Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM

The concepts of normalized irreducible tensorial matrices (NITM ) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M (α₁ × α₂). Elements [X] ∈ M (α₁ α₂) transform under G_a ∈ G according to [G_a] [X][G^?1_a], where [G_a] and [G^?1_a] belong to irreducible representations of G . The usual properties of NITM and the Wigner–Eckart theorem follow from these results, which are valid for both finite and compact unitary groups. The NITM span M (α₁ × α₂) are orthonormal under the trace and transform irreducibly with respect to G . This NITM basis of M (α₁ × α₂) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffith's V coefficients for the octahedral group are tabulated and used to construct a six-coordinate superposition Hamiltonian.     

Subgroup‐chain symmetry‐adapted irreducible matrices and CG coefficients of point groups

The irreducible matrices and Clebsch–Gordan coefficients of any crystallographic point group adapted to all possible canonical subgroup chains are calculated ab initio for both single‐valued and double‐valued representations and tabulated with exact values in the form of or and with components labeled by the irrep labels of the group chain in Koster notation. The phases and ordering of the components of irreducible bases for the cubic point groups are properly chosen so that irreducible matrices for all subgroup chains of G=T_d, O, O_h obey the associated relations D(G)=D(G)D(G), i=4, 6, and the complex conjugation relation for the group T, D(T)=D(T)*. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 67–80, 1999     

Convergence accelerator approach for the evaluation of some three‐electron integrals containing explicit rij factors

A simplified analysis is presented for the evaluation of the three‐electron one‐center integrals of the form ∫rrrrrred r ₁d r ₂d r ₃, for the cases i, j, k, ≥−2, l=−2, m≥−1, n≥−1. These integrals arise in the calculation of lower bounds for energy levels and certain relativistic corrections to the energy when Hylleraas‐type basis sets are employed. Convergence accelerator techniques are employed to obtain a reasonable number of digits of precision, without excessive CPU requirements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 93–99, 1999     

The role of d orbitals in tetrahedral hybrid orbitals of oxy-ions

The hybrid orbitals of tetrahedral oxy-ions containing some d character have been calculated by maximum overlap method. The d characters of hybrid orbitals increase in the order of SiO, PO, SO, ClO, and decrease in order of GeO, AsO, SeO, BrO. The bond strengths are also obtained for these ions. The hybrid Orbital of VO, CrO, and MnO are of the type d³s as the result of calculation.     

Ab initio studies of negative ion-molecule(s) clusters present in the atmosphere. II. OH−(H2O)n for n = 0, 2, 3, 4

Ab initio calculations are performed with 6–31G basis set to study the geometry and binding of the H₃O, H₅O, H₇O, and H₉O complexes. The H₃O complex is also investigated with the 6–31 G* basis set and MP 2 (Moller–Plesset perturbation theory of second order).     

Ab initio MO calculations on the stability of the CF,PF, SF,and MoF ions with the F1s core hole

For the CF, PF, SF, and MoF ions appearing after the F1s photoionization, the possibility of dissociation has been shown by the ab initio MO LCAO method within the Z + 1 core equivalent model. According to the calculations, the decay channel AF → AF + F(1s¹2p⁶) is energetically open for the ions. So the interpretation of the gas-phase emission FKα spectra, in which the bands are assigned to the discrete transition energies, can be unacceptable for these ions. The conditions and signs of such failure are discussed.     

The spin-density-functional formalism for quantum mechanical calculations: Test on diatomic molecules with an efficient numerical method

We have applied the spin-density-functional (SDF ) formalism with the local-spin-density (LSD ) approximation to a number of small molecules with the primary aim of testing the approximation for molecular applications. A new numerical method to solve the one-electron wave equation is developed, utilizing the special features of the SDF formalism. We have calculated energy curves, dissociation energies, and equilibrium distances for some diatomic molecules [H (²Σ, ²Σ), H₂(¹Σ, ³Σ), He (¹Σ), and He₂(¹Σ)] and the vibrational frequencies of H₂. The deviations from the experimental results are typically 1/2 eV for the energies and ≤ 0.1 Å for the distances. We discuss the LSD approximation using the concept of an exchange-correlation hole and make predictions about the applicability to other molecules. The LSD approximation is compared with the Hartree-Fock and multiple-scattering-Xα methods and some difficulties in the latter methods are pointed out. It is argued that the SDF formalism within the LSD approximation has physical advantages compared to the Hartree-Fock and Xα methods and that it should provide a simple and useful method for a broad range of applications.     

The proton affinity and the structure of protonated species of methylsilane

An ab initio LCAO-MO-SCF calculation was made on the proton affinity (PA ) of methylsilane (CH₃SiH₃) by using STO -3G, MIDI -1, and MIDI -1* basis sets. Three types of protonated methylsilane are taken into account, and their geometrical parameters are optimized. The calculated PA of CH₃SiH₃ is 160.5 kcal/mol, which exceeds that of SiH₄ by 11.5 kcal/mol. The protonated species (I) which refers to Si—C bond protonation is shown to be most favorable, and to be a weak σ-complex between CH₄ and SiH. Other two species are also σ-complexes between H₂ molecule and SiH₃CH or CH₃SiH, and similar to CH, SiH, GeH, and C₂H.     

