Atomic Correlation Energies Research Articles

Estimation of atomic correlation energies from the electron density at the nucleus and atomic additivity of the correlation energy in molecules

A new method based on electron density at the nucleus is presented to determine electron correlation energy in atomic species. Using the nuclear-electron coalescence cusp condition, we have shown that the atomic correlation energy can be well represented by different power of atomic number for different isoelectronic series. The computed correlation energies have been achieved chemical accuracy with average absolute deviation in the order of 1 kcal/mol for 17 isoelectronic series. Also it is shown that the correlation energy for a molecule can be expressed in terms of the correlation energies of its constituent atoms and its number of electrons. The proposed formula confirms that in a molecule the electron correlation is larger than the sum over electron correlations of its corresponding constituent atoms.

Linear density response function within the time-dependent exact-exchange approximation

We have calculated the frequency-dependent exact-exchange (EXX) kernel of time-dependent (TD) density-functional theory employing our recently proposed computational method based on cubic splines. With this kernel we have calculated the linear density response function and obtained static polarizabilites, van der Waals coefficients, and correlation energies for all spherical spin-compensated atoms up to argon. Some discrete excitation energies have also been calculated for Be and Ne. As might be expected, the results of the TDEXX approximation are close to those of TD Hartree-Fock theory. In addition, correlation energies obtained by integrating over the strength of the Coulomb interaction turn out to be highly accurate.

Energy density analysis for second‐order Møller‐Plesset perturbation theory and coupled‐cluster theory with singles and doubles: Application to C2H4CH4 complexes

Grid-based energy density analysis (EDA), in which numerical integration is performed for two-electron integrals by the pseudospectral method (Imamura et al., J. Chem. Phys. 2007, 126, 034103), is extended to correlated methods: second-order Møller-Plesset (MP2) perturbation and coupled-cluster singles and doubles (CCSD). Using EDA for MP2 and CCSD, we estimate atomic correlation energy differences and correlation energy density difference maps for C(2)H(4)--CH(4) complexes. The analyses confirm that polarization and diffuse functions essentially contribute to the descriptions of weak interaction around the nuclei and in the area between C(2)H(4) and CH(4), respectively.

The principle-quantum-number (and the radial-quantum-number) expansion of the correlation energy of two-electron atoms

The second-order correlation energy of two-electron ions is studied in terms of an expansion in minimal approximations to the first-order natural orbitals (NOs). The non-linear parameters of these NOs are determined by minimization of the second-order energy. An approximation to the total second-order correlation energy is obtained as a sum of increments e(lp), depending on the angular quantum number l and the radial quantum number p. (Either l or p can be eliminated in favor of the principal quantum number n = l + p.) Closed expressions for these energy increments are derived. For fixed p the increments go as (l + 1)(-5). This is consistent with the behavior of the exact partial wave increments (that depend on the parameter l only) as (l + 1/2)(-4). While the partial wave increments correspond to a summation of e(lp) over p, other partial summations of the two-parameter increments lead to either the principal-quantum-number expansion (PQNE) with energy increments approximately n(-4), or the radial-quantum-number expansion, with a less transparent convergence pattern. Unfortunately these partial summations can neither be done in closed form nor from the asymptotic expansion, but some insight is obtained from a numerical summation. The hope to find a rigorous derivation of the PQNE has not been fulfilled.

Multireference many-electron correlation energies from two-electron reduced density matrices computed by solving the anti-Hermitian contracted Schrödinger equation

