On the computation of matrix elements between numerical wave functions: The canonical functions method

The problem of the computation of the matrix elements is considered when Ψ_v(r) and Ψ_v(r) are eigenfunctions related to a diatomic potential of the RKR type (defined by the coordinates of its turning points P_i with polynomial interpolations). The eigenfunction Ψ(r) is computed by the canonical functions method making use of the abscissas r_i of P_i uniquely. This limited number of points allows the storage of ψ_v(r_i) for all the required levels v, and reduces greatly the computational effort when v, ν′, and k are varying. The present method maintains all the advantages of a highly accurate numerical method (even for levels near the dissociation), and reduces greatly the computing time. Furthermore, it is shown that it may be extended to analytical potentials like Morse and Lennard-Jones functions, to vibrational-rotational eigenfunctions and to matrix elements between eigenfunctions related to two different potentials. Numerical applications are presented and discussed.     

Kirchhoff index of linear hexagonal chains

The resistance distance r_ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L_n consists of the Laplacian spectrum of path P_2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed‐form formula for Kirchhoff index of L_n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L_n is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including L_n. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Kirchhoff index of linear pentagonal chains

The resistance distance r_ij between two vertices v_i and v_j of a connected graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this article, following the method of Yang and Zhang in the proof of the Kirchhoff index of liner hexagonal chain, we obtain the closed‐form formulae of the Kirchhoff index of liner pentagonal chain P_n in terms of its Laplacian spectrum. Finally, we show that the Kirchhoff index of P_n is approximately one half of its Wiener index. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Crosslinking,filler, or transition constraint of polymer networks. I

The basic theory of modulus/swelling is developed to allow for limited extensibility, filler reinforcement or transition effects, and steric hindrance of aligned segments by extended chains or filler particles. Filler forms an effective hard fraction C_h per cubic centimeter of compound with v_c a new (compound) index of swelling. For 1/M_c + σ fix points having ratio φ to gum values 1/F₀(v_r) and with F(v_c) replacing the Flory function F(v_r): where σ denotes entanglement. Linkage reinforcement φ does not vary with sulfur crosslinking of SBR. Vacuoles invalidate φ from mass-increment F₀(v_r)/F(v_r) for inert fillers. Then, or for Graphon, with negligible φ ≈ 1: The effective C_h includes rubber stretched hard on Graphon by swelling or trapped inside hard aggregates. Only the right-hand equation fits normal blacks. In theory, C_h can always be obtained from swollen moduli G by linear slopes (1 + 1.4C_h) relating F(v_c) and (1 ? C)ρRT/G. For filler fractions C ≥ 0 cm^?3 and low strains α = 1.5?2.0 below prestretch the modulus G is given a new basic definition: Here C₂* ≈ 0.7 corresponds to Mooney-Rivlin C₂ and the effective crosslinking 1/[M_c] = (ρRT)^?1G is equal to (1 ? C)(1/M_c + σ) for unswollen prestretched rubber (v_r = 1). For higher strains a hypothesis of strain hardening is proposed. This is distinct and opposite in character to the initial prestretch softening (Mullins effect). Nonlinear effects of crosslinks are expressed by a fractional stress-upturn Ω (1/M_c + σ), effective mesh wieght (1/M_c + σ)^?1 ? Ω, and hard fraction Ω(1/M_c + σ). For μ_h characterizing strain hardening up to the prestretch (α_h ? 1) their contribution is: The sixth-power refinement has J = j(α_b ? 1)^1/2 with j ≈ 0.4. The hard phase is augmented by filler and grows with increasing strain up to the prestretch.     

Graphical Models of Characters of Groups

A graphical method of generating one- and (some) two-dimensional characters () has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G _i )]={h₁,...,h _d–1,h _d } (d=|G|/|G _i |) has been generated from a regular body of G so that it has been governed by the coset representation G(/G _i ). The homomer set has been reduced into a reduced homomer set []={h₁,...,h _d–1}, where we have placed h _d –(h₁++h _d–1) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set [] has graphically generated a one- or (some) two-dimensional character (). The versatility of the graphical method has been tested by using C _3v , D _2h , C _2h , C _2v , D _3h , and C _3h as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., =G(/G _i )–G(/G).     

