Optimal diffuse augmented atomic basis sets for extrapolation of the correlation energy

We seek correlation-consistent diffuse-augmented double-zeta and triple-zeta basis sets that perform optimally in extrapolating the correlation energy to the one-electron complete basis set limit, denoted oAVXZ and oAV(X + d)Z. The novel basis sets are method-dependent in that they are trained to perform optimally for the correlation energy at each specific level of theory. They are shown to yield accurate results in calculating both the energy and tensorial properties such as polarizabilities while not significantly altering the Hartree-Fock energy. Quantitatively, complete basis set limit (CBS)-/(oAVdZ,oAVtZ)-extrapolated correlation energies typically outperform, by 3- to 5-fold, the ones calculated with traditional ansatzes of similar flexibility. Attaining energies of CBS/(AVtZ,AVqZ) type or better accuracy, they frequently outperform expensive raw explicitly correlated ones. Promisingly, a limited test on CBS-extrapolated energies based on conventional basis sets has shown that they compare well even with extrapolated explicitly correlated ones. Calculated atomization and dissociation energies, molecular geometries, ionization potentials, and electron affinities also tend to outperform the ones obtained with traditional Dunning's ansatzes from which the new basis sets have been determined. The method for basis set generation is simple, and there is no reason of principle why the approach could not be adapted for handling other bases in the literature.     

Economical geometry optimization in first row molecules with bond function augmented basis sets

Ab initio MO geometry optimization studies on a number of molecules containing first row atoms show that the use of gaussian bond functions in conjunction with the standard 4-31G basis sets enables geometry calculations closely approaching the accuracy of calculations using more extensive basis sets.     

Overcomplete multicenter basis sets

The distribution of eigenvalues of the overlap matrix is analyzed for large and, in the limit, infinite overcomplete multicenter basis sets. It is shown that in the limit the eigenvalues become infinitely degenerate. A numerical example is given and discussed in terms of a semiclassical distribution in phase space.     

Comments on relativistic basis sets

 Adding the tight and diffuse Gaussian-type functions (GTFs), Faegri's variationally determined double-zeta-quality basis sets for molecular relativistic calculations are examined. An example atom is Cm. When the tight s-type GTF is added the total energy increases, whereas when diffuse GTFs are added the total energy decreases. The reasons for these findings are clarified. It is also pointed out that not only the Faegri's sets but also other variationally determined basis sets would show similar behavior so far as the expansion terms are not sufficient. Received: 22 July 2002 / Accepted: 21 October 2002 / Published online: 31 January 2003 Correspondence to: H. Tatewaki e-mail: htatewaki@nsc.nagoya.cu.ac.jp     

Relativistic well-tempered gaussian basis sets

The effectiveness of well-tempered Gaussian basis sets has been tested for the atoms He-Ar, Ag, and Xe. It has been found that orbital exponents optimized through the well-tempered scheme for non-relativistic atoms can be carried over to relativistic calculations to produce wavefunctions close to the relativistic Hartree-Fock limit.     

Uniform quality gaussian basis sets for molecular calculations. III. Charge optimized basis sets

Gradient optimized constrained (2s ≠ 2p) and unconstrained (2s ≠ 2p) Gaussian 3G basis sets are reported for the first-row atoms and ions X^O, for Q = ?2 to +4. Analytic equations have been fitted to the logarithm of the exponents as a function of the nuclear charge Z and formal charge Q. Consequently only two parameters Z and Q have to be specified in order to completely define a basis set.     

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.     

Gaussian basis sets for molecular applications

The choice of basis set in quantum chemical calculations can have a huge impact on the quality of the results, especially for correlated ab initio methods. This article provides an overview of the development of Gaussian basis sets for molecular calculations, with a focus on four popular families of modern atom‐centered, energy‐optimized bases: atomic natural orbital, correlation consistent, polarization consistent, and def2. The terminology used for describing basis sets is briefly covered, along with an overview of the auxiliary basis sets used in a number of integral approximation techniques and an outlook on possible future directions of basis set design. © 2012 Wiley Periodicals, Inc.     

