Electron correlation described by extended geminal models: The EXGEM4 and EXGEM5 models

Within the framework of the general extended geminal model, two new approximate models EXGEM 4 and EXGEM 5 are introduced. The models are tested against full CI calculations on the water molecule for three different nuclear configurations and a full CI potential energy curve for the LiH molecule in the ground state. On the basis of these calculations, it is suggested that the models will yield electronic correlation energies with an accuracy of 1–2% of the corresponding full CI result.     

Electron correlation described by extended geminal models: The EXGEM7 and EXRHF3 models

Within the framework of the general extended geminal model, two new approximate models, EXGEM 7 and EXRHF 3, are introduced. Compared with previous models, these new models imply a more sophisticated approximation of the four-electron terms {?_KL}. The models are tested by calculations on the beryllium atom and the HF molecule.     

Gaussian geminal basis set optimization with crude SCF reference state

Gaussian geminal basis functions for second-order correlation energy calculations according to the Sinanogˇlu method are optimized with reference to rather crude SCF functions. The optimized geminal basis set is then used in a one-step calculation of the correlation energy with respect to the near-Hartree-Fock reference State. The numerical results for the beryllium atom indicate the usefulness of the proposed technique.     

Quintuple-zeta quality coupled-cluster correlation energies with triple-zeta basis sets

The explicitly-correlated coupled-cluster method CCSD(T)(R12) is extended to include F12 geminal basis functions that decay exponentially with the interelectronic distance and reproduce the form of the average Coulomb hole more accurately than linear-r(12). Equations derived using the Ansatz 2 strong orthogonality projector are presented. The convergence of the correlation energy with orbital basis set for the new CCSD(T)(F12) method is studied and found to be rapid, 98% of the basis set limit correlation energy is typically recovered using triple-zeta orbital basis sets. The performance for reaction enthalpies is assessed via a test set of 15 reactions involving 23 molecules. The title statement is found to hold equally true for total and relative correlation energies.     

孪函数N电子基组的组态

分析了各类孪函数Ｎ电子基组态展开式的特点以及它们对体系相关能的贡献，提出了一种在享函数Ｎ电子基矢下进行多组态自洽场计算时的组态选取方法，并依此方法在ＳＴＯ－６－３１Ｇ基组下对ＬｉＨ分子的基态能量做了计算，结果表明，用该组态选取方法只需选取少量的组态波函数便可得到相当精确的计算结果。     

A critical study of basis set effects and the use of approximate natural orbitals in SCF-CI calculations of molecular geometries and heats of reaction

SCF-CI calculations have been performed on a number of chemical reactions between closed shell molecules in order to determine the heats of reaction. Contracted Gaussian type atomic basis sets of three different qualities were used and the CI calculations were performed in a truncated approximate natural orbital space. The conclusions to be drawn from these calculations are rather pessimistic. For heats of reaction, errors up to 6 kcal/mole are obtained on the SCF-level with a double zeta plus polarization atomic basis. A further improvement is only possible if extended basis sets are used. Correlation effects on heats of reaction are of the same size and CI calculations are therefore only meaningful with large atomic basis sets.For the CI calculations a one-electron space of approximate natural orbitals, obtained from second order RS perturbation theory, was used. Different truncations, using the occupation number as criterion, were tested. The general conclusion is that errors in energy differences obtained with a truncated basis set are of the same magnitude as the error in the total correlation energy. In practice this means that not more than 20–30% of the approximate natural orbitals can be deleted if the error is to be kept less than a few kcal/mole.Finally the truncation error in calculations of bond distances was tested for a few cases. Errors of around 10% of the total change due to correlation were found when 30% of the lowest occupied natural orbitals were deleted.     

Local configuration interaction: An efficient approach for larger molecules

The first full implementation of the localized configuration interaction technique at the variational CI, CEPA-2 and variational CEPA levels is described. Timings are presented for a double-zeta plus polarization calculation on butadiene. The restriction of the correlation space to local basis functions results in a spectacular enhancement of the efficiency of the CI loop. The loss in the correlation energy is only a few percent; we argue that most of the loss is due to the exclusion of intramolecular basis set superposition artifacts.     

Approaching the full CI limit with MRD CI calculations: The X 1A1 state of water with a double-zeta basis

Results of MRD CI calculations with varying numbers of reference configurations for the water molecule employing a double-zeta basis set are compared with the corresponding full CI results of Harrison and Handy as well as with those of other methods. For the three geometries considered a highly uniform percentage (99.8±0.1%) of the available correlation energy in this AO basis is obtained by solving secular equations in the 13–15000 range, i.e. only 5% of the full CI space. Extrapolation of the full CI energy through the use of various correction formulae is found to be unreliable for large bond distances, although such an approach is successful at the equilibrium geometry.     

