Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules

For arbitrary values of n and l quantum numbers, we present a simple exact analytical solution of the D-dimensional (D ≥ 2) hyperradial Schr?dinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact bound state energy eigenvalues (E _nl ) are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions (ψ _nl (r)) are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for a few typical LiH, CH, HCl, CO, NO, O₂, N₂ and I₂ diatomic molecules for various values of n and l quantum numbers. Numerical tests using the energy calculations for the inter dimensional degeneracy (D = 2 − 4) for I₂, LiH, HCl, O₂, NO and CO are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

S5 graphs as model systems for icosahedral Jahn–Teller problems

The degeneracy of the eigenvalues of the adjacency matrix of graphs may be broken by non-uniform changes of the edge weights. This symmetry breaking is the graph-theoretical equivalent of the molecular Jahn–Teller effect (Ceulemans et al. in Proc Roy Soc 468:971–989, 2012). It is investigated for three representative graphs, which all have the symmetric group on 5 elements, S ₅ , as automorphism group: the complete graph K₅, with 5 nodes, the Petersen graph, with 10 nodes, and an extended K₅ graph with 20 nodes. The spectra of these graphs contain fourfold, fivefold, and sixfold degenerate manifolds, respectively, and provide model systems for the study of the Jahn–Teller effect in icosahedral molecules. The S ₅ symmetries of the distortion modes of the quintuplet in the Petersen graph yield a resolution of the product multiplicity in the corresponding $ H \otimes \left( {g + 2h} \right) $ icosahedral Jahn–Teller problem. In the extended Petersen graph with 20 nodes, a selection rule prevents the Jahn–Teller splitting of the sextuplet into two conjugate icosahedral triplets.     

Imminant polynomials of graphs

Summary The imminant polynomials of the adjacency matrices of graphs are defined. The imminant polynomials of several graphs [linear graphs (L _n ), cyclic graphs (C _n ) and complete graphs (K _n )] are obtained. It is shown that the characteristic polynomials and permanent polynomials become special cases of imminant polynomials. The connection between the Schur-functions and imminant polynomials is outlined.Cammile & Henry Dreyfus Teacher-scholar     

Ab initio study of the structure and stability of LaCl n(3 ? n)+ complexes (n = 1?8)

The structural and energetic characteristics of LaCl _n ^{(3 − n)+} (n = 1−8) complexes have been calculated by the ab initio MP2 method. Original Russian Text ? V.Yu. Buz’ko, Kh.B. Kushkhov, M.B. Buz’ko, V.T. Panyushkin, 2008, published in Zhurnal Neorganicheskoi Khimii, 2008, Vol. 53, No. 11, pp. 1899–1905.     

Quantum-chemical study of the stereoelectronic structure of 1-fluorosilatrane, 1,1-difluoroquasisilatrane, 1,1,1-trifluorohyposilatrane,and cations formed thereof

Quantum-chemical study of the electronic structure and the equilibrium geometry of the molecules X_4−n M(OCH₂CH₂) _n NH_3−n and cations X_3−n [M(OCH₂CH₂) _n NH_3−n ]⁺ (M = Si, Ge; X = F, H; n = 1–3) is performed by the B3LYP method using the cc-PVDZ basis set. It is shown that for X = F the strength of the coordination bond N→M increases with the number of the cycles (n), while for X = H, on the contrary, decreases, that is, the strength of the N→M bond increases with the total electronegativity of the substituents surrounding atom M. Effect of the number of the coordination cycles on the strength of the N→M bond in the cations is negligible. The obtained results agree with the experimental data on the structure and spectral properties of the studied compounds.     

Topological analysis of eigenvectors of the adjacency matrices in graph theory: The concept of internal connectivity

The topological properties of eigenvectors of adjacency matrices of a graph have been analyzed. Model systems studied are n-vertex-m-edge (n-V-m-E) graphs where n = 2–4, m = 1–6. The topological information contained in these eigenvectors is described using vertex-signed and edge-signed graphs. Relative ordering of net signs of edge-signed graphs is similar to that of eigenvalues of the adjacency matrix. This simple analysis has also been applied to naphthalene, anthracene and pyrene. It provides a sound basis for the application of graph theory to molecular orbital theory.     

Structures and aromaticity of the planar Al<Subscript>2</Subscript>P<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis>−</Superscript> (<Emphasis Type="Italic">n</Emphasis>=1–4) clusters

Clusters Al₂P₂ ⁿ⁻ (n = 1–4) were theoretically investigated using density functional theory (DFT) methods at the B3LYP/6-311+G* and B3PW91/6-311+G* levels of theory. The calculated results showed that the planar structure (D _2h symmetry) of Al₂P₂ ⁿ⁻ (n = 1–4) species was the global minimum. And the negative nucleus-independent chemical shift (NICS) value of Al₂P₂ ⁿ⁻ (n = 1–4) species indicated the existence of a ring current in the planar structure (D _2h symmetry). A detailed molecular orbital (MO) analysis revealed that the planar structures (D _2h symmetry) had π aromaticity, which further exhibited the strongly aromatic character for Al₂P₂ ⁿ⁻ (n = 1–4) species.     

