Systematic Sequences of Geometric Relativistic Basis Sets. I:s- and p-Block Elements up to Xe

In this work a scheme for constructing systematic sequences of relativistic SCF basis sets at a reasonable computational cost is presented and applied to atoms of the s- and p-block up to Xe. This scheme, which couples simplex optimization and the use of geometric series given by four-term polynomial expressions for the logarithm of the exponents, allows for the construction of basis sets that exhibit very regular patterns of convergence to the numerical reference values of atomic total energies, spinor energies and radial expectation values. This regularity, together with the broad range of basis set sizes presented, enables these sets to be used as building blocks for basis sets applicable in both routine and benchmark relativistic calculations on atomic and molecular systems.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Efficient polarized basis sets for evaluation of static and dynamic molecular electric properties

New medium size Gaussian‐type basis set R‐ORP for evaluation of static and dynamic electric properties in molecular systems is presented. It is obtained in a close resemblance to the original ORP basis set, from the source basis set through addition of two first‐order polarization functions whose exponent values are optimized with respect to the finite field restricted open‐shell Hartree–Fock (ROHF) atomic polarizabilities. As the source set the VTZ basis set of Ahlrichs and coworkers, augmented with additional diffuse functions and contracted to the form [6s/3s] for hydrogen and [11s7p/4s3p] for carbon through fluorine, is chosen. The resulting basis set is of the form [6s2p/3s2p] for hydrogen and [11s7p2d/4s3p2d] for other atoms. Presented basis set is next tested in the CCSD static and dynamic molecular polarizability and hyperpolarizability calculations for a set of ten and four test molecules, respectively, for which very accurate reference data exist. Additionally, the recently developed ORP basis set is employed in the calculations to examine the limits of its applicability. Results are compared to the literature data obtained in both, large and diffuse, as well as reduced‐size basis sets. In the case of polarizability calculations, the aug‐pc‐1 and R‐ORP are the optimal choices among the investigated smaller basis sets, with the overall performance of the aug‐pc‐1 set being better. Among the larger sets, the ORP performs better in the case of average polarizability, while the RMSE values for polarizability anisotropy are practically identical for d‐aug‐cc‐pVDZ and ORP sets. Finally, the R‐ORP and ORP basis sets compete other small bases in the evaluation of the first hyperpolarizability in investigated systems. © 2016 Wiley Periodicals, Inc.     

Optimized Slater-type basis sets for the elements 1-118

Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.     

Even-tempered Slater-type orbitals revisited: from hydrogen to krypton

Even-tempered Slater-type orbital basis sets were developed in 1973, based on total atomic energy optimization. Here, we revisit ET STOs and propose new sets based on past experience and recent computational studies. From preliminary atomic and molecular tests, these sets are shown to be very well balanced and to perform, at lower cost, almost as well as a very large (close to complete) basis set.     

A technique for orbital exponent optimization in ab initio HF?SCF?LCAO?MO calculations

A technique for Slater orbital exponent optimization in an HF? SCF? LCAO? MO calculation is proposed in which orbital exponent variation is incorporated into the SCF scheme. This is accomplished by rewriting Slater's rules so that the shielding terms depend on the molecular charge distribution through the elements of the population matrix. The SCF scheme then includes a calculation of a new set of orbital exponents from the coefficients of self-consistent molecular orbitals obtained from the previous set of exponents. The process is iterated until the energy attains its lowest value. The technique is illustrated by minimal basis calculations on LiH, BH, and HF. Near optimization is obtained with considerably less effort than is necessary for other reported techniques. Aside from interesting properties, the technique can be important for extended basis calculations where exponent optimization is a difficult task.     

