Contracted Basis Sets Research Articles

Efficient Calculation of Molecular Integrals over London Atomic Orbitals.

The use of London atomic orbitals (LAOs) in a nonperturbative manner enables the determination of gauge-origin invariant energies and properties for molecular species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calculations for molecular systems is the evaluation of molecular integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiencies of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP), and Rys quadrature schemes are compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and weights, enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalized algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta; thus, a simple mixed scheme is put forward that selects the most efficient approach to calculate the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm.

Segmented Contracted Error-Consistent Basis Sets of Double- and Triple-ζ Valence Quality for One- and Two-Component Relativistic All-Electron Calculations.

Segmented contracted Gaussian basis sets optimized at the one-electron exact two-component (X2C) level - including a finite size model for the nucleus - are presented for elements up to Rn. These basis sets are counterparts for relativistic all-electron calculations to the Karlsruhe "def2" basis sets for nonrelativistic (H-Kr) or effective core potential based (Rb-Rn) treatments. For maximum consistency, the bases presented here were obtained from the latter by modification and reoptimization. Additionally we present extensions for self-consistent two-component calculations, required for the splitting of inner shells by spin-orbit coupling, and auxiliary basis sets for fitting the Coulomb part of the Fock matrix. Emphasis was put both on the accuracy of energies of atomic orbitals and on the accuracy of molecular properties. A large set of more than 300 molecules representing (nearly) all elements in their common oxidation states was used to assess the quality of the bases all across the periodic table.

Estimation of ΔR/R values by benchmark study of the Mössbauer Isomer shifts for Ru, Os complexes using relativistic DFT calculations

The present study applies all-electron relativistic DFT calculation with Douglas-Kroll-Hess (DKH) Hamiltonian to each ten sets of Ru and Os compounds. We perform the benchmark investigation of three density functionals (BP86, B3LYP and B2PLYP) using segmented all-electron relativistically contracted (SARC) basis set with the experimental Mössbauer isomer shifts for 99Ru and 189Os nuclides. Geometry optimizations at BP86 theory of level locate the structure in a local minimum. We calculate the contact density to the wavefunction obtained by a single point calculation. All functionals show the good linear correlation with experimental isomer shifts for both 99Ru and 189Os. Especially, B3LYP functional gives a stronger correlation compared to BP86 and B2PLYP functionals. The comparison of contact density between SARC and well-tempered basis set (WTBS) indicated that the numerical convergence of contact density cannot be obtained, but the reproducibility is less sensitive to the choice of basis set. We also estimate the values of ΔR/R, which is an important nuclear constant, for 99Ru and 189Os nuclides by using the benchmark results. The sign of the calculated ΔR/R values is consistent with the predicted data for 99Ru and 189Os. We obtain computationally the ΔR/R values of 99Ru and 189Os (36.2 keV) as 2.35×10−4 and −0.20×10−4, respectively, at B3LYP level for SARC basis set.

Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals.

An integral scheme for the efficient evaluation of two-center integrals over contracted solid harmonic Gaussian functions is presented. Integral expressions are derived for local operators that depend on the position vector of one of the two Gaussian centers. These expressions are then used to derive the formula for three-index overlap integrals where two of the three Gaussians are located at the same center. The efficient evaluation of the latter is essential for local resolution-of-the-identity techniques that employ an overlap metric. We compare the performance of our integral scheme to the widely used Cartesian Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials such as standard Coulomb, modified Coulomb, and Gaussian-type operators, which occur in range-separated hybrid functionals, are also included in the performance tests. The speed-up with respect to the OS scheme is up to three orders of magnitude for both integrals and their derivatives. In particular, our method is increasingly efficient for large angular momenta and highly contracted basis sets.

