Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S _N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S _N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.Killam Research Fellow 1987–89.     

A necessary and sufficient condition for the existence of real coupling coefficients for a finite group

We establish a theorem which gives a necessary and sufficient condition for a set of matrix irreps of a finite group to admit real coupling (Clebsch–Gordan) coefficients. The proof is based on the method used by Feit to prove that a full set of coupling coefficients for a finite group determines the group up to isomorphism. A consequence of the theorem is that a finite group with real coupling coefficients is necessarily quasiambivalent. The theorem is used to demonstrate that real coupling coefficients do not exist for the point-group hierarchies T ? D₂ and I ? T or for the double-group hierarchies I* ? D₃*, I* ? D₅*, and O* ? D₃*.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

Spin-orbit coupling coefficients for icosahedral molecules

Summary A self-consistent set of real spin-orbit coupling coefficients for the icosahedral double group is derived. Construction of the basis, resolution of multiplicities by spherical operator techniques, and typical applications to Zeeman and crystal-field problems for transition-metal atoms and lanthanides in icosahedral environments are described.     

Coupling coefficients for the octahedral group with orthorhombic components

A tabulation is given of the vector coupling coefficients for the octahedral group, where the components of the degenerate representations have been defined with respect to a two-fold octahedral axis. Coefficients are given for both real and complex choices of the components, and should be useful in describing nearly octahedral transition metal complexes with orthorhombic perturbations.     

The coupling algebra of trigonally subduced crystal field eigenvectors

The general theory of subduction of eigenvectors between infinite groups is used to derive a finite group subduction operator and define the corresponding subduction coefficients. The coupling behaviour of these subduced eigenvectors can then be described in terms of 3 Γ symbols. These symbols, defined only in relation to complex basis sets are all fully real and have all phases fixed by the subduction operator. They differ from V coefficients in two phase relationships and have the advantage, unlike V coefficients, of retaining all the symmetry properties and selection rules of Wigner 3-j symbols. Appropriate label systems which render these properties in terms of simple algebras are given for all quantizing axes available in O _h . The specific set of 3Γ symbols for each quantization is determined by the orientation of the coordinate axes in the Hamiltonian. The four possible orientations for trigonal quantization are examined and the operator chosen which produces eigenvectors with conventional conjugate phases and a fully real set of 3Γ symbols.     

Symmetrization of eigenfunctions of angular momentum in point groups

The eigenfunctions |jm〉 of angular momentum can combine linearly to make basis functions of irreducible representations of point groups. We surmount the projection operator and find a new method to calculate the combination coefficients. It is proven that these coefficients are components of eigenvectors of some hermitian matrices, and that for all pure rotation point groups, the coefficients can be made real numbers by properly choosing the azimuth angles of symmetry elements of point groups in the coordinate system. We apply the coupling theory of angular momentum to obtain the general formulas of the basis functions of point groups. By use of our formulas, we have calculated the basis functions with half‐integers j from 1/2 to 13/2 of double‐valued irreducible representations for the icosahedral group. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 286–302, 2001     

Long-range dispersion coefficients for like centrosymmetric linear molecules and an application to H2-H2

The real spherical tensor theory of long-range intermolecular coefficients developed in previous papers is applied to derive explicit formulae for the first three dispersion coefficients for like centrosymmetric linear molecules. The expansion of angle-dependent coefficients in associated Legendre polynomials allows one to identify the isotropic and anisotropic components of the dispersion interaction in terms of London dispersion constants, the treatment of higher coefficients being simplified by the coupling of the elementary (l, l′)-polarizations to resultant angular momenta L_A and L_B onto each molecule. The contributions from all coupling schemes are given explicitly for C₆, C₈, C₁₀, and numerical results are presented for H₂-H₂ using two-term reduced spectra values from the Kaiserlautern group.     

