Algebraic solutions for point groups: Cubic groups G in the group chain G⊃T⊃D2⊃C2

Concise algebraic expressions of the symmetry‐adapted functions (SAFs) for both single‐valued and double‐valued representations are derived for the group chain O⊃T⊃D₂⊃C₂ and O⊃D₄⊃D₂⊃C₂, which are functions of only the quantum numbers of the respective group chain without involving any irreducible matrix elements. It is shown that the SAFs of the cubic groups G=O,T_d,T_h,O_h can be expressed in a simple way in terms of the SAFs of the group T. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 585–599, 2000     

Algebraic solutions for point groups: The octahedral group for the group chain O⊃T⊃C3

Using the right‐induced technique and the eigenfunction method, concise algebraic expressions of the projection operators for both single‐valued and double‐valued representations are found for the group chain O?T?C₃ in terms of the projection operators of T?C₃. Extremely simple relations are discovered between the symmetry adapted functions (SAFs) of the groups T and O; namely the SAFs of the subgroup T which have proper symmetry are the SAFs of the group O. The projection operators and SAFs are functions of only the quantum numbers of the group chain [the analogy of ( j,m) for the group chain SO₃?SO₂], without involving any irreducible matrix elements. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 259–270, 2001     

Algebraic expressions of irreducible bases for molecules C20H20, C80, and C240

Using the symmetrized boson representation technique, concise algebraic expressions of the irreducible bases symmetry adapted to the group chain I_h⊃C₅ for the fullerene molecules C₂₀H₂₀, C₈₀, and C₂₄₀ are derived for the most general cases and those for any specific case can be derived from them easily without a projection procedure. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 283–297, 1999     

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     

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     

Symmetry adapted formulation of the generalized Brillouin theorem

The singly excited functions satisfying Brillouin theorem are expressed as linear combinations of configuration-state functions for any spin and spatial symmetries (atomic or molecular) and for any reference wave function. The generality of the formulation is ensured by the use of the irreducible tensor method that can be adapted to any symmetry point group of interest. The expansion coefficients are simply written as products of fractional parentage coefficients, spin- and orbit-recoupling coefficients, and phase factors. The formalism is illustrated for some atomic (K_h) and molecular (C_∞v, C_3v, and T_d) configurations. Group theoretical techniques are also used to correlate the Brillouin conditions within a chain of groups.     

Bases for the irreducible representations of O(3) symmetry adapted to a crystallographic point group

We construct bases for the irreducible representations of the rotation group O(3) which are symmetry adapted to a Crystallographic point group. We obtain explicit expressions for the cubic groups, which are valid for arbitrary values of the angular momentum quantum number l. Our method yields an efficient algorithm for both analytical and numerical work. An explicit formula for the multiplicities of an irreducible representation for the cubic groups in an arbitrary angular momentum term l is also derived.     

Analytical irreducible representation bases of the single and double icosahedral groups and their applications

An equivalent basis of icosahedral molecules is introduced in which the basis functions can be transformed under the operations in the icosahedral group (I_h). In this equivalent basis, the irreducible representation basis (IRB) of I_h, including the double‐valued IRB of I, is deduced analytically based on the method introduced in the literature [J. Comput. Chem. 17 , 851 (1996)]. Therefore the concepts of symmetry‐matrix and symmetry‐supermatrix can be used in the single‐ and multiconfiguration self‐consistent field methods (including relativistic effects) to reduce the storage of two‐electron integrals and calculations of Fock matrix during iterations by a factor of ca. 10,000. In addition, the equivalent basis of I_h can also be used to reduce the calculations of atoms and representations of rank ≥ 2 tensors. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 615–624, 2000     

The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions

In relativistic quantum chemical calculation of molecules, where the spin-orbit interaction is included, the electron orbitals possess both the double point group symmetry and the time-reversal symmetry. If symmetry adapted functions are employed as the basis functions of electron orbitals, it would allow a significant reduction of the computational expense. The point group symmetry adapted functions can be obtained by the group projection operators via its actions on the atomic orbital functions. We have proposed an efficient and simple method to obtain all irreducible representation matrices, which are the basis of the group projection operators, of any finite double point group. Both double point group symmetry and time-reversal symmetry are automatically imposed on the representation matrices. This is achieved by the symmetrized random matrix (SRM) approach, where the SRM is constructed in the regular representation space of a finite group and the eigenfunctions of SRM provide all irreducible representation matrices of the given point group.     

