Momentum density and molecular geometry. Bent BH 2 − and linear BH 2 +

Based on a special form of the molecular virial theorem, the recently proposed method of momentum density for interatomic interactions is here applied to the problem of molecular geometry. Two molecules BH ₂ ^– and BH ₂ ⁺ , which have the same nuclear framework but favor respectively bent and linear conformations, are comparatively studied. Using an approximate Hartree-Fock momentum density, the total molecular energy (including the nuclear repulsion) is partitioned into orbital components, and a geometry correlation diagram is derived. An atom-bond partitioning of the total energy is also examined based on the one- and two-center decomposition of the momentum density.     

Experimental and calculated momentum densities for the valence orbitals of carbon tetrafluoride

Carbon tetraflouoride has been investigated by binary (e,2e) spectroscopy at 1200 eV impact energy. Binding energy spectra (10–60 eV) at azimuthal angles of 0° and 8° are reported and are found to be in quantitative agreement with a previous Green's function calculated spectrum. Momentum distributions corresponding to individual orbitals are also reported and compared with theoretical momentum distributions evaluated using double-zeta quality SCF wavefunctions. Excellent agreement between experimental and theories is found for the strongly bonding 3t₂ orbital and the antibonding 4a₁ orbital but agreement is less good for the outermost non-bonding orbitals. Intense structure due to molecular density (bond) oscillation is observed experimentally in the region above 1.0 a_o^?1 in the case of the non-bonding 4t₂ orbital. It is also notable that the measured 4a₁ momentum distribution exhibits an extremely well-defined “p” character with clear separation between the s and p components. Contour maps of the position-space and momentum-space orbital densities in the F-C-F plane of the molecule are used to provide a qualitative interpretation of the features observed in the momentum distribution. In order to further extend momentum-space chemical concepts to three-dimensional systems, constant density surface plots are also used to give a more comprehensive view of the density functions of the CF₈ molecule.     

Substituent effects in second-row molecules: Basis set performance in calculations of normal valency phosphorus and sulfur compounds

Ab inito molecular orbital calculations of the phosphorus- and sulfur-containing series PH₂X, PH₃X⁺, SHX, and SH₂X⁺ (X = H, CH₃, NH₂, OH, F) have been carried out over a range of Gaussian basis sets and the results (optimized geometrical structures, relative energies, and electron distributions) critically compared. As in first-row molecules there are large discrepancies between substituent interaction energies at different basis set levels, particularly in electron-rich molecules; use of basis sets lower than the supplemented 6-31G basis incurs the risk of obtaining substituent stabilizations with large errors, including the wrong sign. Only a small part of the discrepancies is accounted for by structural differences between the optimized geometries. Supplementation of low level basis sets by d functions frequently leads to exaggerated stabilization energies for π-donor substituents. Poor performance also results from the use of split valence basis sets in which the valence shell electron density is too heavily concentrated in diffuse component of the valence shell functions, again likely to occur in electron-rich molecules. Isodesmic reaction energies are much less sensitive to basis set variation, but d function supplementation is necessary to achieve reliable results, suggesting a marginal valence role for d functions, not merely polarization of the bonding density. Optimized molecular geometries are relatively insensitive to basis set and electron population analysis data, for better-than-minimal bases, are uniform to an unexpected degree.     

Variational solution of the single-particle Dirac equation in the field of two nuclei using relativistically adapted Slater basis functions

A variational method for solving the time-independent single-particle Dirac equation in the Coulomb field of two nuclei is described. A minimax variational principle and basis functions that have the proper analytic behavior, i.e. behave like r ^γ,γ non-integer, in the neighborhood of a nucleus, are used. A momentum space integration scheme for computing the necessary two-center integrals is described. Results are given for a standard test problem on two nuclei with Z=90 with an internuclear separation of 2.0/Z. The results confirm those of a previous calculation [F.A. Parpia and A.K. Mohanty, Chem Phys Lett 238: 209 (1995)]. Received: 13 May 1998 / Accepted: 22 June 1998 / Published online: 28 August 1998     