On the applicability of time‐dependent density functional theory (TDDFT) and semiempirical methods to the computation of excited‐state potential energy surfaces of perylene‐based dye‐aggregates

AbstractThe excited states of perylene‐based dye aggregates were investigated with emphasis on the description of the potential energy surfaces for the intermolecular motion in a dimer, which are of importance for exciton trapping. The aim is to investigate the applicability of TDDFT and semiempirical methods. The longitudinal displacement of two π‐stacked molecules against each other is used as model and the methods are benchmarked against SCS‐CC2 and SCS‐ADC(2). Furthermore a simple character analysis is applied. In the case of the perylene dimer, CAM‐B3LYP and ωB97XD provide accurate results. However, while CAM‐B3LYP fails for the PBI (perylene bisimide) dimer the performance of ωB97XD is highly dependent on the used monomer geometry due to close‐lying diabatic states. A change in geometry changes the character of the excited states even qualitatively. The accuracy of several other long‐range corrected functionals (LC‐BLYP, LC‐ωPBE, ωB97X) and the effect of empirically tuning the range‐separation parameter of ωB97XD are also presented for this interesting case. Furthermore, it is shown that the semiempirical method OM2 fails to provide a correct description of the excited states when the orbital overlap plays a significant role. This is attributed to the too contracted basis set.

Gaussian basis sets for atomic and molecular calculations obtained from stochastic optimization

A fully-optimized Gaussian basis set is generated from a Monte Carlo simulated annealing procedure. The reliability of this stochastic procedure is assessed in generating of Gaussian basis sets for a set of atoms. After to confirm the reliability of the method, the basis set is expanded to include H, He, and the rest of the first- and second-rows atoms of the Periodic Table. Finally, a contraction scheme is chosen, and polarization and diffuse functions are added to the basis set. The proposed methodology has proved to be reliable and flexible to generate atom-optimized Gaussian basis sets with relatively low computational cost. The contracted basis set, augmented with polarization and diffuse functions, is used to calculate correlation energies of a training set of 106 molecular systems and it presents better performance than other basis sets of similar size. The proposed methodology can be used to generate Gaussian basis sets of increasing sizes.

Semi-segmented contraction of generally contracted basis sets by property minimization

A method is presented for reducing the number of primitives in a generally contracted basis set, to improve the efficiency of integral evaluation in a program that is designed for segmented contractions. The method involves a linear transformation of the generally contracted functions to minimize the value of a property that is evaluated over a subset of the primitives in the general contraction. The transformed orbital is truncated by removing the subset of primitives, and a cutoff on the property is used to determine the size of the subset. For the example of Pb in a double-zeta basis contracted for ZORA calculations, a reduction in the number of primitives of a factor of 2 in the s and p spaces and 1.3 in the d space was obtained with an error of 10 microhartrees in the total energy. The method is also compared with the P-orthogonalization method of Jensen.

Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the light elements H–Ar

Relativistic basis sets of double-zeta, triple-zeta, and quadruple-zeta quality have been optimized at the self-consistent field level with a Gaussian nuclear charge distribution for the light elements H–Ar. Correlating functions were optimized for all shells in MR-CISD calculations on the ground configuration. Energies and properties evaluated with the basis sets are compared with those produced by several other basis sets. Prescriptions are given for constructing contracted basis sets. The basis sets are available as an internet archive.

Exploring the Interaction Natures in Plutonyl (VI) Complexes with Topological Analyses of Electron Density.

The interaction natures between Pu and different ligands in several plutonyl (VI) complexes are investigated by performing topological analyses of electron density. The geometrical structures in both gaseous and aqueous phases are obtained with B3LYP functional, and are generally in agreement with available theoretical and experimental results when combined with all-electron segmented all-electron relativistic contracted (SARC) basis set. The Pu– bond orders show significant linear dependence on bond length and the charge of oxygen atoms in plutonyl moiety. The closed-shell interactions were identified for Pu-Ligand bonds in most complexes with quantum theory of atoms in molecules (QTAIM) analyses. Meanwhile, we found that some Pu–Ligand bonds, like Pu–OH−, show weak covalent. The interactive nature of Pu–ligand bonds were revealed based on the interaction quantum atom (IQA) energy decomposition approach, and our results indicate that all Pu–Ligand interactions is dominated by the electrostatic attraction interaction as expected. Meanwhile it is also important to note that the quantum mechanical exchange-correlation contributions can not be ignored. By means of the non-covalent interaction (NCI) approach it has been found that some weak and repulsion interactions existed in plutonyl(VI) complexes, which can not be distinguished by QTAIM, can be successfully identified.