Two-electron reduced density matrices (2-RDMs) have recently been directly calculated by solving the anti-Hermitian contracted Schr\"odinger equation (ACSE) to obtain $95$--$100\phantom{\rule{0.3em}{0ex}}%$ of the ground-state correlation energy of atoms and molecules with the accuracy increasing with the size of the one-electron basis set [Mazziotti, Phys. Rev. Lett. 97, 143002 (2006).] In this paper, the ACSE method is extended to treat strong multireference correlation effects that are often important at nonequilibrium molecular geometries. While previous ACSE calculations have employed an initial 2-RDM from the Hartree-Fock method, we initialize the solution of the ACSE with a 2-RDM guess from a multiconfiguration self-consistent field calculation. Applications are made to multireference correlation in the potential energy surfaces of the molecules $\mathrm{HF}$, ${\mathrm{H}}_{2}\mathrm{O}$, and ${\mathrm{C}}_{2}$ in polarized valence double-zeta basis sets while ${\mathrm{N}}_{2}$ is treated in polarized valence double- and triple-zeta basis sets. Accurate ground-state energies and 1-RDM occupation numbers are obtained at both equilibrium and nonequilibrium geometries where the energies are within a few millihartrees of those from full configuration interaction.

A relationship between atomic correlation energy and Tsallis entropy

AbstractA new relationship between electron correlation energy and Tsallis entropy is presented. This relationship is a generalization of previous equations which correlate the atomic correlation energy and the Shannon entropy. The results, relatively to the atoms with atomic number 2 < Z < 29, put in evidence the crucial role of the p‐parameter in terms of representation of the long‐range interaction contribution in the correlation energy. Moreover, the p‐values, which reproduce the experimental values of the correlation energy, indicate that the atomic wave functions are more localized with respect to those calculated in the limit of p → 1. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

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.

Atomic Correlation Energy from the Electron Density at the Nucleus

Ground-state atomic correlation energies, and their kinetic energy and potential energy components, are shown to be well-represented by empirical formulas of the form CNrho(0)Z(-gamma), where C and gamma are constants that are largely invariant within various sets of atoms and positive ions, Z is the atomic number, N is the number of electrons, and rho(0) is the electron density at the nucleus. Results are given for neutral atoms, singly charged positive ions, and many isoelectronic series-315 atomic species in all.

Extrapolating to the one-electron basis-set limit in electronic structure calculations

A simple, yet reliable, scheme based on treating uniformly singlet-pair and triplet-pair interactions is suggested to extrapolate atomic and molecular electron correlation energies calculated at two basis-set levels of ab initio theory to the infinite one-electron basis-set limit. The novel dual-level method is first tested on extrapolating the full correlation in single-reference coupled-cluster singles and doubles energies for the closed-shell systems CH2((1)A1), H2O, HF, N2, CO, Ne, and F2 with correlation-consistent basis sets of the type cc-pVXZ (X=D,T,Q,5,6) reported by Klopper [Mol. Phys. 6, 481 (2001)] against his own benchmark calculations with large uncontracted basis sets obtained from explicit correlated singles and doubles coupled-cluster theory. Comparisons are also reported for the same data set but using both single-reference Moller-Plesset and coupled-cluster doubles methods. The results show a similar, often better, accordance with the target results than Klopper's extrapolations where singlet-pair and triplet-pair energies are extrapolated separately using the popular X(-3) and X(-5) dual-level laws, respectively. Applications to the extrapolation of the dynamical correlation in multireference configuration interaction calculations carried out anew for He, H2, HeH+, He2 ++, H3+(1 (1)A'), H3+(1 (3)A'), BH, CH, NH, OH, FH, B2, C2, N2, O2, F2, BO, CO, NO, BN, CN, SH, H2O, and NH3 with standard augmented correlation-consistent basis sets of the type aug-cc-pVXZ (X=D,T,Q,5,6) are also reported. Despite lacking accurate theoretical or experimental data for comparison in the case of most diatomic systems, the new method also shows in this case a good performance when judged from the results obtained with the traditional schemes which extrapolate using the two largest affordable basis sets. For the Hartree-Fock and complete-active space self-consistent field energies, a simple pragmatic extrapolation rule is examined whose results are shown to compare well with the ones obtained from the best reported schemes.

Two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation: Enhanced energies and properties with larger basis sets

Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.

Anti-Hermitian part of the contracted Schrödinger equation for the direct calculation of two-electron reduced density matrices

A recent advance in the theory of the contracted Schroedinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM's) that yield 95%-100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schroedinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM's, the role of Nakatsuji's theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including H{sub 2}O, CH{sub 2}, NH{sub 4}{sup +}, HF, and N{sub 2}, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-{zeta} basis set. The computed 2-RDM's very closely satisfy known N-representability conditions.

