Differential nephelauxetic reduction of the F2 and F4 Slater—Condon integrals

The sensitivity of the F₂ and F₄ Slater—Condon integrals to changes in the radial distribution due to covalency has been investigated for 3dⁿ cations using SCF atomic orbitals. It is demonstrated that either F₂ or F₄ may show the greater nephelauxetic effect, although the former is probably the more likely situation.     

The use of the CNDO method in spectroscopy

The Pariser approximation for the two center Coulomb repulsion integrals Γ _rshas been replaced by the Nishimoto-Mataga approximation in the original CNDO/S method. This modification has significantly improved the calculated position of the benzene ¹ B _1u ←¹ A _1g(¹ L _a ) electronic transition in benzenoid compounds. The calculation of transition moments of n — π ^* transitions is also considered. These moments vanish formally in any theory employing the ZDO approximation since integrals of the form 〈2s¦er2p〉 vanish even when the 2s and 2p atomic orbitals are on the same center. In this work the ZDO approximation is abandoned in the evaluation of the electronic transition moment resulting in calculated intensities for n — π ^*, ¹W←¹A, transitions which are in good agreement with experiment.     

An application of double exponential formula to radial quadrature grid in density functional calculation

We report an application of the double exponential formula to the numerical integration of the radial electron distribution function for atomic and diatomic molecular systems with a quadrature grid. Three types of mapping transformation in the double exponential formula are introduced into the radial quadrature scheme to generate new radial grids. The double exponential grids are examined for the electron-counting integrals of He, Ne, Ar, and Kr atoms which include occupied orbitals up to the 4p shell. The performance of radial grid is compared for the double exponential formula and the formulas proposed in earlier studies. We mainly focus our attention on the behavior of accuracy by the quadrature estimation for each radial grid with varying the mapping parameter and the number of grid points. The convergence behavior of the radial grids with high accuracy for atomic system are also examined for the electron-counting integrals of LiH, NaH, KH, Li₂, Na₂, K₂, HF, HCl, HBr, F₂, Cl₂, Br₂, LiF, NaCl, KBr, [ScH]⁺, [MnH]⁺, and [CuH]⁺ molecules. The results reveal that fast convergence of the integrated values to the exact value is achieved by applying the double exponential formula. It is demonstrated that the double exponential grids show similar or higher accuracies than the other grids particularly for the Kr atom, Br₂ molecule, alkali metal hydrides, alkali metal halogenides, and transition metal hydride cations, suggesting that the double exponential transformations have potential ability to improve the reliability and efficiency of the numerical integration for energy functionals.     

The inclusion of a correlation function in the calculation of the ZFS of benzene

We calculated the one-center atomic integral of the spin—spin interaction with respect to Slater orbitals where we have included a two-electron correlation function that we used previously to calculate the hyperfine splitting of the 2³P state of He. The ZFS of the benzene molecule was calculated from this value of the one-center integral and by using the Mulliken approximation combined with a previously introduced approximation for extimating the other integrals. The calculation was performed for two different non-hexagonal configurations I and II, the results are D₁ = 0.132 cm^?1 and D_II = 0.143 cm^?1. The experimental value is D = 0.158 cm^?1.     

Electric and magnetic multipole integrals in STO basis sets: an analytical approach

A new method for the evaluation of one- and two-centre magnetic and electric multipole integrals for Slater-type functions is presented. The method is strictly analytical in that no approximations of any kind are involved. Two simple functions, ℐ₁ ^aug and ℐ₂ ^aug, are introduced, which employ only functions that are well known in electronic structure theory. With the use of augmentation exponents these functions apply to multipole integrals as well as other one-electron integrals, such as nuclear attraction integrals. The proposed method includes the analytic determination of derivatives of the integrals with respect to atomic displacements. Some illustrative test calculations are presented and compared to results from the literature. Received: 20 April 1998 / Accepted: 13 October 1998 / Published online: 1 February 1999     

On the hyperfine structure in the configurations 4f n 6s 2, 4f n 5d6s, 4f n 6s6p,and 4f n−15d6s 2 of the lanthanides

