Thomas–Fermi–Dirac models of atoms constrained by nuclear cusp and long-range conditions

The traditional Thomas–Fermi–Dirac model of the electronic structure for a neutral atom is deficient in that it predicts an infinite electron density at the nucleus and a sharp cutoff of the electron density at a finite radius. This study was carried out to remedy these faults in the model. Extending an idea used earlier in Thomas–Fermi (TF ) theory [Proc. Natl. Acad. Sci. U.S.A. 83 , 3577 (1985)], the Thomas–Fermi–Dirac (TFD ) energy functional is minimized under constraints ∫ρ( r ) d r = N, ∫e^?2kr▽²ρ( r )d r < ∞ and ∫(1 ? e^?k′r)ρ^4/3( r )d r < ∞, with k and k′ determined by the nuclear cusp condition and the correct asymptotic behavior. Optimum coordinate scaling also is considered. It is found that the TFD model is substantially improved by constraining the minimization search domain of the energy functional in this way. Energies are given for five noble gas atoms, and Compton profiles for these atoms are calculated. The behavior of electrons in momentum space is improved in both this modified TFD model and in the corresponding modified TF model.     

Relativistic theory of binding energies of heavy positive ions

The self-consistent relativistic Thomas–Fermi theory of heavy positive ions with N electrons and nuclear charge Ze is shown to lead to a chemical potential μ which has the scaling property with ? = α²Z², α being the fine structure constant. Combining this with the Layzer–Bahcall expansion for the total energy E(Z, N), namely, it is proved that the coefficients E_nm (N) at large N have the asymptotic behavior N^n–2m/3#1/3. The corresponding result for the scaling of the relativistic Thomas–Fermi energy is Scaling properties of the higher order terms in E_nm (N) and E(Z, N) are also proposed.     

Relativistic total energy of heavy atomic ions: Dimensionality dependent scaling

The scaling property of the relativistic total energy E (Z, N) established by Marconi and the writer for heavy positive ions with N electrons and atomic number Z is demonstrated, by setting up the self-consistent relativistic Thomas–Fermi equation in d dimensions, to be a special case of the scaling property α being the fine structure constant.     

β‐Y2Si2O7, a new thortveitite‐type compound,determined at 100 and 280 K

A new form of Y₂Si₂O₇ (diyttrium heptaoxodisilicate) has been synthesized which is isotypic with thortveitite, Sc₂Si₂O₇, and crystallizes in the centrosymmetric space group C2/m, both at 100 and 280 K. The Y³⁺ cation occupies a distorted octahedral site, with Y—O bond lengths in the range 2.239 (2)–2.309 (2) Å. The SiO₄ tetrahedron is remarkably regular, with Si—O bond lengths in the range 1.619 (2)–1.630 (2) Å. The bridging O atom of the Si₂O₇ pyrosilicate group shows a large anisotropic displacement perpendicular to the Si—O bond. Changes in lattice and structural parameters upon cooling are small with, however, a distinct decrease of the anisotropic displacement of the briding O atom. Structure solution and refinement in the non‐centrosymmetric space group C2 are possible but do not yield a significantly different structure model. The Si—O—Si bond angle of the isolated Si₂O₇ groups is 179.2 (1)° at 280 K in C2 and 180° per symmetry in C2/m. The C2/m structure model is favoured.     

Information entropies of many-electron systems

The Boltzmann–Shannon (BS ) information entropy S_ρ = ∫ ρ(r)log ρ(r)dr measures the spread or extent of the one-electron density ρ(r), which is the basic variable of the density function theory of the many electron systems. This quantity cannot be analytically computed, not even for simple quantum mechanical systems such as, e.g., the harmonic oscillator (HO ) and the hydrogen atom (HA ) in arbitrary excited states. Here, we first review (i) the present knowledge and open problems in the analytical determination of the BS entropies for the HO and HA systems in both position and momentum spaces and (ii) the known rigorous lower and upper bounds to the position and momentum BS entropies of many-electron systems in terms of the radial expectation values in the corresponding space. Then, we find general inequalities which relate the BS entropies and various density functionals. Particular cases of these results are rigorous relationships of the BS entropies and some relevant density functionals (e.g., the Thomas–Fermi kinetic energy, the Dirac–Slater exchange energy, the average electron density) for finite many-electron systems. © 1995 John Wiley & Sons, Inc.     

Addendum to “variational principle for obtaining approximate analytical solutions of the temperature-perturbed Thomas–Fermi equation for compressed atoms”

In the present work the total energy of a Ne atom at T = 0 K is calculated as a function of a spherical container radius. The calculation is based on the Thomas–Fermi (TF ) equation, which is solved approximately by an equivalent variational principle. The effect of an approximate exchange correction on the variational TF energy values is investigated.     

