Atomic Correlation Energies Research Articles

New functionals for correlation energy deduced in the framework of the correlation factor approach

The general characteristics of two-body density functionals (TBDF) are explored and two new correlation energy functionals are derived using the correlation factor approach. The optimization of the parameters entering the above functionals requires exact and accurate atomic correlation energies (ACE). We revised the ACE values in the literature and obtained a new set of “exact” ACE for atoms with 2 ≤ Z ≤ 10. Unfortunately, there exist some inaccuracies in the ACE values of the second-row atoms, which make unsuitable the inclusion of them in the optimization. The ACE calculated for the first period with the above functionals, using the optimized sets of parameters, are in excellent agreement with the exact ones, while the corresponding values calculated for the second-row atoms are between the precision margins estimated by us for the exact values. © 1997 John Wiley & Sons, Inc.

Accurate quantum-chemical calculations: The use of Gaussian-type geminal functions in the treatment of electron correlation

We investigate augmenting conventional Gaussian-type one-electron orbital basis sets with two-electron functions that have a Gaussian dependence on the interelectronic distance. We observe substantial improvements in calculated correlation energies for helium and neon atoms and for the water molecule. A feature of our approach is that there is no nonlinear optimization of the two-electron basis function parameters at all.

Variation of the ground-state correlation energy of atoms with atomic number

The empirical correlation energy of atoms and singly charged positive and negative ions in their ground state shows a roughly linear dependence with respect to the number of electrons. However, the deviations from this linear behaviour follow a well defined trend. A formulation based on density functional theory allows us to understand these features. This formulation focuses on the number of pairings between antiparallel-spin electrons in the same main shell.

Correlation energies in polyatomic molecules modelled in terms of bond order

Earlier work has used the concepts of density functional theory to achieve useful estimates of both atomic and diatomic molecular correlation energies. In the present work, some of these ideas are combined with more conventional chemical ideas relating to bond order to evaluate correlation energies in some thirty polyatomic molecules. There are a number of empirical parameters in the present modelling process, but once these are fixed on some 16 molecules the correlation energies in the remaining molecules follow quantitatively, without any adjustments.

Valence correlation energies from pseudopotential calculations

Total valence correlation energies of first-row atoms, their anions and cations as well as their homonuclear dimers have been determined in all-electron and [He]-core pseudopotential calculations using the coupled-cluster method and large uncontracted basis sets containing up to g-type functions. The inaccuracies mainly caused by the pseudo-orbital transformation of the 2s valence orbitals are discussed. The relative errors in total valence correlation energies are found to be 6% at most. Correlation contributions to excitation energies, ionization potentials, electron affinities or binding energies are accurate with absolute errors of 0.1 eV or less.

Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing

A new dynamical correlation functional is constructed subject to a small number of simple, yet key, requirements not all satisfied by existing functionals in the literature. The new functional gives good atomic correlation energies, and, in conjunction with previous gradient-corrected exchange functionals and exact-exchange mixing, excellent thermochemistry in the G2 benchmarks of Pople and co-workers.

Scaling properties of correlation energies of homonuclear diatoms

In our continuing study of correlation energies in both hetero- and homo-nuclear diatomics, contact is established here with the first-principles study of S. Kais, S. M. Sung and D. R. Herschbach (1994, Int. J. Quantum Chem., 49, 657) on atomic correlation energies. Their results derive from renormalization of the large-dimension limit. Similarities of scaling are revealed for light homonuclear diatomics.

Note on the choice of basis set in density functional theory calculations for electronic structures of molecules (test on the atoms from the first three rows of the periodic table (2 ⩽ N ⩽ Z ⩽ 18), water, ammonia and pyrrole)

The effect of basis set in Kohn-Sham theory on estimating correlation energy was investigated using a density functional subroutine and a Hartree-Fock SCF one-electron density program (working with arbitrary basis set). Lee-Yang-Parr density functional estimation, MP2 perturbation calculation and Davidson's accurate configuration interactions calculations for correlation energies of atoms were used for this test. In the density functional estimation a simple 3-21G basis set has produced the same results as the huge triple-zeta basis set with polarization functions within the chemical accuracy (1 kcal/mol).