A möbius inversion of the ulam subgraphs conjecture

The generalized Eichinger matrices are defined asE = _j ⁿ ₁( _j S ^T S)^–1, where _j M denotes the matrixM withj th row and column deleted.S is the incidence matrix andM ^T is the transposed matrix. The conjectureS ^T SE = S _K ^TS _K , where SK is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture isS _G ^T S _G (E _G -I) = S _G ^TS _G , whereG is the complementary graph ofG. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.     

Hosoya polynomials of TUC4C8(S) nanotubes

The Hosoya polynomial of a chemical graph G is , where d _G (u, v) denotes the distance between vertices u and v. In this paper, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(S) nanotubes. Accordingly, the Wiener index, obtained by Diudea et al. (MATCH Commun. Math. Comput. Chem. 50, 133–144, (2004)), and the hyper-Wiener index are derived. This work is supported by the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Grant No. LZULL200809).     

The aromatic diboracyclopropenyl (diboriranyl) anion; CB2H3 −: An ab initio study

The most stable structure of CB₂H₃ ^–, as established computationally, is the aromatic diboracyclopropenyl (diboriranyl) anion (5), while open-chainC _2v, isomer H₂BCBH (7) is only 3 kcal/mol higher in energy at the QCISD(T)/6-311 +G^**//MP2/6-31+G^*+ZPE (HF/6-31 +G^*). The 47-kcal/mol barrier between cyclic,5, and open-chain,7, structures suggests that both of them may be observed. The aromatic stabilization energy of the diboriranyl anion (18 kcal/mol) is half the value in the isoelectronic cyclopropenium ion, C₃H₃ ⁺. The computed, by IGLO method (5a), and experimental (6a) chemical shifts,(¹³C) and(¹¹B), agree within 4 ppm range. The theoretical vibrational frequencies of the most stable isomers,5 and7, are presented for experimental verification of these species.     

Estimation of the parameters in the Kirkwood-Buff theory of solution using Percus-Yevick fluid mixtures

Parameters G _ij in the Kirkwood-Buff theory of solution were calculated for binary fluid mixtures of Lennard-Jones (LJ) 6–12 molecules by using the Percus-Yevick theory. Calculations were carried out for various parameters in the LJ potential. Under the Lorentz-Berthelot rules, G ₁₁ and/or G ₂₂ -composition curves do not show a maximum for any parameters in the LJ potential. When the intermolecular interaction between different species becomes much weaker than that expected from the Berthelot rule, G ₁₁ and G ₂₂ show a maximum and G ₁₂ a minimum. The pressure effect on G _ij was examined and calculations at constant pressure were also carried out. G _ij is almost independent of the pressure when the ratio of the molecular volume of two components is in the range 1.0 to 2.5. Comparison was made between experimental and calculated G _ij for cyclohexane-2,3-dimethylbutane and acetonitrile-toluene systems. For the latter system, the quantitative agreement between the calculated and experimental could not be obtained but showed that the characteristics of G _ij -composition curves can be explained qualitatively by using the PY theory.Adjunct Associate Professor of Institute for Molecular Science (April 1982–March 1984)     

Eigenvalues of large even annulenes in terms ofK 3,S 4 andS 5

Squares of the adjacency matrices of bipartite cycles (C_v) can be block-factored into matrices which correspond to vertex-weighted complete graphs forv = 6, vertex-weighted strongly regular graphs forv = 8 and 10, and vertex-weighted metrically regular graphs forv > 10. Using this fact and some properties of strongly and metrically regular graphs, it is shown that eigenvalues of large bipartite C _v graphs (i.e. large even annulenes) can be expressed by the general formula ± (2 ± (2 ± (... ± (2 +r _p)) ...), wherev = 2 ⁿ ×p,n is the number of surd () signs required andp = 3, 4 and 5. Here,r ₃,r ₄, andr ₅, are the eigenvalues of the complete graphK ₃ and the strongly regular graphsS ₄ andS ₅ respectively. The procedure does not require construction of characteristic polynomials for the determination of eigenvalues, and brings out a common topological origin for the two-fold degeneracies observed in the eigenvalue spectra of all even cycles and many odd cycles.     