Automatic algorithms for completeness‐optimization of Gaussian basis sets

We present the generic, object‐oriented C++ implementation of the completeness‐optimization approach (Manninen and Vaara, J. Comput. Chem. 2006, 27, 434) in the freely available ERKALE program, and recommend the addition of basis set stability scans to the completeness‐optimization procedure. The design of the algorithms is independent of the studied property, the used level of theory, as well as of the role of the optimized basis set: the procedure can be used to form auxiliary basis sets in a similar fashion. This implementation can easily be interfaced with various computer programs for the actual calculation of molecular properties for the optimization, and the calculations can be trivially parallelized. Routines for general and segmented contraction of the generated basis sets are also included. The algorithms are demonstrated for two properties of the argon atom—the total energy and the nuclear magnetic shielding constant—and they will be used in upcoming work for generation of cost‐efficient basis sets for various properties. © 2014 Wiley Periodicals, Inc.     

All-electron scalar relativistic basis sets for the elements Rb–Xe

Segmented all-electron relativistically contracted (SARC) basis sets are presented for the elements ₃₇Rb–₅₄Xe, for use with the second-order Douglas–Kroll–Hess approach and the zeroth-order regular approximation. The basis sets have a common set of exponents produced with established heuristic procedures, but have contractions optimized individually for each scalar relativistic Hamiltonian. Their compact size and loose segmented contraction, which is in line with the construction of SARC basis sets for heavier elements, makes them suitable for routine calculations on large systems and when core spectroscopic properties are of interest. The basis sets are of triple-zeta quality and come in singly or doubly polarized versions, which are appropriate for both density functional theory and correlated wave function theory calculations. The quality of the basis sets is assessed against large decontracted reference basis sets for a number of atomic and ionic properties, while their general applicability is demonstrated with selected molecular examples.     

Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases

We present a library of Gaussian basis sets that has been specifically optimized to perform accurate molecular calculations based on density functional theory. It targets a wide range of chemical environments, including the gas phase, interfaces, and the condensed phase. These generally contracted basis sets, which include diffuse primitives, are obtained minimizing a linear combination of the total energy and the condition number of the overlap matrix for a set of molecules with respect to the exponents and contraction coefficients of the full basis. Typically, for a given accuracy in the total energy, significantly fewer basis functions are needed in this scheme than in the usual split valence scheme, leading to a speedup for systems where the computational cost is dominated by diagonalization. More importantly, binding energies of hydrogen bonded complexes are of similar quality as the ones obtained with augmented basis sets, i.e., have a small (down to 0.2 kcal/mol) basis set superposition error, and the monomers have dipoles within 0.1 D of the basis set limit. However, contrary to typical augmented basis sets, there are no near linear dependencies in the basis, so that the overlap matrix is always well conditioned, also, in the condensed phase. The basis can therefore be used in first principles molecular dynamics simulations and is well suited for linear scaling calculations.     

Linear Approximation of Basis Set Orbital Exponents for Isoelectronic Series of Atoms

The orbital exponents of Slater type atomic orbitals (AOs) in isoelectronic series of atoms may be approximated by the linear dependence on the nuclear charge using a technique developed for optimization of AO basis sets in Hartree–Fock–Roothaan calculations. This approach yields the analytical Hartree–Fock wave functions for any ion in the isoelectronic atomic series without optimization of orbital exponents. The approximated linear equations for atomic orbital basis sets of B, C, O, and F in the ground state are presented as an example.     

Double even tempering of orbital exponents: Application to Roothaan-Hartree-Fock calculations for He through Xe in Slater-type basis sets

Summary Double even tempering (DET) of orbital exponents is proposed as a useful generalization of even tempering (ET). The DET scheme uses two sets of basis functions for each angular momentum. The two sets have different principal quantum numbers and their exponents are generated by two different geometric sequences. Roothaan-Hartree-Fock (RHF) calculations on the atoms from He through Xe using both ET and DET Slater-type basis sets of the same size are carried out to demonstrate the substantial improvement offered by the DET scheme. The DET scheme reduces the maximum deviation of the RHF energies relative to the Hartree-Fock limit from 1.4 to 0.3 millihartrees.     