Charge renormalization at the large-D limit for diatomic molecules

The charge renormalization procedure for the calculation of the correlation energy of atoms utilizing the analytically known large-D limit solutions for the exact and Hartree–Fock equations is extended to diatomic molecules. This procedure is based on the variation of the nuclear charge, Z, and internuclear distance, R, of the Hartree–Fock Hamiltonian such that the Hartree–Fock energy will be significantly closer to the exact energy. We calculate to first order in δZ the leading contribution to the correlation energy by changing the nuclear charge to some renormalized nuclear charge, ZRi→Zi+δZi. To first order in δZ, this leads to an approximate expression, Ecorr(Za,Zb,R)=(∂EHF/∂Za)δZa+ (∂EHF/∂Zb)δZb, for the correlation energy based on the charge renormalization parameter δZ, which is fixed systematically from the large-D limit. The theory is applied to diatomic molecules. Near the equilibrium, we are predicting the correlation energy typically with 80% or greater accuracy in a completely self-consistent and systematic way with no additional cost to the Hartree–Fock calculation. An improved approach to estimating the correlation energy for all R is outlined.

Hartree-Fock soft Coulomb hole to estimate correlation energies in atoms, molecules and polymers

We have shown that the empirical correction introduced into the Hartree-Fock method to calculate correlation energies for atoms and therefore to remove the error caused by the so-called Coulomb hole can be extended from atoms to molecules and polymers. A reformulation was required of the necessary parameter representation. The reparametrization has been performed staying as close as possible to the original expressions for atoms reported by Chakravorty and Clementi (S.J. Chakravorty and E. Clementi, Phys. Rev. A, 39 (1989) 2290). In addition to their work, where the correlation energy has been calculated with the self-consistent Hartree-Fock wavefunction and the correction integrals, we have performed investigations, including the perturbation operator in the Fock operator, so that the total energy also contains the correlation energy. The applications of this approach to atoms and molecules show that the total electron correlation energies and ionization potentials calculated as differences of total energies can be obtained very satisfactorily. On the basis of the reported calculations it turns out that one obtains better agreement with reference values of more sophisticated calculations when the correction integrals are used to build up the Fock matrix. Furthermore we have found that the magnitude of the correlation energy depends only weakly on the size of the basis sets, which makes this empirical method very attractive for its application to large molecular and polymeric systems.

Coulomb-hole-Hartree-Fock functional for molecular systems

We have extended to molecules a density functional previously parametrized for atomic computations. The Coulombhole-Hartree-Fock functional, introduced by Clementi in 1963, estimated the dynamic correlation energy by the computation of a Hartree-Fock type single-determinant wavefunction, where the Hartree-Fock potential was augmented with an effective potential term, related to a hard Coulomb hole enclosing each electron. The method was later revised by S. Chakravorty and E. Clementi, Phys. Rev. A, 38 (1989) 2290, so that a Yukawa-type soft Coulomb hole replaced the previous hard hole. Atomic correlation energies, computed for atoms with Z = 2 to 54, as well as for a number of excited states, validated the method. In this work we have parametrized for molecules a function which controls the width of the soft Coulomb hole by fitting the first and second atomic ionization potentials of atoms with 1 ⩽ Z ⩽ 18 and the binding energies of a few diatomic molecules. The parametrization was successfully validated by computing the dissociation energy for a number of molecules. A few-determinant version of the Coulomb-Hartree-Fock method (CHF- N) necessary to account for the non-dynamic correlation correction and to ensure proper dissociation products, is briefly discussed with reference to a previous proposal by G.C. Lie and E. Clementi, J. Chem. Phys., 60 (1974) 1275 and 60 (1974) 1288.