Graphs corresponding to reference polynomial or to circuit characteristic polynomial

A vertex-weighted graphG ^* is studied which is obtained by deleting edgee _rs in a circuit of a graphG and giving two vertices _r and _s weightsh _r = 1 andh _s = -1, respectively. It is shown that if subgraphG - _r is identical with subgraphG - _s, then the reference polynomial ofG ^* is identical with that ofG and the characteristic polynomial ofG ^* contains the contributions due to only a certain part of the circuits found in the original graphG. This result gives a simple way to find a graph whose characteristic polynomial is equal to the reference polynomial in the topological resonance energy theory or to the circuit characteristic polynomial in the circuit resonance energy theory. This approach can be applied not only to Hilckel graphs but also to Möbius graphs, provided that they satisfy a certain condition. The significances of this new type of reference graph thus obtained are pointed out.     

Molecular structure of ethynylbenzene from electron diffraction and ab initio molecular orbital calculations

The molecular structure and benzene ring distortions of ethynylbenzene have been investigated by gas-phase electron diffraction and ab initio MO calculations at the HF/6-31G^* and 6-3G^** levels. Least-squares refinement of a model withC _2v, symmetry, with constraints from the MO calculations, yielded the following important bond distances and angles:r _g(C _i -C _o )=1.407±0.003 Å,r _g(C _o -C _m )=1.397±0.003 Å,r _g(C _m -C _p )=1.400±0.003 Å,r _g(Cr _i -CCH)=1.436 ±0.004 Å,r _g(C=C)=1.205±0.005 Å, C _o -C _i -C _o =119.8±0.4°. The deformation of the benzene ring of ethynylbenzene given by the MO calculations, including o-C_i-C_o=119.4°, is insensitive to the basis set used and agrees with that obtained by low-temperature X-ray crystallography for the phenylethynyl fragment, C₆H₅-CC-, in two different crystal environments. The partial substitution structure of ethynylbenzene from microwave spectroscopy is shown to be inaccurate in the ipso region of the benzene ring.     

Hylleraas method for many‐electron atoms. II. Integrals over wave functions with one interelectronic coordinate

A procedure is proposed to evaluate matrix elements containing r linked with angular functions. Using this procedure, the different types of two‐, three‐, and four‐electron radial and angular integrals that appear in a five‐electron atom, in the case of only one r_ij coordinate per basis function, are written in a compact form, separated in radial coordinates of one electron. The general formulas with which to obtain the integrals for powers ν ≥ 1 are developed, based on the orthogonality of the Legendre polynomials. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Crosslinking,filler, or transition constraint of polymer networks. II

The theory of Part I is developed by application to filler reinforcement of NR and SBR. For unswollen but prestretched networks it quantifies entire stress–strain curves and applies new concepts of extensibility and strain hardening. Constraint of swelling is expressed by a constant φ, termed linkage reinforcement, and by an effective hard fraction C_h per cubic centimeter of compound. For rubber–filler swelling v_c the modified Flory functions F(v_c) in part 1 need 3% correction. Then, relative to gum fix points 1/F₀(v_r): where C_h ≤ 1.15C for filler concentrations C per cubic centimeter of compound. The effective C_h comprises the volume fraction C* of bonded particles and 5–10 Å of surface–bound rubber that has been stretched hard by swelling. When needed, the actual crosslink density and intrinsic linkage reinforcement φ₀ can be obtained by dividing by (1 + 2.5C_φ) where C_φ = (C ln φ)/(1 + 2.5C). The case C_h ≤ 1.15C with Graphon or inert fillers is identified and assessed by equations: Results C_h > 1.15C are invalid, but then C_h ≈ 1.15C* ≈ 1.15C, e.g., for carbon blacks. Even Graphon is distinguished from inert fillers at low concentrations C by substantial constraint reinforcement F₀(v_r)/F(v_c) > 1. For prestressed dry rubber a modulus G, network extensibility α_b ? 1, and upturn coefficient μ_h express the whole curve; G and μ_h show identical constraint strength distributions. Network extensibility α_b ? 1 is the microbreaking strain (prestretch); for pure elastomer it is elongation at break. The relation of stress F to extension ratio α is: where C₂* = 0.7 and j = 0.4 from NR/MPC data. Strain-hardening coefficients h are obtained from μ_h by the theory given in Part I. Hard modulus components G_h = 0.7 ln (h/h₀) vanish as h → h₀ (gum) = 110. After high prestresses the residual ln-(h^*/h₀) due to strong carbon-rubber linkages implies G_h* = 0.42 kg/cm², i.e., ca. 10% of the normal cure crosslinks.     

