Attractive electron–electron interactions within robust local fitting approximations

An analysis of Dunlap's robust fitting approach reveals that the resulting two‐electron integral matrix is not manifestly positive semidefinite when local fitting domains or non‐Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four‐center two‐electron integrals based on the resolution‐of‐the‐identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair‐atomic resolution‐of‐the‐identity (PARI) approach, atomic‐orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree–Fock and Kohn–Sham calculations, the indefinite integral matrix causes nonconvergence in the self‐consistent‐field iterations. In these cases, the two‐electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb‐metric RI method. The speed‐up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple‐zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky‐decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations. © 2013 Wiley Periodicals, Inc.     

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Software news and updates. Basis-set completeness profiles in two dimensions

A two-electron basis-set completeness profile is proposed by analogy with the one-electron profile introduced by D. P. Chong (Can J Chem 1995, 73, 79). It is defined as Y(alpha, beta) = sigmam sigman (Galpha(1)Gbeta(2)/(1/r12)/ psim(1)psin(2)) (psim(1)psin(2)/r12/Galpha(1)Gp(2)) and motivated by the expression for the basis-set truncation correction that occurs in the framework of explicitly correlated methods (Galpha is a scanning Gaussian-type orbital of exponent alpha and [psim] is the orthonormalized one-electron basis under study). The two-electron basis-set profiles provide a visual assessment of the suitability of basis sets to describe electron-correlation effects. Furthermore, they provide the opportunity to assess the quality of the basis set as a whole--not only of the individual angular momentum subspaces, as is the case for the one-electron basis-set profiles. The two-electron completeness profiles of the cc-pVXZ (X = D, T, Q), aug-cc-pVTZ, cc-pCVTZ, and SVP-auxiliary basis sets for the carbon atom are presented as illustrative examples.     

Orbital connections for perturbation-dependent basis sets

Summary The use of perturbation-dependent basis sets is analysed with emphasis on the connection between the basis sets at different values of the perturbation strength. A particular connection, the natural connection, that minimizes the change of the basis set orbitals is devised and the second quantization realization of this connection is introduced. It is shown that the natural connection is important for the efficient evaluation of molecular properties and for the physical interpretation of the terms entering the calculated properties. For example, in molecular Hessian calculations the natural connection reduces the size of the relaxation term, leading to faster convergence of the response equations. The physical separation of the terms also means that first-order non-adiabatic coupling matrix elements can be obtained in a very simple way from a molecular Hessian calculation.     

Second-order Møller-Plesset calculations on the water molecule using Gaussian-type orbital and Gaussian-type geminal theory

The Gaussian-type orbital and Gaussian-type geminal (GGn) model is applied to the water molecule, at the level of second-order M?ller-Plesset (MP2) theory. In GGn theory, correlation factors are attached to all doubly-occupied orbital pairs (GG0), to all doubly-occupied and singly-excited pairs (GG1), or to all orbital pairs (GG2). Optimizing the GG2 model using a weak-orthogonality functional, we obtain the current best estimate of the all-electron MP2 correlation energy of water, -361.95 mE(h). In agreement with previous observations, the GG1 model performs almost as well as the GG2 model (-361.26 mE(h)), whereas the GG0 model is poorer (-351.36 mE(h)). For the barrier to linearity of water, we obtain an MP2 correlation contribution of -463 +/- 5 cm(-1).     

Electron correlation: the many-body problem at the heart of chemistry

The physical interactions among electrons and nuclei, responsible for the chemistry of atoms and molecules, is well described by quantum mechanics and chemistry is therefore fully described by the solutions of the Schr?dinger equation. In all but the simplest systems we must be content with approximate solutions, the principal difficulty being the treatment of the correlation between the motions of the many electrons, arising from their mutual repulsion. This article aims to provide a clear understanding of the physical concept of electron correlation and the modern methods used for its approximation. Using helium as a simple case study and beginning with an uncorrelated orbital picture of electronic motion, we first introduce Fermi correlation, arising from the symmetry requirements of the exact wave function, and then consider the Coulomb correlation arising from the mutual Coulomb repulsion between the electrons. Finally, we briefly discuss the general treatment of electron correlation in modern electronic-structure theory, focussing on the Hartree-Fock and coupled-cluster methods and addressing static and dynamical Coulomb correlation.     

Accurate quantum-chemical calculations using Gaussian-type geminal and Gaussian-type orbital basis sets: applications to atoms and diatomics

We have implemented the use of mixed basis sets of Gaussian one- and two-electron (geminal) functions for the calculation of second-order M?ller-Plesset (MP2) correlation energies. In this paper, we describe some aspects of this implementation, including different forms chosen for the pair functions. Computational results are presented for some closed-shell atoms and diatomics. Our calculations indicate that the method presented is capable of yielding highly accurate second-order correlation energies with rather modest Gaussian orbital basis sets, providing an alternative route to highly accurate wave functions. For the neon atom, the hydrogen molecule, and the hydrogen fluoride molecule, our calculations yield the most accurate MP2 energies published so far. A critical comparison is made with established MP2-R12 methods, revealing an erratic behaviour of some of these methods, even in large basis sets.     

Theoretical studies of nuclear magnetic resonance parameters for the proton-exchange pathways in porphyrin and porphycene

The nuclear magnetic resonance (NMR) parameters in porphyrin and porphycene have been calculated to investigate their changes during the process of proton exchange, using density-functional theory (DFT) for both the spin-spin coupling constants and the shielding constants. In addition, in calculations on the smaller 1,3-bis(arylimino)isoindoline molecule, we have tested the performance of our computational approach against experimental data. The calculated nuclear spin-spin coupling constants and shielding constants have been analyzed as functions of the progress of the proton transfer between two nitrogen atoms. The one-bond couplings between proton and nitrogen, dominated by the Fermi-contact term, decay steeply as the internuclear distance increases. The small changes in the intramolecular J(HH) coupling between two inner protons are mainly determined by the sum of relatively large spin-orbit terms. The isotropic shielding constant shows a strong deshielding of the nitrogen nuclei as the proton migrates away. Both the isotropic shielding of the exchanged protons and the shielding anisotropy exhibit a minimum close to the transition states.