Basis Sets Of Gaussian-type Functions Research Articles

Accurate dipole polarizabilities of small silicon clusters from ab initio and density functional theory calculations

We report conventional ab initio and density functional theory calculations for the static dipole polarizability for the small silicon clusters Si3 to Si7. Our effort relies on the design of flexible basis sets of Gaussian-type functions. The dependence of the calculated mean polarizability per atom (ᾱ/n) on basis set type and level of theory is brought forth and discussed. The effect of electron correlation is found to be small for all studied clusters. The density functional theory based methods are seen to predict reliable values for (ᾱ/n). By introducing and analyzing the differential mean polarizability per atom, ᾱdiff/n=ᾱ(Sin)/n−ᾱ(Si), we show that in fact ab initio and density functional theory calculations yield distinctly different pictures of the polarizability of small silicon clusters. Computational strategies are proposed for the extension of theoretical studies to larger structures.

Electric Multipole Moments and (Hyper)Polarizability of X–C≡C–X, X = F, Cl, Br and I

We have calculated self-consistent field (SCF) and second-order Møller-Plesset perturbation theory (MP2) for the dihaloethynes X–C≡C–X, X = F, Cl, Br and I. All calculations have been performed with carefully optimized, flexible basis sets of gaussiantype functions. Our best values for the quadrupole moment (Θ/ea02) are -0.6524 (FCCF), 3.6612 (ClCCCl), 5.8143 (BrCCBr) and 8.3774 (ICCI). The dipole polarizability is strongly anisotropic. For the mean (α /e2a02Eh-1) and the anisotropy (Δα/e2a02Eh-1) we obtain 23.58 and 15.09 (FCCF), 51.75 and 48.30 (ClCCCl), 66.53 and 60.04 (BrCCBr), 93.79 and 78.91 (ICCI). The mean dipole hyperpolarizability (γ /e4a04Eh-3) increases rapidly as 2932 (FCCF), 9924 (ClCCCl), 17409 (BrCCBr) and 35193 (ICCI). The transversal component of the hyperpolarizability is larger than the longitudinal one for FCCF, γxxxx > γzzzz but this is reversed for the other molecules in the series. Difluoroethyne is less (hyper)polarizable than ethyne.

Ab initio determination of the electric multipole moments and static (hyper)polarizability of HCCX, X = F, Cl, Br, and I.

We report electric multipole moments and (hyper)polarizabilities for the haloethynes HCCX, X = F, Cl, Br, and I. The molecular properties have been obtained from finite-field self-consistent field, Møller-Plesset perturbation theory and coupled cluster calculations with large, carefully optimized basis sets of gaussian-type functions. The mean dipole (hyper)polarizability and the mean quadrupole polarizability near the Hartree-Fock limit are alpha/e(2)a(0) (2)E(h) (-1) = 23.74 (HCCF), 37.26 (HCCCl), 43.97 (HCCBr), 56.44 (HCCI), beta/e(3)a(0) (3)E(h) (-2) = -73.9 (HCCF), -67.0 (HCCCl), -39.5 (HCCBr), 42.7 (HCCI), gamma/e(4)a(0) (4)E(h) (-3) = 4,914 (HCCF), 6,554 (HCCCl), 9,328 (HCCBr), 14,949 (HCCI), and C/e(2)a(0) (4)E(h) (-1) = 160.3 (HCCF), 317.1 (HCCCl), 471.2 (HCCBr), 671.2 (HCCI). Electron correlation has a small effect on the dipole polarizability but affects strongly the hyperpolarizability. Agreement with the available experimental data is more or less fair for HCCF, HCCCl, and HCCBr but less satisfactory for HCCI.

Relativistic MR RCI Calculation of Energy Levels and Transition Probabilities of Boron Isoelectronic Sequence

Transition energies, oscillator strengths and lifetimes of the excited states for boron-like ions, Ne+5, Na+6 and Mg+7, have been calculated using multireference relativistic configuration interaction method based on the no-pair Dirac–Coulomb–Breit 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 and relativistic MR CI procedures. Transition energies are compared with experiment and other results.

