Basis Set Size Research Articles

All-electron Gaussian basis sets of double zeta quality for the actinides.

For the actinides, two segmented all-electron basis sets of valence double zeta quality plus polarization functions (DZP) are developed. One of them must be used along with the non-relativistic Hamiltonian, whereas the other with the Douglas-Kroll-Hess (DKH) one. Adding diffuse functions of s, p, d, f, and g symmetries to the non-relativistic and relativistic sets, augmented basis sets are developed. These functions are essential to describe correctly electrons far away from the nuclei. For some compounds, geometric parameters, atomic charges and valence orbital populations of the actinides, and bond dissociation energies are calculated using the Becke 3-parameter (exchange) and the Lee, Yang, and Parr (correlation) functional in conjunction with the DZP-DKH basis set. For Am and No, the static electric mean dipole polarizabilities are also reported. Comparison with benchmark theoretical and experimental values found in the literature is carried out. It is verified that the performances of the relativistic compact size basis sets generated in this work are regular, efficient, and reliable. They will be extremely helpful in molecular property calculations that need explicitly to consider the core electrons.

Calibrating Reaction Enthalpies: Use of Density Functional Theory and the Correlation Consistent Composite Approach in the Design of Photochromic Materials.

Acquisition of highly accurate energetic data for chromium-containing molecules and various chromium carbonyl complexes is a major step toward calibrating bond energies and thermal isomerization energies from mechanisms for Cr-centered photochromic materials being developed in our laboratories. The performance of six density functionals in conjunction with seven basis sets, utilizing Gaussian-type orbitals, has been evaluated for the calculation of gas-phase enthalpies of formation and enthalpies of reaction at 298.15 K on various chromium-containing systems. Nineteen molecules were examined: Cr(CO)6, Cr(CO)5, Cr(CO)5(C2H4), Cr(CO)5(C2ClH3), Cr(CO)5(cis-(C2Cl2H2)), Cr(CO)5(gem-(C2Cl2H2)), Cr(CO)5(trans-(C2Cl2H2)), Cr(CO)5(C2Cl3H), Cr(CO)5(C2Cl4), CrO2, CrF2, CrCl2, CrCl4, CrBr2, CrBr4, CrOCl2, CrO2Cl2, CrOF2, and CrO2F2. The performance of 69 density functionals in conjunction with a single basis set utilizing Slater-type orbitals (STO) and a zeroth-order relativistic approximation was also evaluated for the same test set. Values derived from density functional theory were compared to experimental values where available, or values derived from the correlation consistent composite approach (ccCA). When all reactions were considered, the functionals that exhibited the smallest mean absolute deviations (MADs, in kcal mol-1) from ccCA-derived values were B97-1 (6.9), VS98 (9.0), and KCIS (9.4) in conjunction with quadruple-ζ STO basis sets and B97-1 (9.3) in conjunction with cc-pVTZ basis sets. When considering only the set of gas-phase reaction enthalpies (ΔrH°gas), the functional that exhibited the smallest MADs from ccCA-derived values were B97-1 in conjunction with cc-pVTZ basis sets (9.1) and PBEPBE in conjunction with polarized valence triple-ζ basis set/effective core potential combination for Cr and augmented and multiple polarized triple-ζ Pople style basis sets (9.5). Also of interest, certainly because of known cancellation of errors, PBEPBE with the least-computationally expensive basis set combination considered in the present study (valence double-ζ basis set/effective core potential combination for Cr and singly-polarized double-ζ Pople style basis sets) also provided reasonable accuracy (11.1). An increase in basis set size was found to have an improvement in accuracy for the best performing functional (B97-1).

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.

Effect of mutation on the stabilization energy of HIV-1 zinc fingers: a hybrid local self-consistent field/molecular mechanics investigation.

Metal-ligand interactions give rise to a wide variety of metal complexes with various physical properties and chemical behaviours and numerous practical applications. The ability of the zinc ion to enhance the structural stability of many proteins by electrostatic interactions or by co-ordination with surrounding amino acids makes it the most important metal ion found in biological systems. In this paper, we highlight the importance of non-covalent interaction established between a metal ion and its environment in stabilizing biomolecules. Specifically, we are interested in understanding the stabilization role of the zinc ion in a native and point-mutated zinc-activated site. We have adapted a new quantum mechanics/molecular mechanics (QM/MM) strategy to describe a large molecular system; it is referred to as the local self-consistent field/MM method. It involves the use of frozen doubly occupied strictly localized bonding orbitals to link the QM subsystem to the one treated at the MM level. The B3LYP method, combined with LanL2DZ and 6-311G basis sets, was used to describe the QM region that comprised the zinc ion and the lateral chains of the four co-ordinating amino acids. For the surroundings (the backbone), CHARMM27 force field was used to describe MM interactions. The influence of the basis set size on the quality of the structural parameters of the zinc-binding site and the effects of mutation on the structural stabilization energy are analysed and reported.

