Gaussian-type Functions Research Articles

A Linear-Scaling MP2 Method for Large Molecules by Rigorous Integral-Screening Criteria

Abstract A brief review of our linear-scaling method for atomic-orbital (AO) second-order Møller-Plesset perturbation theory (MP2) is given. The key feature of our method is the rigorous preselection of numerically significant four-center two-electron integrals based on multipole-based integral estimates (MBIE) that do not only account for the exponential coupling between the Gaussian-type basis functions forming charge distributions, but also for the 1/R coupling between the charge distributions. This coupling turns for the required integral products in AO-MP2 into at least a 1/R4 or even a 1/R6 decay behavior. Using MBIE we attain linear scaling, which is illustrated for DNA fragments with up to 1052 atoms and 10 674 basis functions as computed on a single processor. The largest molecule calculated in our present work at the scaled-opposite spin (SOS-) AO-MP2 level is an RNA system comprising 1664 atoms and 19 182 basis functions. Furthermore, we present results for the use of Cholesky-decomposed pseudo-density matrices in Laplace-based MP2, that offers the advantage of exploiting occupied/virtual blocking both with and without auxiliary basis sets.

INFRARED SPECTRA OF CALCIUM PHOSPHATE GLASSES

Infrared absorption spectra of calcium phosphate glasses, x CaO ·(100-x) P 2 O 5 have been investigated by FT-IR spectroscopy. By increasing CaO content, the characteristic bands due to the stretching and bending vibrations were identified and analyzed. The deconvolution of each experimental spectrum was made using a Gaussian-type function. Based on this, the specific structural units of these glasses were identified.

Systematic Improvement of Energy-Components by Simultaneous Optimization of Exponents and Centers of Gaussian-Type Function Basis Sets for Molecular Self-Consistent-Field Wave Functions

We have applied the fully variational molecular orbital (FVMO) method to systematically improve the values of energy-components, that is, the kinetic energy of electron and potential energy, as with that of total energy. The FVMO method enables us to describe accurate values of energy-components, since the quantum-mechanical Virial and Hellmann-Feynman theorems are completely satisfied due to the optimization of the exponents and centers in Gaussian-type functions (GTFs) basis sets. In the calculation of H2 molecule, we have found that the energy components and total energy with FVMO method using only [6s] GTFs are actually improved more than those with the conventional MO using a much larger basis set including the multiple polarization functions of [6s3p2d1f]. Additionally, drastic improvement of virial ratios for several hydride molecules with the FVMO method is demonstrated in the current work. The energy-component analysis using the FVMO method would be a powerful tool to elucidate various chemical problems.

Optimized Molecular Exponents on Gaussian Basis Sets for Hybrid Orbitals of Hydrocarbon Molecules

To elucidate the adequate molecular exponents for sp3-, sp2-, and sp-hybrid characters, we have performed the optimization for both geometry and Gaussian-type function (GTF) exponents using various hydrocarbon molecules. We have found that the scale factor 1.2 is significant for the p-type GTFs in sp3-carbon, as well as hydrogen. The optimized molecular exponents give a flexible description of σ and π bonds in C-C chemical bonds induced by the difference of hybrid states. We also confirmed the efficiency of our calculation concerning the total energy and geometrical parameters in comparison with the results using the high quality basis sets.

Determination of lateral modulation apodization functions using a regularized, weighted least squares estimation.

Recently, work in this group has focused on the lateral cosine modulation method (LCM) which can be used for next-generation ultrasound (US) echo imaging and tissue displacement vector/strain tensor measurements (blood, soft tissues, etc.). For instance, in US echo imaging, a high lateral spatial resolution as well as a high axial spatial resolution can be obtained, and in tissue displacement vector measurements, accurate measurements of lateral tissue displacements as well as of axial tissue displacements can be realized. For an optimal determination of an apodization function for the LCM method, the regularized, weighted minimum-norm least squares (WMNLSs) estimation method is presented in this study. For designed Gaussian-type point spread functions (PSFs) with lateral modulation as an example, the regularized WMNLS estimation in simulations yields better approximations of the designed PSFs having wider lateral bandwidths than a Fraunhofer approximation and a singular-value decomposition (SVD). The usefulness of the regularized WMNLS estimation for the determination of apodization functions is demonstrated.