Thomas–Fermi theory from a perturbative treatment of the hydrogenic limit energy density functional: A possible link to local density functional theory

A perturbation expansion which connects the hydrogenic limit energy density functional to the Thomas–Fermi functional is discussed. This perturbation series, where the Coulomb energy density functional is treated as the perturbation to the hydrogenic limit functional, is, in fact, the q = (N/Z) expansion of Thomas–Fermi theory. A truncated form of the first-order correction to the functional provides further insight into the model which treats the ground state energy as a local functional of the electron density.     

Di­methyl (1‐methyl‐1,3‐benzimidazol‐5‐yl)­amino­methyl­ene­propane­dioate monohydrate

In the title compound, C₁₄H₁₅N₃O₄·H₂O, there is a strong conjugation push–pull effect across the central double bond, as reflected in the molecular dimensions and the planarity of the en­amino­ne portion of the mol­ecule. The mol­ecule has an intramolecular hydrogen bond between the NH and CO groups in the Z configuration, adopting the chelated form. The two π systems of the mol­ecule (1‐methyl­benz­imidazole and en­amino­ne) are deconjugated and tilted with respect to each other by 15.6 (2)°. The solvent water mol­ecule is hydrogen bonded to the N¹ atom of the 1‐methyl­benzimidazolyl group.     

(Z)‐2‐[(1‐Phenylsulfonyl‐1H‐indol‐3‐yl)methylene]‐1‐azabicyclo[2.2.2]octan‐3‐one semicarbazone

In crystals of the title compound, C₂₃H₂₃N₅O₃S, the indole system is planar and the phenyl ring of the phenylsulfonyl group makes a dihedral angle with the best plane of the indole system of 77.18 (4)°. The olefinic bond connecting the azabicyclic and indole systems has Z geometry. The geometry adopted by the C=O bond with respect to the N—N bond is trans. The O atom of the carbonyl group of each molecule is hydrogen bonded to the hydrazidic H atom of an adjacent molecule to form an eight‐membered‐ring dimeric structure.     

The molecular and electronic structure features of silatranes, germatranes, and their carbon analogs

Molecular geometries of fifty-six metallatranes N(CH₂CH₂Y)₃M-X and fifty-six carbon analogs HC(CH₂CH₂Y)₃M-X (M = Si, Ge; X = H, Me, OH, F; Y = CH₂, O, NH, NMe, NSiMe₃, PH, S) were optimized by the DFT method. Correlations between changes in the bond orbital populations, electron density ρ(r), electron density laplacian ∇²ρ(r), |λ₁|/λ₃ ratio, electronic energy density E(r), bond lengths, and displacement of the central atom from the plane of three equatorial substituents and the nature of substituents X and Y were studied. As the number of electronegative substituents at the central atom increases, the M←N, M-X, and M-Y bond lengths decrease, while the M←N bond strength and the electron density at critical points of the M←N, M-X, and M-Y bonds increase. An increase in electronegativity of a substituent (X or Y) is accompanied by a decrease in the ionicities of the other bonds (M-X, M-Y, and M←N) formed by the central atom (Si, Ge). A new molecular orbital diagram for bond formation is proposed, which takes into account the interaction of all five substituents at the central atom (M = Si, Ge) in atrane molecules. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 448–460, March, 2006.     

Octa(ɛ-caprolactam)erbium(III) hex(isothiocyanato)chromate(III): Synthesis and crystal structure

The ionic mixed-ligand bimetallic complex [Er(?-C₆H₁₁NO)₈][Cr(NCS)₆] has been synthesized and studied by X-ray diffraction analysis. The crystals are monoclinic, space group C2/c, a = 39.627(2) Å, b = 22.3406(11) Å, c = 23.7155(10) Å, β = 107.687(2)°, V = 20 002.9(16) ų, Z = 12, ρ_calcd = 1.467 g/cm³. The coordination polyhedron of the erbium atom is a distorted square antiprism formed by the oxygen atoms of the organic ligands. The Er-O bond lengths vary within 2.29–2.44 Å. The coordination polyhedron of the chromium atom is a slightly distorted octahedron, and the Cr-N bond lengths range from 1.99 to 2.01 Å.     

