Clebsch-Gordan coefficients for chains of groups of interest in quantum chemistry

The algebra of the representation of the special unitary group SU(2), the universal covering of the proper rotation group SO(3), is studied in a nonstandard basis. We are using a basis adapted to a chain of type SU(2) ? …? ? G″ ? G′ ? G. The introduction of such a chain enables us to label, at least partially, the elements of the irreducible tensorial sets under SU(2) with irreducible representations of G, G″ G″, …. We are thus led to introduce the restriction SU(2) → …? → G″ → G′ → G in the Wigner-Racah algebra of the group SU(2). The physical interest of this machinery lies in the fact that the double group of any point symmetry group belongs, up to an isomorphism, to the considered chain. The formalism described in this paper thus appears to be useful in molecular and solid-state calculations. It is particularly efficient in the fields of vibrational-rotational and electronic spectroscopy of molecules. In Appendix A the master formulae, principally the Wigner-Eckart-Racah theorem, for the Wigner-Racah algebra of a chain of compact topological groups (discrete or continuous) are briefly discussed. Lastly, a programme for computing Clebsch-Gordan coefficients for a chain SU(2) ? …? ? G″ ? G′ ? G and numerical results for chains isomorphic to SU(2) ? O′ ? D′₄ ? D′₂ are described in Appendix B.     

Projection operators and Clebsch–Gordan coefficients for projective representations

Previous discussions of the bases and projection operators for projective representations are analyzed and alternatives are proposed. Detailed proofs are provided for a result which is often assumed or proved within unacceptable limitations, namely that the form of the projection operators which is standard for vector representations is also valid for unitary projective representations. These proofs provide necessary conditions for this result, and they are constructed in terms of the definition given for the bases of projective representations. The calculation of Clebsch–Gordan coefficients for projective representations by means of the projection operators is discussed. Whereas in the method of Dirl the work is carried out entirely in terms of the matrix representations, and the symmetrization of the bases has to be considered in a second step, all the work of this paper is conducted starting from the symmetrization of the bases of the projective representations, so that those two steps are carried out simultaneously.     

Derivation of the Clebsch–Gordan coefficients by means of projection operators

The Clebsch–Gordan coefficients for the coupling of two angular momenta are derived by using the projection operator technique, developed by Löwdin. The derivation is done in two steps; first for the so-called principal case, i.e. k = m, then for the general case with an arbitrary m. Two different derivations are given for the principal case, a direct one and one based on a recursion procedure. The general case is obtained from the principal case with a step-down operator.     

Clebsch–Gordan coefficients for the group chains O ⊃ T ⊃ D2 and Td ⊃ T ⊃D2 by the method of projective representations

The spinor representations for the groups forming the chains O ? T ? D ₂ and T _d ? T ? D ₂ are constructed as projective representations. The Clebsch–Gordan coefficients are then calculated using a standard method. Projective factor systems, irreducible representations, canonical bases, and tables of Clebsch–Gordan coefficients are given. The subduction from O to D ₃ is discussed.     

Algebraic expressions of Clebsch–Gordon coefficients for the group chain O† ⊃ C

Using the algebraic expressions of the projection operators for the group chain O ? C, concise algebraic expressions of the Clebsch–Gordon (CG) coefficients are derived in the group chain O ? C for both single‐valued and double‐valued representations. The simplicity of the expressions is that they are merely functions of the quantum numbers of the group chain O ? C. The symmetry of the CG coefficients is also derived. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Tables of spectroscopic coefficients for magnetic point group symmetry

For the first time symmetrized Clebsch-Gordan coefficients for corepresentations, the so called 2D- and 3D-symbols (analogous to the Wigner 2j- and 3j-symbols correspondingly) are calculated, and tabulated for all single- and double-valued corepresentations of all 90 antiunitary magnetic point groups.     

Symmetry properties of graphs of interest in chemistry. II. Desargues–Levi graph

The Desargues–Levi graph represents important chemical transformations: (1) isomerization routes for some carbonium ion rearrangements, (2) isomerization of trigonal bipyramidal structures, and (3) some pseudorotations of octahedral complexes. The symmetry properties of this graph have not been fully investigated in the past. Using the concept of the smallest binary code, all permutations which form the symmetry operations in the graph are registered. The resulting symmetry group can be represented as the direct product of S₅ (the full symmetric permutation group on five objects) and C_i (the inversion in the center). There are 14 classes belonging to the following partitionings: 1²⁰(1), 1⁸2⁶(1), 1⁴2⁸(1), 1²3⁶(1), 1²3²6²(1), 2 6³(2), 2²4⁴(2), 2¹⁰(3), 5⁴(1), and 10²(1). The total of 240 symmetry operations are distributed among the above 14 classes as follows: 1, 10, 15, 20, 20, 20, 20, 30, 30, 15, 10, 1, 24, and 24, respectively. Since partitioning cannot uniquely characterize a class, it is suggested that the distance between vertices in a cycle be introduced as an additional parameter to discriminate among classes having identical partitioning. Also, a suggestion to a generalization of the Mulliken notation for irreducible representations of the point molecular groups valid for more versatile symmetry groups of graphs is indicated.     