Interaction induced electric dipole moment and (hyper)polarizability in the dihydrogen–helium pair

We have calculated the interaction dipole moment, polarizability and first hyperpolarizability of the dihydrogen–helium pair using flexible basis sets of Gaussian-type functions and high-level ab initio methods. Around the equilibrium position of the linear configuration (the helium-atom at a distance of R e / a 0 = 6.43 from the centre of the dihydrogen molecule) the interaction mean ( α ¯ int ) and anisotropy (Δ α int) of the dipole polarizability and the mean hyperpolarizability ( β ¯ int ) vary as [ α ¯ int ( R ) - α ¯ int ( R e ) ] / e 2 a 0 2 E h - 1 = 0.0069 ( R - R e ) - 0.0125 ( R - R e ) 2 + 0.0052 ( R - R e ) 3 [ Δ α int ( R ) - Δ α int ( R e ) ] / e 2 a 0 2 E h - 1 = - 0.0601 ( R - R e ) - 0.0052 ( R - R e ) 2 + 0.0042 ( R - R e ) 3 + 0.0020 ( R - R e ) 4 [ β ¯ int ( R ) - β ¯ int ( R e ) ] / e 3 a 0 3 E h - 2 = - 1.22 ( R - R e ) + 0.68 ( R - R e ) 2 - 0.15 ( R - R e ) 3 + 0.01 ( R - R e ) 4 at the CCSD/[6s4p3d1f/6s4p3d1f] level of theory.

Relativistic contracted Gaussian-type basis functions for atoms K through Xe

For 36 atoms from K to Xe, compact relativistic Gaussian-type function basis sets are developed in a segmented contraction form: ( 843 2 / 74 / 7 ) for the third row atoms and ( 843 3 / 743 / 74 ) for the fourth row atoms. Spin-free relativistic effects are considered through the third-order Douglas–Kroll approximation. The present basis sets are tested for all-electron calculations of four homo- and 12 heteronuclear diatomics in their ground states. The calculated spectroscopic constants are in good agreement with the experimental values.

An adaptive wavelet network for function learning

In this article, a wavelet neural network (WNN) model is proposed for approximating arbitrary nonlinear functions. Our WNN model structure comes from the idea of adaptive neuro-fuzzy inference system (ANFIS) which is used for obtaining fuzzy rule base from the input–output data of an unknown function. The WNN model which is called in this study as adaptive wavelet network (AWN) consists of wavelet scaling functions in its processing units whereas in an ANFIS, mostly Gaussian-type membership functions are used for a function approximation. We present to train an AWN by a hybrid-learning method containing least square estimation (LSE) with gradient-based optimization algorithm to obtain the optimal translation and dilation parameters of our AWN for model accuracy. Simulation examples are also given to illustrate the effectiveness of the method.

AB INITIO INVESTIGATION OF ELECTRONIC AND VIBRATIONAL PROPERTIES OF ZnS AND ZnSe CRYSTALS BY DIFFERENT XC-FUNCTIONALS

Ground state properties of zinc-blende structures ZnS and ZnSe are investigated from first principles using plane wave basis, projector augmented wave method and local orbital-dependent potential, as well as a basis set of Gaussian-type functions with hybrid functionals for the approximation of exchange-correlation energy. The results of DFT calculations with the LDA, LDA+U, GGA, GGA+U, BLYP, B3P86, P3PW, BH-HP, KMLYP and B3LYP approximations are compared. It is shown that the hybrid functional B3LYP provides an accurate quantitative description of the structural, electronic and vibrational properties of ZnS and ZnSe in comparison with experimental data.

