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The nonperturbative calculation of molecular properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London atomic orbitals (LAOs). With these orbitals, the calculation of four-centre electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resolution-of-the-identity (RI) approximation for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approximation is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-centre integrals in the RI approximation remain real-valued, and the three-centre integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calculation of four-centre ERIs over LAOs, using the RI approximation thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an additional factor of at least two. By using other well-established methods such as Cauchy–Schwarz screening, the difference-density approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calculations in magnetic fields can be increased even further. [ABSTRACT FROM AUTHOR]

We have investigated the correlation factors exp(-ζr12), r12 exp(-ζr12), erfc(ζr12), and r12 erfc(ζr12) in place of the linear-r12 term for use in explicitly correlated electronic-structure methods. The accuracy obtained with all of these correlation factors is significantly greater than that obtained with the plain correlation factor r12. Polarization functions that are more diffuse than those of standard basis sets give even better results. The correlation factor exp(-ζr12) is very close to the optimum correlation factor for helium and outperforms the others. [ABSTRACT FROM AUTHOR]

A systematic way to construct compact complementary auxiliary basis sets (ABSs) from and for atomic natural orbital basis sets for the use as resolution-of-the-identity sets in explicitly correlated F12 calculations is presented. The performance of these newly constructed ABSs has been benchmarked in explicitly correlated calculations of absolute energies, ionisation potentials and electron affinities. [ABSTRACT FROM PUBLISHER]

Explicitly correlated coupled-cluster theory has developed into a valuable computational tool for the calculation of electronic energies close to the limit of a complete basis set of atomic orbitals. In particular at the level of coupled-cluster theory with single and double excitations (CCSD), the space of double excitations is quickly extended towards a complete basis when Slater-type geminals are added to the wave function expansion. The purpose of the present article is to demonstrate the accuracy and efficiency that can be obtained in computational thermochemistry by a CCSD model that uses such Slater-type geminals. This model is denoted as CCSD(F12), where the acronym F12 highlights the fact that the Slater-type geminals are functions f( r12) of the interelectronic distances r12 in the system. The performance of explicitly correlated CCSD(F12) coupled-cluster theory is demonstrated by computing the atomization energies of 73 molecules (containing H, C, N, O, and F) with an estimated root-mean-square deviation from the values compiled in the Active Thermochemical Tables of σ = 0.10 kJ/mol per valence electron. To reach this accuracy, not only the frozen-core CCSD basis-set limit but also high-order excitations (connected triple and quadruple excitations), core–valence correlation effects, anharmonic vibrational zero-point energies, and scalar and spin–orbit relativistic effects must be taken into account. [ABSTRACT FROM AUTHOR]

The analytical computation of nuclear gradients has been derived and implemented for the explicitly correlated second-order Møller-Plesset method (MP2-F12). The implementation has been accomplished in the TURBOMOLE program package for ansatz MP2-F12/2*A. A Slater-type geminal expanded in six Gaussian geminals (STG-6G), a complementary auxiliary basis set (CABS), and robust density fitting approximations are used. In addition, a second-order perturbation theory correction for single excitations into the complementary auxiliary basis set (CABS singles) is included to reduce the Hartree-Fock error. Smooth convergence towards the basis set limit is observed for a selection of molecules. For computations on dimers of weakly interacting molecules in small basis sets, explicitly correlated second-order Møller-Plesset theory outperforms conventional second-order Møller-Plesset theory because basis set superposition errors are largely avoided at the MP2-F12/2*A level. [ABSTRACT FROM AUTHOR]

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]

The introduction of the resolution-of-the-identity (RI) approximation for electron repulsion integrals in quantum chemical calculations requires in addition to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Møller–Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys Chem Chem Phys, 2005, 7, 3297–3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb–Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resolution-of-the-identity calculations with second-order Møller–Plesset perturbation theory (MP2) and related methods can now be performed for molecules containing elements from H to Rn. The error of the RI approximation has been evaluated for a test set of 385 small and medium sized molecules, which represent the common oxidation states of each element, and is compared with the one-electron basis set error, estimated based on highly accurate explicitly correlated MP2–R12 calculations. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table. [ABSTRACT FROM AUTHOR]