New Noncentrosymmetric Tetrel Pnictides Composed of Square-Planar Gold(I) with Peculiar Bonding

Three novel isostructural equiatomic gold tetrel pnictides, AuSiAs, AuGeP, and AuGeAs, were synthesized and characterized. These phases crystallize in the noncentrosymmetric (NCS) monoclinic space group Cc (no. 9), featuring square-planar Au within cis-[AuTt₂Pn₂] units (Tt=tetrel, Si, Ge; Pn=pnictogen, P, As). This is in drastic contrast to the structure of previously reported AuSiP, which exhibits typical linear coordination of Au with Si and P. Chemical bonding analysis through the electron localization function suggests covalent two-center two-electron Tt−Pn bonds, and three-center Au−Tt−Au and Au−Pn−Au bonds with 1.6 e⁻ per bond. X-ray photoelectron spectroscopy studies support the covalent and nonionic nature of Au−Pn and Au−Tt bonds. The title materials were found to be n-type narrow-gap semiconductors or semimetals, with nearly temperature-independent electrical resistivities and low thermal conductivities. A combination of the semimetallic properties with tunable NCS structure provides opportunities for the development of materials based on gold tetrel pnictides.     

Methods for the calculation of spherically averaged Compton profiles with GTOs

For the first time, an analytical and efficient algorithm for the evaluation of spherically averaged reciprocal form factors B(s)=〈B( s )〉_Ω using Gauss-type basis functions is presented. The spherically averaged Compton profile is available by Fourier transformation of the reciprocal form factor. The algorithm has been successfully implemented in connection with the quantum chemistry codes GAMESS and CRYSTAL92, which perform Hartree–Fock calculations for molecules and solids. In addition, an analytical algorithm for the direct evaluation of spherically averaged Compton profiles and the moments 〈p^ν〉 (ν≥−1) via the momentum density is proposed for Gauss-type basis functions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 213–223, 1997     

A CNDO/INDO molecular orbital formalism for the elements H to Br. theory

A CNDO and INDO formalism is presented that can be used for any atom combination up to bromine under inclusion of the first transition metal series. The semiempirical parameters were chosen to reproduce results ofab initio calculations on metalorganic compounds. The calculational results are invariant to rotations of the coordinate system but not to a general transformation into other basis functions. The one-center Coulomb-expressions were selected in order to include intraatomic correlation contributions. Within the CNDO model this could be achieved by the scaled monopole termF ₀, while in the INDO framework the one-center Coulomb integrals are given as a sum of the monopole-contributionF ₀ and higher multipole contributions expressed as a linear combination of Slater-Condon parameters. The invariance problem in the case of local rotations within the INDO approximation was solved by considering the combination of one-center Coulomb and exchange integrals as a function ofl but independent ofm. The two-center electron-electron interaction terms were calculated via the Dewar-Sabelli, Ohno-Klopman relation. Penetration effects were treated according to Fischer and Kollmar. For the resonance integralH _v ^AB parameters are used which carry information related to the directed nature of the chemical bond by using optimized Klondyke functions. The core-core repulsion is constructed as a superposition of a soft potential function, describing polarization effects of the atomic cores, and a hard repulsion function, avoiding the collapse of the atomic cores with decreasing distance.     

Basis set quality. II. Information theoretic appraisal of various s- orbitals

The expectation values 〈r^k〉 (?2 ? k ? 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21 , 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.     

Spherical tensor gradient operator method for integral rotation: a simple, efficient, and extendable alternative to Slater-Koster tables

We present a novel alternative to the use of Slater-Koster tables for the efficient rotation and gradient evaluation of two-center integrals used in tight-binding Hamiltonian models. The method recasts the problem into an exact, yet implicit, basis representation through which the properties of the spherical tensor gradient operator are exploited. These properties provide a factor of 3 to 4 speedup in the evaluation of the integral gradients and afford a compact code structure that easily extends to high angular momentum without loss in efficiency. Thus, the present work is important in improving the performance of tight-binding models in molecular dynamics simulations and has particular use for methods that require the evaluation of two-center integrals that involve high angular momentum basis functions. These advances have a potential impact for the design of new tight-binding models that incorporate polarization or transition metal basis functions and methods based on electron density fitting of molecular fragments.     