Study of quark distribution amplitudes of 1S and 2S heavy quarkonium states

In this paper, the quark distribution amplitudes of 1S and 2S heavy quarkonium states are studied in terms of Gaussian-type wave functions. The transverse momenta $p_\perp$ integrals of the formulae for the decay constant are performed analytically. Then the quark distribution amplitudes are obtained. In addition, the $\xi$-moments are also calculated. After fixing the relevant parameters appearing in the quark distribution amplitude, the curves of the quark distribution amplitude for 1S and 2S heavy quarkonium states are plotted. Finally, the numerical results of this approach are compared with the other theoretical predictions.

Ab initio simulations on the atomic and electronic structure of single-walled BN nanotubes and nanoarches

To simulate the perfect single-walled boron nitride nanotubes and nanoarches with armchair- and zigzag-type chiralities and uniform diameter of ∼5 nm, we have constructed their one-dimensional (1D) periodic models. In this study, we have compared the calculated properties of nanotubes with those for both hexagonal and cubic phases of bulk: bond lengths, binding energies per B–N bond, effective atomic charges as well as parameters of total and projected one-electron densities of states. For both phases of BN bulk, we have additionally verified their lattice constants. In the density functional theory (DFT), calculations performed using formalism of the localized Gaussian-type atomic functions as implemented in the CRYSTAL-06 code we have applied Hamiltonians containing either PWGGA or hybrid (DFT+HF) B3PW exchange-correlation functionals. After calculation of Hessian matrix for the optimized structures of BN bulk (both phases) and nanotubes (both chiralities) using the CRYSTAL code we have estimated their normal phonon modes within the harmonic approximation. Applying both atomistic and continuum models we have calculated the elastic energies and moduli for SW BN nanoarches. Our calculations clearly show a reproducibility of the atomic structure, effective charges and total energy, as well as phonon and elastic properties when using either PWGGA or hybrid B3PW Hamiltonians. On other hand, there is a high sensitivity of the discrete energy spectra parameters (including band gap) to the choice of the first principles approach (the hybrid method reproduce them noticeably better).

Relativistic multireference configuration interaction calculations of lifetime of 2s22p 2P3/2 level along boron isoelectronic sequence

Lifetime of 2s22p 2P3/2 level along with fine structure separations for the lowest 2P doublet of boron-like ions, from Z=14 to Z=30 have been calculated using multireference relativistic configuration interaction (MR RCI) method based on the no-pair Dirac-Coulomb-Breit (DCB) Hamiltonian. Analytic basis sets of Gaussian-type functions are employed to expand the upper and lower components of the Dirac four-spinors in the matrix Dirac-Fock self-consistent field (DF SCF) and relativistic MR CI procedures. Error with respect to variation of basis set parameters was estimated. QED corrections were estimated by evaluation of electron self-energy and vacuum polarisation.

Third-order Douglas–Kroll self-consistent field energies for the neutral atoms H to Uuo

Third-order Douglas-Kroll self-consistent field (DK3-SCF) calculations with a finite-nucleus model were carried out for the neutral atoms H to Uuo, and DK3-SCF energies were obtained for them. The basis set used was the B-spline set. The parameters of the B-spline set were determined so as to reproduce the SQR-SCF limit energies given by Gaussian-type functions (GTFs). The SQR-SCF is a self-consistent field calculation with a simplified first-order Douglas-Kroll Hamiltonian (the "SQR" Hamiltonian) and can be carried out exactly with GTFs. The DK3-SCF energies given by this B-spline set should be highly accurate. A comparison is made with the DK3-SCF energies of Nakajima and Hirao [J. Chem. Phys. 116, 8270 (2002).

Electron affinity of lead: An ab initio four-component relativistic study

The electron affinity of lead is calculated to be 0.403 eV, using four-component relativistic multiconfigurational quasidegenerate perturbation theory. Uncontracted (27 s 23 p 16 d 10 f 2 g) Gaussian-type functions are used to generate valence basis functions (7 s 9 p − 10 p + 8 d ± 10 f ± 2 g ±) in the reduced frozen core approximation, where the KLM shells are frozen. The electrons in the remaining shells are correlated in the perturbation calculations with the (6 s, 6 p) complete active space. The calculated value is within 10% of the experimental value of 0.364 eV. The energy levels of the low-lying excited states of Pb and Pb − are also calculated.

