Graphical unitary group approach to arbitrary spin representations

A graph theoretic approach to the representation of N particle systems involving arbitrary single-particle spin is presented. The method is a generalization of the Distinct Row Table (DRT ) technique employed by Shavitt in developing the graphical unitary group approach (GUGA ) to electronic spin-orbitals. A detailed analysis of the DRT and GUGA representations is presented based on several theoretical considerations and computer-tested implementations. The use of the representation to establish rules concerning the evaluation of the matrix elements of the group generators is also discussed.     

Computation of overlap integrals over Slater-type orbitals using auxiliary functions

Analytical expressions through the binomial coefficients and recursive relations are derived for the expansion coefficients of overlap integrals in terms of a product of well-known auxiliary functions A_k and B_k. These formulas are especially useful for the calculation of overlap integrals for large quantum numbers. Accuracy of the computer results is satisfactory for the values of quantum numbers up to 50 and for the arbitrary values of screening constants of atomic orbitals and internuclear distances. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 199–204, 1998     

Efficient method for computation of representation matrices of the unitary group generators

The representation matrices of the unitary group generators E_k1 are equivalent to the representation matrices of cyclic permutations (k, k ± l, …, l ? 1, l). A method is presented for simultaneous computation of matrices corresponding to all different generators at a cost of less than one multiplication per nonzero element. The number of operations necessary for calculation of individual matrices for single generators or for products of two generators is at most proportional to the number of matrix elements of the final matrix. This approach eliminates the need to store the representation matrices in CI calculations.     

Multicomponent Molecular Orbital–Climbing Image–Nudged Elastic Band Method to Analyze Chemical Reactions Including Nuclear Quantum Effect

To analyze the H/D isotope effects on hydrogen transfer reactions in XHCHCHCHY?XCHCHCHYH (X, Y=O, NH, or CH₂) including the nuclear quantum effect of proton and deuteron, we propose a multicomponent molecular orbital‐climbing image‐nudged elastic band (MC_MO–CI–NEB) method. We obtain not only transition state structures but also minimum‐energy paths (MEPs) on the MC_MO effective potential energy surface by using MC_MO–CI–NEB method. We find that nuclear quantum effect affects not only stationary‐point geometries but also MEPs and electronic structures in the reactions. We clearly demonstrate the importance of including nuclear quantum effects for H/D isotope effect on rate constants (k_H/k_D).     

Vibrational spectra of icosahedral (Ih and I) fullerenes

On the bases of the topological structures of the three big classes of icosahedral fullerenes: (1) C_n(I_h, n=60h²; h=1, 2,…), (2) C_n(I_h, n=20h²; h=1, 2,…), and (3) C_n(I, n=20(h²+hk+k²), h>k; h, k=1, 2,…), we derived formulas for the decomposition of their nuclear motions into irreducible representations. Hence, we obtained the infrared and Raman active modes for all of the icosahedral (I_h and I) fullerenes theoretically. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 113–117, 1998     

Comment on Dual-Hard-Sphere Models for Liquid Semiconductors Si And Ge

Abstract

Various generalized dual-hard-sphere (DHS) models are reviewed on calculating the liquid structure factor for semiconductor elements Si and Ge. It is found that the model generalized by Canessa, Mariani and Vignolo gives the best fitting of experimental structure factor S _exp(k) in the range k > 2k_F , (k_F , the Fermi wave vector), and all previous models including a new generalized model by the author fail to reproduce the experimental structure factor S _exp(k) of Si and Ge in the whole range of k vector.     

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).      

Multiconfigurational SCF approach for high-symmetry molecules and its applications to fullerene trianion

A symmetry-adapted multiconfiguration self-consistent field (MC SCF) approach aimed at calculations of high-symmetry molecules is proposed. The self-consistency procedure applicable to the molecular terms of any symmetry and multiplicity is developed. It holds the symmetry transformation properties of varied molecular orbitals, thus taking advantage of the relationships within the set of two-electron integrals through molecular invariants. For orbital optimization, a unified coupling operator is constructed on the basis of the pseudosecular method providing for efficient convergence to energy minimum. Based on the group-theory technique, computer codes have been developed for straightforward determination of the invariant expansions for two-electron integrals and configuration interaction (CI) matrix elements. Calculated in this way, the expansion coefficients are presented for the three-electron states that originate from joint t_1u and t_1g shells of an icosahedral fullerene C₆₀, the case important for the calculations of anion C₆₀³⁻ representing the charge state of the fullerene molecule in the superconducting ionic solids K₃C₆₀ or Rb₃C₆₀. The results of MC SCF calculations for lowest quasi-π-electronic states of C₆₀³⁻ are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 293–304, 1998     

