Atomic Correlation Energies Research Articles

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.

Validity criteria for rotor-vibrator and independent-particle models of atoms: overlaps, oscillator strengths, and angular deviations

A comparison is made of Hartree-Fock, independent-particle (HF) and collective, molecule-like rotor-vibrator (RV) models for atoms with twovalenceelectrons. The two models are constructed from the same pseudopotential Hamiltonian. The criteria for the comparison are (a) overlap with well-converged configuration interaction (CI) wave functions, (b) comparison of oscillator strengths with values from well-converged CI functions (and, wherever possible, from experiment), and (c) comparison of the root-mean-square deviation of the angular distribution, Le., ((B12)2 - (812~))'/~. In the comparison of the overlaps, neither model is obviously superior to the other. The RV wave functions yield better oscillator strengths in more cases than the HF functions, despite the results of the overlap calculations. Qualitative comparisons of the spatial distributions of probability density suggest that the RV functions have angular distributions much more like the CI functions than do the HF functions. A simple model calculation of the effect of a small amount of configuration mixing elucidates this similarity. A brief comparison is made of energy optimization and overlap optimization. Many methods of modeling atoms use an independent-particle approach as a first approximation. In the independent-particle model, the electrons are assumed to maintain their individual angular momenta and energies, independent of all the other electrons in the atom. A traditional measure of how well the independent-particle model describes an atom is the correlation energy. The correlation energy in atoms, strictly the difference between the exact, nonrelativistic energy and the Hartree-Fack energy, can be fairly accurately determined by comparing the energy expectation values of well-converged CI (configuration interaction) and HF (Hartree-Fack) wave functions. Two new ways of classifying ground and low-lying excited states of pseudotwo-electron atoms have suggested some different zero-order approximations to the wave functions.Id One approach uses the hyperspherical coordinate system typically used in the adiabatic appro~imation.3.~ other uses the electron-nucleus-clectron angle and the electron-nucleus distances in rovibrational functi0ns5,~ as distinct from the functions developed by Hylleraas7q8 which include the interelectron distance explicitly and so correctly describe the cusp region. This paper shows the results of a comparison of the second of these two types of correlated wave functions, the RV functions with the independent-particle model. The HF method takes into account the mean field of the probability distribution of the charge of the electrons on each electron but does not include the individual interactions between electrons. Thus, the HF method, apart from the effects of particle symmetry, omits the correlation between the electrons present in any atom or molecule involving more than one electron. In this paper, we use newly calculated HFSCF (self-consistent field) wave functions as the independent-particle wave functions. Wave functions developed by Hunter and BerryS in the form of overlap optimized rotor-vibrator (RV) wave functions are used as the extreme of correlated wave functions. Although the CI wave functions are constructed from an independent-particle basis, the completeness of the expansion set allows a formally exact convergence, and tests performed previously6 justify the assump tion that the CI wave functions are well converged. The CI wave functions used are those calculated with Sturmian basis functions by Krause and Berry6 and use a p~eudopotential~ for the core electrons of atoms with more than two electrons. Overlaps between CI and RV wave functions have previously been calculated5 for several states, as have the oscillator strengthsl0

Are High Heats of Atomization for Many Oxides and Fluorides of Z above 37 due to Enhanced Correlation Energy?

Heats of atomization Ha (per atom) are compared for for 28 diatomic oxides, 14 diatomic fluorides, 26 solid and 2 polyatomic gaseous oxides, 9 solid and 5 polyatomic gaseous fluorides. There is no universal trend toward lower Ha with increasing Z in a column of the Periodic Table; d-group, post-d-group, and partly filled 4f and 5f shells encourage smaller Ha but barium, lanthanum, lutetium, hafnium, tantalum, (to the largest extent thorium), and uranium show unexpected high Ha. This 'Barium-Thorium Effect' can be rationalized by known facts of correlation energy in atoms and (light) compounds, possibly including indirect relativistic effects due to distorted AO in LCAO.

Origin of the linear dependence with atomic number of correlation energies in neutral atoms

Using only the ground-state electron density, an interpretation is proposed, and supported by numerical calculations on the Kr and Xe atoms, for the approximately linear dependence of the correlation energies of neutral atoms with the nuclear charge Z. The results of the Colle-Salvetti correlation energy are in quantitative agreement with experiment in the range of nuclear charge Z ˇ- 18. The scaling of the Thomas-Fermi density with Z is invoked for larger values of Z.

Comparison and assessment of different forms of open shell perturbation theory

Existing methods for the calculation of the correlation energy of open shell atoms and molecules using Mo/ller–Plesset perturbation theory are discussed with special emphasis on recent advances in methods based on the restricted Hartree–Fock wave function. The convergence of the Mo/ller–Plesset series is examined to high order for a number of small systems and it is shown that some of the methods offer significant advantage over traditional unrestricted Mo/ller–Plesset theory. The optimized bond length and vibrational frequencies of CN, NO, and O2 are also compared for second and fourth order Mo/ller–Plesset theory, and again it is found that the same methods give much better results than the corresponding unrestricted results. Further supporting calculations are presented on the electron affinity of CN and the dissociation energy of HCN. The new methods involve relatively minor adaptation of existing unrestricted Mo/ller–Plesset (UMP) codes, and in the light of their superior performance are recommended instead of UMP theory for open shell molecules.