Relation between molecular and atomic correlation energies via overlap population

A relationship is explored between correlation energies of light diatomic molecules, both hetero and homo, correlation energies of the constituent atoms and the overlap population S. While S is quantitatively sensitive to the choice of basis set, it is demonstrated that the general trends revealed are largely independent of such a choice. A weak residual dependence on molecular chemical potential is also indicated by the present study.

Coulomb‐Hole–Hartree–Fock functional

AbstractWe have extended to molecules a density functional previously parametrized for atomic computations. The Coulomb‐hole–Hartree–Fock functional, introduced by Clementi in 1963, estimates the dynamical correlation energy by the computations of a Hartree–Fock‐type single‐determinant wave function, where the Hartree–Fock potential was augmented with an effective potential term, related to a hard Coulomb hole enclosing each electron. The method was later revisited by S. Chakravorty and E. Clementi [Phys. Rev. A 39, 2290 (1989)], where a Yukawa‐type soft Coulomb hole replaced the previous hard hole; atomic correlation energies, computed for atoms with Z = 2 to Z = 54 as well as for a number of excited states, validated the method. In this article, we parametrized a function, which controls the width of the soft Coulomb hole, by fitting the first and second atomic ionization potentials of the atoms with 1 ⩽ Z ⩽ 18. The parametrization has been preliminarily validated by computing the dissociation energy for a number of molecules. A few‐determinant version of the Coulomb‐hole–Hartree–Fock method, necessary to account for the nondynamic correlation corrections, is briefly discussed. © 1994 John Wiley & Sons, Inc.

Correlated electronic structure models suggested by the large-dimension limit

We describe three simple models for electronic structure in many-electron atoms and ions. Each model is parametrized by the spatial dimensionality D, which is ordinarily set to 3. All procedures are motivated by the solution for D → ∞, a classical limit which can be solved exactly either with or without electron correlation. Each model modifies the D → ∞ limit treatment to reflect important finite-D effects, but retains the classical character of the limit; the modifications, in order of increasing realism, are designated post-scaling, pre-scaling, and pre-structuring. The three models yield pointlike electronic structures somewhat reminiscent of pre-quantal atomic models. However, all electronic positions have components outside the D-dimensional physical subspace, a feature which significantly enhances the ability of a localized structure to represent the true solution and to model it quantitatively. Specific calculations reported are total and correlation energies for atoms with Z ≤ 92 (computed by post-scaling and pre-scaling), correlation corrections to ionization potentials and electron affinities for Z ≤ 54 (post-scaling), and the asymptotic behavior of correlation energies for Z → ∞ (post-scaling).

The 1/Z expansion and renormalization of the large-dimension limit for many-electron atoms

Analytic expressions for the large-dimension limit, when renormalized by introducing a suitable effective nuclear charge ζ yield accurate D=3 nonrelativistic energies for ground states of many-electron atoms. Using Hartree–Fock data to estimate ζ, which typically differs from the actual charge Z by ∼1% or less, we find this dimensional renormalization method (denoted DR-0) gives results substantially better than the HF input. Comparison of the 1/Z expansion for the large-D limit with that for D=3 atoms provides expressions for the leading error terms in the renormalized total energy and correlation energy. When configuration mixing occurs in the Z→∞ limit (as for Be and many other atoms), we find the renormalization procedure is markedly improved by including the zeroth-order mixing (denoted DR-1); this contributes a term linear in Z. Including the Z-independent term (DR-2) also improves the accuracy when zeroth-order mixing is absent (e.g., ground-state atoms with N=2, 3, and 7–11) but not otherwise. Correlation energies for atoms and cations with N=2–18 electrons and Z=2–28 are obtained with a mean error of 26% using just the large-D limit or HF input (DR-0); the mean error improves to only 5% when the leading 1/Z term is included (either DR-1 or DR-2). Results much better than the HF approximation are likewise obtained for the ionization potentials and electron affinities of neutral atoms.

