Supplementary d and f functions in molecular wave functions at large and small internuclear separations

The CO, CO₂, CS, CIF, and SO₂ molecules were used to test the dependence of supplementary d and f function exponents to changes in bond lengths and bond angles in MO calculations utilizing Gaussian basis sets in Hartree–Fock and Moller–Plesset calculations. Using Dunning–Hay double zeta basis sets, optimizations were performed at internuclear separations from 100–200 pm and beyond. The energy cost of not reoptimizing d function exponents when bonds are stretched or compressed is much smaller for correlated calculations than for those at the Hartree–Fock level and is greatest at the lower end of the range of internuclear distances. The problem is much less serious at all levels when multiple sets of d functions are used. © 1993 John Wiley & Sons, Inc.     

The energetic effects of p,d, and f Gaussian polarization functions on closed-shell AHn oxygen and sulfur hydrides

A series of LCAO -MO -SCF calculations, using various basis sets of Gaussian-type functions, has been made in order to study the effects of p, d, and f polarization functions for a 10-electron isoelectronic series of oxygen hydrides and for an 18-electron isoelectronic series of sulfur hydrides. Conclusions from these results suggest that meaningful proton affinities cannot be calculated without the inclusion of a d function on the heavy atom and a p function on the hydrogen atoms.     

Polarization functions for gaussian basis sets for the first row atoms

Exponent optimization was performed for a single set ofd-type Gaussians on the first row atoms C, N, and O in fifteen small molecules. The hydrogenp-exponents were kept at the fixed value of 1.0. For the underlying valence shell basis sets, Dunning's double zeta basis sets were used. Standard exponents of polarization functions are suggested for the most common valence states of the C, N, and O atoms.     

The basis set dependence of structures and energies of various states of cyclodisiloxane

Several common basis sets, ranging from minimal to double-zeta, are applied to study the neutral singlet and triplet as well as positive- and negative-ion doublet states of cyclodisiloxane. The effect of d-polarization function exponents on the equilibrium geometries and energies is analyzed. The d-type functions seem to be essential in the basis set of silicon, whereas their presence on oxygen is less critical. The optimum exponents (with respect to SCF energy) are determined to be 0.45 for Si and 0.60 for O, very close to those recommended for the 6–31G^** basis set. The best structural predictions are obtained with the 6–31G(2d, p) basis set, which contains two sets of d functions on the heavy atoms. The predicted Si? O bond length is 166 pm; the Si? Si and O? O distances are 237 and 232 pm, respectively, which correspond to an O—Si? O angle of 88.6°. The ground state is found to be a singlet. All higher states have longer Si? O bonds and Si—Si distances, whereas O—O distances are shorter. The energy separation between the singlet and other states is modified by electron correlation (MP treatment) by only a few kcal/mol.     

Contracted polarization functions for B to Ar

Using optimal exponents for B through Ne given by Dunning and those for Al through Ar by Woon and Dunning, d-type contracted polarization functions (2d/1d), (3d/1d), and (3d/2d) are generated from natural orbitals of atomic single and double excitation configuration interaction (SDCI) calculations, where the numbers before and after the slash are those of the primitive and contracted Gaussian type functions. The resulting contracted functions are tested on N₂ and P₂ molecules by self-consistent field and SDCI calculations, which clarify characteristics of the present polarization functions. Received: 5 June 1997 / Accepted: 20 August 1997     

Combined bond-polarization basis sets for accurate determination of dissociation energies. II. Basis set superposition error as a function of the parent basis set

The effect of the parent basis set on the basis set superposition error caused by bond functions is investigated systematically. An important difference between BSSE at the SCF and correlated levels is pointed out. Three new basis sets are defined, denoted 6-311 + G(d,p)B, 6-311 + G(2d,p)B, and 6-311 + G(2df,p)B. BSSE for the first-row hydrides seems to increase uniformly with increasing atomic number of the central atom. Expansion of the valence part of the basis set from 6-31G to 6-311G, as well as adding f functions, has a significant effect on the BSSE. Additional BSSEs incurred by bond functions are less than or equal to 1 kcal/mol for the 6-311 + G(2df,p)B basis set. For the dissociation energies of the first-row hydride species, agreement with experiment within only a few kcal/mol can be obtained even without resorting to isogyric reaction cycles. For high-quality calculations, adding bond functions seems to have definite advantages over expanding the polarization space beyond the [2d1f] level.     

Structure and properties of transition-metal compounds. A systematic study of basis set effects in ab initioSCF calculations

The recently developed Gaussian basis functions [2] were used in calculations on the ground electronic states of molecules containing transition-metal atoms: ScF₃, TiCl₄, ZrCl₄, Cr(CO)₆, Ni(CO)₄, CuF, CuCl, Zn(CH₃)₂, and Cd(CH₃)₂. The usefulness of minimal basis sets, the importance of splitting of the valence part of the minimal basis sets, the role of the triple splitting of the d-block functions, and the need for p-, d-, and f-type polarization functions were discussed in the context of the geometrical structure and the firstorder electronic properties of the transition-metal atom compounds.     