DFT and Local-MP2 Periodic Study of the Structure and Stability of Two Proton-Ordered Polymorphs of Ice

The equilibrium geometry and the formation energy of two periodic polymorphs of Ice have been theoretically studied: the former (Ice XI, crystal group Cmc2(1)) is experimentally observed as the most stable structure at low temperature and pressure; the latter (crystal group Pna2(1)) is the simplest proton-ordered model of ordinary ice. With the Crystal code, the problem is solved using Hartree-Fock (HF), pure Kohn-Sham (PW91), or hybrid (B3LYP) one-electron Hamiltonians. The B3LYP results are those in best agreement with the experiment. Using the B3LYP-optimized geometry and starting from the corresponding HF Crystal solution, the energetics of the two polymorphs have been investigated at an ab initio MP2 level using the Cryscor code, based on a local-correlation approach: these calculations have allowed us not only to confirm the excellent B3LYP results as concerns the formation energy and the relative stability of the two structures but also to analyze the role in this respect of the intra- and intermolecular contributions to the correlation energy. Since both Crystal and Cryscor adopt a basis set of localized Gaussian-type functions and since very small energy differences are involved, utter attention has been paid to correcting for the basis set superposition error in the calculation of formation energies.

Atomic and electronic structure of single-walled BN nanotubes containing N vacancies as well as C and O substitutes of N atoms

Defective single-walled BN nanotubes of armchair- and zigzag-type chiralities with uniform diameter can be simulated using a total geometry optimization for the 1D-periodic model. For calculations, we have applied the formalism of localized Gaussian-type atomic functions using the Hamiltonian containing hybrid (DFT+HF) non-local exchange-correlation functional B3PW as implemented in CRYSTAL code. Single N vacancy as well as C and O substitutes of N atom cause an appearance of the energy levels inside the BN NT band gap accompanied by relaxation of the nearest atomic spheres closest to the point defect and electronic charge redistribution around it.

The mechanism of the first stage of formation of magnetite nanoparticles obtained by the chemical precipitation method

The first stage of the formation of nanoparticles was described by a kinetic equation. The solution to the kinetic equation was a Gauss-type function. The critical size of nanoparticles below which their solution and above which their growth could occur was determined. Changes in the size of nanoparticles with time and the rate of size changes were calculated.

Density matrix formulation of the nuclear-electronic orbital approach with explicit electron-proton correlation

The density matrix formulation of the nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) approach for including electron-proton correlation in mixed nuclear-electronic wave functions is presented. This approach is based on a general ansatz for the nuclear-electronic wave function that includes explicit dependence on the nuclear-electronic distances with Gaussian-type geminal functions. The NEO-XCHF approach is extended to treat multielectron, multiproton systems and to describe a broader class of systems that require a more general form of the wave function, such as open-shell and multireference wave functions. General expressions are derived for the one-particle and two-particle densities, as well as the total energy. In addition, expressions for the total energy and Fock matrices in an atomic orbital basis are derived for the special case of a closed-shell electronic system. The resulting Hartree-Fock-Roothaan equations can be solved iteratively to self consistency. An advantage of the density matrix representation is that it facilitates the development of approximate NEO-XCHF methods in which specified high-order density terms are neglected to decrease the computational expense. Another advantage of the density matrix representation is that it provides the foundation for the development of electron-proton functionals within the framework of density functional theory, thereby enabling a consistent treatment of both electron-electron and electron-proton correlation in a computationally practical manner.

On the basis‐set dependence of local and integrated electron density properties: Application of a new computer program for quantum‐chemical density analysis

A new computer program for post-processing analysis of quantum-chemical electron densities is described. The code can work with Slater- and Gaussian-type basis functions of arbitrary angular momentum. It has been applied to explore the basis-set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple-zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis-set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple-zeta/higher level with Gaussian/Slater functions.

Photoionization cross sections of H2+ and H2with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.