Local quantum chemistry: The local space approximation for Møller–Plesset perturbation theory

The local space approximation is an accurate technique for describing a relatively small cluster embedded within an extended system. It has previously been developed for the Hartree-Fock, local density functional, configuration interaction, and coupled cluster electronic structure methods. Here it is extended to Møller-Plesset perturbation theory. © 1995 John Wiley & Sons, Inc.     

On the validity of the mass–velocity operator in quantum chemistry

The problem of the validity of the mass–velocity operator in computational quantum chemistry is discussed. The opinion that the mass–velocity operator is incorrect is shown not to be well founded.     

Subgroup‐chain symmetry‐adapted irreducible matrices and CG coefficients of point groups

The irreducible matrices and Clebsch–Gordan coefficients of any crystallographic point group adapted to all possible canonical subgroup chains are calculated ab initio for both single‐valued and double‐valued representations and tabulated with exact values in the form of or and with components labeled by the irrep labels of the group chain in Koster notation. The phases and ordering of the components of irreducible bases for the cubic point groups are properly chosen so that irreducible matrices for all subgroup chains of G=T_d, O, O_h obey the associated relations D(G)=D(G)D(G), i=4, 6, and the complex conjugation relation for the group T, D(T)=D(T)*. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 67–80, 1999     

The new pictorial structural covariance method for qualitative quantum chemistry. III: Fused polycyclics and their ions

The number of non-bonding (NBMOs), bonding, and anti-bonding MOs, the HOMO-LUMO types, reactivity to electro- or nucleophiles, qualitative relative stabilities of the neutral species, their anions or cations, are readily deduced for pi-polycyclics on the blackboard directly from the structural formulae utilizing the recently presented pictorial quantum rules. Among others, the bicyclo [p.q.0] pi-hydrocarbons are treated extensively and are found to fall into ten distinct homolog classes. In addition to the customary pi-aromatic or pi-anti-aromatic types, classes of compounds with other behaviour (e.g. one here called anti-Kekulé) are found. When an anti-aromatic ring is fused with another anti-aromatic or with a pi-aromatic ring, the resulting bicyclo-molecule is not anti-aromatic. In another series, the heptalene and other (4k + 3), (4k + 3) bicyclics have two electrons in a single NBMO, making them anti-Kekule. The method is able to distinguish between the properties of pairs of isomers such as (s-indacene; as-indacene) and (1.2,5.6-dibenzpentalene; 1.2,4.5-dibenzpentalene).     

Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Method of local-scaling transformations and density functional theory in quantum chemistry. III. The energy density functional: Spin-restricted approach

The rigorous derivation of the energy density functional is proposed within the framework of the spinfree, or spin-restricted formulation of the energy density functional theory. It is shown particularly that the kinetic energy density functional is given by a sum of the Weizsacker term and the so-called “modified” Thomas–Fermi one. The variational principle is formulated for the energy density functional theory in terms of the Euler–Lagrange equation, and the virial theorem is proposed.     

A density functional representation of quantum chemistry. III. Rigorous realization of the program in lattice space

A mathematically rigorous reformulation of molecular quantum mechanics in terms of the particle density operator and a canonically conjugated phase field is given. Using a momentum cutoff, it is shown that the usual molecular Hamiltonian can be expressed in terms of the particle density operator and a rigorously defined phase operator. It is shown that this Hamiltonian converges strongly to the cutoff-free Hamiltonian. In spite of the fact that this Hamiltonian is of second order in the phase operators, all hitherto published expressions are not correct. Unfortunately, the correct formulation destroys the intuitive appeal of using the particle density operator as a coordinate for the many-body problems of quantum chemistry. Unless somebody provides an essential new and clever idea, we propose to resist the fascination of a local quantum field theory of molecular matter in terms of the particle density operator.     

Spin-free quantum chemistry. XIV. The infinite interaction range model for ferromagnetism

The infinite interaction range model (IIRM ) for ferromagnetic systems is presented in its spin-free formulation. In this formulation the states are labelled by partitions which provide a natural variable for thermodynamic computation. We have extended the calculations of Kittel and Shore by computing to a practical thermodynamic limit (N ～ 100,000). The heat capacity, magnetic susceptibility and the magnetization of the first two functions exhibit a critical temperature while the magnetization is zero at zero field for all temperatures. Spontaneous magnetization is obtained by linear extrapolation from high field or equivalently by a polarized partition function. Relationships are explored among IIRM , the Heisenberg model and the mean field model. Application to IIRM of the Yang-Lee condition for a phase transition yields a critical temperature identical to that obtained by the direct calculation.