Die horizontale Korrelation in π-Elektronensystemen und ihre Beschreibung durch Elektronenpaarfunktionen

Different types of pair functions (geminal products and their linear combinations) are tested with respect to their ability to describe the “horizontal correlation” of the π-electrons of butadiene. The validity of the π-electron approximation is not discussed and “full configuration interaction” within the limited LCAO basis is used as the standard to which the model calculations are referred. An APSG-function (APSG = antisymmetrized product of strongly orthogonal geminals) built up from equivalent (localized) geminals, which contains only one variational parameter is able to account for about 90% of the “horizontal correlation energy”. Both APSG and APIG functions constructed from delocalized geminals, are much less favorable. Criteria of the goodness of an approximate wave function are a) the energy b) comparison of its one- and two-particle density matrices with those obtained from “full CI”. The good results with the localized APSG function are related to the fact that electron correlation between electrons of opposite spin is (in this molecule) essential only within either of the “double bonds” of the “canonical structure”. The pertinent results are quite insensitive to different parametrization of the integrals.     

An energy decomposition analysis of intermolecular interactions

Within the context of extended geminal models the concepts of charge centroids and charge ellipsoids of the geminal one-electron densities, and an energy decomposition of the intermolecular potential, are introduced as tools of analysis. The intermolecular potential can within this framework be written as a sum of the distortion energies of the subsystems and the interaction energies between the distorted subsystems. The interaction energy is further partitioned into a Coulombic, exchange and correlation contribution. Three classes of complexes are studied: hydrogen bonded systems (HF)₂, H₂OHF, (H₂O)₂; strongly bonded electron donor-acceptor (EDA) complexes: BH₃NH₃, BH₃CO; and weakly bonded EDA complexes: F₂NH₃, Cl₂NH₃ and ClFNH₃. The main results of the calculations, using basis sets consisting of [9s, 6p, 2d](Cl), [7s, 4p, 2d] (B, N, O, F), [4s, 2p](H) contracted Gaussian-type functions, and the numerical models EXRHF3 and EXGEM7, are as follows. The bonding in these complexes is essentially due to a lone pair of the donor subsystems approaching the vacant space in the vicinity of a nucleus of the acceptor system. The interaction energy is therefore dominated by the Coulombic term. However, the sum of the distortion terms is larger than the magnitude of the Coulombic term. Hence, the exchange and correlation terms give a substantial contribution to the intermolecular potential. If the components of the decomposition of the potential are resealed by using the magnitude of the interaction energy as the energy unit, a remarkable similarity between the three classes of complexes is disclosed.     

A natural linear scaling coupled-cluster method

It is shown that using an appropriate localized molecular orbital (LMO) basis, one is able to calculate coupled-cluster singles and doubles (CCSD) wave functions and energies for very large systems by performing full CCSD calculations on small subunits only. This leads to a natural linear scaling coupled-cluster method (NLSCC), in which total correlation energies of extended systems are evaluated as the sum of correlation energy contributions from individual small subunits within that system. This is achieved by defining local occupied orbital correlation energies. These are quantities, which in the LMO basis become transferable between similar molecular fragments. Conventional small scale existing molecular CCSD codes are all that is needed, the local correlation effect being simply transmitted via the appropriate LMO basis. Linear scaling of electronic correlation energy calculations is thus naturally achieved using the NLSCC approach, which in principle can treat nonperiodic extended systems of infinite basis set size. Results are shown for alkanes and several polyglycine molecules and the latter compared to recent results obtained via an explicit large scale LCCSD calculation. (c) 2004 American Institute of Physics.     

孪函数多电子基组在LiH分子方面的应用

选择从头算STO-6G基组,以孪函数Ⅳ电子基组计算LiH分子的基态(s=0)和激发态(s=1).结果表明,无需全部孪函数N电子基组,只要适当选择少数基组即可达到所需精度.     

Applicability of the supermolecule MP2 approach to intermolecular interactions: He2 and Ne2

Arguments are given for applying the full counterpoise correction to eliminate properly the basis set superposition error in the correlation energy. Comparison is made with the intermolecular perturbation theory and more complete treatments (MP4, CEPA). The results for the dimers studied suggest that MP2 is a poor approximation. Even if very extended basis sets are used, hardly more than 65% of the stabilization energy is recovered.     