The (n,n)-graphs with the first three extremal Wiener indices

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e., , where d _G (u, v) is the distance between vertices u and v in G. In this paper, we first give a new formula for calculating the Wiener index of an (n,n)-graph according its structure, and then characterize the (n,n)-graphs with the first three smallest and largest Wiener indices by this formula.     

Determination of the molecular structure of diacetylene from high-resolution FTIR spectroscopy

The high-resolution infrared absorption spectra of eight²H or¹³C substituted isotopomers of diacetylene have been recorded, and the bands corresponding to thev ₄ fundamental andv ₆ combination of the major isotopomer have been analyzed using a Loomis-Wood-type program. Effective ground-state rotational constants have been obtained from ground-state combination differences. A number ofr ₀,r _s,r _m , and (r _m )_corr structures have been calculated from the available data and are compared to those obtained by ab initio methods. The (r _m )_corr structure, which is a reliable near-equilibrium structure of diacetylene, isr _C–H=106.131(13) pm;r _C–C=137.081(16) pm;r _C-C=120.964(14). (r _m )_corr structures of the related molecules cyanogen, cyanoacetylene, and cyanodiacetylene have been calculated, and near-equilibrium structures of triacetylene and dicyanoacetylene have been predicted.     

Theoretical study with rovibrational and dipole moment calculation of sextet states of the CrCl molecule

The potential energy curves have been investigated for the 13 lowest sextet electronic states in the representation below 53,000 cm^?1 of the molecule CrCl via CASSCF and MRCI (single and double excitation with Davidson correction) calculations. The harmonic frequency ω_e, the internuclear distance r_e, the rotational constant B_e, the electronic energy with respect to the ground state T_e, and the permanent dipole moment μ have been calculated. By using the canonical functions approach, the eigenvalues E_v, the rotational constant B_v, and the abscissas of the turning points r_min and r_max have been calculated for the considered electronic states up to the vibrational level v = 16. Nine electronic states have been studied theoretically here for the first time. The comparison of these values with the theoretical and experimental results available in the literature shows a good agreement. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Use of Kirkwood–Buff Integrals for Extracting Distinct Diffusion Coefficients in Macromolecule–Solvent Mixtures

The possibility to extract velocity correlation quantities from fluctuation thermodynamic properties is explored in the case of macromolecule–solvent mixtures. Indeed, Kirkwood–Buff integrals, G_ij, together with self‐diffusion and viscosity data can provide an approximation for distinct diffusion coefficients (DDCs), D^d_ij. Herein, D^d_ij for binary polyethyleneglycol (PEG)(i)–water(0) systems is calculated. These systems show positive values of D^d_ii coefficients, indicating strong PEG–PEG interaction, and providing marker of water mediated PEG–PEG networks. The efficiency of several standard DDCs present in literature for D^d_ij analysis is compared, summarizing the usefulness of each one, depending on the nonideality degree.

     

Multipole analysis of electron repulsion energies in many-electron atoms

The repulsion energy W between two electrons located at r ₁ and r ₂ can be expressed by the sum of the interaction energies W _k between an electron located at and linear electric multipoles located at the coordinate origin along the vector , where and are the vectors with smaller and larger moduli, respectively, of the two vectors r ₁ and r ₂. All the existing multipole contributions W _k to the Hartree–Fock electron repulsion energy W are examined for the 102 atoms He through Lr in their ground states. It is found that |W _k | decreases rapidly with increasing k, W ₀ > W, and W _k with k ≥ 1 work to reduce W ₀. The effect of electron correlation is also discussed for some helium-like atoms.     

Some upper bounds for the energy of graphs

Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v ₁,v ₂,...,v _n }. The 2-degree of v _i , denoted by t _i , is the sum of degrees of the vertices adjacent to . Let σ _i be the sum of the 2-degree of vertices adjacent to v _i . In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, t _i , and σ _i , from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve some known results for trees.