Application of Angular Momentum Theory to Constructing Basis Functions of Irreducible Representations of Icosahedral Group

A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e. , coefficients S, are the components of the eigenvectors of some Hermitian matrices, and can be made as real numbers for all pure rotation point groups. The general formula for coefficient S was deduced, and applied to constructing the basis functions of single-valued irreducible representations of icosahedral group from the spherical harmonics with angular momentum j≤7.     

Calculation of transition matrix elements by nonsingular orbital transformations

A general strategy is described for the evaluation of transition matrix elements between pairs of full class CI wave functions built up from mutually nonorthogonal molecular orbitals. A new method is proposed for the counter‐transformation of the linear expansion coefficients of a full CI wave function under a nonsingular transformation of the molecular‐orbital basis. The method, which consists in a straightforward application of the Cauchy–Binet formula to the definition of a Slater determinant, is shown to be simple and suitable for efficient implementation on current high‐performance computers. The new method appears mainly beneficial to the calculation of miscellaneous transition matrix elements among individually optimized CASSCF states and to the re‐evaluation of the CASCI expansion coefficients in Slater‐determinant bases formed from arbitrarily rotated (e.g., localized or, conversely, delocalized) active molecular orbitals. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Calculation of transition density matrices by nonunitary orbital transformations

An efficient scheme for calculating one- and two-electron transition density matrices for two wave functions is described. The method applies to CAS (complete active space) wave functions and certain multireference CI expansions. The orbital sets of the two wave functions are not assumed to be equal. They are transformed to a biorthonormal basis, and the corresponding transformation of the CI coefficients is carried out directly, using the one-electron coupling coefficients.     

Femtosecond electron-transfer reactions in mono- and polynucleotides and in DNA

Quenching of redox active, intercalating dyes by guanine bases in DNA can occur on a femtosecond time scale both in DNA and in nucleotide complexes. Notwithstanding the ultrafast rate coefficients, we find that a classical, nonadiabatic Marcus model for electron transfer explains the experimental observations, which allows us to estimate the electronic coupling (330 cm(-1)) and reorganization (8070 cm(-1)) energies involved for thionine-[poly(dG-dC)](2) complexes. Making the simplifying assumption that other charged, pi-stacked DNA intercalators also have approximately these same values, the electron-transfer rate coefficients as a function of the driving force, DeltaG, are derived for similar molecules. The rate of electron transfer is found to be independent of the speed of molecular reorientation. Electron transfer to the thionine singlet excited state from DNA obtained from calf thymus, salmon testes, and the bacterium, micrococcus luteus (lysodeikticus) containing different fractions of G-C pairs, has also been studied. Using a Monte Carlo model for electron transfer in DNA and allowing for reaction of the dye with the nearest 10 bases in the chain, the distance dependence scaling parameter, beta, is found to be 0.8 +/- 0.1 A(-1). The model also predicts the redox potential for guanine dimers, and we find this to be close to the value for isolated guanine bases. Additionally, we find that the pyrimidine bases are barriers to efficient electron transfer within the superexchange limit, and we also infer from this model that the electrons do not cross between strands on the picosecond time scale; that is, the electronic coupling occurs predominantly through the pi-stack and is not increased substantially by the presence of hydrogen bonding within the duplex. We conclude that long-range electron transfer in DNA is not exceptionally fast as would be expected if DNA behaved as a "molecular wire" but nor is it as slow as is seen in proteins, which do not benefit from pi-stacking.     

Palladium‐Catalyzed Asymmetric Benzylic Alkylation of Active Methylene Compounds with α‐Naphthylbenzyl Carbonates and Pivalates

A Pd/(R)‐H₈‐BINAP‐catalyzed asymmetric benzylic alkylation of active methylene compounds has been developed. The reaction proceeds without the use of an external base, and the starting racemic diarylmethyl carbonates are converted into the optically active coupling products which contain the benzylic chiral stereocenter by a dynamic kinetic asymmetric transformation (DYKAT). Additionally, with suitable carbonates bases, the same palladium catalysis allows the corresponding pivalates to be adopted in the same DYKAT process.     