Accurate analytical self-consistent field wave functions for Tm2+ and Tm3+

The analytical expansion self-consistent field method was employed to perform ab initio calculations for the ground states of the rare-earth ions, Tm²⁺, 4f¹³, ²F, and Tm³⁺, 4f¹², ³H, (Z = 69). In each case the total number of basis functions used in the analytical expansions was 29, distributed as follows: 10, 8, 5, and 6, for the symmetries s, p, d, and f, respectively. All of the orbital exponents of the basis functions were optimized repeatedly, to the extent of the single-precision computer representation. Values of 〈rⁿ〉 for the 4f orbital of both ions are also presented, for the convenience of experimentalists.     

Quality of correlating functions generated from commonly used basis sets

Tests have been performed on the quality of correlating functions generated from commonly used Gaussian basis sets, such as the 4-31G and MIDI-4 sets. The atoms tested were carbon, nitrogen, and oxygen. Self-consistent field and configuration interaction (CI) calculations were performed for the ground and lower excited states of neutral atoms as well as for positive and negative ions, using the original sets. Next, after adding (1) one d, and (2) two d and one f primitive Gaussian-type functions (GTFs) to the original sets, the CI calculations were repeated. In order to investigate the quality of the correlating orbitals generated from the GTF sets, parallel calculations to those for the GTF sets were carried out with an extended set of Slater-type functions. It was found that the excitation energies change in a stepwise manner as the basis sets changed from the original sets to the original set + 1d and the original set +2d1f. The improvements in excitation energies and ionization energies were almost independent of the original sets and were found to be strongly dependent on the augmented correlation functions. © 1996 by John Wiley & Sons, Inc.     

Combined bond polarization function basis sets for accurate ab initio calculation of the dissociation energies of AHn molecules (A=LI to F)

An alternative route toward developing basis sets for post-Hartree-Fock calculations, the hybrid bond polarization function method, is investigated. Two new basis sets, denoted 6-31G(d, p)+ B and 6-31 + G(d,p)+B, are defined for the first-row hydrides. The dissociation energies of the first-row hydride species in their respective ground states are computed using full fourth-order Møller-Plesset theory, and compared with results obtained with large polarized basis sets containing no bond functions. It is shown that results are competitive even with basis sets as large as 6-311++G(3df,3pd), while computation times are reduced by a factor of 4 to 20. On empirical grounds, the basis set superposition error should be neglected entirely.     

On the additivity of basis set effects in some simple fluorine containing systems

The reactions F + H₂ → HF + H, HF → H + F, F → F⁺ + e^? and F + e^? → F^? were used as simple test cases to assess the additivity of basis set effects on reaction energetics computed at the MP4 level. The 6-31G and 6-311G basis sets were augmented with 1, 2, and 3 sets of polarization functions, higher angular momentum polarization functions, and diffuse functions (27 basis sets from 6-31Gd, p) to 6-31 ++ G(3df, 3pd) and likewise for the 6-311G series). For both series substantial nonadditivity was found between diffuse functions on the heavy atom and multiple polarization functions (e.g., 6-31 + G(3d, 3p) vs. 6-31 + G(d, p) and 6-31G(3d, 3p)). For the 6-311G series there is an extra nonadditivity between d functions on hydrogen and multiple polarization functions. Provided that these interactions are taken into account, the remaining basis set effects are additive to within ±0.5 kcal/mol for the reactions considered. Large basis set MP4 calculations can also be estimated to within ±0.5 kcal/mol using MP2 calculations, est. E_MP4(6-31 ++ G(3df, 3pd)) ≈ E_MP4(6-31G(d, p)) + E_MP2(6-31 ++ G(3df, 3pd)) – E_MP2(6-31G(d, p)) or E_MP4(6-31 + G(d, p) + E_MP2(6-31 ++ G(3df, 3pd)) – E_MP2(6-31 + G(d, p)) and likewise for the 6-311G series.     

Optimized gaussian bond functions for CO

Optimum bond function parameters of ξ_1s = 1.12 and ξ_2p = 0.70 placed at 0.44 of the bond distance from the oxygen atom are reported for the CO molecule. Using these parameters, the total ground-state energy is lower than that obtained by Neumann and Moskowitz using two sets of 3d type polarization functions on each atomic center with exponents of 0.5 and 1.5. The one-electron properties, however, are slightly inferior to those calculated using the 3d functions.     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Generally contracted basis sets for the first row transition metal atoms Sc-Zn have been constructed using the atomic natural orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANOs are constructed by averaging over the three electronic configurationsd ⁿ ,d ^n–1 s, andd ^n–2 s ² for the neutral atom as well as the ground state for the cation and the ground state atom in an external electric field. The primitive sets are 21s15p10d6f4g. Contraction to 6s5p4d3f2g yields results that are virtually identical to those obtained with the corresponding uncontracted basis sets for the atomic properties, which they have been designed to reproduce. Slightly larger deviations are obtained with the 5s4p3d2f1g for the polarizability, while energetic properties still have only small errors. The design objective has been to describe the ionization potential, the polarizability and the valence spectrum as accurately as possible. The result is a set of well-balanced basis sets for molecular calculations, which can be used together with basis sets of the same quality for the first and second row atoms.     