Theoretical studies on anionic clusters of sulfate anions and carbon dioxide, SO 4?1/?2 (CO2)n, n?=?1?4

Geometry optimizations were performed on monoanionic and dianionic clusters of sulfate anions with carbon dioxide, SO₄^−1/−2(CO₂) _n , for n = 1–4, using the B3PW91 density functional method with the 6-311 + G(3df) basis set. Limited calculations were carried out with the CCSD(T) and MP2 methods. Binding energies, as well as adiabatic and vertical electron detachment energies, were calculated. No covalent bonding is seen for monoanionic clusters, with O₃SO–CO₂ bond distances between 2.8 and 3.0 ?. Dianionic clusters show covalent bonding of type [O₃S–O–CO₂]⁻², [O₃S–O–C(O)O–CO₂]⁻², and [O₂C–O–S(O₂)–O–CO₂]⁻², where one or two oxygens of SO₄⁻² are shared with CO₂. Starting with n = 2, the dianionic clusters become adiabatically more stable than the corresponding monoanionic ones. Comparison with SO₄^−1/−2(SO₂) _n and CO₃^−1/−2(SO₂) _n clusters, the binding energies are smaller for the present SO₄^−1/−2(CO₂) _n systems, while stabilization of the dianion occurs at n = 2 for both SO₄⁻²(CO₂) _n and SO₄⁻²(SO₂) _n , but only at n = 3 for CO₃⁻²(SO₂) _n .     

Theoretical and experimental study of the structure and stability of multiply Na-substituted glucose and 2,4,6-trihydroxyacetophenone derivatives

The equilibrium geometric parameters and the energetic and spectroscopic characteristics of low lying conformers for series of polyhydroxyl molecules and ions in which sodium atoms are successively substituted for the hydroxyl hydrogen atoms have been calculated by the density functional theory B3LYP method with the 6−31G* and 6−311+G** basis sets. The glucose derivatives [Glu − nH + nNa] and [Glu − nH + (n + 1)Na]⁺ (n = 1−5) and the 2,4,6-trihydroxyacetophenone derivatives [THAP − nH + nNa] and [THAP − nH + (n + 1)Na]⁺ (n = 1−4) have been considered. The affinities of the neutral [Glu − nH + nNa] and [THAP − nH + nNa] molecules for adding Na⁺ cations, as well as the energies of successive substitution of Na atoms for H atoms in the Glu and THAP molecules and the Glu⁺ and THAP⁺ ions in their reaction with sodium acetate molecules, have been estimated. Computations show that the first substitution of Na for H in ions is slightly exothermic and, presumably, can spontaneously occur under common conditions. Further substitutions are endothermic, but the required energy inputs are small. Therefore, successive substitutions for two, three, or more hydroxyl H atoms in the molecules and ions under consideration are possible at relatively low energy inputs. The computation results and conclusions are compared with the MALDI TOF mass spectral data for Na-substituted glucose and 2,4,6-trihydroxyacetophenone derivatives in the [glucose + CH₃COONa + THAP] system where, in addition to common Glu · Na⁺ and THAP · Na⁺ ion-molecular complexes, multiply substituted positive ions of the [Glu − nH + (n + 1)Na]⁺ (n = 1−4) and [THAP − nH + (n + 1)Na]⁺ (n = 1−3) type have been identified.     

DFT study of the structure and stability of Pu(III) and Pu(IV) chloro complexes

The structural characteristics and energies of PuCl _n ^{(3 − n)+} and PuCl _n ^{(4 − n)+} complexes (n = 2–8) have been studied by the density functional theory (DFT) method in the SVWN5 local functional approximation.     

Structure and stability of oligomeric anions [M n X3 n + 1]? (M = Al, Ga, In; X = F, Cl, Br, I; n = 2–4): A quantum-chemical study

The structural and thermodynamic properties of oligomeric anions [M _n X_{3n+ 1}]⁻ (M = Al, Ga, In; X = F, Cl, Br, I; n = 2, 3, 4) have been obtained by the density functional theory B3LYP method with the LAN2DZ(d) and LAN2DZ(d)+ basis sets. A wide diversity of structural isomers was found for trimeric fluoride anions M₃F₁₀⁻. Among the trimers, except In₃F₁₀⁻, the most stable is a linear isomer composed of two MX₃ molecules coordinated to the MX₄⁻ anion. The formation of tetrameric anions M₄X₁₃⁻ was demonstrated to be thermodynamically allowed at low temperatures at MX₃: X⁻ > 4: 1. The existence of higher oligomers is less probable. The affinity of oligomer halides (MX₃) _n for halide ions increases with an increase in n. The propensity to form oligomeric anions decreases in the series F > Cl ≥ Br > I. The fluoride systems show a tendency to form structures with CN = 5 and 6, these structures for In being the most stable. Original Russian Text ? A.Yu. Timoshkin, 2009, published in Zhurnal Neorganicheskoi Khimii, 2009, Vol. 54, No. 1, pp. 87–100.     