A correlation-energy functional from a correlation-factor model

In this work, a way to approximate the correlation energy functional starting from a model correlation factor is shown. The problem is addressed by using formally exact properties of the second-order density matrix and actual values of correlation energies for atoms. An Ansatz for the correlation factor is proposed that allows one to derive some known and some new correlation energy density functionals. Results for atomic systems show the reliability of the approach. © 1994 John Wiley & Sons, Inc.

Local density expressions for the correlation energy of atoms

Approximate local density expressions for the correlation energy of closed shell atomic systems are derived in the framework of the density functional method based on perturbation theory.

High precision atomic computations from finite element techniques: Second-order correlation energies of rare gas atoms

The p-version finite element method for atomic computations [J. Chem. Phys. 91, 7030 (1989)] has been implemented within the frame of second-order Mo/ller–Plesset theory and used to calculate correlation energies of the rare-gas atoms from He to Rn. The calculation proceeds through a sequence of levels of computation that is systematic and hierarchic in nature and helps to estimate the error of the final values. It is possible to include virtual orbitals of very high angular momentum (lmax=12) even for the heaviest elements; so very accurate results can be obtained. Comparison with the most accurate values found in the literature indicates that our FEM-MP2 method competes very well with any other method, even with Kutzelnigg’s MP2-R12 method [J. Chem. Phys. 94, 2002 (1991)], especially in the case of heavy atoms. The results presented here could be the most accurate published so far.

Correlation energies of Be-like atoms: A multimodel space many-body perturbation calculation with finite basis sets.

With finite basis sets constructed from B splines, a systematical many-body perturbation calculation of the Be-like atoms with Z from 3 to 36 is performed. A two-configuration model space (2s+2p${)}^{2}$ is adopted in the calculation in order to speed the perturbation convergence. With perturbations up to the second order, we obtain 93--99.8 % of the total correlation energies given by Davidson et al. [Phys. Rev. A 44, 7071 (1991)] for Z from 4 to 20. The discrepancies between experimental and theoretical values approach a constant of 0.0005 while atomic number Z increases.

Accuracy of the Boys and Bernardi function counterpoise method

The performance of the Boys and Bernardi function counterpoise (FCP) method in eliminating the basis set superposition error (BSSE) is studied for He2, at R=5.6 a.u., within the supermolecular coupled electron pair approximation (CEPA-1) method. A series of one-electron Gaussian basis sets is designed that allows a systematic approach to the basis set limit value of the interaction energy. Every basis set contains a part suitable to reproduce the atomic correlation energy and a second part optimized for the dispersion interaction in He2. BSSE-free correlated first-order interaction energies [E(1)], calculated using perturbation theory, are reported for each of these sets. Extrapolation to the basis set limit yields a new value of 33.60±0.02 μH for E(1) at R=5.6 a.u. Extending previous work, the supermolecular CEPA-1 interaction energies for each set are then compared to the total of E(1) and the BSSE-free Mo/ller–Plesset second-order dispersion energy reported previously. While for some basis sets the uncorrected ΔE values deviate up to 43 K from the perturbation estimate, the FCP-corrected results always agree within 0.4 K. A virtuals-only counterpoise procedure is considered as well, but fails badly. The remaining discrepancies in the FCP results are ascribed to a failure of the Mo/ller–Plesset approach to precisely model the dispersion energy at the CEPA level. This problem is removed in a further, more stringent test where supermolecular EintCEPA-intra results, in which only the intra-atomic correlation (at the CEPA-1 level) is taken into account, are directly compared to the BSSE-free E(1) values. In this test the FCP-corrected supermolecular results agree, for the larger sets, to within 0.001 K with the results expected on the basis of E(1). These findings demonstrate, for the first time, that at least in He2 the FCP recipe yields interaction energies that correspond precisely (to machine precision) to the basis set and correlation method at hand.