Spherically symmetric axial Gaussian lobe 3d and 4f orbitals

The axial Gaussian lobe orbital (AGLO ) representations of 3d and 4f orbitals proposed by LeRouzo and Silvi have been angularly optimized to ensure spherical symmetry of filled 3d and 4f shells. The functions have been tested on the hydrogen atom in the presence of high quality s and p basis sets and found to provide excellent minimal Gaussian representations of polarization functions. Exact orbital degeneracy is not obtained within each shell, however. Tabulated values are given to allow arbitrary scaling of the 3d and 4f lobe mimic orbitals.     

Basis sets for geometry optimizations of second-row transition metal inorganic and organometallic complexes

Modest-sized basis sets for the second-row transition metal atoms are developed for use in geometry optimization calculations. Our method is patterned after previous work on basis sets for first-row transition metal atoms. The basis sets are constructed from the minimal basis sets of Huzinaga and are augmented with a set of diffuse p and d functions. The exponents of these diffuse functions are chosen to minimize both the difference between the calculated and experimental equilibrium geometries and the total molecular energies for several second-row transition metal inorganic and organon etallic complexes. Slightly smaller basis sets, based on the same Huzinaga minimal sets but augmented with a set of diffuse s and p functions rather than diffuse p and d functions, are also presented. The performance of these basis sets is tested on a wide variety of second-row transition metal inorganic and organometallic complexes and is compared to pseudopotential basis sets incorporating effective core potentials.     

Hylleraas–Configuration Interaction calculations on the 11S ground state of helium atom

Hylleraas–Configuration Interaction (Hy–CI) calculations on the ground 1¹S state of helium atom are presented using s-, p-, d-, and f-Slater orbitals of both real and complex form. Techniques of construction of adapted configurations, optimization of the orbital exponents, and structure of the wave function expansion are explored. A new method to evaluate the two-electron kinetic energy integrals occurring in the Hy–CI method has been tested in this work and compared with other methods. The non-relativistic Hy–CI energy values are ≈10 picohartree accurate, about 2.2 × 10^?6 cm^?1. The Hy–CI calculations are compared with Configuration Interaction (CI) and Hylleraas (Hy) calculations employing the same orbital basis set, same computer code, and same computer machines. The computational required times are reported.     

Configuration interaction calculations on the 2P ground state of boron atom and C+ using Slater orbitals

Configuration Interaction (CI) calculations on the ground ²P state of boron atom are presented using a wave function expansion constructed with L‐S eigenfunction configurations of s‐, p‐, and d‐Slater orbitals. Two procedures of optimization of the orbital exponents have been investigated. First, CI(SD) calculations including few types of configurations and full optimization of the orbital exponents led to the energy ?24.63704575 a.u. Second, full‐CI (FCI) calculations including a large number of configuration types using a fixed set of orbital exponents for all configurations gave ?24.63405222 a.u. using the basis [4s3p2d] and 2157 configurations, and to an improved result of ?24.64013999 a.u. for 3957 configurations and a [5s4p3d] basis. This last result is better than earlier calculations of Schaefer and Harris (Phys Rev 1968, 167, 67), and compares well with the recent ones from Froese Fischer and Bunge (personal communication). In addition, using the same wave functions, CI calculations of the boron isoelectronic ion C⁺ have been performed obtaining an energy of ?37.41027598 a.u. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Is tetrahedral H42+ a minimum? Anomalous behavior of popular basis sets with the standard p exponents on hydrogen

The nature of the tetrahedral H₄²⁺ stationary point (minimum or triply degenerate saddle) depends remarkably upon the theoretical level employed. Harmonic vibrational analyses with, e.g., the 6-31G** (and 6-31 + +G**) and Dunning's [4s2p1d;2s1p] [D95(d,p)] basis sets using the standard p exponent suggest (erroneously) that the T_d geometry is a minimum at both the HF and MP2 levels. This is not the case at definitive higher levels. The C₃H₄²⁺ structure with an apical H is another example of the failure of the calculations with the 6-31G**, 6-311G**, and D95(d,p) basis sets. Even at MP2/6-31G** and MP2/ cc-pVDZ levels, the C_3v structure has no negative eigenvalues of the Hessian. Actually, this form is a second-order saddle point as shown by the MP2/6-31G** calculation with the optimized exponent. The D_4h methane dication structure is also an example of the misleading performance of the 6-31G** basis set. In all these cases, energy-optimized hydrogen p exponents give the correct results, i.e., those found with more extended treatments. Optimized values of the hydrogen polarization function exponents eliminate these defects in 6-31G** calculations. Species with higher coordinate hydrogens may also be calculated reliably by using more than one set of p functions on hydrogen [e.g., the 6-31G(d,2p) basis set]. Not all cases are critical. A survey of examples, also including some boron compounds, provides calibration. © 1993 John Wiley & Sons, Inc.