Search

The connection between the random-phase approximation and the ring-coupled-cluster-doubles method bridges the gap between density-functional and wave-function theories and the importance of the random-phase approximation lies in both its broad applicability and this linking role in electronic-structure theory. In this contribution, we present an explicitly correlated approach to the random-phase approximation, based on the direct ring-coupled-cluster-doubles ansatz, which overcomes the problem of slow basis-set convergence, inherent to the random-phase approximation. Benchmark results for a test set of 106 molecules and a selection of 10 organic complexes from the S22 test set demonstrate that convergence to within 99% of the basis-set limit is reached for triple-zeta basis sets for atomisation energies, while quadruple-zeta basis sets are required for interaction energies. Corrections due to single excitations into the complementary auxiliary space reduce the basis-set incompleteness error by one order of magnitude, while contributions due to the coupling of conventional and geminal amplitudes are in general negligible. We find that a non-iterative explicitly correlated correction to first order in perturbation theory exhibits the best ratio of accuracy to computational cost. [ABSTRACT FROM AUTHOR]

Within the framework of density-functional theory, the correlation energy is computed in the random-phase approximation (RPA) using spinors obtained from a two-component relativistic Kohn-Sham calculation accounting for spin-orbit interactions. Ring-coupled-cluster equations are solved to obtain the two-component RPA correlation energy. Results are presented for the hydrides of the halogens Br, I, and At as well as of the coinage metals Cu, Ag, and Au, based on two-component relativistic exact-decoupling Kohn-Sham calculations. [ABSTRACT FROM AUTHOR]

Within the framework of density-functional theory, the basis-set convergence of energies obtained from the random-phase approximation to the correlation energy is equally slow as in wavefunction theory, as for example in coupled-cluster or many-body perturbation theory. Fortunately, the slow basis-set convergence of correlation energies obtained in the random-phase approximation can be accelerated in exactly the same manner as in wavefunction theory, namely by using explicitly correlated two-electron basis functions that are functions of the interelectronic distances. This is demonstrated in the present work. [ABSTRACT FROM AUTHOR]

To reduce remaining basis-set errors in the determination of molecular equilibrium geometries, a basis-set extrapolation (BSE) scheme is suggested for the forces used in geometry optimizations. The proposed BSE scheme is based on separating the Hartree-Fock and electron-correlation contributions and uses expressions obtained by straightforward differentiation of well established extrapolation formulas for energies when using basis sets from Dunning’s hierarchy of correlation-consistent basis sets. Comparison with reference data obtained at the R12 coupled-cluster level [CCSD(T)-R12] demonstrates that BSE significantly accelerates the convergence to the basis-set limit, thus leading to improvements comparable to or even better than those obtained by increasing the cardinal number in the used basis set by one. However, BSE alone is insufficient to improve agreement with experiment, even after additional consideration of inner-shell correlation and quadruple-excitation effects (mean error and standard deviation with extrapolation are -0.014 and 0.047 pm in comparison with mean error and standard deviation of -0.002 and 0.036 pm without extrapolation). Improvement is obtained only when other contributions of similar magnitude as the BSE contributions (e.g., pentuple-excitation effects and relativistic effects) are also considered. A rather large discrepancy (of the order of a few tenths of a picometer) is observed for the F2 molecule indicating an enhanced basis-set requirement for the various contributions in this case. [ABSTRACT FROM AUTHOR]

The analytic calculation of first-order properties has been implemented in the DALTON program at the level of explicitly correlated second-order Mo\ller-Plesset perturbation theory (MP2-R12). The implementation has been accomplished for MP2-R12 theory based on standard approximations A, A′, and B, using an auxiliary basis for the resolution-of-the-identity approximation, with and without a frozen core. MP2-R12 first-order molecular properties have been calculated analytically for a few small test molecules. For BH and HF, the MP2-R12 results were supplemented with explicitly correlated coupled-cluster calculations (but at this level from numerical derivatives) including vibrational and relativistic corrections. [ABSTRACT FROM AUTHOR]

