Wim Klopper

BASIS-SET CONVERGENCE OF CORRELATED CALCULATIONS ON WATER

The basis-set convergence of the electronic correlation energy in the water molecule is investigated at the second-order Mo/ller–Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2 mEh by means of (1) extrapolations from sequences of calculations using correlation-consistent basis sets and (2) from explicitly correlated calculations employing terms linear in the interelectronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a+bX−3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calculations involving as many as 492 atomic orbitals are reported.

BASIS-SET CONVERGENCE IN CORRELATED CALCULATIONS ON NE, N2, AND H2O

Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.

The Dalton quantum chemistry program system

Dalton is a powerful general‐purpose program system for the study of molecular electronic structure at the Hartree–Fock, Kohn–Sham, multiconfigurational self‐consistent‐field, Møller–Plesset, configuration‐interaction, and coupled‐cluster levels of theory. Apart from the total energy, a wide variety of molecular properties may be calculated using these electronic‐structure models. Molecular gradients and Hessians are available for geometry optimizations, molecular dynamics, and vibrational studies, whereas magnetic resonance and optical activity can be studied in a gauge‐origin‐invariant manner. Frequency‐dependent molecular properties can be calculated using linear, quadratic, and cubic response theory. A large number of singlet and triplet perturbation operators are available for the study of one‐, two‐, and three‐photon processes. Environmental effects may be included using various dielectric‐medium and quantum‐mechanics/molecular‐mechanics models. Large molecules may be studied using linear‐scaling and massively parallel algorithms. Dalton is distributed at no cost from http://www.daltonprogram.org for a number of UNIX platforms.

Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory

The matrix elements needed in a CI‐SD, CEPA, MP2, or MP3 calculation with linear r12‐dependent terms for closed‐shell states are derived, both exactly and in a consistent approximate way. The standard approximation B guarantees that in the atomic case the error due to truncation of the basis at some angular momentum quantum number L goes as ∼L−7, at variance with L−3 in conventional calculations (without r12 terms). Another standard approximation A has errors ∼L−5, but is simpler and—for moderate basis sets—somewhat better balanced. The explicit expressions for Mo/ller–Plesset perturbation theory of second and third order with linear r12 terms (MP2‐R12 and MP3‐R12, respectively) are explicitly given in the two standard approximations.

Basis-set convergence of the energy in molecular Hartree–Fock calculations

Abstract The basis-set convergence towards the numerical limit of the Hartree–Fock total energy and binding energy is investigated for the correlation-consistent cc-pVXZ basis sets. For both energies, solid improvements are obtained with each increment in X . The basis-set errors for the total energy (Δ E ) fit an exponential form better than a power form, and the total energy is better fitted than the binding energy. It is difficult to find generally reliable extrapolation schemes for the total energy. In most cases, the most successful scheme gives results extrapolated beyond a given X that are comparable to the cc-pV(X+1)Z results, but occasionally it fails dramatically for large X . Indeed, explicit calculation of the energy in a larger basis set, especially the cc-pV6Z set for which Δ E ⩽0.1 mE h , gives the most reliable estimate of the basis-set limit.

R12 methods in explicitly correlated molecular electronic structure theory

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: Kutzelniggs 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(−αr 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. Contents PAGE 1. Introduction 429  1.1. The origin of the problem 429  1.2. Two-electron systems 430  1.3. Explicitly correlated MP2 methods 430  1.4. Gaussian geminals 431  1.5. Exponentially correlated Gaussians 432  1.6. The transcorrelated method 433 2. R12 wavefunctions 433  2.1. Definition 434  2.2. Correlation factors 435  2.3. Projection operators 437  2.4. Levels of theory 439  2.5. Methods for open shells 440 3. Approximations of many-electron integrals 441  3.1. Exact evaluation 442  3.2. Approximations: GBC, EBC and 443  3.3. Resolution of the identity 445  3.4. Numerical quadrature 447  3.5. Density fitting 449  3.6. DF combined with RI 451 4. Examples from second-order perturbation theory 452  4.1. Technical details 453  4.2. R12 results in comparison with extrapolated values 454  4.3. Comparison between R12 and F12 results 458 5. Perspectives 461  5.1. Higher level methods 461  5.2. Local approximations 461  5.3. Conclusions 462   5.3.1. Correlation factor 462   5.3.2. Projection operator 462   5.3.3. Formulation of intermediate B 463   5.3.4. Approximating integrals 463   5.3.5. Efficiency improvements 463 Acknowledgements 463 References 464

Explicitly correlated second-order Møller–Plesset methods with auxiliary basis sets

In explicitly correlated Moller–Plesset (MP2-R12) methods, the first-order wave function is expanded not only in terms of products of one-electron functions—that is, orbitals—but also in terms of two-electron functions that depend linearly on the interelectronic coordinates rij. With these functions, three- and four-electron integrals occur, but these integrals can be avoided by inserting a resolution of the identity (RI) in terms of the one-electron basis. In previous work, only one single basis was used for both the electronic wave function and the RI approximation. In the present work, a new computational approach is developed that uses an auxiliary basis set to represent the RI. This auxiliary basis makes it possible to employ standard basis sets in explicitly correlated MP2-R12 calculations.

Basis set convergence of the interaction energy of hydrogen-bonded complexes

The Hartree-Fock and correlation contributions to the interaction energy of the hydrogen-bonded complexes (HF)2, (HCl)2, H2OHF, HCNHF, and (H2O)2 are computed in conventional calculations employing the aug-cc-pVXZ series of basis sets at the levels of Hartree-Fock theory, second-order perturbation theory, and coupled-cluster theory with single and double excitations augmented by a perturbative triples correction. The basis set convergence of the interaction energy is examined by comparison with results obtained with an explicitly correlated wave function model. The counterpoise-corrected and uncorrected Hartree-Fock interaction energies both converge very unsystematically. The convergence of the uncorrected correlation contribution is also very unsystematic because the basis set superposition error and the error from the incomplete description of the electronic Coulomb cusp both are present. Once the former has been effectively removed by the counterpoise correction, the cusp dominates and the convergence...

Explicitly Correlated Electrons in Molecules

Explicitly Correlated Electrons in Molecules Christof H€attig, Wim Klopper,* Andreas K€ohn, and David P. Tew Lehrstuhl f€ur Theoretische Chemie, Ruhr-Universit€at Bochum, D-44780 Bochum, Germany Abteilung f€ur Theoretische Chemie, Institut f€ur Physikalische Chemie, Karlsruher Institut f€ur Technologie, KIT-Campus S€ud, Postfach 6980, D-76049 Karlsruhe, Germany Institut f€ur Physikalische Chemie, Johannes Gutenberg-Universit€at Mainz, D-55099 Mainz, Germany School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom

Møller-plesset calculations taking care of the correlation CUSP

Abstract Moller-Plesset calculations to second order have been carried out on the ten-electron systems Ne, HF and H 2 O with a new functional, including r 12 -dependent pair correlation functions, which takes care of the correlation cusp. The calculated second-order pair energies are accurate to within a few millihartree in comparison with the estimated exact values. In particular, second-order energies of 384.2, 380.1 and 362.9 m E h , have been obtained for Ne, HF and H 2 O respectively.