A survey of hyperfine analyses in the low-lying configurations 4f ⁿ 6s ², 4f ⁿ 5d6s, 4f ⁿ 6s6p, and 4f ^n?1 5d6s ² of the lanthanides is given. Experimental hf radial integrals 〈r ^?3〉 _nl ^kj are indicated for the configurations under investigation. From a comparison of experimental and theoretical hf radial integrals configuration-interaction contributions Δ _nl ^kj to the hfs have been evaluated for the configurations for which corresponding integrals were available. With nuclear electric quadrupole moments from muonic x-ray investigations and radial integrals 〈r ^?3〉 _nl ⁰¹ and 〈r ^?3〉 _nl ⁰² from experimental hfs quadrupole shielding correctionsR _nl ⁰¹ have been obtained. The variation of hyperfine radial integrals, configuration-interaction contributions and quadrupole shielding corrections over the 4f shell in the configurations under study is discussed. Trends of these quantities in different configurations are compared.     

Determination of many-particle integrals by the method of distance functions

An interpolative method for determining many-particle integrals needed in quantumchemical calculations is proposed. The basic idea is that integrals of the form ∫dr _I ∫dr _J?(r _I ,r _J )r _IJ ^2h (h=01,2,...,)(r _I and r _J are coordinates of the I-th and the J-th particle, respectively, r _IJ is their distance and ?(r _I ,r _J ) is a charge distribution) can be easily calculated and can be used to determine integrals∫dr _I ∫dr _J?(r _I ,r _J )r _IJ ^{2h ? 1} by interpolation (or for h = 0 by extrapolation). By making use of the identityΔ _I Δ _J r _IJ ^h+4 = (h + 2) (h + 3) (h + 4) (h + 5) r _IJ ^h (where Δ is the Laplacian operator) the extrapolation can be replaced by interpolation definitely improving the accuracy. As a probably most natural tool for interpolation the use of an auxiliary quantity called the distance function is discussed. The method can be considerably generalized (e.g. to many-particle integrals). Numerical results obtained by applying a preliminary form of the method to two-center integrals resulting from a Slater-type base are given.     

An analysis of the integrand occurring in correlation energy functionals

An analysis of the integrand occurring in current density functional calculations is presented, concentrating attention on correlation energy functionals and the atomic regions, i.e., the regions of space surrounding atomic centers. The analysis follows the structure of a previously proposed numerical integration scheme for three-dimensional integrals occurring in electronic structure calculations. The scheme is based on the choice of density-based weight functions that naturally partition the space into «atomic» volumes (in which the integration is performed in terms of spherical coordinates) and «diatomic» volumes (in which the integration is performed in terms of confocal elliptical coordinates). From this analysis, a simplified procedure for the atomic «internal» integrations is developed, whereas preliminary results are discussed for the atomic and diatomic «external» integrations. The numerical tests are performed on the C₆₀ molecule in the symmetrical configuration. © John Wiley & Sons, Inc.     

Hyperfine structure in the configurations4f 4 6s 2 and4f 4 5d 6s of Nd I

The high resolution laser-atomic-beam technique was used to investigate the hyperfine structure in Nd I 4f ⁴6s ^{2 5} I,⁵ F,⁵ S and 4f ⁴5d6s ⁷ L,⁷ K,⁷ I,⁷ H. The metastable states were populated by an arc discharge burning in the atomic beam. The measured hyperfine constantsA andB of the levels of 4f ⁴6s ² and 4f ⁴5d6s allow a parametric analysis to be performed using the effective tensor operator formalism. The experimental radial integrals of the 4f and 5d electrons fit with those of the other lanthanides. The 4f radial integrals are in agreement with values of optimized Hartree-Fock-Slater calculations. The spectroscopic quadrupole moments of¹⁴³Nd and¹⁴⁵Nd are deduced from the 4f parameters:Q _I =?0.610(21) b and ?0.314(12) b, respectively. TheQ _I resulting from the 5d parameter are in satisfactory agreement with these values. The hyperfine anomaly due to thes electron in 4f ⁴5d 6s amounts to about 1%.     

Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

Optical Study of Two-Dimensional Organic Metal (EO-TTP)2AsF6 (EO-TTP=2-(4,5-Ethylenedioxy-1,3-Dithiol-2-Ylidene)-5-(1,3-Dithiol-2-Ylidene)-1,3,4,6-Tetrathiapentalene)

Polarized reflectance spectra were measured on the conductive (010) plane of metallic (EO-TTP)₂AsF₆ single crystal. At room temperature, well-defined plasma edges appeared in both directions parallel (E||a) and perpendicular (E⊥a) to the molecular stack. The intra- and inter-stack transfer integrals were estimated from the plasma frequencies in the framework of tight-binding model. Based on these transfer integrals, we suggested that the Fermi surface was open in the k_c direction. Using a generalized Drude model, we obtained the frequency dependence of relaxation rate, which conformed to γ(ω)=γ₀+bω² in the frequency range 1800-5500 cm⁻¹.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

Linear dependence ofXα eigenvalues on occupation number application to the determination of the ligand field integrals

A linear relation between theXα eigenvalues and the occupation number of the ligand field states in CrCl₄ is established and used to calculate mono- and bimolecular integrals of ligand field interest. A formula for calculating 10 Dq and Koide-Pryce's parameter (ε) from theXα eigenvalues is given. The method is generalized for anyd ^N system.     

A modified MNDO procedure for calculating the heats of formation of organic anions

A modification of the MNDO method is suggested based on the relation ζ_A=ζ_A ⁰(1 + ϰ_AQ_A) for Slater parameters that takes into account the dependence of overlap integrals and the MNDO parameters (expressed via these integrals) on atomic charges Q_A. The core parameters U_μμ include corrections for nonorthogonality of basis functions of atoms in a molecule. For free atoms, the parameters were determined from the condition of reproducible electron affinities. The method is advantageous for calculating the heats of formation of anions, particularly those with the charge concentrated on one atom. Institute of Chemical Kinetics and Combustion, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 1, pp. 3–11, January-February, 1994. Translated by O. Kharlamova     

Step structure in the atomic Kohn-Sham potential

In this work we analyze the exchange-correlation potentialv _xc within the Kohn-Sham approach to density functional theory for the case of atomic systems. The exchange-correlation potential is written as the sum of two potentials. One of these potentialsv _xc,scr is the long-range. Coulombic potential of the coupling constant integrated exchange-correlation hole which represents the screening of the two-particle interactions due to exchange-correlation effects. The other potentialv _xc,scr ^resp contains the functional derivative with respect to the electron density of the coupling constant integrated pair-correlation function representing the sensitivity of this exchange-correlation screening to density variations. As explicit expression of the exchange-part of this functional derivative is derived using an approximation for the Greens function of the Kohn-Sham system and is shown to display a distinct atomic shell structure. The corresponding potentialv _xc,scr ^resp has a clear step structure and is constant within the atomic shells and changes rapidly at the atomic shell boundaries. Numerical examples are presented for the Be and Kr atoms using the Optimized Potential Model (OPM).     

Electronic structure of free and doped actinides: N and Z dependences of energy levels and electronic structure parameters

Theoretical study of electronic structure of antinide ions and its dependence on N and Z are presented in this paper. The main 5f^N and excited 5f^Nn′l′^N′ configurations of actinides have been studied using Hartree-Fock-Pauli approximation. Results of calculations of radial integrals and the energy of X-ray lines for all 5f ions with electronic state AC⁺¹−AC⁺⁴ show approximate dependence on N and Z. A square of N and cubic of Z are ewalized for the primary electronic parameters of the actinides. Theoretical values of radial integrals for free actinides and for ions in a cluster AC⁺ⁿ:[L]_k are compared, too.     

The calculation of the molecular orbital parameters for the square coplanar Cu(NH3)2+4 cation in Cu(NH3)4PtCl4

The calculation of the molecular orbital parameters of the σ-bonding square-planar Cu(NH₃)²⁺₄ cation in Cu(NH₃)₄PtCl₄ is reported, involving metal—ligand and ligand—ligand overlap integrals. In order to obtain a reasonable correlation of the magnetic g- and A-values and the electronic transition energies it is necessary to use a Cu⁺ wavefunction to represent the radially expanded Cu²⁺ wavefunction in this complex.