The past few years have seen a particularly rich period in the development of the explicitly correlated R12 theories of electron correlation. These theories bypass the slow convergence of conventional methods, by augmenting the traditional orbital expansions with a small number of terms that depend explicitly on the interelectronic distance r 12 . Amongst the very numerous discoveries and developments that we will review here, two stand out as being of particular interest. First, the fundamental numerical approximations of the R12 methods withstand the closest scrutiny: Kutzelnigg's use of the resolution of the identity and the generalized Brillouin condition to avoid many-electronic integrals remains sound. Second, it transpires that great gains in accuracy can be made by changing the dependence on the interelectronic coordinate from linear ( r 12 ) to some suitably chosen short-range form (e.g., exp(-α r 12 )). Modern R12 (or F12) methods can deliver MP2 energies (and beyond) that are converged to chemical accuracy (1  kcal/mol) in triple- or even double-zeta basis sets. Using a range of approximations, applications to large molecules become possible. Here, the major developments in the field are reviewed, and recommendations for future directions are presented. By comparing with commonly used extrapolation techniques, it is shown that modern R12 methods can deliver high accuracy dramatically faster than by using conventional methods. [ABSTRACT FROM AUTHOR]

In a previous contribution, we established the suitability of density functional theory (DFT) for the calculation of molecular anharmonic force fields. In the present work, we have assessed a wide variety of basis sets and exchange-correlation functionals for harmonic and fundamental frequencies, equilibrium, and ground-state rotational constants, and thermodynamic functions beyond the rigid rotor-harmonic oscillator (RRHO) approximation. The fairly good performance of double-zeta plus polarization basis sets for frequencies results from an error compensation between basis set incompleteness and the intrinsic error of exchange-correlation functionals. Triple-zeta plus polarization basis sets are recommended, with an additional high-exponent d function on second-row atoms. All conventional hybrid generalized gradient approximation (GGA) functionals perform about equally well: high-exchange hybrid GGA and meta-GGA functionals designed for kinetics yield poor results, with the exception of of the very recently developed BMK functional, which takes a middle position along with the HCTH/407 (second-generation GGA) and TPSS (meta-GGA) functionals. Second-order Møller–Plesset perturbation theory (MP2) performs similarly to these functionals but is inferior to hybrid GGAs such as B3LYP and B97-1. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [ABSTRACT FROM AUTHOR]

The second-order Møller-Plesset (MP2) correlation energies of the valence shells of C 2 H 4 and C 2 H 6 (with fixed nuclear coordinates) were computed as accurately as technically feasible. The pair energies for singlet and triplet pairs are reported for localized molecular orbitals. An accuracy of the order of 0.01m E h is achieved for all of the individual pair energies. The best estimates of the total valence-shell MP2 correlation energies of C 2 H 4 and C 2 H 6 in the chosen geometries are -373.6±0. and -410.0±0.1m E h , respectively. In the aug-cc-pVQZ, aug-cc-pV5Z and aug-cc-pV6Z basis sets, 96.2, 98.0 and 98.8% of the these values are recovered, respectively. The percentages are improved to 99.8% by means of spin-adapted two-point extrapolations from the aug-cc-pVQZ and aug-cc-pV5Z results. An extrapolation based on the aug-cc-pV5Z and aug-cc-pV6Z correlation energies yields 99.9%. [ABSTRACT FROM AUTHOR]

We present the extension of a recently proposed universal explicit electron correlation (F12) correction for multi-reference perturbation theories to general complete active spaces and arbitrary choices of complete active space self-consistent field (CASSCF) orbitals. This F12 correction is applied to Mukherjee's multi-reference second-order perturbation theory (Mk-MRPT2). Pilot examples show the expected reduction of the basis sets incompleteness error of about two cardinal numbers. [ABSTRACT FROM AUTHOR]