Pseudosymmetry analysis of molecular orbitals

We introduce a pseudosymmetry analysis of molecular orbitals by means of the newly proposed irreducible representation measures. To do that we define first what we consider as molecular pseudosymmetry and the relationships of this concept with those of approximate symmetry and quasisymmetry. We develop a general algorithm to quantify the pseudosymmetry content of a given object within the framework of the finite group algebra. The obtained mathematical expressions are able to decompose molecular orbitals by means of the irreducible representations of any reference symmetry point group. The implementation and usefulness of the pseudosymmetry analysis of molecular orbitals is demonstrated in the study of σ and π orbitals in planar and nonplanar polycyclic aromatic hydrocarbons and the t_2g and e_g character of the d‐orbitals in the [FeH₆]^3? anion in its high spin state along the Bailar twist pathway. © 2013 Wiley Periodicals, Inc.     

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     

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.     

Symmetry functions adapted to the subgroup chain U(7) ⊃ SO(7) ⊃ G2 ⊃ SO(3) ⊃ G

Summary In this paper the Lie algebra technique is used to construct symmetry functions adapted to the subgroup chain U(7) SO(7) G ₂ SO(3) G, which is one of symmetry group chains appearing in the weak ligand field scheme for f ^N ions. The functions are expressed in terms of the Gelfand states.     

Sequence-adapted molecular tensors: Algebraic methods and application to crystal field theory

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, T_d, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d¹ ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.     

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.     

Complete TDA and RPA Calculations on the Electronic Transitions of Fullerene‐C60 in the CNDO/S and INDO/S Approximations*

Complete single‐excitation mixing calculations on the electronic transitions of the icosahedral C₆₀ molecule have been carried out with the Tamm–Dancoff approximation (TDA) and random‐phase approximation (RPA) schemes in the CNDO/S and INDO/S approximations. The complete space of 14,400 (1p–1h) pairs is partitioned into subspaces classified according to the irreducible representations of the Ih group. For this purpose, matrix representations of the group generators are obtained on a fixed set of basis functions and are used to construct the projection operators. Degenerate molecular orbitals in each energy level are symmetry‐adapted to these projection operators. Degenerate (1p–1h) pairs or singly excited configuration wave functions are similarly symmetrized. In addition, the Clebsch–Gordan coefficients are obtained and listed in an Appendix. The TDA and RPA equations are then solved for each irreducible representation separately. Both schemes with the projection operators and with the Clebsch–Gordan coefficients gave the same results as expected, indicating that the calculations were correctly done. The transition energies from the ground state 1¹A_g to low‐lying singlet and triplet excited states and the oscillator strengths for the allowed transitions (n¹T_1u–1¹A_g) are given in tables. A proper way to normalize is discussed for the eigenvectors of the RPA‐type matrix equation. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Symmetry Adapted Analysis of Linear Molecules

Symmetry groups of the linear molecules belong to the C_∞v and D_∞h infinite groups. The symmetry adapted analysis of such types of molecule, is usually not systematically performed in the text book or paper. Since the standard formulas of symmetry adapted analysis are usually applicable for the finite groups only, one has to analyze the different subgroups of the linear molecules indirectly and correlates them with the irreducible representation of D_∞h and C_∞v. In this work, a systematic symmetry adapted analysis are introduced for the C_∞v and D_∞h molecules. It is a uniquely convenient way for molecular orbital calculations and vibrational normal mode analysis of the linear molecules.     

Alternative implementation of the unitary group approach to the atomic shell theory

Subduction coefficients adapted to the group chain, which appeared in Racah's treatment of fⁿ configurations, are defined and calculated in the unitary group approach. The coefficients are then utilized to construct successively adapted term functions and evaluate other interesting coefficients. In addition the simplified expressions for the Coulomb and spin-orbit operators are obtained in terms of generators.     

Unitary group approach to spin-adapted open-shell coupled cluster theory

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with n_c doubly occupied and n_v unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(n_c) ? U(n_V) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(n_c + n_v). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group S_M. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(n_c) ? U(n_s) tensors, can be shown to be connected, the irreducible tensor operators of U(n_c) ? U(n_v) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=n_c + n_v + n_s is not simply reducible under the group U(n_c) ? U(n_s) ? U(n_v). The multiplicity problem is resolved using the group chain U(n) ? U(n_c + n_v) ? U(n_s) ? U(n_c) ?U(n_s)? U(n_v) ? U(n_v). The labeling of the resulting configuration-state functions (which, in general, are not GT states when n_c > 1) by the irreducible representations of the intermediate group U(n_c + n_v) ?U(n_s) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.     

Unique Euler angles and self‐consistent multiplication tables for double point groups

The standard Euler angle parameterization of rotations is not unique. This is a particular problem when considering spinor representations. We enlarge the domain of the Euler angles from an SO₃ covering to an SU₂ covering, 0≤α<2π, 0≤β≤π, 0≤γ<4π. With this modification we can find unique Euler angles for operations of the double groups and thus construct self‐consistent group tables for those groups. Factor systems can then be described for the projective representations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 1–9, 1999