Improved Segmented All-Electron Relativistically Contracted Basis Sets for the Lanthanides.

Improved versions of the segmented all-electron relativistically contracted (SARC) basis sets for the lanthanides are presented. The second-generation SARC2 basis sets maintain efficient construction of their predecessors and their individual adaptation to the DKH2 and ZORA Hamiltonians, but feature exponents optimized with a completely new orbital shape fitting procedure and a slightly expanded f space that results in sizable improvement in CASSCF energies and in significantly more accurate prediction of spin-orbit coupling parameters. Additionally, an extended set of polarization/correlation functions is constructed that is appropriate for multireference correlated calculations and new auxiliary basis sets for use in resolution-of-identity (density-fitting) approximations in combination with both DFT and wave function based treatments. Thus, the SARC2 basis sets extend the applicability of the first-generation DFT-oriented basis sets to routine all-electron wave function-based treatments of lanthanide complexes. The new basis sets are benchmarked with respect to excitation energies, radial distribution functions, optimized geometries, orbital eigenvalues, ionization potentials, and spin-orbit coupling parameters of lanthanide systems and are shown to be suitable for the description of magnetic and spectroscopic properties using both DFT and multireference wave function-based methods.

On the zeroth-order hamiltonian for CASPT2 calculations of spin crossover compounds.

Complete active space self-consistent field theory (CASSCF) calculations and subsequent second-order perturbation theory treatment (CASPT2) are discussed in the evaluation of the spin-states energy difference (ΔH(elec)) of a series of seven spin crossover (SCO) compounds. The reference values have been extracted from a combination of experimental measurements and DFT + U calculations, as discussed in a recent article (Vela et al., Phys Chem Chem Phys 2015, 17, 16306). It is definitely proven that the critical IPEA parameter used in CASPT2 calculations of ΔH(elec), a key parameter in the design of SCO compounds, should be modified with respect to its default value of 0.25 a.u. and increased up to 0.50 a.u. The satisfactory agreement observed previously in the literature might result from an error cancellation originated in the default IPEA, which overestimates the stability of the HS state, and the erroneous atomic orbital basis set contraction of carbon atoms, which stabilizes the LS states.

Further studies into the photodissociation pathways of phenylperoxyl radicals

In this work, we report performed a high level ab initio study on the low-lying electronic states of C6H5OO, utilizing complete active space self-consistent field (CASSCF) and multiconfiguration second-order perturbation theory (CASPT2) method, and the contracted ANO-L basis set have been taken into account. The potential energy curves for the four lowest states associated with the lowest dissociation limit of C6H5OO radical. The calculated results clearly assigned the experimentally observed photodissociation channels leading to C6H5O (X2A, 22A)+O(3Pg) and C6H5 (X2A)+O2(X3Σg−, 11Δg, 11Σg+).

An efficient method for energy levels calculation using full symmetry and exact kinetic energy operator: tetrahedral molecules.

A simultaneous use of the full molecular symmetry and of an exact kinetic energy operator (KEO) is of key importance for accurate predictions of vibrational levels at a high energy range from a potential energy surface (PES). An efficient method that permits a fast convergence of variational calculations would allow iterative optimization of the PES parameters using experimental data. In this work, we propose such a method applied to tetrahedral AB4 molecules for which a use of high symmetry is crucial for vibrational calculations. A symmetry-adapted contracted angular basis set for six redundant angles is introduced. Simple formulas using this basis set for explicit calculation of the angular matrix elements of KEO and PES are reported. The symmetric form (six redundant angles) of vibrational KEO without the sin(q)(-2) type singularity is derived. The efficient recursive algorithm based on the tensorial formalism is used for the calculation of vibrational matrix elements. A good basis set convergence for the calculations of vibrational levels of the CH4 molecule is demonstrated.