Thermal and Spectral Studies of Y(III) and Lanthanide(III) Complexes with Mesaconic Acid

Y(III) and lanthanide(III) mesaconates were prepared as crystalline solids with general formula Ln₂(C₅H₄O₄)₃⋅nH₂O, where n=7 for La−Pr, n=4 for Y,Nd−Ho, n=8 for Er−Lu. IR spectra of the prepared mesaconates suggest that carboxylate groups are bidentate bridging anf chelating. During heating the hydrated complexes are dehydrated in one (Y, Nd−Lu) or two steps (La−Pr) and then decompose directly to oxides (Y, Ce, Pr, Sm, Gd−Lu) or with intermediate formation Ln₂O₂CO₃ (La, Nd, Eu). This revised version was published online in August 2006 with corrections to the Cover Date.     

Augmentation of the Generalised n×n Eigenvalue Equation to a Generalised (n+1)×(n+1) Eigenvalue Equation

Generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) where B and S ^b are n×n Hermitian matrices while S ^b is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H| _k = _k S| _k (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and S ^b , respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues _k and the eigenvectors | _k of the augmented eigenvalue equation can be expressed in terms of the eigenvalues _i and the eigenvectors | _i of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n ²). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n ³) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.     

Processes of dissociative electron capture by 20-hydroxyecdysone molecules

The peculiarities of dissociative electron capture by 20-hydroxyecdysone molecules with the formation of fragment negative ions were studied. In the region of high electron energies (5–10 eV), processes of skeleton bond rupture are accompanied by the elimination of H₂O and H₂ molecules. In the region of thermal energies of electrons (≈0 eV), the mass spectrum is formed mainly by the [M−nH₂O]^.− (n=1–3) and [M−H₂−nH₂O]^.− (n=0−3) ions that are generated exclusively by the rearrangement. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 709–712, April, 2000.     

Synthesis and mesomorphism of diacetylene-bridged triphenylene discotic liquid crystal dimers

Connecting two discotic mesogens via a spacer not only stabilizes the columnar mesophase but also leads to the formation of glass columnar phase, and therefore improves the physical properties of discotic liquid crystals as organic semiconductor. Here, we report the synthesis of eight diacetylene-bridged triphenylene discotic liquid crystal dimers, [C18H6(OCnH2n+1)4(OMe)O2C-C8H16-C≡≡ C-]2, 3(n), (n = 4-8), [C18H6(OC6H13)5O2C-C8H16-C≡≡ C-]2, 6 and [C18H6(OC6H13)5O-(CH2)m-C≡≡ C-]2, 8(m), (m = 1, 3) by Eglinto...     

Structure and coordination of organometallic groups on a chemically modified SiO2 surface

The structural, electronic, and energy parameters of the metal-containing clusters (H₃SiO)₃Si−O−XMe _n (X=H, B, Al, or Zn;n=0, 1, or 2), which model organometallic groups on a SiO₂ surface modified with B-, Al-, and Zn-containing alkyls, have been studied by quantum-chemical methods. Full geometry optimization for these clusters was carried out by the SCF MO LCAO method taking into account the electron correlation within the frameworks of the MP2 and B3LYP schemes using the 6-31G(d) (6-311G(d) for Zn) basis set. The effect of the crystal environment was taken into account in calculations of siliconoxygen clusters containing 10 and 30 silicon atoms using theab initio SCF/6-31G(d) and semiempirical MNDO-PM3 methods. Various modes of coordination and interactions of organometallic groups with oxygen atoms of surface groups were studied. For the organoaluminum group on the surface, two stable conformations were found, namely, the three-coordinate structure with the chain −O−AlMe₂ ligand and the four-coordinate (quasicyclic) structure with the Al atom that forms two nonequivalent bonds with the O atoms at the same Si atom. The four-coordinate structure is energetically more favorable. No stable structures were found for the organoboron and organozinc fragments. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 7, pp. 1296–1303, July, 1998.     

Generalized graphs and the Sinanoğlu graphical rules

A comparison of Sinano?lu's VIF (Ref. 1) and generalized graph is presented. Generalized graphs have vertex and edge weights. An abridged history of generalized graphs in theoretical chemistry is given. VIF 's are generalized graphs and therefore have adjacency matrices. The “graphical” rules of Sinano?lu can be represented by congruent transformations on the adjacency matrix. Thus the method of Sinano?lu is incorporated into the broad scheme of graph spectral theory. If the signature of a graph is defined as the collection of the number of positive, zero, and negative eigenvalues of the graph's adjacency matrix, then it is identical to the all-important {n₊, n₀, n_?}, the {number of positive, zero, and negative loops of a reduced graph} or the {number of bonding, nonbonding, and antibonding MO s}. A special case of the Sinano?lu rules is the “multiplication of a vertex” by (?1). In matrix language, this multiplication is an orthogonal transformation of the adjacency matrix. Thus, one can multiply any vertex of a generalized graph by ?1 without changing its eigenvalues.