Applications of fourier transforms in molecular orbital theory. Calculation of optical properties and tables of two-center one-electron integrals

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular part of the integrals to be treated by means of projection operators. The radial parts of the integrals are treated by the methods of Harris. The results are obtained in a simple closed form which avoids the usual transformation to local coordinates. The two-center integrals evaluated include matrix elements of the momentum operator, the dipole moment operator, the tensor operator , the quadrupole moment operator, and the angular momentum operator. These are evaluated between 1s, 2s, and 2p Slater-type atomic orbitals located on different atoms. The results are expressed as functions of the Slater exponents and of the relative coordinates of the two atoms.     

Diatomic interactions in momentum space. The 1sσg and 2pσ u states of H 2 + system

The method of momentum electron density for interatomic interactions has been applied to the two lowest σ states of the H ₂ ⁺ system. For attractive (1sσ_g) and repulsive (2pσ _u ) interactions, the behaviour of momentum density and its effect on the stabilization energy of the system are examined quantitatively. The concept of contraction and expansion of the momentum density is shown to form an important guiding principle in this approach. The origin of covalent bonding is discussed based on the energy partitioning proposed previously.     

Electron correlation and the compton profile of atomic neon

The radial momentum distribution I_o(p) and the Compton profile J_o(q) are determined for atomic neon from several restrictid Hartree-Fock (RHF) wavefunctions and two configuration interaction (CI) wavefunctions. The CI functions are the well correlated (full“second-order”) function of Viers, Schaeffer and Harris, and the Ahlrichs-Hinze multi-configuration Hartree-Fock (MCHF) function which includes only L-shell correlation. It is found for this completely closed shell system that the effects of electron correlation are quite small. This contrasts with the results for systems such as Be(²S) and B(²P) where the semi-internal and internal correlation effects were responsible for significant discrepancies between the RHF and CI results. These results indicate that a wavefunction which carefully includes the semi-internal, orbital polarization, and internal correlations beyond the RHF wavefunction (i.e., a “first-order” or “charge-density” function), should account for the principal correlation effects on the Compton profiles and momentum distributions.     

The Fredholm method in low energy electron-molecule scattering: Static exchange

The Fredholm method using a two-center, L² basis is presented for electron-hydrogen molecule scattering including exchange, but neglecting polarization. At low energies, phase shifts computed by analytically continuing the Fredholm determinant agree well with J = 0 close-coupling results.     

On the additivity of basis set effects in some simple fluorine containing systems

The reactions F + H₂ → HF + H, HF → H + F, F → F⁺ + e^? and F + e^? → F^? were used as simple test cases to assess the additivity of basis set effects on reaction energetics computed at the MP4 level. The 6-31G and 6-311G basis sets were augmented with 1, 2, and 3 sets of polarization functions, higher angular momentum polarization functions, and diffuse functions (27 basis sets from 6-31Gd, p) to 6-31 ++ G(3df, 3pd) and likewise for the 6-311G series). For both series substantial nonadditivity was found between diffuse functions on the heavy atom and multiple polarization functions (e.g., 6-31 + G(3d, 3p) vs. 6-31 + G(d, p) and 6-31G(3d, 3p)). For the 6-311G series there is an extra nonadditivity between d functions on hydrogen and multiple polarization functions. Provided that these interactions are taken into account, the remaining basis set effects are additive to within ±0.5 kcal/mol for the reactions considered. Large basis set MP4 calculations can also be estimated to within ±0.5 kcal/mol using MP2 calculations, est. E_MP4(6-31 ++ G(3df, 3pd)) ≈ E_MP4(6-31G(d, p)) + E_MP2(6-31 ++ G(3df, 3pd)) – E_MP2(6-31G(d, p)) or E_MP4(6-31 + G(d, p) + E_MP2(6-31 ++ G(3df, 3pd)) – E_MP2(6-31 + G(d, p)) and likewise for the 6-311G series.     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