Erratum: Relativistic effect on total energies for determination of correlation energies of atoms from their experimental total energies [J. Chem. Phys. 91, 4738 (1989)

Erratum: Relativistic effect on total energies for determination of correlation energies of atoms from their experimental total energies [J. Chem. Phys. <b>91</b>, 4738 (1989)

Core correlation and the binding energy of Sc2

The dissociation energy of the Sc2 dimer has been computed using a multireference CI treatment and including the effects of core(3p) correlation. Special emphasis has been put on the contraction of the basis set to minimize atomic correlation energy loss and molecular superposition errors. Core(3p) correlation gives a contraction of the 4s orbital resulting in a reduced ratio between the radial extents of the 4s and 3d orbitals. This results in a decreased bond distance (by 0.20 a0) and improved 3d–3d overlap. The binding energy is increased by 0.11 eV through core correlation. The finally obtained De is 0.77 eV (1.1±0.2 eV expt.). An extension of the core-polarization potential formalism to include field-gradients and quadrupole core polarizabilities is presented and compared with the results of explicit core correlation for both the atom and Sc2.

Hierarchic computation of atomic correlation energies using a p-version finite element method

A new method is presented, based on a p-version finite element formalism, for calculating atomic second-order correlation energies using the Møller—Plesset approach. The method permits the definition of a hierarchic sequence of computations; this feature greatly simplifies the calculations via a simple adaptive technique. Results for He, Be and Ne are given and compared with the most accurate values available.

Atomic valence correlation energies from ab initio model potential calculations

Ab initio model potential calculations on X −, X, and X + (XF, Cl, Br, and I) using CISD wavefunctions have been performed in order to test a recent conclusion on the irrelevance of the valence orbital internal nodes in the values of atomic valence correlation energies. The results show that the ab initio model potential orbitals, showing the correct nodal structure, provide valence correlation energies much closer to the corresponding all-electron values than nodeless pseudo-orbitals do.

Rates of convergence of the partial-wave expansions of atomic correlation energies

The coefficients of the leading terms of the partial-wave expansion of atomic correlation energies in powers of (l+1/2)−1 are derived for the second- and third-order perturbed energies in the 1/Z expansion for all possible states of two-electron atoms, and for second-order Mo/ller–Plesset (many-body perturbation) theory for arbitrary n-electron atoms. The expressions for these coefficients given in Table I involve simple integrals over the zeroth-order wave functions (for the third order energies first-order wave functions are also involved). The leading term of E(2) goes as (l+1/2)−4 for natural parity singlet states, as (l+1/2)−6 for triplet states, and as (l+1/2)−8 for unnatural parity singlet states. There are no odd powers of (l+1/2)−1 present in E(2), and the coefficient of the (l+1/2)−6 term for natural parity singlet states of two-electron systems in the 1/Z expansion is generally −5/4 times the coefficient of the (l+1/2)−4 term. In E(3) there are terms that go as odd powers of (l+1/2)−1; the leading term is expressible in terms of the zeroth-order wave function and goes as (l+1/2)−5 for natural parity singlet states, as (l+1/2)−7 for triplet states, and as (l+1/2)−9 for unnatural parity singlet states. Numerical values of the coefficients for E(2) in two-electron atoms are given in Table II. The relation of our results to those obtained by Hill for variational calculations on the He ground state is discussed and generalizations of Hill’s formulas are conjectured. The unexpected results for unnatural parity singlet states are traced back to the behavior of their wave functions at the point of coalescence of two electrons. In terms of the relative coordinate of the two electrons they represent d waves, in contrast to s waves for natural parity singlet states and p waves for triplet states of either parity.

Density-functional approach to the calculation of correlation energies of two-electron atoms and ions

The correlation-energy density functional is studied here through an exact solution of a two-electron system bound by a harmonic-oscillator external potential. Since the correlation energy is a universal functional of density in density-functional theory, the same functional form is valid for two-electron atoms and ions as well. Using this fact we have proposed here density-functional schemes, within a local-density approximation as well as nonlocal considerations, for the study of correlation energies of two-electron atomic and ionic systems. The nonlocal scheme leads to significant improvement over the local ones, yielding results that show good agreement with the standard values.

Diagrammatic many-body perturbation expansion for atoms and molecules: VII experiments in vector and parallel processing for fourth-order energy terms involving triply excited intermediate states

The efficient evaluation of the fourth-order energy terms corresponding to linked diagrams involving triply excited intermediate states which arise in the many-body perturbation expansion for the electron correlation energy of closed-shell atoms and molecules on vector processing and parallel processing computers is discussed. The linked-diagram theorem decouples the many-electron problem allowing efficient implementation on parallel processing machines. Furthermore, the computation associated with each of the resulting subproblems is very well suited to vector processing machines. A “dynamic load balancing” technique is employed in both a dedicated and a multi-user environment. Timing tests are reported for the CRAY X-MP/416 computer. Execution rates in excess of 828 Mflops are observed.