Convergence of third order correlation energy in atoms and molecules

We have investigated the convergence of third order correlation energy within the hierarchies of correlation consistent basis sets for helium, neon, and water, and for three stationary points of hydrogen peroxide. This analysis confirms that singlet pair energies converge much slower than triplet pair energies. In addition, singlet pair energies with (aug)-cc-pVDZ and (aug)-cc-pVTZ basis sets do not follow a converging trend and energies with three basis sets larger than aug-cc-pVTZ are generally required for reliable extrapolations of third order correlation energies, making so the explicitly correlated R12 calculations preferable.

Relevance of the Slowly Varying Electron Gas to Atoms, Molecules, and Solids

We present a Thomas-Fermi-inspired density scaling under which electron densities of atomic, molecular, or condensed matter become both large and slowly varying, so that semiclassical approximations and second-order density gradient expansions are asymptotically exact for the kinetic and exchange energies. Thus, even for atoms and molecules, density-functional approximations should recover the universal second-order gradient expansions in this limit. We also explain why common generalized gradient approximations for exchange do not.

OEP atomic-structure calculations with the renormalized Møller–Plesset correlation energy

We perform the atomic-structure calculations by means of the optimized effective potential (OEP) method. The expression for the correlation energy is developed on the basis of the density functional perturbation theory. The third- and higher-order perturbation terms are renormalized into the constant factor multiplied by the Møller–Plesset second-order (MP2) perturbation energy. Exchange and correlation energies for He and Ne atoms are much improved as compared with those of the conventional local density approximation and those of the OEP-MP2 calculations.

Ab initio asymptotic-expansion coefficients for pair energies in MP2 perturbation theory for atoms

The leading asymptotic expansion (AE) coefficients for two partial-wave (PW) expansions in powers of (l+1/2)−1 are derived for the second-order symmetry-adapted pair (SAP) energies defining the second-order Møller–Plesset (MP2) correlation energies for closed-shell N-electron atoms. In the main expansion, the PW increments (PW/m) are directly defined by the pair of angular momenta of the orbitals used in their computation. In the auxiliary expansion the PW increments (PW/a) are defined by the order, l, of the Legendre polynomial Pl (cos ϑ12) in the expansion of the pair function. Our approach differs in several respects from that put forward a decade ago by Kutzelnigg and Morgan, who derived the leading AE coefficients in the PW/a expansion for all states of He-like ions within the framework of the 1/Z expansion perturbation theory. However, we use their results concerning many technical details. It is shown that the expressions for the AE coefficients of the main expansion can be obtained from those for the He-like ions if some terms are multiplied by simple rational factors and the radial integrals are expressed in terms of Hartree–Fock orbitals defining the pair instead of hydrogen-like wave functions. To obtain the formulae for the AE coefficients for the main expansion, we have derived expressions relating the PW/m and PW/a energy increments. The AE coefficients for the main expansion are calculated for the Ne atom and compared with their counterparts determined from computed MP2 energy increments. Very close agreement is found.

Towards benchmark second-order correlation energies for large atoms. II. Angular extrapolation problems

We have studied the use of the asymptotic expansions (AEs) for the angular momentum extrapolation (to l --> infinity) of atomic second-order Moller-Plesset (MP2) correlation energies of symmetry-adapted pairs (SAPs). The AEs have been defined in terms of partial wave (PW) increments to the SAP correlation energies obtained with the finite element MP2 method (FEM-MP2), as well as with the variational perturbation method in a Slater-type orbital basis. The method employed to obtain AEs from PW increments is general in the sense that it can be applied to methods other than MP2 and, if modified, to molecular systems. Optimal AEs have been determined for all types of SAPs possible in large atoms using very accurate FEM PW increments up to lmax = 45. The impact of the error of the PW increments on the coefficients of the AEs is computed and taken into account in our procedure. The first AE coefficient is determined to a very high accuracy, whereas the second involves much larger errors. The optimum l values (lopt) for starting the extrapolation procedures are determined and their properties, interesting from the practical point of view, are discussed. It is found that the values of the first AE coefficients obey expressions of the type derived by Kutzelnigg and Morgan [J. Chem. Phys. 96, 4484 (1992); 97, 8821(E) (1992)] for He-type systems in the bare-nucleus case provided they are modified by fractional factors in the case of triplet and unnatural singlet SAPs. These expressions give extremely accurate values for the first AE coefficient both for the STO and the FEM Hartree-Fock orbitals. We have compared the performance of our angular momentum extrapolations with those of some of the principal expansion extrapolations performed with correlation consistent basis sets employed in the literature and indicated the main sources of inaccuracy.