On the quality of basis sets

An analysis of the quality of a given basis set is presented in terms of the three fundamental parts of total energy. This analysis clearly displays three types of error occasionally present in the components of total energy. As an illustration of the application of these concepts, several STO and Gaussian basis sets for the Ni atom are analyzed.     

Spatially restricted Double Z-Simplified Box Orbital basis sets: Optimization and comparison with some standard basis sets

As part of previous studies, we introduced a new type of basis function named Simplified Box Orbitals (SBOs) that belong to a class of spatially restricted functions which allow the zero differential overlap (ZDO) approximation to be applied with complete accuracy. The original SBOs and their Gaussian expansions SBO-3G form a minimal basis set, which was compared to the standard Slater-type orbital basis set (STO-3G). In the present paper, we have developed the SBO basis functions at double-zeta (DZ) level, and we have assessed the option of expanding the SBO-DZ as a combination of Gaussian functions. Finally, we have determined the quality of the new basis set by comparing the molecular properties calculated with SBO-nG with those achieved with some standard basis sets.     

Relativistic effects determined using the Douglas-Kroll contracted basis sets and correlation consistent basis sets with small-core relativistic pseudopotentials

The coupled cluster approximation with single, double, and quasiperturbative triple excitations [CCSD(T)] was used in combination with the Douglas-Kroll contracted correlation consistent basis sets [cc-pVnZ-DK, where n = D(2), T(3), Q(4), and 5] and small-core relativistic pseudopotentials (PP) with correlation consistent polarized valence basis sets (cc-pVnZ-PP and aug-cc-pVnZ-PP) to investigate the impact of scalar relativistic corrections on energetic and structural properties of small molecules containing third-row (Ga-Kr) atoms. These molecules were taken from the Gaussian-2 extended test set for third-row atoms. Atomization energies, ionization energies, electron affinities, and proton affinities for molecules in the test set were determined and compared with nonrelativistic results which were obtained in a recent study in which the standard and augmented correlation consistent basis sets were used in combination with CCSD(T). Several schemes were used to extrapolate the energies to the complete basis set limit.     

Auxiliary basis sets for density-fitted correlated wavefunction calculations: weighted core-valence and ECP basis sets for post-d elements

We report optimised auxiliary basis sets for the resolution-of-the-identity (or density-fitting) approximation of two-electron integrals in second-order M?ller-Plesset perturbation theory (MP2) and similar electronic structure calculations with correlation-consistent basis sets for the post-d elements Ga-Kr, In-Xe, and Tl-Rn. The auxiliary basis sets are optimised such that the density-fitting error is negligible compared to the one-electron basis set error. To check to which extent this criterion is fulfilled we estimated for a test set of 80 molecules the basis set limit of the correlation energy at the MP2 level and evaluated the remaining density-fitting and the one-electron basis set errors. The resulting auxiliary basis sets are only 2-6 times larger than the corresponding one-electron basis sets and lead in MP2 calculations to speed-ups of the integral evaluation by one to three orders of magnitude. The density-fitting errors in the correlation energy are at least hundred times smaller than the one-electron basis set error, i.e. in the order of only 1-100 μH per atom.     

Atomic orbital basis sets for molecular interactions

We describe a general approach to the parametrization of linear combinations of Gaussian atomic orbitals, useful for atomic and molecular interactions. We use a Gaussian transform method and Gauss-Legendre quadratures to express hydrogenic atomic orbitals, with varying effective charges, in terms of Gaussian-type orbitals. This procedure provides well-defined rules for calculating exponent factors and combination coefficients of the linear combinations of Gaussians in problems where nuclear distances may vary over large ranges during interactions. © 1994 by John Wiley & Sons, Inc.     