Ground-state correlation energy of Ne

A series of basis sets and configuration interaction (CI ) wave functions, both of which were constructed so as to systematically approach to the complete set limit and the full CI limit, were used for the ground state of Ne. These calculations yielded an estimated correlation energy of ?0.3891 au, which is 99.6% of a recent theoretical estimate of ?0.3905 au. The CI value, ?0.3821 au, was obtained by SDCI calculation with seven reference configurations by using Slater-type orbitals (STO s) from s to h functions. © 1994 John Wiley & Sons, Inc.     

Counterpoise correction is not useful for short and Van der Waals distances but may be useful at long range

This article investigates the errors in supermolecule calculations for the helium dimer. In a full CI calculation, there are two errors. One is the basis set superposition error (BSSE), the other is the basis set convergence error (BSCE). Both of the errors arise from the incompleteness of the basis set. These two errors make opposite contributions to the interaction energies. The BSCE is by far the largest error in the short range and larger than (but much closer to) BSSE around the Van der Waals minimum. Only at the long range, the BSSE becomes the larger error. The BSCE and BSSE largely cancel each other over the Van der Waals well. Accordingly, it may be recommended to not include the BSSE for the calculation of the potential energy curve from short distance till well beyond the Van der Waals minimum, but it may be recommended to include the BSSE correction if an accurate tail behavior is required. Only if the calculation has used a very large basis set, one can refrain from including the counterpoise correction in the full potential range. These results are based on full CI calculations with the aug-cc-pVXZ (X = D, T, Q, 5) basis sets.     

Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

SCF-CI studies of the equilibrium structure and the proton transfer barrier H3O 2 −

Large-scale configuration interaction (CI) calculations have been performed in order to study the effect of the correlation energy on the equilibrium geometrical structure, the stability, and on the energy barrier of the proton transfer reaction in the hydrogen bonded system HO^– · HOH. An extended Gaussian basis set including polarization functions on each nuclear centre has been employed to approximate the molecular Orbitals. All possible single and double replacements resulting from a single determinant Hartree-Fock reference state have been taken into account in the CI wavefunction. Compared to the SCF results the equilibrium oxygen/oxygen distance has been obtained from the CI calculations to be smaller by about 0.08 Å and the correlation energy has been found to stabilize the composed system by 3.6 kcal/mole. An almost symmetric equilibrium structure with the hydrogen bonding H-atom midway between the two oxygen centres has been obtained in the CI treatment, whereas SCF calculations yield an asymmetric geometrical configuration with a small energy barrier of 1.4 kcal/mole for the proton transfer process.     

Configuration interaction benchmark for Be ground state

A set of 432 energy-optimized Slater-type radial orbitals together with spherical harmonics up to ? = 30 is used to approximate the corresponding full configuration interaction (CI) expansion for Be ground state. An analysis of radial and angular patterns of convergence for the energy yields a basis set incompleteness error of 8.7 μhartree of which 85% comes from radial basis truncations for ? ≤ 30. Select-divide-and-conquer CI (Bunge in J Chem Phys 125:014107, 2006; Bunge and Carbó-Dorca in J Chem Phys 125:014108, 2006) produces an energy upper bound 0.02(1) μhartree above the full CI limit. The energy upper bound E = ?14.6673473 corrected with these two truncation energy errors yields E = ?14.6673560 a.u. (Be) in fair agreement with the latest explicitly correlated Gaussian results of E = ?14.66735646 a.u. (Be). The new methods employed are discussed. It is acknowledged that at this level of accuracy traditional atomic CI has reached a point of diminishing returns. Modifications of conventional (orbital) CI to seek for significantly higher accuracy without altering a strict one-electron orbital formalism are proposed.     

N-band Hubbard models for copper oxides and isoelectronic systems. New models for organic and organometallic magnetic conductors and superconductors

The electronic structures of undoped and doped copper oxides and other related oxides are investigated on the basis of the N-band Hubbard models. The Hubbard Hamiltonians for clusters of transition metal oxides are exactly diagonalized by the full valence-bond (VB ) configuration interaction (CI) method in order to elucidate populations of doped holes, electronic excitation energies, etc. Possible mechanisms of the high-T_c superconductivity for oxide superconductors are discussed on the basis of the calculated results, together with available experiments. The analysis of correlation and spin correlation effects on doped copper oxides indicates theoretical possibilities of new models for organic and organometallic magnetic conductors and superconductors. Organic and organometallic analogs to copper oxides are therefore proposed on the basis of these results.