A family of intracules, a conjecture and the electron correlation problem

We present a radical approach to the calculation of electron correlation energies. Unlike conventional methods based on Hartree-Fock or density functional theory, it is based on the two-electron phase-space information in the Omega intracule, a three-dimensional function derived from the Wigner distribution. Our formula for the correlation energy is isomorphic to the Hartree-Fock energy expression but requires a new type of four-index integral. Preliminary results, obtained using a model that is based on the known correlation energies of small atoms, are encouraging.

Compact and efficient basis sets of s- and p-block elements for model core potential method

We propose compact and efficient valence-function sets for s- and p-block elements from Li to Rn to appropriately describe valence correlation in model core potential (MCP) calculations. The basis sets are generated by a combination of split MCP valence orbitals and correlating contracted Gaussian-type functions in a segmented form. We provide three types of basis sets. They are referred to as MCP-dzp, MCP-tzp, and MCP-qzp, since they have the quality comparable with all-electron correlation consistent basis sets, cc-pVDZ, cc-pVTZ, and cc-pVQZ, respectively, for lighter atoms. MCP calculations with the present basis sets give atomic correlation energies in good agreement with all-electron calculations. The present MCP basis sets systematically improve physical properties in atomic and molecular systems in a series of MCP-dzp, MCP-tzp, and MCP-qzp. Ionization potentials and electron affinities of halogen atoms as well as molecular spectroscopic constants calculated by the best MCP set are in good agreement with experimental values.

Towards benchmark second-order correlation energies for large atoms: Zn2+ revisited

To provide very accurate reference results for the second-order Møller-Plesset (MP2) energy and its various components for Zn(2+), which plays for 3d-electron systems a similar role as Ne for smaller atoms and molecules, we have performed extensive calculation by two completely different implementations of the MP2 method: the finite element method (FEM) and the variation-perturbation (VP) method. The FEM and VP calculations yield partial wave contributions up to l(max)=45 and 12, respectively. Detailed comparison of all FEM and VP energy components for l(max)=12 has disclosed an extraordinary similarity, which justifies using the present results as benchmarks. The present correlation energies are compared with other works. The dependability of an earlier version of FEM, already applied to very large closed-shell atoms, is confirmed. It has been found that for larger atoms the accuracy of the analytical Hartree-Fock results has an impact on the accuracy of the MP2 energies greater than for smaller atoms. Fields of applications of the present results in studies of various electron correlation effects in 3d-electron atoms and molecules are indicated.

Correlation energies of light atoms related to pairing between antiparallel spinelectrons

Early theoretical work emphasized the gross trend of thetotal correlation energy of neutral atoms with atomic numberZ. Thiswas later improved quantitatively, by focusing on the number of pairings betweenantiparallel-spin electrons in the same main shell (i.e., K, L, M...shells).Here, this viewpoint is pressed, and correlation energies associated with s–s, s–pand p–p pairings are extracted. The s–s energy turns out to be rather insensitiveto a change from K to L shells, but a small increase is found to occur in the Mshell. Also, the s–p and p–p pairing energies show a small increase from the L tothe M shells. The s–s and s–p energies are ≈1 eV, thes–s contribution being the larger. The p–p energy is ≈1/3 eV.The effect of removing or adding one electron to the neutral atom has also beenanalysed. Wavefunction overlap and wavefunction localization appear to controlthe values of the pairing energies.