Hartree-Fock exchange fitting basis sets for H to Rn

For elements H to Rn (except Lanthanides), a series of auxiliary basis sets fitting exchange and also Coulomb potentials in Hartree–Fock treatments (RI-JK-HF) is presented. A large set of small molecules representing nearly each element in all its common oxidation states was used to assess the quality of these auxiliary bases. For orbital basis sets of triple zeta valence and quadruple zeta valence quality, errors in total energies arising from the RI-JK approximation are below ∼1 meV per atom in molecular compounds. Accuracy of RI-JK-approximated HF wave functions is sufficient for being used for post-HF treatments like Møller–Plesset perturbation theory, MP2. Compared to nonapproximated treatments, RI-JK-HF leads to large computational savings for quadruple zeta valence orbital bases and, in case of small to midsize systems, to significant savings for triple zeta valence bases. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2008     

Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn

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.     

Accurate orbital amplitudes at nuclei frog gaussian basis sets

A simple extrapolation procedure combining wavefunctions obtained from gaussian basis sets with exact solutions of the nuclear cusp equations is proposed for computing orbital amplitudes at nuclei. Comparison with exact results for atoms and diatomic molecules indicates that the procedure is capable of giving Hartree—Fock amplitudes with errors of at most 10^?2 for the low amplitude outer orbitals and errors of less than 10^?3 for the important inner orbitals. The resulting errors in the total densities are around 10^?2. These accuracies are comparable with those obtained with energy-optimized Slater basis sets.     

Accurate extrapolation of electron correlation energies from small basis sets

A new two-point scheme is proposed for the extrapolation of electron correlation energies obtained with small basis sets. Using the series of correlation-consistent polarized valence basis sets, cc-pVXZ, the basis set truncation error is expressed as deltaE(X) proportional, variant(X + xi(i))(-gamma). The angular momentum offset xi(i) captures differences in effective rates of convergence previously observed for first-row molecules. It is based on simple electron counts and tends to values close to 0 for hydrogen-rich compounds and values closer to 1 for pure first-row compounds containing several electronegative atoms. The formula is motivated theoretically by the structure of correlation-consistent basis sets which include basis functions up to angular momentum L = X-1 for hydrogen and helium and up to L = X for first-row atoms. It contains three parameters which are calibrated against a large set of 105 reference molecules (H, C, N, O, F) for extrapolations of MP2 and CCSD valence-shell correlation energies from double- and triple-zeta (DT) and triple- and quadruple-zeta (TQ) basis sets. The new model is shown to be three to five times more accurate than previous two-point schemes using a single parameter, and (TQ) extrapolations are found to reproduce a small set of available R12 reference data better than even (56) extrapolations using the conventional asymptotic limit formula deltaE(X) proportional, variantX(-3). Applications to a small selection of boron compounds and to neon show very satisfactory results as well. Limitations of the model are discussed.     

Accurate estimation of correlation energies using locally dense basis sets

Locally dense basis sets using the mixed 6-311G(d, p)/3-21G basis can be used to reproduce total energies and correlation energies after empirical adjustment to 2–4 kcal/mol for a variety of small and medium size molecules containing hydrog en, carbon, and oxygen. Post-Hartree\–Fock methods can be calculated faster by this method by factors of 2–3, in general, and higher in the presence of high molecular symmetry; density functional approaches take longer and are impractical in th e locally dense basis set approach. It is shown that the correlation energy in two of the better characterized density functional approaches is generally significantly larger than that of the post-Hartree–Fock treatments studied here and appears to be insensitive to the basis set employed. © 1996 by John Wiley & Sons, Inc.     

Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy

Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 molecules representing (nearly) all elements-except lanthanides-in their common oxidation states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, density functional theory and correlated methods, for which we had chosen M?ller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.     

Polarized basis sets of Slater-type orbitals: H to Ne atoms

We present three Slater-type atomic orbital (STO) valence basis (VB) sets for the first and second row atoms, referred to as the VB1, VB2, and VB3 bases. The smallest VB1 basis has the following structure: [3, 1] for the H and He atoms, [5, 1] for Li and Be, and [5, 3, 1] for the B to Ne series. For the VB2 and VB3 bases, both the number of shells and the number of functions per shell are successively increased by one with respect to VB1. With the exception of the H and Li atoms, the exponents for the VB1 bases were obtained by minimizing the sum of the Hartree-Fock (HF) and frozen-core singles and doubles configuration interaction (CISD FC) energies of the respective atoms in their ground state. For H and Li, we minimized the sum of the HF and CISD FC energies of the corresponding diatoms (i.e., of H(2) or Li(2)) plus the ground-state energy of the atom. In the case of the VB2 basis sets, the sum that was minimized also included the energies of the positive and negative ions, and for the VB3 bases, the energies of a few lowest lying excited states of the atom. To account for the core correlations, the VBx (x = 1, 2, and 3) basis sets for the Li to Ne series were enlarged by one function per shell. The exponents of these extended (core-valence, CV) basis sets, referred to, respectively, as the CVBx (x = 1, 2, and 3) bases, were optimized by relying on the same criteria as in the case of the VBx (x = 1, 2, and 3) bases, except that the full CISD rather than CISD FC energies were employed. We show that these polarized STO basis sets provide good HF and CI energies for the ground and excited states of the atoms considered, as well as for the corresponding ions.     