The vertical ionisation energies of the molecules of the GW27 (27 molecules) and GW100 (100 molecules) test sets are computed in a polarised triple-zeta-valence basis set in the framework of coupled-cluster theory with single, double, and non-iterative triple substitutions. The molecular geometries were kept fixed to those of the two test sets. To demonstrate the usefulness of the coupled-cluster reference values, they are compared with quasi-particle energies obtained in the G0W0approximation for functionals commonly used in Kohn–Sham density-functional theory (DFT). Furthermore, an approximation is assessed, in which only exchange contributions are added to the DFT orbital energies. [ABSTRACT FROM PUBLISHER]

We present a new method for the evaluation of four-centre electron-repulsion integrals that combines density fitting (DF) with numerical integration by quadrature (QD). In this composite DF-QD approach, the orbital product density is fitted by a truncated spectral set of auxiliary functions and the fitting residual is expanded in a discrete space of functions localised at the eigenvalues of the coordinate operator. The new DF-QD approach has the advantage of DF of reducing computational effort and storage requirements of conventional quantum chemical algorithms, and allows for high accuracy and error control by means of QD. Systematic improvement of the accuracy is achieved for a chosen DF basis set by enlarging the number of grid points used for QD. Test calculations on a set of small molecules reveal that errors in absolute energy of the DF-QD approximation with less than 1000 and 3000 grid points per atom do not exceed one milli- and one microhartree, respectively, even if standard (non-optimised) AO basis sets are used for DF. [ABSTRACT FROM AUTHOR]

Abstract: We report a critical analysis and comparison of a variety of random-phase-approximation (RPA) based approaches to determine the electronic ground-state energy. Interrelations between RPA variants are examined by numerical examples with particular attention paid to the role of spin-flipped excitations and the behaviour of the adiabatic-connection integrands where appropriate. In general, it is found that RPA variants that include Hartree–Fock exchange contributions are unsuitable as generally applicable methods for the determination of correlation energies. Of the remaining methods only the direct RPA and RPA with second-order screened exchange are recommended for general use. [Copyright &y& Elsevier]

One-electron basis sets for F12 explicitly-correlated molecular electronic-structure methods are assessed by analysing the accuracy of Hartree-Fock energies and valence-only second-order correlation energies of a test set of 106 small molecules containing the atoms H, C, N, O and F. For these molecules, near Hartree-Fock-limit energies and benchmark second-order correlation energies (accurate to within 99.95% of the basis-set limit) are provided. Absolute energies are analysed as well as the Hartree-Fock and second-order correlation contributions to the atomisation energies of the molecules. Standard basis sets such as the Karlsruhe def2-TZVPP and def2-QZVPP sets and the augmented correlation-consistent polarised valence X-tuple zeta (aug-cc-p VXZ, X = D, T, Q, 5) sets are compared with the specialised cc-pVXZ-F12 (X = D, T, Q) sets that were recently optimised by Peterson and co-workers [J. Chem. Phys. 128, 084102 (2008)] for use in F12 theory. The results obtained from F12 explicitly-correlated molecular electronic-structure calculations are compared with those that are obtained by standard electronic-structure calculations followed by basis-set extrapolation based on the X-3 convergence behaviour of the aug-cc-pVXZ basis sets. The most important conclusions are that the cc-pVXZ-F12 sets are the preferred basis sets for F12 theory and that the X-3 extrapolation from the aug-cc-pVQZ and aug-cc-pV5Z is slightly more accurate than F12 theory in the cc-pVTZ-F12 basis but less accurate than F12 theory in the cc-pVQZ-F12 basis. [ABSTRACT FROM AUTHOR]

We review the current status of quantum chemistry as a predictive tool of chemistry and molecular physics, capable of providing highly accurate, quantitative data about molecular systems. We begin by reviewing wave-function based electronic-structure theory, emphasizing the N-electron hierarchy of coupled-cluster theory and the one-electron hierarchy of correlation-consistent basis sets. Following a discussion of the slow basis-set convergence of dynamical correlation and basis-set extrapolations, we consider the methods of explicit correlation, from the early work of Hylleraas in the 1920s to the latest developments in such methods, capable of yielding high-accuracy results in medium-sized basis sets. Next, we consider the small corrections to the electronic energy (high-order virtual excitations, vibrational, relativistic, and diagonal Born-Oppenheimer corrections) needed for high accuracy and conclude with a review of the composite methods and computational protocols of electronic-structure theory. [ABSTRACT FROM AUTHOR]

