio methods to be strongly bound and were suggested to be important intermediates during mercury depletions events observed in the polar troposphere. In the present work accurate near-equilibrium potential energy surfaces (PESs) of these species are reported. The PESs are determined using accurate coupled cluster methods and a series of correlation consistent basis sets with subsequent extrapolation to the complete basis set limit. Additive corrections for both core-valence correlation energy and relativistic effects are also included. The anharmonic ro-vibrational spectra of HgBrO and HgClO have been calculated in variational calculations. Strong infrared band strengths are predicted for all fundamentals in these species. The spin-orbit splitting dominates over the vibronic coupling effect in both HgClO and HgBrO. The Renner-Teller vibronic energy levels corresponding to the bending mode of these molecules are calculated via perturbation theory.

Core-polarization potentials for Si and Ti

We introduce a method for constructing core-polarization potentials (CPPs) which describe core-valence correlation effects. Core-valence correlation energies are defined in terms of restricted configuration multiconfiguration Hartree-Fock energies and parametrized CPPs are fitted to reproduce the core-valence correlation energies as well as possible. We test our CPPs on Si and Ti, for which they work well.

Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited

Correlation consistent basis sets for accurately describing core–core and core–valence correlation effects in atoms and molecules have been developed for the second row atoms Al–Ar. Two different optimization strategies were investigated, which led to two families of core–valence basis sets when the optimized functions were added to the standard correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence–electron correlated calculations, i.e., for the core–core plus core–valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms [D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995)]. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell (core–valence) correlation energy converges more slowly than the intrashell (core–core) correlation energy. Since the effect of including the core electrons on the calculation of molecular properties tends to be dominated by core–valence correlation effects, a second scheme for determining the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core–valence (intershell) correlation energy, except that a small amount of core–core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core–valence basis sets (cc-pwCVnZ), significantly improve the convergence of many molecular properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster [CCSD(T)] calculations on a series of second row homonuclear diatomic molecules (Al2, Si2, P2, S2, and Cl2), as well as on selected diatomic molecules involving first row atoms (CO, SiO, PN, and BCl). For the calculation of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones.

Theoretical determination of the heat of formation of methylene

The heat of formation of CH 2 has been calculated using high accuracy ab initio results for its dissociation energy. MRCI calculations were carried out with correlation consistent polarized valence basis sets (cc-p VXZ, X=2–6). Core-valence correlation energies were computed using CCSD(T) with correlation consistent polarized core–valence basis sets (cc-pCVXZ, X=2–5). For the ground state CH 2 ( 3 B 1 ) our best estimate of the heat of formation at 0 K is Δ f,0 H E =388.7±0.6 kJ mol -1 . This is in agreement with experimental results, in the range 384.5–395.8 kJ mol -1 . Our result implies that for the first excited state CH 2 (a 1 A 1 ), Δ f,0 H E =426.4±0.6 kJ mol -1 .

Configuration-interaction study of the differential correlation energies among the lowest 2S, 2D, and 2Po states of Ca+

In order to analyze the differential correlation energies among the lowest $^{2}$S(4${\mathit{s}}^{1}$), $^{2}$D(3${\mathit{d}}^{1}$), and $^{2}$${\mathit{P}}^{\mathit{o}}$(4${\mathit{p}}^{1}$) states of ${\mathrm{Ca}}^{+}$, configuration-interaction calculations have been carried out for the three states. The results reveal that the $^{2}$D state has a larger core-valence correlation energy by 0.66 and 0.91 eV and a smaller core-core correlation energy by 0.38 and 0.39 eV than the $^{2}$S and $^{2}$${\mathit{P}}^{\mathit{o}}$ states, respectively. Based on an analysis of the computed correlation energies in terms of contributions by various excitation types, the differential core-core correlation energy is interpreted as the difference in the contribution from the 3${\mathit{p}}^{2}$\ensuremath{\rightarrow}dd' excitation between the $^{2}$D state and the others. The 3s intrashell correlation energies as well as the 3s-3p intershell correlation energies are uniformly equal among the three states.

Large atomic core—core correlation effects

The effects of core—valence and core—core correlation on valence ionization potentials (IP) and excitation energies of Ca +, neutral Sc and Sc 2+ are studied. Core—valence correlation has a large effect, particularly on 3d excitations, but results in overestimated (by up to 0.8 eV) IPs. Core—core correlation reduces the effect of core—valence correlation and may reduce the computed IP by as much as 1 eV compared with the core—valence CI result. Computing only the valence and core—valence correlation energy while excluding the core—core contribution in molecules containing Ca and Sc is thus not advisable.

A relativistic C.I. study on theJ=1/2 even spectrum of Sr II

We present a method for performing relativistic CI calculations in ground and excited atomic and ionic levels. An electron occupying a relativistic shellnκ in a given electronic configuration is described by a single numerical four-component Dirac-Fock orbital having the samen and κ quantum numbers to those of the shellnκ. Application of this method yields estimates for the I.P. (88 741 cm−1) and the core correlation energy (−30916 cm−1) for Sr II and for the total correlation energy in Sr III (−30916 cm−1). Core-valence correlation energies for the |core 5s〉 (−4379 cm−1), |core 6s〉 (−1191 cm−1) and |core 13s〉 (−32 cm−1) levels have been calculated for Sr II. Estimates for the total relativistic, Breit, vacuum polarization and self energy corrections for these levels are also reported. Configurations in which the core is not fully occupied have been found to yield significant contributions to the correlation energies of both ground and excited levels. Our results show that full scale relativistic CI calculations using numerical four-component Dirac-Fock orbitals are feasible and provide a useful ab-initio tool for the investigation of atomic properties within the framework of fully relativistic theories.