Medium-size polarized basis sets for high-level-correlated calculations of molecular electric properties

Summary The basis set polarization method is used to derive the first-order polarized basis sets for Ge through Br for calculations of atomic and molecular electric properties. The performance of the [15.12.9/9.7.4] GTO/CGTO basis sets generated in this study is verified in calculations of atomic dipole polarizabilities and dipole moments and polarizabilities of the third-row atom hydrides. Whenever accurate reference data are available for comparison, the excellent performance of the derived first-order polarized basis sets is demonstrated. The role of the core polarization and relativistic contributions to atomic and molecular is also investigated. The detailed basis set data for Ge through Br are given in Appendix.     

Double- and triple-zeta Slater-type basis sets with common exponents

Double- and triple-zeta basis sets of Slater-type functions (STFs) are developed for the 17 atoms from He to Ar. For computational economy, the exponents of STFs corresponding to the same atomic subshell are restricted to be common. Instead, the principal quantum numbers of the STFs are thoroughly optimized within the framework of integer values to reduce the energy loss due to the common exponent restriction. Received: 10 November 1999 / Accepted: 25 January 2000 / Published online: 19 April 2000     

Property-optimized gaussian basis sets for molecular response calculations

With recent advances in electronic structure methods, first-principles calculations of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for molecules with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calculations since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H-Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small number of moderately diffuse basis functions. The exponents are determined variationally by maximization of atomic Hartree-Fock polarizabilities using analytical derivative methods. The performance of the resulting basis sets is assessed using a set of 313 molecular static Hartree-Fock polarizabilities. The mean absolute basis set errors are 3.6%, 1.1%, and 0.3% for property-optimized basis sets of split-valence, triple-zeta, and quadruple-zeta valence quality, respectively. Density functional and second-order M?ller-Plesset polarizabilities show similar basis set convergence. We demonstrate the efficiency of our basis sets by computing static polarizabilities of icosahedral fullerenes up to C(720) using hybrid density functional theory.     

Optimization of complex slater‐type functions with analytic derivative methods for describing photoionization differential cross sections

The complex basis function (CBF) method applied to various atomic and molecular photoionization problems can be interpreted as an method to solve the driven‐type (inhomogeneous) Schrödinger equation, whose driven term being dipole operator times the initial state wave function. However, efficient basis functions for representing the solution have not fully been studied. Moreover, the relation between their solution and that of the ordinary Schrödinger equation has been unclear. For these reasons, most previous applications have been limited to total cross sections. To examine the applicability of the CBF method to differential cross sections and asymmetry parameters, we show that the complex valued solution to the driven‐type Schrödinger equation can be variationally obtained by optimizing the complex trial functions for the frequency dependent polarizability. In the test calculations made for the hydrogen photoionization problem with five or six complex Slater‐type orbitals (cSTOs), their complex valued expansion coefficients and the orbital exponents have been optimized with the analytic derivative method. Both the real and imaginary parts of the solution have been obtained accurately in a wide region covering typical molecular regions. Their phase shifts and asymmetry parameters are successfully obtained by extrapolating the CBF solution from the inner matching region to the asymptotic region using WKB method. The distribution of the optimized orbital exponents in the complex plane is explained based on the close connection between the CBF method and the driven‐type equation method. The obtained information is essential to constructing the appropriate basis sets in future molecular applications. © 2017 Wiley Periodicals, Inc.     

The ellipsoidal Gaussian basis in molecular orbital theory

The ellipsoidal Gaussian basis function used in a minimal valence atomic orbital representation is compared with the double-zeta spherical Gaussian basis orbital representation for some seventeen molecules made up of first row atoms and hydrogen. Except for acetylene the double-zeta basis gives consistently better total electronic energies and generally better property values than the optimized ellipsoidal single zeta basis. Difference molecular density contour maps comparing the two basis sets, as well as other one-electron property values, indicate that the ellipsoidal basis exaggerates the transfer of charge from the atomic regions to the interatomic and lone pair regions of molecules. Apparently, the forced complete elliptization of the valence atomic orbital in the single-zeta representation does not allow the basis set sufficient flexibility to simultaneously represent both the basically spherical atomic part of these orbitals and the non-spherical molecular bond formation. Other properties and aspects of the ellipsoidal Gaussian basis are also discussed.