The ab initio calculation of molecular geometrical properties in the Hellmann-Feynman approximation is discussed in which the atomic orbitals are fixed at the positions of the nuclei at the reference geometry, thereby avoiding the calculation of derivatives of the molecular integrals with respect to the positions of the atomic orbitals. For the molecular gradient, the molecular Hessian, and the molecular dipole gradient, the convergence of the calculated properties is studied for a large number of basis sets at the Hartree-Fock level and at the CCSD(T)-R12 level. In the Hellmann-Feynman approximation, it is found to be necessary to impose explicitly rotational and translational invariance. Although small basis sets perform poorly in the Hellmann-Feynman approximation (compared with the standard approach where the atomic orbitals are moving with the displaced nuclei), satisfactory convergence is obtained for geometries and harmonic frequencies (to within 1% of the standard approximation) with the larger of the correlation-consistent core-valence cc-pCVXZ basis sets. For the infrared intensities, the agreement with the standard approach is still poor (only within 15% for the largest correlation-consistent basis). The best results are obtained with an R12 basis previously developed for the calculation of energies in the explicitly correlated R12 approximation. In this basis, the geometrical parameters and harmonic frequencies are within 0.5% of the standard approach and the infrared intensities within 5% , suggesting that the Hellmann-Feynman approximation may be useful for applications at the highly accurate MP2-R12, CCSD-R12, and CCSD(T)-R12 levels of theory. [ABSTRACT FROM AUTHOR]

The equations of the original ab initio scalar-relativistic zeroth-order regular approximation (ZORA) and the infinite-order regular approximation (IORA) are expanded in orders of 1/c[sup 2]. It is shown that previous ZORA/IORA implementations in ab initio quantum chemistry programs were not correct to order 1/c[sup 2], but contained imperfections leading to fictitious self-interactions. These errors can be avoided by adding exchange-type terms (coupling the large and small components) to the relativistic ZORA correction to the Hamiltonian, yielding improved ab initio relativistic zeroth- and infinite-order regular approximations that are correct to order 1/c[sup 2]. The new methods have been tested numerically by computing the total energies, orbital energies, and static electric dipole polarizabilities of the rare gas atoms He through Xe. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Abstract: This article is concerned with the calculation of the two-electron Darwin term (D2). At the level of explicitly correlated second-order perturbation theory (MP2-F12), the D2 term is obtained as an analytic energy derivative; at the level of explicitly correlated coupled-cluster theory, it is obtained from finite differences. To avoid the calculation of four-center integrals, a density-fitting approximation is applied to the D2 two-electron integrals without loss of accuracy, even though the absolute value of the D2 term is typically about 0.1mE h. Explicitly correlated methods provide a qualitatively correct description of the short-range region around the Coulomb hole, even for small orbital basis sets. Therefore, explicitly correlated wave functions remedy the otherwise extremely slow convergence of the D2 contribution with respect to the basis-set size, yielding more accurate results than those obtained by two-point basis-set extrapolation. Moreover, we show that the interference correction of Petersson’s complete-basis-set model chemistry can be used to compute a D2 basis-set correction at the MP2-F12 level to improve standard coupled-cluster singles-and-doubles results. [Copyright &y& Elsevier]

Abstract: In 1985 Kutzelnigg showed that a large percentage of the electron correlation energy for helium can be recovered using a single explicitly correlated basis function, chosen to fit the cusp at the correlation hole. In particular the simple wave function returned more than 80% of the correlation energy. In this paper we return to Kutzelnigg’s simple ansatz and remove the conventional double excitations in explicitly correlated CC2 theory (denoted as CCS(F12)), applying all established developments in modern F12 theory, such as replacing linear r 12 with f(r 12)=exp(−γr 12) and the use of auxiliary basis sets for the standard RI approximation in R12 theory. Analysing different approximations we show that in general the CCS(F12) approach yields 80–95% of the CC2 correlation energy, which is astonishingly large considering the small number and restricted form of the geminal basis functions. [Copyright &y& Elsevier]