Numerical test of SAC-CI methods for calculating vertical ionization energies

Valence, vertical ionization energies of a representative set of closed-shell molecules were calculated with the symmetry-adapted-cluster, configuration-interaction (SAC-CI) method using ten basis sets for its level 1 and level 2 operator inclusion criteria, whereas for its more stringent level 3 scheme, 15 basis sets were used. SAC-CI level 3 is capable of producing mean unsigned errors of approximately 0.2 eV with quadruple $$\zeta$$ correlation-consistent basis sets. Fortuitously better results may be obtained when smaller basis sets are used. Anomalous behavior with respect to the basis set size may occur when the level 1 and level 2 options are employed.

DFT and ab initio composite methods: Investigation of oxygen fluoride species

The capability of ccCA, G3, and G3B3 for the prediction of the enthalpies of formation of oxygen fluoride species was evaluated. In addition to these composite methods, the performance of M06 and M06-2X in conjunction with the correlation consistent basis sets (aug-cc-pVnZ), where n=D, T, Q, was also examined for predicting the structures and enthalpies of formation of oxygen fluoride species. A set of various oxygen fluorides were considered, including FOx radicals, FOxH, and FOxF, where x=1–3. The effects of basis set size and spin contamination were also considered.

Scaling of collision strengths for highly-excited states of ions of the H- and He-like sequences

Emission lines from highly-excited states (n >= 5) of H- and He-like ions have been detected in astrophysical sources and fusion plasmas. For such excited states, R-matrix or distorted wave calculations for electron-impact excitation are very limited, due to the large size of the atomic basis set needed to describe them. Calculations for n >= 6 are also not generally available. We study the behaviour of the electron-impact excitation collision strengths and effective collision strengths for the most important transitions used to model electron collision dominated astrophysical plasmas, solar, for example. We investigate the dependence on the relevant parameters: the principal quantum number n or the nuclear charge Z. We also estimate the importance of coupling to highly-excited states and the continuum by comparing the results of different sized calculations. We provide analytic formulae to calculate the electron-impact excitation collision strengths and effective collision strengths to highly-excited states (n >= 8) of H- and He-like ions. These extrapolated effective collision strengths can be used to interpret astrophysical and fusion plasma via collisional-radiative modelling.

Communication: Convergence of many-body wave-function expansions using a plane-wave basis in the thermodynamic limit.

Basis set incompleteness error and finite size error can manifest concurrently in systems for which the two effects are phenomenologically well-separated in length scale. When this is true, we need not necessarily remove the two sources of error simultaneously. Instead, the errors can be found and remedied in different parts of the basis set. This would be of great benefit to a method such as coupled cluster theory since the combined cost of nocc (6)nvirt (4) could be separated into nocc (6) and nvirt (4) costs with smaller prefactors. In this Communication, we present analysis on a data set due to Baardsen and co-workers, containing 2D uniform electron gas coupled cluster doubles energies for rs = 0.5, 1.0, and 2.0 a.u. at a wide range of basis set sizes and particle numbers. In obtaining complete basis set limit thermodynamic limit results, we find that within a small and removable error the above assertion is correct for this simple system. We then use this method to obtain similar results for the 3D electron gas at rs = 1.0, 2.0, and 5.0 a.u. and make comparison to the Ceperley-Alder quantum Monte Carlo results. This approach allows for the combination of methods which separately address finite size effects and basis set incompleteness error.

A comparative examination of density functional performance against the ISOL24/11 isomerization energy benchmark

The isomerization energy (ΔisomE) performance of a wide range of density functionals was tested against the set of 24 individual intramolecular reactions collectively comprising the ISOL24/11 benchmark database. Among 50 non-dispersion corrected functionals examined, the M06-2X single-component density functional offered the best estimation capacity, yielding a mean unsigned error (MUE) of 2.4kcal/mol. The addition of molecular mechanics dispersion-like terms to the M06-2X functional offers minimal improvement in its capacity to reliably estimate isomerization energies, whereas the choice of basis set played a significant role. Triple zeta Dunning, Ahlrichs, and Pople basis sets offer effectively equivalent ΔisomE performance, modestly better than double zeta equivalent basis sets, with ΔisomE performance subsequently rapidly declining at basis set sizes below this level. Among a reduced set of dispersion corrected functionals considered, the APFD functional displayed ΔisomE performance (MUE=1.8kcal/mol) superior to both the non-dispersion corrected M06-2X functional and the wB97X-D and PBE0-D3 dispersion corrected functionals (1.9kcal/mol). At this accuracy, the APFD functional is competitive with computationally expensive high-accuracy methods such as MC3MPWB and MC3MPW and nearing that of composite methods and dispersion corrected double-hybrid density functionals.

