The influence of electron correlation on reaction energies. The dimerization energies of BH3 and LiH

Results of rigorous computations employing extended Gaussian-type basis sets are reported for BH₃, B₂H₆, LiH, and Li₂H₂ in their respective equilibrium geometries. The dimerization energy of BH₃ is calculated as −20.7 kcal/mol within the Hartree-Fock approximation and as −36.6 kcal/mol if electron correlation is included. The corresponding results for the dimerization of LiH are −47.3 kcal/mol and −48.3 kcal/mol. Partitioning of the correlation energy contributions allows to attribute the effect of electron correlation to the increase of next neighbour bond interactions on the dimerization of BH₃ and LiH. The difficulties of accurate computations of reaction energies are discussed in detail.     

The estimation of electron affinities fromab initio 1s orbital energies

Summary It has been found that the electron affinities of alkoxy-radicals can be estimated using a correlation with the 1s orbital energy of the oxygen on the associated alkoxy-anion, EA=–0.64503 * (1s orbital energy) –351.58. The method assumes that the species of interest accepts the electron into an orbital which is localized on the oxygen.     

Accurate calculation of N1s and C1s core electron binding energies of substituted pyridines. Correlation with basicity and with Hammett substituent constants

Substituent shifts of the energetics of four related ionization processes of pyridines and benzoic acids (Fig. 1) were investigated. The first process is core-electron ionization of gas-phase pyridines (Fig. 1A), while the second concerns gas-phase acid-base reaction between a substituted pyridine and a conjugated acid (Fig. 1B), and the third and fourth processes are the acid dissociation of substituted benzoic acids in aqueous solution (Fig. 1C) and in vacuum (Fig. 1D), respectively. Core-electron binding energies for the first process were calculated using density-functional theory with the scheme ΔE_KS (PW86x-PW91c/TZP+C_rel)//HF/6-31G*. Average absolute deviation of calculated core electron binding energy shifts at N atom in substituted pyridines from experiment was 0.08 eV. The shift at N coincides highly with that at a ring carbon atom. The four shifts corresponding to the four processes shown in Figs. 1A–D correlate strongly with one another, with numerical values fairly close to each other when expressed in unit of electron volts.     

Dyson-corrected orbital energies for the perturbative treatment of electron correlation

The effect of replacing the Hartree–Fock one-particle energies with ionization potentials obtained from inverse Dyson equation when calculating electron correlation energies perturbatively is investigated. Though the energy shifts vary from system to system, the slight decrease of the resulting excitation energies at around equilibrium geometries leads to a slight increase of the correlation energies in most cases. In the dissociation limit the inverse Dyson equation opens the gap, thus nondiverging potential curves emerge even at the restricted Hartree–Fock (RHF)+RS2 level. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 713–719, 1998     

Accurate extrapolation of electron correlation energies from small basis sets

A new two-point scheme is proposed for the extrapolation of electron correlation energies obtained with small basis sets. Using the series of correlation-consistent polarized valence basis sets, cc-pVXZ, the basis set truncation error is expressed as deltaE(X) proportional, variant(X + xi(i))(-gamma). The angular momentum offset xi(i) captures differences in effective rates of convergence previously observed for first-row molecules. It is based on simple electron counts and tends to values close to 0 for hydrogen-rich compounds and values closer to 1 for pure first-row compounds containing several electronegative atoms. The formula is motivated theoretically by the structure of correlation-consistent basis sets which include basis functions up to angular momentum L = X-1 for hydrogen and helium and up to L = X for first-row atoms. It contains three parameters which are calibrated against a large set of 105 reference molecules (H, C, N, O, F) for extrapolations of MP2 and CCSD valence-shell correlation energies from double- and triple-zeta (DT) and triple- and quadruple-zeta (TQ) basis sets. The new model is shown to be three to five times more accurate than previous two-point schemes using a single parameter, and (TQ) extrapolations are found to reproduce a small set of available R12 reference data better than even (56) extrapolations using the conventional asymptotic limit formula deltaE(X) proportional, variantX(-3). Applications to a small selection of boron compounds and to neon show very satisfactory results as well. Limitations of the model are discussed.     