Structure and potential energy function of PuNO molecules

Density functional (B3LPY) method has been utilized to optimize the possible structures of PuN, PuO, NO and PuNO molecules using the contracted valence basis set (LANL2 DZ) for Pu atom, and the AUG-cc-pVTZ basis set for N and O atoms. It is shown that the ground state of the PuNO molecules has Cv (Pu-N-O) symmetry and the ground electronic state is 6-. The equilibrium nuclear distances for Pu-N and N-O bonds in the PuNO molecules are RPuN=0.22951 nm and RNO=0.12257 nm, and the dissociation energy is De=8.10537 eV. Furthermore, the other two metastable states of the PuNO molecules are also obtained, and the electronic states of the two configurations are 6- and A with Cv (Pu-O-N) and Cs (O-Pu-N) symmetry, respectively. Then the Murrell-Sorbie potential energy functions of the PuN, PuO and NO molecules have been simulated and the analytical potential energy function of the PuNO molecules has been derived using the many-body expansion theory. The contours of the potential energy functions reproduce exactly the most stable equilibrium structures, the two metastable state structures as well as the dissociation energy of the PuNO molecules. The molecular static reaction pathway, based on the potential energy function, is also discussed.

Development of new auxiliary basis functions of the Karlsruhe segmented contracted basis sets including diffuse basis functions (def2-SVPD, def2-TZVPPD, and def2-QVPPD) for RI-MP2 and RI-CC calculations

We report optimized auxiliary basis sets for use with the Karlsruhe segmented contracted basis sets including moderately diffuse basis functions (Rappoport and Furche, J. Chem. Phys., 2010, 133, 134105) in resolution-of-the-identity (RI) post-self-consistent field (post-SCF) computations for the elements H-Rn (except lanthanides). The errors of the RI approximation using optimized auxiliary basis sets are analyzed on a comprehensive test set of molecules containing the most common oxidation states of each element and do not exceed those of the corresponding unaugmented basis sets. During these studies an unsatisfying performance of the def2-SVP and def2-QZVPP auxiliary basis sets for Barium was found and improved sets are provided. We establish the versatility of the def2-SVPD, def2-TZVPPD, and def2-QZVPPD basis sets for RI-MP2 and RI-CC (coupled-cluster) energy and property calculations. The influence of diffuse basis functions on correlation energy, basis set superposition error, atomic electron affinity, dipole moments, and computational timings is evaluated at different levels of theory using benchmark sets and showcase examples.

Segmented contracted basis sets optimized for nuclear magnetic shielding.

A family of segmented contracted basis sets is proposed, denoted pcSseg-n, which are optimized for calculating nuclear magnetic shielding constants. For the elements H-Ar, these are computationally more efficient than the previously proposed general contracted pcS-n basis sets, and the new basis sets are extended to also include the elements K-Kr. The pcSseg-n basis sets are optimized at the density functional level of theory, but it has been shown previously that these property-optimized basis sets are also suitable for calculating shielding constants with correlated wave function methods. The pcSseg-n basis sets are available in qualities ranging from (unpolarized) double-ζ to pentuple-ζ quality and should be suitable for both routine and benchmark calculations of nuclear magnetic shielding constants. The ability to rigorously separate basis set and method errors should aid in developing more accurate methods.

Vibrational frequency scaling factors for correlation consistent basis sets and the methods CC2 and MP2 and their spin-scaled SCS and SOS variants.