Matrix Isolation Infrared Spectroscopy of an O–H···π Hydrogen-Bonded Complex between Formic Acid and Benzene

Mid-infrared spectra of an O-H···π hydrogen-bonded 1:1 complex between formic acid and benzene were measured by isolating the complex in an argon matrix at a temperature of 8 K. The O-H stretching fundamental of formic acid (νO-H) undergoes a red shift of 120 cm(-1), which is the largest among the known π-hydrogen bonded complexes of an O-H donor with respect to benzene as acceptor. Electronic structure theory methods were used extensively to suggest a suitable geometry of the complex that is consistent with a recent study performed at CCSD(T)/CBS level by Zhao et al. (J. Chem. Theory Comput. 2009, 5, 2726-2733), as well as with the measured IR spectral shifts of the present study. It has been determined that density functional theory (DFT) D functionals as well as parametrized DFT functionals like M06-2X, in conjunction with modestly sized basis sets like 6-31G (d, p), are sufficient for correct predictions of the spectral shifts observed in our measurement and also for reproducing the value of the binding energy reported by Zhao et al. We also verified that these low-cost methods are sufficient in predicting the νO-H spectral shifts of an analogous O-H···π hydrogen-bonded complex between phenol and benzene. However, some inconsistencies with respect to shifts of νO-H arise when diffuse functions are included in the basis sets, and the origin of this anomaly is shown to lie in the predicted geometry of the complex. Natural bond orbital (NBO) and atoms-in-molecule (AIM) analyses were performed to correlate the spectral behavior of the complex with its geometric parameters.

Composite vibrational spectroscopy of the group 12 difluorides: ZnF2, CdF2, and HgF2.

The vibrational spectra of group 12 difluorides, MF2 (M = Zn, Cd, Hg), were investigated via coupled cluster singles, doubles, and perturbative triples, CCSD(T), including core correlation, with a series of correlation consistent basis sets ranging in size from triple-zeta through quintuple-zeta quality, which were then extrapolated to the complete basis set (CBS) limit using a variety of extrapolation procedures. The explicitly correlated coupled cluster method, CCSD(T)-F12b, was employed as well. Although exhibiting quite different convergence behavior, the F12b method yielded the CBS limit estimates closely matching more computationally expensive conventional CBS extrapolations. The convergence with respect to basis set size was examined for the contributions entering into composite vibrational spectroscopy, including those from higher-order correlation accounted for through the CCSDT(Q) level of theory, second-order spin-orbit coupling effects assessed within four-component and two-component relativistic formalisms, and vibrational anharmonicity evaluated via a perturbative treatment. Overall, the composite results are in excellent agreement with available experimental values, except for the CdF2 bond-stretching frequencies compared to spectral assignments proposed in a matrix isolation infrared and Raman study of cadmium difluoride vapor species [Loewenschuss et al., J. Chem. Phys. 50, 2502 (1969); Givan and Loewenschuss, J. Chem. Phys. 72, 3809 (1980)]. These assignments are called into question in the light of the composite results.

N-H bond cleavage of ammonia on graphene-like B36 borophene: DFT studies.

Ammonia N-H bond cleavage at metal-free substrates has attracted great attention because of its industrial importance. Here, we investigate the dissociative adsorption of ammonia onto the surface of a B36 borophene sheet by means of density functional theory calculations. We show that the N-H bond may be broken at the edges of B36 even at room temperature, regarding the small energy barrier of 14.1-19.3kcalmol(-1) at different levels of theory, and more negative Gibbs free energy change. Unlike basis set size, the kind of exchange correlation functional significantly affects the electronic properties of the studied systems. Also, by increasing the percentage of Hartree Fock (HF) exchange of density functionals, the activation and adsorption energies are lowered. A linear relationship between the highest occupied molecular orbital or lowest unoccupied molecular orbital of B36 borophene and the %HF exchange of functionals is predicted. Our work reveals that pure whole boron nanosheets may be promising metal-free materials in N-H bond cleavage, which would raise the potential application of these sheets.

Efficient Heat-Bath Sampling in Fock Space.