Assessing the performance of ab initio methods on static (hyper)polarizability predictions for silicon clusters. Si4 as a test case

We have calculated static electric dipole (hyper)polarizabilities for a small silicon cluster, Si4. The molecular properties have been obtained from finite-field restricted-Hartree–Fock, Møller–Plesset perturbation theory, coupled-cluster and density functional theory calculations performed with carefully designed basis sets of gaussian-type functions. A large [8s6p5d2f] basis set is thought to provide near-Hartree–Fock quality results: = 142.77 and Δα = 77.93 e2a20E−1h for the mean and the anisotropy of the dipole polarizability and = 87.5 × 103e4a40E−3h for the mean hyperpolarizability. Electron correlation has a rather small effect on but a relatively strong one on Δα and . Our best values were obtained at the CCSD(T) level of theory with a [5s4p3d1f] basis set: = 140.35, Δα = 83.78 e2a20E−1h and = 106.3 × 103e4a40E−3h. At the same level of theory the differential (hyper)polarizability is (Si4) − 4Si = −9.21 e2a20E−1h and is (Si4) − 4Si = −65.3 × 103e4a40E−3h. To our knowledge, this is the first complete calculation of the dipole (hyper)polarizability of this cluster to be reported in the literature.

Relativistic density functional theory using Gaussian basis sets

A four-component formulation of relativistic density functional theory is presented together with the details of its implemention using a G-spinor basis set. The technical features of this approach are compared to those found in the nonrelativistic density functional theory of quantum chemistry which employ scalar basis sets of Gaussian-type functions. Numerical results of the G-spinor expansion method are presented for a sequence of closed-shell atoms, and for a selection of relativistic density functionals, and are compared with finite difference benchmarks.

Modification of nonrelativistic Gaussian basis sets for relativistic calculations

A simple method is proposed in which basis sets of Gaussian-type functions (GTFs), suitable for relativistic Dirac–Fock–Roothaan (DFR) calculations, are derived from their nonrelativistic analogs. The relativistic basis set is obtained through augmenting the nonrelativistic one by several GTFs determined from relativistic calculations for hydrogen-like atoms. The usefulness and reliability of the method is illustrated by DFR calculations of the ground-state energies of lanthanide and actinide atoms.

Static electric polarizability andhyperpolarizability of BCl(X 1Σ+)

We have obtained accurate electric properties for the ground state ofboron chloride. Our work relies on finite-field self-consistent field (SCF),Møller-Plesset perturbation theory (MP) and coupled-cluster (CC)calculations with very large, systematically optimized basis sets ofGaussian-type functions. Our best values were obtained at the CCSD(T)level of theory with a [9s5p4d2f/9s7p5d2f] basis set. For the dipoleproperties we report: dipole moment µ = 0.5689 ea0, mean andanisotropy of the dipole polarizability ᾱ = 32.61 andΔα = 12.13 e2a02Eh-1, mean first β̄ = 107.0 e3a03Eh-2 and second γ̄ = 105 29 e4a04Eh-3 hyperpolarizability. For the quadrupoleproperties we report the following: quadrupole moment Θ = -2.90 ea02 andmean quadrupole polarizability C̄ = 267.47 e2a04Eh-1. The mean first and second hyperpolarizabilitydecreases monotonically around the equilibrium internuclear separation ofRe = 3.2414 a0,(dβ̄/dR)e<0 and (dγ̄/dR)e<0.Various aspects of the relative performance of the Møller-Plessetperturbation theory and coupled-cluster methods on this molecule arebrought forth and discussed.

Finite-field Møller–Plesset perturbation theory and coupled cluster calculations of the electric multipole moments and the dipole polarizability of As 2

We have calculated the quadrupole ( Θ) and hexadecapole ( Φ) moment and the static dipole polarizability ( α αβ ) of As 2. The electric properties have been extracted from finite-field Møller–Plesset and coupled cluster calculations with large, carefully optimized basis sets of Gaussian-type functions. A very large (21s16p10d4f) basis set is thought to provide near-Hartree–Fock values for the electric moments, the mean and the anisotropy of the dipole polarizability: Θ=1.46 ea 0 2, Φ=−52.4 ea 0 4, α=64.95 and Δα=40.37 e 2 a 0 2 E h −1. At the SDQ-MP4 level of theory the R-dependence of the electric properties has been calculated with a [7s6p5d2f] basis set as [ Θ(R)− Θ(R e )]/ea 0 2 =2.84(R−R e )−0.87(R−R e ) 2−0.32(R−R e ) 3−0.21(R−R e ) 4 [ Φ(R)− Φ(R e )]/ea 0 4 =21.0(R−R e )+8.7(R−R e ) 2−1.6(R−R e ) 3−2.0(R−R e ) 4 [ α(R)− α(R e )]/e 2a 0 2E h −1 =17.42(R−R e )−0.048(R−R e ) 2−0.33(R−R e ) 3+1.71(R−R e ) 4 [Δα(R)−Δα(R e )]/e 2a 0 2E h −1 =22.62(R−R e )+6.68(R−R e ) 2+2.79(R−R e ) 3+1.04(R−R e ) 4 around the experimental bond length of R e=2.1026 Å. Basis set effects have been studied at all levels of theory. Comparing the electric properties of the sequence N 2, P 2, As 2 it is seen that the quadrupole moment is negative for N 2 but positive for P 2 and As 2 while the hexadecapole is negative for all three diatomics. The mean dipole polarizability increases with size, α( N 2)< α( P 2)< α( As 2) and the same trend is observed for the anisotropy as well, Δα(N 2)< Δα(P 2)< Δα(As 2).