Self-consistent Thomas–Fermi–Dirac theory,extended by Gell-Mann and Brueckner correlation,in terms of density n and its two reduced gradients ∇2n/n and ∇n/n

We prove the following results, relevant for the density functional theory: the Thomas–Fermi–Dirac theory, generalized to include the contribution due to the high electron density result of Gell-Mann and Brueckner for the correlation energy, is shown to lead to a differential equation for the self-consistent ground-state density n( r ) in atoms and molecules in the form F(n, { ∇ n/n}², ∇²n/n)=1, where the function F is given explicitly. A straightforward extension yields a similar result for the equation determining the Pauli plus exchange–correlation potential and for the divergence of the many-electron force. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 145–149, 1998     

trans‐Tetrachlorobis(N,N′‐dimethyl­imidazolidine‐2‐thione)tellurium(IV), a thiourea complex of tellurium with asymmetric Te—S bonds

The structure of the title compound, [TeCl₄(C₅H₁₀N₂S)₂] or C₁₀H₂₀Cl₄N₄S₂Te, has been solved in order to study the stereochemical activity of the lone pair of electrons on Te^IV. The two crystallographically independent mol­ecules in the asymmetric unit both show a distorted octahedral coordination of the Te atom. The two Te—S bonds are trans to each other in both mol­ecules and are greatly asymmetric, with bond lengths of 2.5686 (7) versus 2.8557 (8) Å and 2.5859 (7) versus 2.8165 (9) Å. The Te—Cl bond lengths lie in the range 2.5236 (7)–2.5589 (8) Å. The asymmetric Te—S bonds and a large S—Te—Cl angle of ca 97° involving the long Te—S bonds indicate stereochemical activity of the lone pair of electrons on Te.     

Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

Simulation of metal ion coordination sphere in crystals with fluorite structure

Using quantum-chemical methods, it has been found that the structure of fluorite coincides with the symmetry and coordination number of central calcium atom in the (O_h)-Ca₇F₁₄ cluster. The Ca–F and F–F interatomic distances in the cluster are 3–4% shorter than in the crystal. The symmetry of the (O_h)–(ScCa₆F₁₄)⁺ cluster mimicking the cationic defect after the β-decay of ⁴⁵Ca does not correspond to the energy minimum. The increase in the cation charge from 1.79 to 2.80 a. u. reduces the radius of the first coordination sphere by 0.14 Å. For ytterbium dihalides, the bond lengths Yb–F 2.344, Yb–Cl 2.897 Å and the cation charges 1.81, 1.64 a. u., respectively, have been found.

     

Synthese und Struktur von zwei- und dreikernigen Heterometallkomplexen mit Nitridobrücken zwischen Re und Mo

Synthesis and Structure of Two- and Threenuclear Heterometallic Complexes with Nitrido Bridges between Re and Mo The reaction of ReNCl₂(PMe₂Ph)₃ with MoCl₄(NCEt)₂ yields the heterometallic threenuclear complex [{(Me₂PhP)₃(EtCN)ClRe≡N–}₂MoCl₄][MoNCl₅]. The anion [MoNCl₅]^2– presumably results from a transfer of the nitrido ligand from the Re to the Mo atom. The air-sensitive compound is paramagnetic with μ_eff = 2.87 B. M. at room temperature. A reduction of the magnetic moment to 1.74 B.M at 20 K starts at 140 K. The complex crystallizes in the orthorhombic space group Pca2₁ with a = 2430(1), b = 1328(1), c = 2436.3(2) pm, Z = 4. With bond angles Re–N–Mo of 164° and 167° the nitrido bridges are almost linear. The distances Re–N of 169 and 170 pm can be interpreted with triple bonds. The Mo–N bond lengths of 210 and 211 pm correspond to single bonds. In the anion [MoNCl₅]^2– the distance Mo≡N is 167 pm. Hydrolysis of the threenuclear complex results in a cleavage of one of the nitrido bridges to yield (Me₂PhP)₃(EtCN)ClRe≡N–MoOCl₄. The compound is paramagnetic with μ_eff = 1.71 B.M. at room temperature. It crystallizes in the orthorhombic space group Pbca with a = 1718.5(4), b = 2037(1), c = 2041.1(7) pm, Z = 8. In the dinuclear complex the [MoOCl₄]^– unit is only weakly coordinated to the nitrido ligand with Mo–N = 246.5 pm, while the distance of the Re≡N bond of 168.1 pm is almost unchanged in comparison with a terminal bond. The bond angle Re≡N–Mo is 165.6°.     