Atomic integrals containing rrr with λ, μ, ν ≥ −2

In this paper, the efficient evaluation of the atomic integrals I =∫rrrrrre^{?αr1?βr2?γr3}dτ with one or two factors r is described. These integrals are necessary for a lower-bound calculation for Li-like systems using the method of variance minimization or Temple's formula. © 1993 John Wiley & Sons, Inc.     

SCF and MC–SCF Study of CO2 with bond angle variation

Multiconfiguration (MC ) SCF calculations are reported for CO₂ for bond angles between 60° and 180°. The ground state configuration is found to be …?5a4b∣b∣a for small bending angles and …?6a3b∣b∣a for large bending angles, the change in ground state character occurring at a bond angle of about 100°. The force constant for bending obtained from the MC –SCF function is about 8.0% lower than the corresponding SCF value, and in considerably better agreement with experiment.     

Dipole,quadrupole, octupole,and dipole–dipole–quandrupole polarizabilities and second hyperpolarizabilities at real and imaginary frequencies for helium in the 23S state: Dispersion-energy coefficients for interactions between He(23S) and H(12S), He(11S), He(23S), or H2(X1∑g+)

We have determined the dynamic dipole (α₁), quadrupole (α₂), octupole (α₃), and dipole–dipole–quadrupole (B) polarizabilities and the second hyperpolarizability tensor (γ) for the helium atom in its lowest triplet state (2³S). We have done so for both real and imaginary frequencies: in the former case, for a range of frequencies (ω) between zero and the first electronic-transition frequency, and in the latter case for a 32-point Gauss–Legendre grid running from zero to ?ω = 20 E_h. We have also determined the dispersion-energy coefficients C₆, C₈, and C₁₀ for the systems H(1²S)? He(2³S), He(1¹S)? He(2³S), and He(2³S)? He(2³S) and the C, C, C, C, and C coefficients for the interaction He(2³S)? H₂(X¹∑). Our values of the higher-order multipolar polarizabilities and of γ for the 2³S state of helium are, we believe, the first to be published. © 1993 John Wiley & Sons, Inc.     

Dipole,quadrupole, octupole,and dipole–octupole polarizabilities at real and imaginary frequencies for H,HE, and H2 and the dispersion-energy coefficients for interactions between them

We have calculated certain dynamic polarizabilities (for both real and imaginary frequencies) for H, He, and H₂ and the dispersion-energy coefficients for long-range interactions between them. We have done so in a sum-over-states formalism with explicitly electron-correlated wave functions to describe the states. To be precise, we have determined the dipole (α₁), quadrupole (α₂), and octupole (α₃) polarizabilities of H and He for real frequencies (ω) in a range between zero and the first electronic-transition frequency and for imaginary frequencies (iω) on a 32-point Gauss-Legendre grid running from zero to ?ω = 20 E_h, and for H₂, we have found the dipole (α), quadrupole (C), and dipole–octupole (E) polarizability tensors for the same real and imaginary frequencies. The dispersion-energy coefficients, obtained by combining the sum-over-states for-malism for the polarizabilities with analytic integration over ω, gave values of C₆, C₈, and C₁₀ for the atom–atom systems; C, C, C, C, and C for the atom–diatom systems; and C₆, C and C for the H₂? H₂ system. Nearly all the results are considered to be more reliable than those hitherto published and some have been obtained for the first time, e.g., C(iω), E(ω), and E(iω) for H₂ and C, C, and C for the H? H₂ system. © 1993 John Wiley & Sons, Inc.     

Electronegativity equalization and the electronic structure of polyhedral clusters of main-group atoms

The concept of the equalization of atomic electronegativities accompanving molecule formation is applied to a study of the electronic structure of polyhedral clusters of main-group atoms such as Ge, Sn, Pb, Tl, and Bi. Emphasis is placed upon charged clusters such as Sn_9?x Pb(x = 0 → 9), Sn_9-xGe, Sn_8?xPb_x Tl^5?, Sn₂Bi, SnTe, etc. The role of the relativistic spin-orbit splitting of an np shell into np_1/2 and np_3/2 subshells in modifying atomic and hence molecular electronegativities is discussed. Correlations are made between calculated charge distributions and observed¹⁹⁹ Sn NMR chemical shifts for clusters of a given size and charge. It is concluded that a useful picture of charge distributions in these clusters may be obtained from electronegativity equalization considerations.     