On the relation between core electron binding energies of ions in solution and their solvation energies

Experimental and computational results from the study of positive and negative ions in solution are presented. The importance of short-range interactions between ion and solvent is studied with regard to core ionization of the ion. Exchange repulsion is found to be a significant factor in the interpretation of data for both cations and anions. Experimental results are presented for the core ionization of the OH^? ion in solution. The data show a strong similarity with corresponding data for the F^? ion, resulting in a large negative solvation energy for the final core hole state. The Be²⁺ ion shows large solvation energies for both ground- and core-ionized states, which is interpreted as due to charge transfer effects between solvent and ion.     

Green's function calculations of ground-state correlation energies

A computationally convenient and numerically stable procedure is reported for the direct calculation of ground-state correlation energies employing one-particle Green's functions.     

Wavelength-decomposition-based embedded cluster density approximation for systems with nonlocal electron correlation

Local correlation methods rely on the assumption that electron correlation is nearsighted. In this work, we develop a method to alleviate this assumption. This new method is demonstrated by calculating the random phase approximation (RPA) correlation energies in several one-dimensional model systems. In this new method, the first step is to approximately decompose the RPA correlation energy to the nearsighted and farsighted components based on the wavelength decomposition of electron correlation developed by Langreth and Perdew. The short-wavelength (SW) component of the RPA correlation energy is then considered to be nearsighted, and the long-wavelength (LW) component of the RPA correlation energy is considered to be farsighted. The SW RPA correlation energy is calculated using a recently developed local correlation method: the embedded cluster density approximation (ECDA). The LW RPA correlation energy is calculated globally based on the system's Kohn-Sham orbitals. This new method is termed λ-ECDA, where λ indicates the wavelength decomposition. The performance of λ-ECDA is examined on a one-dimensional model system: a H₂₄ chain, in which the RPA correlation energy is highly nonlocal. In this model system, a softened Coulomb interaction is used to describe the electron-electron and electron-ion interactions, and slightly stronger nuclear charges (1.2e ) are assigned to the pseudo-H atoms. Bond stretching energies, RPA correlation potentials, and Kohn-Sham eigenvalues predicted by λ-ECDA are in good agreement with the benchmarks when the clusters are made reasonably large. We find that the LW RPA correlation energy is critical for obtaining accurate prediction of the RPA correlation potential, even though the LW RPA correlation energy contributes to only a few percent of the total RPA correlation energy.     

Universal density functional approach to the calculation of correlation energies of atoms

A new local density functional approach for the calculation of correlation energies of many-electron atomic systems is proposed by using the exact results for the correlation energy of a two-electron system bound by a harmonic oscillator external potential. This is motivated by the fact that the correlation energy is a universal functional of the electron density, and the form of this functional is independent of the external potential. The calculated numerical results for the correlation energies show very good agreement with the standard values reported in the literature. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 461–465, 1997     

Relativistic study of tautomerism and core electron binding energies of thio- and selenocytosine

We applied the spin-free relativistic infinite-order two-component method to calculate the core-electron binding energies of different tautomeric structures of thio- and selenocytosine. The importance of relativistic effects in the ionization of 1s electrons was studied for selenocytosine. The present method provides a reasonably simple and reliable computational tool for calculating the core electron binding energies of molecules containing heavy atoms. The method can also be used for characterization of different tautomeric structures of nucleic acid basis.     

Slater-like model for carbon 1s core ionization energies of halomethanes.

Based on the atomic electron affinity EA, the average energy of the valence-shell electrons EI and the polarizability alpha, the charge effect and the relaxation effect were evaluated for the carbon 1s core ionization energies of halomethanes CHnY4-n-mZm (Y, Z=F, Cl, Br, I). The charge effect was scaled by the electronegativity discrepancy (the discrepancy of EA and the discrepancy of EI between the C and H or halogen atom in the C-H or C-halogen chemical bond). The relaxation effect (induced dipole) was scaled by the charge on the carbon atom together with the polarizability of the H and halogen atoms. Further, the electrostatic relaxation shielding DeltaSi of the carbon 1s electron in the halomethane was expressed by the charge effect together with the relaxation effect. By introducing DeltaSi into the Slater model, a Slater-like model was obtained for calculating the carbon 1s core ionization energy E1,C of halomethane, whose correlation coefficient r is 0.99985 and the average absolute error is only 0.041 eV between the calculated and the experimental carbon 1s core ionization energies for 27 halomethanes. Also the cross-correlation was tested by the leave-one-out (LOO) cross-validation method, and the obtained model has good predictive ability and stability (the correlation coefficient rcv is 0.99976, the average absolute error between the predicted and the experimental values is only 0.052 eV). The proposed model perhaps lays a good foundation for computing the core ionization energies of various atoms in more complex molecules.     