An approximate local-density expression for the correlation energy of atoms

An approximation for the correlation energy of closed-shell atomic systems is derived in the framework of the density-functional method based on perturbation theory.

Atomic correlation energies from effective-core-potential and model-potential calculations

Canonical SCF orbitals from effective-core-potential and model-potential calculations were used in CISD calculations on electron correlation energies, electron affinities, and ionization energies of fluorine, chlorine, bromine, and iodine. Results obtained with the two different pseudo-potential approaches are very close to one another in spite of richer nodal structure of the model-potential orbitals. Contamination of higher virtual model-potential orbitals by core states was found to lead to anomalous behaviour of correlation energies.

Relativistic effect on total energies for determination of correlation energies of atoms from their experimental total energies

Relativistic effect Erel upon the total electronic energy of an atom is discussed with particular reference to obtaining the nonrelativistic total energy Eexact from the experimental total energy. Numerical values of this effect obtained by various authors by different nonempirical methods are compared for neutral atoms of rare-gas elements. It is shown that methods either of a Hartree–Fock-type or of a Dirac–Hartree–Fock-type give much the same Erel value for He through Ar. It is pointed out that Erel calculated with Hartree–Fock wave functions is not adequate for use in obtaining Eexact from the experimental total energy and that the Erel value calculated with wave functions including electron correlation should work well, although an actual demonstration can be done only for two-electron systems for lack of data. A semiempirical formula is therefore proposed, which is useful for least-squares fit of experimental total energies of isoelectronic series of atoms to extract nonrelativistic total energies along with the relativistic effect. From nonrelativistic energies thus derived, semiempirical values of correlation energies of atoms are obtained. The results thus obtained are in reasonable agreement with correlation energies derived by Clementi along somewhat different lines. The power series expansion in Z of the fitted formula for the He series shows that numerical values of expansion coefficients agree reasonably well with the corresponding values obtained by accurate relativistic and nonrelativistic Z expansion-type calculations.

Results obtained with the correlation energy density functionals of becke and Lee, Yang and Parr

Two recently published density functionals (A.D. Becke, J. Chem. Phys. 88 (1988) 1053 and C. Lee, W. Yang and R.G. Parr, Phys. Rev. B 37 (1988) 785) are used to calculate the correlation energies of first-row atoms, ions and molecules. The correlation contributions to ionization energies, electron affinities and dissociation energies thus obtained are of comparable quality to those of other density functionals.

Soft Coulomb hole for the Hartree-Fock model to estimate atomic correlation energies.

An empirical model for the correlation energy (${E}_{c}$) of atoms is proposed. The method, a perturbation applied to the Roothaan Hartree-Fock atomic functions, requires the reevaluation of the two-electron integrals due to the introduction of a ``soft'' Coulomb hole whose magnitude is calibrated semiempirically. We have tested it by computing the total correlation energy, the first ionization potentials for atoms helium through xenon, by considering excited states of the ground-state configuration for Z=3 to Z=18 and the electron affinity for fluorine, chlorine, bromine, and iodine ions.

Atomic correlation energy differences by means of a polarization potential.

Correlation energy differences have been evaluated using an empirical polarization potential for the two-, three-, and four-electron atom series. In contrast to local-density approximations the Z dependency of the correlation energies has been well reproduced. Moreover, the quantitative agreement with experimental values is surprisingly good. Deviations are smaller than 0.01 hartree.

Local and non-local spin density functional calculations of the correlation energy of atoms in molecules

Local (including self-correlation corrections) and non-local correlation energy functionals are used to calculate the correlation energy of atoms and diatomic molecules. The reliability of five schemes is tested for both correlation and dissociation energies. Recently developed spin-polarised correlation energy functionals incorporating gradient corrections are found to be generally superior for the properties examined. The theory of atoms in molecules is used to aid in the discussion of the trends in correlation and dissociation energies.

Dynamic correlation for MCSCF wave functions: An effective potential method

A method is suggested which allows the inclusion of dynamic correlation into CASSCF calculations. An effective Coulomb hole potential is added to the Hamiltonian. The potential has a simple form, which allows its implementation into existing LCAO programs using Gaussian integral packages. The parameters appearing in the potential are determined by fitting to empirical valence correlation energies for first row atoms. Calculations of ionization energies and electron affinities show considerable improvement compared to the MCSCF values. Test calculations on three molecules give the following results, H2∶ re=0.745 (0.741) A, De=4.62 (4.75) eV; N2∶ re=1.099 (1.098) A, De= 10.42 (9.91) eV; O2∶ re=1.198 (1.207) A, De=4.73 (5.21) eV. Experimental values within parenthesis.