Correlating basis sets for the H atom and the alkali-metal atoms from Li to Rb

 Contracted Gaussian-type function sets are developed for correlating p, d, and f functions for a valence electron of the hydrogen atom and alkali-metal atoms from Li to Rb. A segmented contraction scheme is used for its compactness and efficiency. Contraction coefficients and exponents are determined by minimizing the deviation from the K orbitals of the atoms. The present basis sets yield an accuracy comparable to the correlation-consistent basis set for the hydrogen atom and also give a similar high accuracy for the alkali-metal atoms. In the calculations of spectroscopic constants of alkali hydrides, the decontraction of the p function plays an important role, especially for LiH. The contributions of d and f functions are nontrivial for KH and RbH. Received: 6 September 2002 / Accepted: 13 November 2002 / Published online: 19 March 2003 Acknowledgements. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. Correspondence to: T. Noro e-mail: tashi@sci.hokudai.ac.jp     

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.     

Automatically generated Coulomb fitting basis sets: design and accuracy for systems containing H to Kr

For intermediate sized chemical systems the use of an auxiliary basis set (ABS) to fit the charge density provides a useful means of accelerating the performance of various quantum chemical methods. As a consequence much effort has been devoted to the design of various ABSs. This paper explores a fundamentally new approach where the ABS is created dynamically based on the specific orbital basis set (OBS) being used. The new approach includes a parameter that is used to coalesce candidate fitting functions together but which can also be used to provide some coarse grain control over the number of functions in the ABS. The accuracy of the new automatically generated ABS (auto-ABS) is systemically studied for a variety of small systems containing the elements H-Kr. Errors in the Coulomb energy computed using auto-ABS and with a variety of OBSs are shown to be small compared to errors in the Hartree-Fock energy due to incompleteness in the OBS. In contrast to fixed size ABSs, the use of auto-ABS is shown to lead to smaller errors as the size (quality) of the OBS is expanded. The performance of auto-ABS is also compared with the use of the recently proposed universal fitting sets [Weigend, Phys. Chem. Chem. Phys. 8, 1057 (2006)] for 180 compounds containing atoms from H to Kr.     

Accurate atomic Gaussian basis functions for second-row atoms: Small split-valence 3-21SP and 4-22SP basis sets

Small split-valence Gaussian 3-21SP and 4-22SP basis sets, previously reported for the first-row atoms [Chem. Phys. Lett., 229 , 151 (1996)], have been extended for the second-row elements of the Periodic Table. The total energies of the ground states of the second-row atoms calculated with the new basis sets are significantly lower than those obtained with the well-known 3-21G (J. Am. Chem. Soc., 104 , 2797 (1982)] and 4-31G [J. Chem. Phys., 56 , 5255 (1972)] basis sets. This is because, as first noted in our previous work for first-row atoms, that the 3-21G and 4-31G basis sets only correspond to a local minimum of the Hartree–Fock energy functional, which is relatively far from its global minimum. The proposed basis sets have been tested by performing geometry optimizations and calculations of normal frequencies in the harmonic approximation of some diatomic and polyatomic molecules at the Hartree–Fock level. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1200–1210     

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.     

Relativistic Gaussian basis sets for the atoms hydrogen to neon

Gaussian basis sets for use in relativistic molecular calculations are developed for atoms and ions with one to ten electrons. A relativistic radial wavefunction coupled to an angular function of l-symmetry is expanded into a linear combination of spherical Gaussians of the form r ^l exp (–r ²). One set of basis functions is used for all large and small components of the same angular symmetry. The expansion coefficients and the orbital exponents have been determined by minimizing the integral over the weighted square of the deviation between the Dirac or Dirac-Fock radial wavefunctions and their analytical approximations. The basis sets calculated with a weighting function inversely proportional to the radial distance are found to have numerical constants very similar to those of their energy-optimized non-relativistic counterparts. Atomic sets are formed by combining l-subsets. The results of relativistic and non-relativistic calculations based on these sets are analyzed with respect to different criteria, e.g. their ability to reproduce the relativistic total energy contribution and the spin-orbit splitting. Contraction schemes are proposed.Dedicated to Prof. Dr. A. Neckel on occasion of his 60th birthday     

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.     

Accurate electric multipole moments for HCN and HCP from CCSD(T) calculations with large Gaussian basis sets

Summary Accurate values of the electric multipole moments of HCN and HCP have been obtained from self-consistent field (SCF) and coupled-cluster (CCSD(T)) calculations. With the origin at the centre of mass and hydrogen along the positive molecular axis in both systems, a [9s5p2d/10s7p5d3f/10s7p5d3f] basis set is expected to predict near-Hartree-Fock values for the dipole (=1.2962ea ₀), quadrupole (=2.1046ea ₀ ² ), octopole (=10.088ea ₀ ³ ) and the hexadecapole (=24.23ea ₀ ⁴ ) moment of HCN. An analogous basis set, [9s5p2d/10s7p5d3f/14s11p7d3f], predicts SCF values of =0.1421ea ₀, =3.8786ea ₀ ² , =19.633ea ₀ ³ and =65.89ea ₀ ⁴ for HCP. Electron correlation reduces the dipole moment of HCN but increases the dipole moment of HCP. At the CCSD(T) level of theory the calculated values are =1.1800ea ₀, =1.6461ea ₀ ² , =9.762ea ₀ ³ and =22.45ea ₀ ⁴ for HCN and =0.1710ea ₀, =3.2312ea ₀ ² , =16.578ea ₀ ³ and =60.87ea ₀ ⁴ for HCP.     

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.