An insight into optical and EPR properties of AgCl and AgF complexes through MS–Xα and SCCEH calculations

MS-Xα and SCCEH calculations on the Ag²⁺ complexes AgF and AgCl (displaying an elongated D_4h symmetry) have been carried out for a better understanding of their experimental optical and EPR properties. As salient features, the present work supports that the unpaired electron in AgCl spends a little more time on ligands than on Ag²⁺, in agreement with the previous analysis of EPR and optical data for KCl:Ag²⁺. Furthermore, the five experimental optical transitions observed in that case are reasonably assigned. The first transition (observed at 12,500 cm^?1) is assigned to a jump involving the 5a_1g orbital built mainly (∽70%) from 3p orbitals of axial ligands, a fact that reflects the distinct level scheme for AgCl when compared to that for more ionic complexes. Calculations on AgF and AgF performed as a function of the equatorial Ag²⁺ –F^? distance led to a reasonable understanding of experimental gyromagnetic and superhyperfine tensors displayed by Ag²⁺ in fluorides. The different relative decrease undergone by g‖– go (8%) and g ? – go (28%) on passing from CsCdF₃:Ag²⁺ to RbCdF₃:Ag²⁺ is shown to be consistent with the formation of AgF and AgF complexes, respectively, related to the different substitutional position of Ag²⁺ in such lattices. The decrement of about 8.5% experienced by both g‖ – go and g? – go values on going from CsCdF₃:Ag²⁺ to NaF:Ag²⁺ is pointed out to reflect the different electrostatic potential (exerted by the rest of the lattice upon the complex) seen by AgF embedded in NaCl or perovskite-type lattices. © 1994 John Wiley & Sons, Inc.     

Ab initio calculations of many-body energy expansion in LiF− clusters

The effects of the basis-set size on many-body energy expansion in LiF^? clusters are investigated and correlated with previously reported values on LiCl^? analogs. Coulomb and non-Coulomb energies in LiF^? at different configurations are also examined. Although at the minimal STO -3G basis V^na(3, 4) and V^na(4, 4) nonadditivity terms were the smallest in the D₃h configuration, they were the largest at the extended 6-311 ++G basis. V(m, n) terms where m = n ≥ 3 were found to be playing a small role in the chemistry and physics of LiF^? clusters compared with V(3, n) terms in LiCl^? clusters.     

Ab initio MO calculations of hyperfine coupling constants. A basis set study for 19F, 35Cl, 19F,and 35Cl

Hyperfine coupling constants (HFCC ) of the ¹⁹F and ³⁵Cl atoms and the ¹⁹F and ³⁵Cl radical anions have been calculated by the unrestricted Hartree–Fock (UHF ) method using polarization and diffuse functions with contracted double-zeta as well as uncontracted basis sets. The A_dip values are fairly insensitive to changes in the basis set and show good accordance with experimental and other theoretical studies. The isotropic HFCCS a_N of ¹⁹F, ¹⁹F, and ³⁵Cl show strong dependence on d functions and the state of contraction of the s, p set. Spin-projected UHF wave functions lead to better agreement with experiment.     

The formation of highly excited H in the reaction H2+(v) + H2 → H + H

A quasiclassical trajectory surface hopping method has been used to study H(v) + H₂ → H + H for v = 0, 3, 7, 10, 13, and 17 with an emphasis on determining the H internal energy and angular momentum distributions for high v. For v = 13 and 17, significant cross sections are found for producing H at energies above its dissociation energy. An average metastable H lifetime of 11.5 ps for v = 13 and 4.7 ps for v = 17 is found, but there is also a much longer lived component to the lifetime distributions that is more important for v = 13 than for v = 17. Some of the longer lived metastables correspond to high angular momentum orbiting states of H, but other sources of metastability are also present.     

Polysila analogs of aromatic hydrocarbon ions: Structures and energies of Si3H3+, Si4H,and Si5H5−

Silicon analogs of aromatic monocyclic ions, (SiH) ( 4 ), (SiH) ( 5 ), and (SiH) ( 6 ) have been studied ab initio at MP 2(full)/6-31G *. The D_3h structure of Si₃H₃⁺ is the global minimum, whereas other two ions are nonplanar. The D_2d structure of (SiH) is less folded than the carbon analog and possesses a higher stabilization energy. Stabilization energies for the monocharged ions are diminished with respect to the corresponding carbons © 1993 John Wiley & Sons, Inc.     

Density functional calculations of structures and ionization energies for heavy group V cluster anions

The structure and electronic structure of heavy-group V cluster anions (Sb, Bi) are calculated with density functional methods within the local spin density approximation (LSDA ). The influence of gradient corrections of the exchange and correlation energy is investigated. The calculated vertical and adiabatic ionization energies are in very good agreement with data from photoelectron spectroscopy (PES ) for Sb, whereas the relatively large deviations for Bi can be reduced by the consideration of relativistic effects in a scalar-relativistic manner. Concerning the structures, a strong similarity to the corresponding P clusters was found. In particular, the negatively charged pentamers are planar rings (with similarities to the aromatic [C₅H₅]^? anion) with especially high ionization energies. © 1995 John Wiley & Sons, Inc.