Quality of contracted Gaussian-type function basis sets

The valence quality of contracted (C) Gaussian-type function (GTF) basis sets in molecular calculations is discussed for the first- through fourth-row atoms. The split-valence basis sets derived from minimal-type CGTF sets are compared with those derived from primitive (P) GTF sets. Using F, Cl, Br, and I atoms and their homonuclear diatomics as test species, we find that the split-valence CGTF sets have almost the same quality as PGTF sets with larger s and p expansion terms: for example, the (53/5), (533/53), (5333/533/5), and (53 333/5333/53) CGTF sets correspond approximately to the [9/5], [15/9], [19/15/5], and [22/18/7] PGTF sets for the first- to fourth-row atoms, respectively, where the slash separates the s, p, and d symmetries. For the main group atoms of the four rows, we recommend using the above-mentioned CGTFs or larger.     

High-quality Gaussian basis sets for fourth-row atoms

Summary Energy-optimized Gaussian basis sets of triple-zeta quality for the atoms Rb-Xe have been derived. Two series of basis sets are developed; (24s 16p 10d) and (26s 16p 10d) sets which we expand to 13d and 19p functions as the 4d and 5p shells become occupied. For the atoms lighter than Cd, the (24s 16p 10d) sets with triple-zeta valence distributions are higher in energy than the corresponding double-zeta distribution. To ensure a triple-zeta distribution and a global energy minimum the (26s 16p 10d) sets were derived. Total atomic energies from the largest basis sets are between 198 and 284E _H above the numerical Hartree-Fock energies.     

Small gaussian basis sets for second-row atoms

6s-type and 4p-type gaussian basis sets are obtained for the second row atoms by fitting, using a least squares criterion, to 12s-type and 9p-type gaussian basis sets which are close to the self-consistent field atomic orbital wave functions. The small gaussian expansions are considered to be more suited for molecular calculations using double basis sets. The differences between these sets and the 10s-type, 6p-type and 9s-type, 5p-type are analysed. For molecular calculations using single gaussian basis sets the 10s-type and 6p-type would seem to be the best compromise.

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Zusammenfassung Ein Basissatz von Gaußfunktionen vom 6s- bzw. 4p-Typ für Atome der zweiten Reihe wird erhalten, indem die Funktionen mit Hilfe des Kriteriums der kleinsten quadratischen Abweichung einem Satz von Gaußfunktionen vom 12s- bzw. 9p-Typ angepaßt werden; dabei ist der letztgenannte Satz der selbstkonsistenten Wellenfunktion aus Atomorbitalen stark angenähert. Die kürzeren Entwicklungen nach Gaußfunktionen werden für geeigneter bei Berechnungen mit zweifachen Basissätzen gehalten. Die Unterschiede zwischen diesen Sätzen und solchen vom 10s- bzw. 6p-Typ sowie vom 9s- und 5p-Typ werden untersucht. Für Molekülrechnungen mit einfachen Basissätzen von Gaußfunktionen scheint der Satz vom 10s- bzw. 6p-Typ den besten Kompromiß darzustellen.

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Résumé On obtient des bases gaussiennes de type 6s et 4p pour les atomes de la seconde ligne par ajustement selon un critère de moindre carré à des bases gaussiennes de type 12s et 9p proches des orbitales atomiques SCF. Les petits développements en gaussiennes sont plus adaptés à des calculs moléculaires en bases doubles. Analyse des différences entre cas bases et les bases de types 10s et 6p, 9s et 5p. Pour des calculs moléculaires à base simple, 10s et 6p semble le meilleur compromis.

     

Simple extended STO basis sets. Helium

The two-parameter function, φ = (C₁ + C₂r^n?1) exp (?ζr), (n = 2–5), has been used as a basis function to determine the independent particle model energy of two-electron atomic systems in their ground state. The best energy is found for n = 3 (He—B³⁺) and for n = 4 (H^?). Our energy values are significantly close to Hartree-Fock results.     

Molecular integrals over generalized gaussian basis sets

A method is given for obtaining the common molecular integrals over generalized gaussian functions: The present algorithms are expected to be more efficient than those given in earlier work by the same author.