Structure and Polymorphism of M(thd)3 (M = Al,Cr, Mn,Fe, Co,Ga, and In)

Formation, crystal structure, polymorphism, and transition between polymorphs are reported for M(thd)₃, (M = Al, Cr, Mn, Fe, Co, Ga, and In) [(thd)^– = anion of H(thd) = C₁₁H₂₀O₂ = 2, 2, 6, 6‐tetramethylheptane‐3, 5‐dione]. Fresh crystal‐structure data are provided for monoclinic polymorphs of Al(thd)₃, Ga(thd)₃, and In(thd)₃. Apart from adjustment of the M–O_k bond length, the structural characteristics of M(thd)₃ complexes remain essentially unaffected by change of M. Analysis of the M–O_k, O_k–C_k, and C_k–C_k distances support the notion that the M–O_k–C_k–C_k–C_k–O_k– ring forms a heterocyclic unit with σ and π contributions to the bonds. Tentative assessments according to the bond‐valence or bond‐order scheme suggest that the strengths of the σ bonds are approximately equal for the M–O_k, O_k–C_k, and C_k–C_k bonds, whereas the π component of the M–O_k bonds is small compared with those for the O_k–C_k, and C_k–C_k bonds. The contours of a pattern for the occurrence of M(thd)₃ polymorphs suggest that polymorphs with structures of orthorhombic or higher symmetry are favored on crystallization from the vapor phase (viz. sublimation). Monoclinic polymorphs prefer crystallization from solution at temperatures closer to ambient. Each of the M(thd)₃ complexes subject to this study exhibits three or more polymorphs (further variants are likely to emerge consequent on systematic exploration of the crystallization conditions). High‐temperature powder X‐ray diffraction shows that the monoclinic polymorphs convert irreversibly to the corresponding rotational disordered orthorhombic variant above some 100–150 °C (depending on M). The orthorhombic variant is in turn transformed into polymorphs of tetragonal and cubic symmetry before entering the molten state. These findings are discussed in light of the current conceptions of rotational disorder in molecular crystals.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Ionization thresholds and positions of avoided crossing of potassium Stark Rydberg states: a Stark-adapted quantum defect orbital treatment

We have calculated the positions of the avoided level crossings between (n+2)s, np states and nd, k Stark states in the Rydberg Stark states of the potassium atom with principal quantum number n comprised between 12 and 17. We have also studied the adiabatic electric field ionization thresholds for the above Rydberg states. Both the ionization thresholds and the positions of avoided crossings have been calculated using the recently developed Stark-adapted quantum defect orbital (SQDO) formalism. The presently reported values appear to be in very good agreement with the available theoretical and experimental data.     

Computation of determinant expansion coefficients within the graphically contracted function method

Most electronic structure methods express the wavefunction as an expansion of N‐electron basis functions that are chosen to be either Slater determinants or configuration state functions. Although the expansion coefficient of a single determinant may be readily computed from configuration state function coefficients for small wavefunction expansions, traditional algorithms are impractical for systems with a large number of electrons and spatial orbitals. In this work, we describe an efficient algorithm for the evaluation of a single determinant expansion coefficient for wavefunctions expanded as a linear combination of graphically contracted functions. Each graphically contracted function has significant multiconfigurational character and depends on a relatively small number of variational parameters called arc factors. Because the graphically contracted function approach expresses the configuration state function coefficients as products of arc factors, a determinant expansion coefficient may be computed recursively more efficiently than with traditional configuration interaction methods. Although the cost of computing determinant coefficients scales exponentially with the number of spatial orbitals for traditional methods, the algorithm presented here exploits two levels of recursion and scales polynomially with system size. Hence, as demonstrated through applications to systems with hundreds of electrons and orbitals, it may readily be applied to very large systems. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

Quantum wave packet and quasiclassical trajectory studies of the reaction H(2S) + CH(X2Π; v = 0, j = 1) → C(1D) + H2(X1 ): Coriolis coupling effects and stereodynamics