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.     

Relativistic contributions to single and double core electron ionization energies of noble gases

We have performed relativistic calculations of single and double core 1s hole states of the noble gas atoms in order to explore the relativistic corrections and their additivity to the ionization potentials. Our study unravels the interplay of progression of relaxation, dominating in the single and double ionization potentials of the light elements, versus relativistic one-electron effects and quantum electrodynamic effects, which dominate toward the heavy end. The degree of direct relative additivity of the relativistic corrections for the single electron ionization potentials to the double electron ionization potentials is found to gradually improve toward the heavy elements. The Dirac-Coulomb Hamiltonian is found to predict a scaling ratio of ～4 for the relaxation induced relativistic energies between double and single ionization. Z-scaling of the computed quantities were obtained by fitting to power law. The effects of nuclear size and form were also investigated and found to be small. The results indicate that accurate predictions of double core hole ionization potentials can now be made for elements across the full periodic table.     

A density functional that accounts for medium-range correlation energies in organic chemistry

It has recently been pointed out that current density functionals are inaccurate for computing stereoelectronic effects and energy differences of isomerization reactions and isodesmic reactions involving alkanes; this has been interpreted as an incorrect prediction of medium-range correlation energies. This letter shows that the recently published M05-2X functional has good accuracy for all three of the recently highlighted problems and should be useful for a wide variety of problems in organic chemistry. [structure: see text]     

The effect of electron correlation on the topological and atomic properties of the electron density distributions of molecules

The topological properties of the electron density and the properties of an atom in a molecule are calculated by means of second-order Møller-Plesset perturbation theory (MP2) and compared with the results of configuration interaction calculations (C12) which include all single and double substitutions from the Hartree-Fock reference configuration. A software package for analyzing the effects of electron correlation on the topological properties of the electron density of molecules is described. H₂CO is used to provide a numerical example and to indicate that the number of bond critical points is unaffected by the inclusion of electron correlation. Correlation leads to only a small shift in the positions of bond critical points and a small change in the electron density at bond critical points. It is further shown that the energy of an atom in a molecule can be calculated to an accuracy of 1 kcal/mol and the electron population of an atom to about 0.001e. A statistical method is used to show that the deviation of the MP2 correlation correction relative to the CI2 correlation correction for a variety of atomic properties is about 25%.     

The electron correlation cusp

The Kais function is an exact solution of the Schrödinger equation for a pair of electrons trapped in a parabolic potential well with r ₁₂ ^?1 electron-electron interaction. Partial wave analysis (PWA) of the Kais function yields E _L = E + C₁(L + \-C ^?1 ₂)^?3 + O(L ^?5) where E is the exact energy and E _L the energy of a renormalized finite sum of partial waves omitting all waves with angular momentum ? > L. Slight rearrangement of an earlier result by Hill shows that the corresponding full CI energy differs from E _L only by terms of order O(L ^?5) with FCI values of C ₁ and \-C ^?1 ₂ identical to PWA values. The dimensionless \-C ₂ parameter is weakly dependent upon the size of the physical system. Its value is 0.788 for the Kais function, and 0.893 for the less diffuse helium atom, and approaches \-C ₂→ 1 in the limit of an infinitely compact charge distribution. The ?th energy increment satisfies an approximate virial theorem which becomes exact in the high ? limit. This analysis, formulated to facilitate use of the Maple system for symbolic computing, lays the mathematical ground work for subsequent studies of the electron correlation cusp problem. The direction of future papers in this series is outlined.     

The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement

Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced density matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F(2), HF, N(2), and a hydrogen chain.     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.