We introduce an algorithm for sampling many-body quantum states in Fock space. The algorithm efficiently samples states with probability approximately proportional to an arbitrary function of the second-quantized Hamiltonian matrix element connecting the sampled state to the current state. We apply the new sampling algorithm to the recently developed semistochastic full configuration interaction quantum Monte Carlo (S-FCIQMC) method, a semistochastic implementation of the power method for projecting out the ground state energy in a basis of Slater determinants. Our new sampling method requires modest additional computational time and memory compared to uniform sampling but results in newly spawned weights that are approximately of the same magnitude, thereby greatly improving the efficiency of projection. A comparison in efficiency between our sampling algorithm and uniform sampling is performed on the all-electron nitrogen dimer at equilibrium in Dunning's cc-pVXZ basis sets with X ∈ {D, T, Q, 5}, demonstrating a large gain in efficiency that increases with basis set size. In addition, a comparison in efficiency is performed on three all-electron first-row dimers, B2, N2, and F2, in a cc-pVQZ basis, demonstrating that the gain in efficiency compared to uniform sampling also increases dramatically with the number of electrons.

Benchmark Calculations of Interaction Energies in Noncovalent Complexes and Their Applications.

Data sets of benchmark interaction energies in noncovalent complexes are an important tool for quantifying the accuracy of computational methods used in this field, as well as for the development of new computational approaches. This review is intended as a guide to conscious use of these data sets. We discuss their construction and accuracy, list the data sets available in the literature, and demonstrate their application to validation and parametrization of quantum-mechanical computational methods. In practical model systems, the benchmark interaction energies are usually obtained using composite CCSD(T)/CBS schemes. To use these results as a benchmark, their accuracy should be estimated first. We analyze the errors of this methodology with respect to both the approximations involved and the basis set size. We list the most prominent data sets covering various aspects of the field, from general ones to sets focusing on specific types of interactions or systems. The benchmark data are then used to validate more efficient computational approaches, including those based on explicitly correlated methods. Special attention is paid to the transition to large systems, where accurate benchmarking is difficult or impossible, and to the importance of nonequilibrium geometries in parametrization of more approximate methods.

A reduced basis approach for calculation of the Bethe–Salpeter excitation energies by using low-rank tensor factorisations

ABSTRACTThe Bethe–Salpeter equation (BSE) is a reliable model for estimating the absorption spectra in molecules and solids on the basis of accurate calculation of the excited states from first principles. Direct diagonalisation of the BSE matrix is practically intractable due to O(N6) complexity scaling in the size of the atomic orbital basis set, N. In this paper, we introduce and analyse a reduced basis approach to computation of the Bethe–Salpeter excitation energies which can lead to a relaxation of the numerical costs down to O(N3). The BSE operator is specified in terms of the two-electron integrals in the Hartree–Fock molecular orbital basis and the respective energies, calculated by the tensor-based solver described in previous works. The reduced basis method includes two steps. First, the diagonal plus low-rank approximation to fully populated blocks in the BSE matrix is calculated, enabling an easier partial eigenvalue solver for a large auxiliary system relying only on matrix–vector multiplications with rank-structured matrices. Second, a small subset of eigenvectors from the auxiliary eigenvalue problem is selected to build a projection of the exact BSE system onto this reduced basis set. Numerical tests demonstrate the ϵ-rank bounds for the blocks of the BSE matrix on examples of some compact molecules. The accuracy of the reduced basis approach vs. the effective matrix rank is illustrated.

Combining the Complete Active Space Self-Consistent Field Method and the Full Configuration Interaction Quantum Monte Carlo within a Super-CI Framework, with Application to Challenging Metal-Porphyrins.

A novel stochastic Complete Active Space Self-Consistent Field (CASSCF) method has been developed and implemented in the Molcas software package. A two-step procedure is used, in which the CAS configuration interaction secular equations are solved stochastically with the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach, while orbital rotations are performed using an approximated form of the Super-CI method. This new method does not suffer from the strong combinatorial limitations of standard MCSCF implementations using direct schemes and can handle active spaces well in excess of those accessible to traditional CASSCF approaches. The density matrix formulation of the Super-CI method makes this step independent of the size of the CI expansion, depending exclusively on one- and two-body density matrices with indices restricted to the relatively small number of active orbitals. No sigma vectors need to be stored in memory for the FCIQMC eigensolver--a substantial gain in comparison to implementations using the Davidson method, which require three or more vectors of the size of the CI expansion. Further, no orbital Hessian is computed, circumventing limitations on basis set expansions. Like the parent FCIQMC method, the present technique is scalable on massively parallel architectures. We present in this report the method and its application to the free-base porphyrin, Mg(II) porphyrin, and Fe(II) porphyrin. In the present study, active spaces up to 32 electrons and 29 orbitals in orbital expansions containing up to 916 contracted functions are treated with modest computational resources. Results are quite promising even without accounting for the correlation outside the active space. The systems here presented clearly demonstrate that large CASSCF calculations are possible via FCIQMC-CASSCF without limitations on basis set size.