On the static electric polarizability and hyperpolarizability of sodium. How good is the agreement between theory and experiment for the dipole polarizability?

We have obtained accurate static electric polarizabilities and hyperpolarizabilities for the sodium atom. Relying on finite-field coupled-cluster (CC) calculations with large, systematically optimized basis sets of Gaussian-type functions we report values of 165.06 e 2 a 0 2 E h −1 for the dipole polarizability α, 1915.4 e 2 a 0 4 E h −1 for the quadrupole polarizability α 2, −66.1×10 3 e 3 a 0 4 E h −2 for the dipole–dipole–quadrupole hyperpolarizability B and 956×10 3 for the dipole hyperpolarizability γ. The present theoretical dipole polarizability value is larger than all three available experimental estimates of this important property. The calculation of the dipole hyperpolarizability appears to be a problem of some difficulty, establishing the importance of reliable computational strategies for the determination of nonlinear optical properties of sodium clusters.

Chemically reliable uncontracted Gaussian-type basis sets for atoms H to Lr

Highly accurate Gaussian-type function basis sets are developed for 103 atoms from H (atomic number Z=1) to Lr ( Z=103). Choosing the last atoms of the seven periods and referring to the numerical Hartree–Fock total energies, the sizes of the present sets are so determined that the total energy errors are less than 1 millihartree. The average total energy error over the 103 atoms is only 0.79 millihartrees and the maximum error is 1.47 millihartrees for the Zn atom, indicating that the new sets are well-balanced throughout the atoms in the periodic table. The basis sets are tested for the GdF molecule.

Theoretical study of the spectroscopic constants of low-lying states of Ga 2

Spectroscopic constants for the ground and low-lying excited electronic states of the Ga 2 molecule have been studied using MRSDCI and MRCPA methods with an extensive basis set of Gaussian-type functions. In this study, we are going to present spectroscopic constants and potential energy curves for the ground 3Π u and the excited 3Σ g −, 1Σ g +, 1Π u , 1Δ g , (2) 1Σ g + and 3Σ u − states, in which electron correlation between semi-valence (3d) and valence (4s,4p) and among valence electrons is taken into account. Our calculated vibrational frequency for the ground state, 3Π u , is 175 cm −1 by CISD with Davidson's correction and 170 cm −1 by MRCPA, which are in good agreement with the experimental value of 180 cm −1. The dissociation energy for the ground state, reported to be 1.28 eV, compares well with the experimental value of 1.4 eV. Avoided crossing between the two lowest 1Σ g + states was properly described by MRCPA.

Structural and electronic properties of pyrope garnet : an ab initio study

The structural and electronic properties of the pyrope garnet have been investigated with the periodic ab initio quantum mechanical program CRYSTAL. An all-electron extended basis set of gaussian-type functions has been used. The geometrical parameters of the unit cell, which contains 80 atoms, have been fully optimized. The calculated static geometry agrees with experiment. The bulk modulus and the enthalpy of formation from atoms (binding energy) and simple oxides (-quartz) are also evaluated and turn out to be in good agreement with experimental results. Densities of states, charge-density maps and Mulliken population data are used for discussing the electronic structure and nature of the different bonds of the system.

On the accuracy of the algebraic approximation in molecular electronic structure calculations: V. Electron correlation in the ground state of the nitrogen molecule

Many-body perturbation theory, formulated within the algebraic approximation, is used to determine the correlation energy of the ground state of the nitrogen molecule. Systematically constructed even-tempered basis sets of Gaussian-type functions, which have been shown to achieve an accuracy approaching Hartree in matrix Hartree - Fock calculations, have been developed to afford basis sets suitable for electron correlation studies using the second-order many-body perturbation expansion. Over 98% of an empirical estimate of the ground-state correlation energy of the molecule at its equilibrium geometry is recovered by using a basis set constructed from even-tempered sets centred on the atoms and on the bond midpoint and containing functions of s, p, d, f, g and h symmetry. The importance of functions with i symmetry is also assessed. It is estimated that the calculated correlation energy corresponds to 99.1% of the exact second-order energy. The correlation energy obtained in the present study is compared with second-order energies obtained by using previously reported basis sets, which are shown to recover, at best, about 10% less of the empirical correlation energy estimate.