Relation between the inverse Laplace transforms of I(tβ) and I(t): Application to the Mittag-Leffler and asymptotic inverse power law relaxation functions

The relation between H(k), inverse Laplace transform of a relaxation function I(t), and H_β(k), inverse Laplace transform of I(t^β), is obtained. It is shown that for β < 1 the function H_β(k) can be expressed in terms of H(k) and of the Lévy one-sided distribution L_β(k). The obtained results are applied to the Mittag-Leffler and asymptotic inverse power law relaxation functions. A simple integral representation for the Lévy one-sided density function L_1/4(k) is also obtained.AMS (MOS) classification: 33E12 Mittag-Leffler functions and generalizations, 44A10 Laplace transform, 60E07 Infinitely divisible distributions; stable distributions     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. II. Kinetic and potential energy integrals and examples of numerical results

This article is concerned with the construction of the general algorithm for evaluating two-center, two- and three-electron integrals occurring in matrix elements of one-electron operators in the basis of variational correlated functions. This problem has been solved here in prolate spherical coordinates, using the modified and extended form of the Neumann expansion of the interelectronic distance function r^k_ij derived in Part I of this series for k = ?1, 0, 1, 2. This work expands the method proposed by one of us in the preceding paper for integrals of the types mentioned above. The results of numerical calculations for different types of the two- and three-electron integrals are presented. The problem of convergence of the proposed procedures used is also discussed.     

Solid‐state 1H → 19F/19F → 1H CP/MAS NMR study of poly(vinylidene fluoride)

Solid‐state ¹H → ¹⁹F and ¹⁹F → ¹H cross‐polarization magic angle spinning (CP/MAS) NMR spectra have been investigated for a semicrystalline fluoropolymer, namely poly(vinylidene fluoride) (PVDF). The ¹H → ¹⁹F CP/MAS spectra can be fitted by five Lorentzian functions, and the amorphous peaks were selectively observed by the DIVAM CP pulse sequences. Solid‐state spin‐lock experiments showed significant differences in T_1ρ^F and T_1ρ^H between the crystalline and amorphous domains, and the effective time constants, T_HF* and T_1ρ*, which were estimated from the ¹H → ¹⁹F CP curves, also clarify the difference in the strengths of dipolar interactions. Heteronuclear dipolar oscillation behaviour is observed in both standard CP and ¹H → ¹⁹F inversion recovery CP (IRCP) experiments. The inverse ¹⁹F → ¹H CP‐MAS and ¹H → ¹⁹F CP‐drain MAS experiments gave complementary information to the standard ¹H → ¹⁹F CP/MAS spectra in a manner reported in our previous papers for other fluoropolymers. The value of N_F/N_H (where N is a spin density) estimated from the CP‐drain curve is within experimental error equal to unity, which is consistent with the chemical structure. Copyright © 2001 John Wiley & Sons, Ltd.     

Density matrices for correlated Gaussians: Helium and dipositronium

Formulas are derived for the density matrices belonging to an n-particle wave function built on the basis of single-center explicitly correlated Gaussian basis functions. An explicit formula for the first-order density matrix, P(r₁, r′₁), is obtained for computing the probability distribution P(r₁, r₁). Other formulas are derived for matrix elements of the first-order density operator P on a basis of single-particle Gaussian orbitals so that natural orbitals (NOs) can be expressed in such a basis. The method is illustrated for the case of the ground state of the helium atom using the 16-term (geminal) wave function by Singer and Longstaff (E = −2.90233 au) and a set of even-tempered Gaussian orbitals. The resulting natural orbitals compare favorably with natural orbitals from Cl expansions. The method is also applied to our 20 term (trimal) wave function for the ground state of dipositronium (E = −0.51560 au). Analysis is made in this case for pair correlation functions of both the electron-electron and the positron-electron pairs; results include the radial distributions of these pairs and their relative angular momentum. © 1996 John Wiley & Sons, Inc.