The quantum mechanics (QM) and quasiclassical trajectory (QCT) calculations have been carried out for the title reaction with the ground minimal allowed rotational state of CH (j = 1) on the 1 ¹A′ potential energy surface. For the reaction probability at total angular momentum J = 0, a similar trend of the QM and QCT calculations is observed, and the QM results are larger than the latter almost in the whole considered energy range (0.1–1.5 eV). The QCT integral cross sections are larger than the QM results with centrifugal sudden approximation, while smaller than those from QM method including Coriolis coupling for collision energies bigger than 0.25 eV. The quantum wave‐packet computations show that the Coriolis coupling effects get more and more pronounced with increasing of J. In addition to the scalar properties, the stereodynamical properties, such as the average rotational alignment factor <P₂( j′?k )>, the angular distributions P(θ_r), P(?_r), P(θ_r,?_r), and the polarization‐dependent generalized differential cross sections have been explored in detail by QCT approach. © 2013 Wiley Periodicals, Inc.     

Indexing scheme for classes of 𝒮N; partitions of N

The classes of the symmetric group 𝒮_N are identified by partitions of N. In this work an indexing scheme is presented which provides a dense enumeration of the classes of 𝒮_N. The method is based on a graphical representation of partitions of N, which also enables the determination of the class corresponding to a given number. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 421–426, 1997     

Graphical properties of polyhexes: Perfect matching vector and forcing

From the viewpoint of graph theory and its applications, subgraphs of the tiling of the plane with unit squares have long been studied in statistical mechanics, In organic chemistry, a much more relevant case concerns subgraphs of the tiling with unit hexagons. Our purpose here is to take a mathematical view of such polyhex graphsG and study two novel concepts concerning perfect matchingsM. First, the forcing number ofM is the smallest number of edges ofM which are not contained in any other perfect matching ofG. Second, the perfect matching vector ofM is written (n ₃,n ₂,n ₁,n ₀), wheren _k is the number of hexagons with exactlyk edges inM. We establish some initial results involving these two concepts and pose some questions.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Products of arbitrary class-sums in the symmetric group

An algorithm for the evaluation of products of arbitrary conjugacy class-sums in the symmetric group is conjectured. This algorithm generalizes a procedure presented sometime ago, which deals with products in which at least one of the class-sums involved consists of a single cycle (and an appropriate number of fixed points). Let the support size of a conjugacy class be the number of indices that are not fixed points. The algorithm proposed implies that the coefficient of the class-sum C in the product of the class-sums A and B is given in terms of a well-defined enumeration problem within the symmetric group S_p, where p is the smallest of the support sizes of A, B, and C. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 429–440, 1998     

Interpretation of the high-energy band within the photoelectro spectra of alkanes both in terms of appropriate orbitals and on the basis of chemical structure

Two generalized interpretations of the high-energy band (HEB ) within the photoelectron spectras (PES ) of alkanes, usually referred to as the C_2s band, have been suggested. The quantum chemical one consists of attributing to the HEB of any alkane (C_nH_2n+2) the chain of n similiar orbitals η_k (k = 1…n), each of them containing mainly one 2s_c AO and some contributions of another four orbitals from its nearest environment. The structure of each orbital η_k has been shown to be similar to that of the lowest totally symmetric MO of methane. The chemical interpretation of the same band involves relating it and the definite subspectrum of the full chemical graph of alkane, associated mainly with its C skeleton. The interrelation of these two complementary interpretations follows from the one-to-one correspondence between the orbitals η_k and the CH₂(CH₃) groups in alkane. © 1995 John Wiley & Sons, Inc.     

A graphical evaluation of characteristic polynomials of huckel trees

Characteristic polynomials of acyclic carbon chains (Huckel trees) are treated in a systematic way. Formulas of coefficients (a_k) of the polynomial are obtained in terms of connectivities that were introduced for dealing with moments in a previous paper. Based on the meaning of a_k, a graph-theoretical analysis is given such that a_k can be expressed as a linear combination of binomial factors specified by a set of graphs containing ½k edges. The numerical relationship is disclosed between each binomial factor and its specified graph. This stimulates the proposal of a novel approach for evaluating ak by simply collecting the graph set of defnite edges. The approach is equally applicable for the evaluation of matching polynomials of cyclic systems and extendable to the investigation of general trees.