We present scaling factors for vibrational frequencies calculated within the harmonic approximation and the correlated wave-function methods coupled cluster singles and doubles model (CC2) and Møller-Plesset perturbation theory (MP2) with and without a spin-component scaling (SCS or spin-opposite scaling (SOS)). Frequency scaling factors and the remaining deviations from the reference data are evaluated for several non-augmented basis sets of the cc-pVXZ family of generally contracted correlation-consistent basis sets as well as for the segmented contracted TZVPP basis. We find that the SCS and SOS variants of CC2 and MP2 lead to a slightly better accuracy for the scaled vibrational frequencies. The determined frequency scaling factors can also be used for vibrational frequencies calculated for excited states through response theory with CC2 and the algebraic diagrammatic construction through second order and their spin-component scaled variants.

Unifying General and Segmented Contracted Basis Sets. Segmented Polarization Consistent Basis Sets.

We propose a method, denoted P-orthogonalization, for converting a general contracted basis set to a computationally more efficient segmented contracted basis set, while inheriting the full accuracy of the general contracted basis set. The procedure can be used for any general contracted basis set to remove the redundancies between general contracted functions in terms of primitive functions. The P-orthogonalization procedure is used to construct a segmented contracted version of the polarization consistent basis sets, which are optimized for density functional theory calculations. Benchmark calculations show that the new pcs-n basis sets provide uniform error control of the basis set incompleteness for molecular systems composed of atoms from the first three rows in the periodic table (H-Kr) and for different exchange-correlation functionals. The basis set errors at a given zeta quality level are lower than other existing basis sets, and the pcs-n basis sets are furthermore shown to be among the computationally most efficient. The pcs-n basis sets are available in qualities ranging from (unpolarized) double-zeta to pentuple zeta quality and should therefore be well suited for both routine and benchmark calculations using density functional theory methods in general.

The solvation radius of silicate melts based on the solubility of noble gases and scaled particle theory.

The existing solubility data on noble gases in high-temperature silicate melts have been analyzed in terms of Scaling Particle Theory coupled with an ab initio assessment of the electronic, dispersive, and repulsive energy terms based on the Polarized Continuum Model (PCM). After a preliminary analysis of the role of the contracted Gaussian basis sets and theory level in reproducing appropriate static dipole polarizabilities in a vacuum, we have shown that the procedure returns Henry's law constants consistent with the values experimentally observed in water and benzene at T = 25 °C and P = 1 bar for the first four elements of the series. The static dielectric constant (ɛ) of the investigated silicate melts and its optical counterpart (ɛ(∞)) were then resolved through the application of a modified form of the Clausius-Mossotti relation. Argon has been adopted as a probe to depict its high-T solubility in melts through an appropriate choice of the solvent diameter σs, along the guidelines already used in the past for simple media such as water or benzene. The σs obtained was consistent with a simple functional form based on the molecular volume of the solvent. The solubility calculations were then extended to He, Ne, and Kr, whose dispersive and repulsive coefficients are available from theory and we have shown that their ab initio Henry's constants at high T reproduce the observed increase with the static polarizability of the series element with reasonable accuracy. At room temperature (T = 25 °C) the calculated Henry's constants of He, Ne, Ar, and Kr in the various silicate media predict higher solubilities than simple extrapolations (i.e., Arrhenius plots) based on high-T experiments and give rise to smooth trends not appreciably affected by the static polarizabilities of the solutes. The present investigation opens new perspectives on a wider application of PCM theory which can be extended to materials of great industrial interest at the core of metallurgical processes, ceramurgy, and the glass industry.

Potential energy curve study on the 3Π electronic states of GaX (X = F, Cl, and Br) molecules

The potential energy curves (PECs) of the 3Π states of GaX (X = F, Cl, and Br) molecules are calculated using the multireference configuration interaction method with a large contracted basis set aug-cc-pV5Z. The PECs are accurately fitted to analytical potential energy functions (APEFs) using the Murrell—Sorbie potential function. The spectroscopic parameters for the states are determined using the obtained APEFs, and compared with the theoretical and experimental data available presently in the literature.