Two new thiophosphoramide structures: N,N′,N′′‐tricyclohexylphosphorothioic triamide and O,O′‐diethyl (2‐phenylhydrazin‐1‐yl)thiophosphonate

The compound N,N′,N′′‐tricyclohexylphosphorothioic triamide, C₁₈H₃₆N₃PS or P(S)[NHC₆H₁₁]₃, (I), crystallizes in the space group Pnma with the molecule lying across a mirror plane; one N atom lies on the mirror plane, whereas the bond‐angle sum at the other N atom has a deviation of some 8° from the ideal value of 360° for a planar configuration. The orientation of the atoms attached to this nonplanar N atom corresponds to an anti orientation of the corresponding lone electron pair (LEP) with respect to the P=S group. The P=S bond length of 1.9785 (6) Å is within the expected range for compounds with a P(S)[N]₃ skeleton; however, it is in the region of the longest bond lengths found for analogous structures. This may be due to the involvement of the P=S group in N—H...S=P hydrogen bonds. In O,O′‐diethyl (2‐phenylhydrazin‐1‐yl)thiophosphonate, C₁₀H₁₇N₂O₂PS or P(S)[OC₂H₅]₂[NHNHC₆H₅], (II), the bond‐angle sum at the N atom attached to the phenyl ring is 345.1°, whereas, for the N atom bonded to the P atom, a practically planar environment is observed, with a bond‐angle sum of 359.1°. A Cambridge Structural Database [CSD; Allen (2002). Acta Cryst. B 58 , 380–388] analysis shows a shift of the maximum population of P=S bond lengths in compounds with a P(S)[O]₂[N] skeleton to the shorter bond lengths relative to compounds with a P(S)[N]₃ skeleton. The influence of this difference on the collective tendencies of N...S distances in N—H...S hydrogen bonds for structures with P(S)[N]₃ and P(S)[O]₂[N] segments were studied through a CSD analysis.     

The structure of tetramethyl μ-monothiopyrophosphate

The compound tetramethyl μ-monothiopyrophosphate (C₄H₁₂O₆P₂S) crystallizes in the monoclinic space group C 2/c, with (at -130°C) a = 10.322 Å, b = 8.229 Å, c = 12.062 Å, β = 98.44°, and D_calc = 1.639 g/mL³ and Z = 4. The crystal structure has been determined by single crystal X-ray diffraction to give a final R value of 0.0329 for 614 independent observed reflections [F_˚ > 2.5σ(F_˚)]. The sulfur atom resides on a crystallographic two-fold axis. The P S P bond angle is 105.4° and the P S bond lengths are 2.093 Å. The bond angles around phosphorus range from 99.1° to 118.2°. The terminal PO bond is 1.465 Å, and the methoxyl P O bond is about 1.556 Å. The H₃C O P bond angle is about 119.5°. Many structural features are interpreted in terms of π-bonding to phosphorus. Comparisons with the structures of pyrophosphate and related compounds indicate that the combined effects of increased acuteness of the P S P bond and the increased length of the P—S bonds lead to an increase of about 0.4 Å in the separation of phosphorus atoms in the sulfur-bridging compound. These facts, together with the weakness of the P S bond, must be taken into account in the interpretation of kinetic data for enzymatic reactions of phosphorothiolates as substrates in place of phosphates.     

Complexes olefiniques du ruthenium: V. Structure cristalline et etude RMN de [RuClH(cod)(pyridine)2]

The complexes [RuClH(cod)(amine)₂] undergo amine exchange reactions with pyridine, γ-picoline or 4-(dimethylamino)pyridine. The crystalstructure of [RuClH(cod)(py)₂] has been determined from three-dimensional X-ray data. It crystallizes in space group P2₁/n of the monoclinic system; a 11.278(5); b 10.619(9); c 14.231(4) Å; β 90.34(3)°; Z = 4. The structure was solved by standard heavy atom methods and has been refined by least squares to a conventional R factor of 0.038 based on 5535 reflections. The coordination geometry around the ruthenium atom is octahedrally distorded with cis-aminepyridine ligands, the chloro and hydrido ligands being trans to each other and the cyclooctadiene moiety bound through the two double bonds. Principal bond lengths are: Ru-H 1.60(3); Ru-Cl 2.584(1), Ru-N 2.167(2) and 2.153(2) Å. ¹H and ¹³C NMR data are presented for the three complexes. The ¹³C NMR shifts of the olefin group indicate a large degree of π-back-bonding.     

Unsymmetrical,Cyclic Diborenes and Thermal Rearrangement to a Borylborylene

Cyclic diboranes(4) based on a chelating monoanionic benzylphosphine linker were prepared through boron–silicon exchange between arylsilanes and B₂Br₄. Coordination of Lewis bases to the remaining sp² boron atom yielded unsymmetrical sp³‐sp³ diboranes, which were reduced with KC₈ to their corresponding trans‐diborenes. These compounds were studied with a combination of spectroscopic methods, X‐ray diffraction, and DFT calculations. PMe₃‐stabilized diborene 6 was found to undergo thermal rearrangement to gem‐diborene 8 . DFT calculations on 8 reveal a polar boron–boron bond, and indicate that the compound is best described as a borylborylene.