Extended Born-Oppenheimer equation for a three-state system

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.     

Some aminophosphinocarboxylic acids and their derivatives

The addition of Schiff bases of α-aminoacid esters to vinylphosphoryl compounds was studied as a method for the synthesis of phosphinothricin and its analogues. The reaction was found to proceed smoothly in DMSO in the presence of strong nitrogen bases and under the conditions of phase transfer catalysis. The Claisen condensation of β-phosphorylated propionitrile with diethyl oxalate was studied; phosphorylated derivatives of hydroxycitraconic acid nitrile were prepared on hydrolysis; transformation of such a derivative into a cyclic imide as well as into trimethylsilyl esters of these acids and their Z → E transformation were investigated. Cyclization reactions between α,ω-dibromoalkanes and phosphoryl compounds containing an active methylene group affording cyclopropane and other cyclic derivatives were studied. The cyclopropane ring is cleaved by amines to give aminophosphinocarboxylic acids.     

对称性轨道的构造

提出1种以配体轨道的旋转性质为基础,并利用群生成元表示矩阵来构造对称性轨道的新方法。讨论了对称性系数之间的相互关系,同时给出群重迭积分的计算过程。     

Finite group theory for large systems. 2. Generating relations and irreducible representations for the icosahedral point group, I(h)

Generating relations and irreducible representations are given for the icosahedral point group, I(h), that are suited to computerized projection of symmetry-adapted bases of arbitrary spaces invariant to the group. With 120 elements I(h ) is a good prototype for symmetry-adaptation to a large finite nonabelian group. The four- and five-dimensional irreducible representations are obtained by coupling direct products of the three-dimensional irreducible representations using the symmetry-adaptation algorithm. The method is applied to the Hückel treatment of icosahedral C(20) fullerene.     

Detection of deletion mutations in DNA using water-soluble cationic fluorescent thiophene copolymer

In this contribution, we designed a fluorescent thiophene copolymer to detect insertion/deletion mutation in DNA by doping aldehyde group in the main chain. The fluorescence of the copolymer could be dramatically quenched on the addition of single-stranded DNA (ssDNA) via strong electrostatic interactions and electronic/energy transfer. Although the complementary ssDNA made the fluorescence recover, the hydrogen bonds and chemical coupling also played a significant role between the unpaired bases and aldehyde group, which could differentiate the subtle differences in such mutant DNA. The influence of buffer pH, concentration of NaCl, heating time and the temperature was systemically investigated and the proposed method was then successfully applied to detect real sample. With the respect to the linearity, limit of detection precision, specificity, this procedure could provide sensitive methodologies for the rapid detection and identification of nucleic acids.     

Improved treatment for matrix elements of spin-dependent operators

In this paper a general method for the evaluation of the matrix elements of spin-dependent operators is proposed to improve the treatment primitively suggesteed by Cooper and Musher. This approach is largely based on the recent results which the present authors have achieved in the representation theory for the inner- and outer-product reduction of the symmetric group. It is shown that the so-called outer-product coupling coefficients (OPCC ) can be used to generalize the method for constructing the irreducible tensor operators of group S_n. Together with the use of inner-product coupling coefficients (IPCC ), an expression for the matrix elements of spin-dependent operators is presented as the product of a Racah coefficient for S_n and a reduced matrix element which can be expressed in terms of IPCC, OPCC , and the related integrals. The treatment for one- and two-electron spin-dependent operators is discussed in detail.     

Ligand‐free Pd/Cu‐catalyzed decarboxylative coupling of aryl iodides with α‐oxocarboxylates

This paper describes a palladium/copper‐catalyzed decarboxylative coupling of aryl iodides with α‐oxocarboxylates. The cross‐coupling reaction gives high chemical yields of aryl ketones and has wide functional group tolerance, making the transformation an attractive alternative to the traditional cross‐coupling approaches for aryl ketones. Copyright © 2014 John Wiley & Sons, Ltd.