A universal basis set for high precision electronic structure studies

The accuracy of the total energies obtained for the series of 14 electron molecules N2, CO, BF, NO+ and CN- by means of matrix Hartree-Fock calculations using a universal basis set of Gaussian-type functions is reaccessed in the light of improved finite difference calculations.

Relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian: Relativistic correlation energies for the noble-gas sequence through Rn (Z=86), the group-IIB atoms through Hg, and the ions of Ne isoelectronic sequence.

Relativistic correlation energies have been computed for the neutral rare-gas atoms Ne, Ar, Kr, Xe, and Rn, the group-IIB atoms Zn, Cd, and Hg, and the ions of the Ne isoelectronic sequence with up to Z=100 by means of a recently developed relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit 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 and relativistic many-body perturbation procedures. The effects of the low-frequency Breit interaction on relativistic many-body effects for high-Z neutral atoms and Ne-like highly ionized ions are examined.

Relativistic configuration interaction calculations for open-shell atomic systems

A relativistic configuration interaction method, which can handle a general class of open-shell multiplets as the reference configurations, has been developed by combining the open-shell matrix Dirac-Fock self-consistent field algorithm implemented by Koc and Ishikawa [Phys. Rev. A 49 (1994) 794] with the relativistic configuration interaction algorithm of Kagawa et al. [Phys. Rev. A 44 (1991) 7092]. We employ basis sets of Gaussian-type functions to expand the large and small components of Dirac four-spinors. Relativistic configuration interaction calculations are tested on the ground and low-lying excited states of He and He-like ions.

Relativistic coupled cluster method based on Dirac—Coulomb—Breit wavefunctions. Ground state energies of atoms with two to five electrons

A relativistic Fock-space coupled cluster theory based on the Dirac—Coulomb—Breit wavefunctions has been developed and implemented, employing analytic basis sets of Gaussian-type functions. Relativistic all-order calculations including single and double excitations were performed for the ground states of the He and Be atoms and for the Ne ( Z = 10), Ar ( Z = 18) and Sn ( Z = 50) ions with 2–5 electrons. Comparison is made with the Dirac—Coulomb and non-relativistic formulation and with available experimental results. The non-additivity of relativistic and correlation effects is discussed.

A universal basis set for high-precision molecular electronic structure studies

High-precision matrix Hartree-Fock calculations are reported for a series of five isoelectronic diatomic systems in their ground states. A universal even-tempered basis set of Gaussian-type functions is developed for the nitrogen molecule at its experimentally derived equilibrium nuclear separation and then used to parameterize the single-particle state functions in the matrix Hartree-Fock description of the neutral molecules CO and BF, and in the ions NO+ and CN- at their respective experimentally determined equilibrium geometries. The calculated energies are compared with the results of previously reported fully numerical calculations which were performed by using finite difference and/or finite element techniques and which are taken to define Hartree-Fock limit. Energies obtained using the universal even-tempered finite basis set expansion are well within what is usually regarded as 'chemical accuracy' of approximately 1 mHartree and are only in error by approximately 2.3, approximately 1.5, approximately 90, approximately 40 and approximately 140 mu Hartree for N2, CO, BF, NO+ and Cn-, respectively.

On the accuracy of the algebraic approximation in molecular electronic structure calculations. III. Comparison of matrix Hartree-Fock and numerical Hartree-Fock calculations for the ground state of the nitrogen molecule

A comparison of matrix Hartree-Fock calculations using basis sets of Gaussian-type functions with numerical Hartree-Fock studies is made for the ground state of the nitrogen molecule. The use of both atom- and bond-centred functions is investigated. It is shown that by employing systematic sequences of even-tempered basis sets the accuracy achieved in the numerical Hartree-Fock calculations can be approached when basis sets of atom-centred functions are employed and can be matched when bond-centred functions are included. An energy of -108.993 823 Hartree is obtained within the algebraic approximation which should be compared with -108.993 808 Hartree from previously reported fully numerical and semi-numerical calculations. The advantages of the matrix Hartree-Fock method are emphasized.