Nuclear–nuclear correlation function from non-Born–Oppenheimer calculations of diatomic rovibrational states with total angular momentum equal to two (N = 2). Charge asymmetry in HD

ABSTRACTThe HD molecule in rovibrational states where the total angular momentum quantum number is equal to two (N = 2) is characterised with quantum mechanical calculations without assuming the Born–Oppenheimer (BO) approximation. Explicitly correlated all-particle Gaussian functions are used in the calculations. The convergence of the total non-BO energies of the considered states with the basis set size is analysed. The calculations of the averaged interparticle distances demonstrate the asymmetry of the electronic charge distribution. The algorithm to calculate the nuclear–nuclear correlation function for the N = 2 states is derived and implemented. Plots of this function for different rovibrational states provide a visual representation of the molecular structure.

Variational tensor approach for approximating the rare-event kinetics of macromolecular systems

Essential information about the stationary and slow kinetic properties of macromolecules is contained in the eigenvalues and eigenfunctions of the dynamical operator of the molecular dynamics. A recent variational formulation allows to optimally approximate these eigenvalues and eigenfunctions when a basis set for the eigenfunctions is provided. In this study, we propose that a suitable choice of basis functions is given by products of one-coordinate basis functions, which describe changes along internal molecular coordinates, such as dihedral angles or distances. A sparse tensor product approach is employed in order to avoid a combinatorial explosion of products, i.e., of the basis set size. Our results suggest that the high-dimensional eigenfunctions can be well approximated with relatively small basis set sizes.

High-level theoretical rovibrational spectroscopy beyond fc-CCSD(T): The C3 molecule

An accurate local (near-equilibrium) potential energy surface (PES) is reported for the C3 molecule in its electronic ground state (X̃(1)Σg (+)). Special care has been taken in the convergence of the potential relative to high-order correlation effects, core-valence correlation, basis set size, and scalar relativity. Based on the aforementioned PES, several rovibrational states of all (12)C and (13)C substituted isotopologues have been investigated, and spectroscopic parameters based on term energies up to J = 30 have been calculated. Available experimental vibrational term energies are reproduced to better than 1 cm(-1) and rotational constants show relative errors of not more than 0.01%. The equilibrium bond length has been determined in a mixed experimental/theoretical approach to be 1.294 07(10) Å in excellent agreement with the ab initio composite value of 1.293 97 Å. Theoretical band intensities based on a newly developed electric dipole moment function also suggest that the infrared active (1, 1(1), 0)←(0, 0(0), 0) combination band might be observable by high-resolution spectroscopy.

Efficient calculation of the density response function from generalized polarizabilities

We present a method to calculate the density response function (DRF), also known as the density response kernel or the polarization propagator in the static limit. Our method uses generalized polarizabilities (GPs) which are second derivatives of the molecular energy with respect to two arbitrary perturbations of the external electrostatic potential. They are generalizations of the common multipole polarizabilities. The latter use solid spherical harmonics as perturbing potentials, while GPs can use any function. We use a sine function expansion of the electrostatic potential. Generalized polarizabilities were originally introduced for a different project, ultrafast quantum/ molecular mechanics calculations. By transforming the GPs to a (discretized) direct space representation, we obtain the DRF in the static limit. The method has been implemented for single-determinant (Hartree–Fock and density functional theory) wavefunctions, but can be generalized to more accurate wavefunctions. The number of coupled- perturbed self-consistent fi eld (CP-SCF) calculations in our method is proportional to the molecular volume at a given spatial resolution, i.e., scales linearly with the system size, and is independent of the basis set size. The best previous method has the number of CP-SCF calculations proportional to the product of the number occupied and virtual orbitals, and therefore scaling quadratically with the system size at constant basis set quality. The diagonal elements of the DRF (local polarizabilities) of water and butadiene in the molecular plane are displayed. An example of non-diagonal DRF is presented for butadiene 0.77 A above the molecular plane, with one point fi xed over the midpoint of a C=C bond. We obtain a fairly localized DRF, even in this conjugated system. The values become quite small if the distance of the two local perturbations exceeds a bond length. By using frequency-dependent polarizabilities, one could readily calculate the dynamic DRF.