Explicit representation of Gel'fand–Tsetlin states in clifford algebra unitary group approach

A explicit expression for the unitary group Clebsch–Gordan coefficients, which couple two fully antisymmetric single-column states into the two-column Gel'fand–Tsetlin states, is given in terms of isoscalar factors for the canonical subgroup chain U(n) ? U(n – 1) ? …? ? U(1). The isoscalar factors are expressed through the step numbers labeling canonical basis states and enable a straightforward construction of Gel'fand–Tsetlin states in the Clifford algebra unitary group approach, without the use of the tables for the symmetric group outer-product reduction coefficients.     

Valence bond approach exploiting Clifford algebra realization of Rumer–Weyl basis

A detailed algorithm is described that enables an implementation of a general valence bond (VB ) method using the Clifford algebra unitary group approach (CAUGA ). In particular, a convenient scheme for the generation and labeling of classical Rumer–Weyl basis (up to a phase) is formulated, and simple rules are given for the evaluation of matrix elements of unitary group generators, and thus of any spin-independent operator, in this basis. The case of both orthogonal and nonrothogonal atomic orbital bases is considered, so that the proposed algorithm can also be exploited in molecular orbital configuration interaction calculations, if desired, enabling a greater flexibility for N-electron basis-set truncation than is possible with the standard Gel'fand–Tsetlin basis. Finally, an exploitation of this formalism for the VB method, based on semiempirical Pariser–Parr–Pople (PPP )-type Hamiltonian and nonorthogonal overlap-enhanced atomic orbital basis, and its computer implementation, enabling us to carry out arbitrarily truncated or full VB calculations, is described in detail.     

Clifford algebra and unitary group formulations of the many-electron problem

An essential role of Clifford algebras for quantum-chemical finite-dimensional orbital models of many-electron systems is pointed out. The relationship between Clifford algebra matric units, the generators of the unitary group approach (UGA) and the higher order replacement or excitation operators, as well as between their first and second quantized realizations, is elucidated. The usefulness of higher order replacement operators in the spin-adaptation of various many-body theories is briefly outlined and illustrated on the orthogonally spin-adapted coupled-pair approach. A natural connection with the Clifford algebra UGA is explored and new possibilities for its exploitation in large scale configuration interaction calculations are suggested.Dedicated to Professor J. Koutecký on the occasion of his 65th birthdayKillam Research Fellow 1987–8     

Approach of the associated Laguerre functions to the su(1,1) coherent states for some quantum solvable models

Using second‐order differential operators as a realization of the su(1,1) Lie algebra by the associated Laguerre functions, it is shown that the quantum states of the Calogero‐Sutherland, half‐oscillator and radial part of a 3D harmonic oscillator constitute the unitary representations for the same algebra. This su(1,1) Lie algebra symmetry leads to derivation of the Barut‐Girardello and Klauder‐Perelomov coherent states for those models. The explicit compact forms of these coherent states are calculated. Also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

BONDED TABLEAU AND UNITARY GROUP APPROACH FOR MANY-ELECTRON PROBLEMS

A unitary group approach based on the so-called bonded tableaux (VB) states is described. Several different realizations of the bonded tableaux are discussed and their relations are pointed out. From a viewpoint of the symmetric group we reveal a simple structure of the matrix elements of unitary group generators and generator products. This structure makes an efficient approach to the matrix element evaluation.     

Generalized P,Q and G conditions

The N-Representability Problem entails characterizing the set of second order reduced states that are contractions of N-electron states of the Fermion Fock algebra. This problem is formulated in the form of finding the conditions that a positive linear functional defined on a subspace of this algebra must satisfy in order to be extended to the whole algebra. As this algebra is a w*-algebra one can utilize a theorem by Kadison that shows it is sufficient to consider the values of linear functionals on projectors contained in the subspace in order to determine whether they have positive extensions. Thus we find the form of projectors belonging to the subspace of one and two particle operators and subsequently show that the extension conditions needed in the N-Representability Problem correspond to generalized P, Q and G conditions plus the additional constraints that the functionals be dispersion free on the number operator and their values on one particle operators determined by their values on two particle operators.     

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.     

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.     

Clifford algebra,symmetries, and vectors

The realization of a Clifford algebra in laboratory space is considered and it is demonstrated that the elements of the algebra cannot, as often assumed, be directly identified with vectors in this space, but, rather, that they form the parametric space of the symmetry operations of the Euclidean group as performed in the laboratory space. Details of this parametrization are established and expressions are given that determine the action of the Euclidean-group operations (screws included) on laboratory-space vectors in terms of the elements of the Clifford algebra. A discussion of Clifford vectors, bivectors, and pseudoscalars and their relation to the Gibbs vectors is provided. The correct definition of axial and polar vectors within the Clifford algebra is carefully discussed. It is shown how simple it is to generate finite point groups in 4-dimensional space by means of the Clifford algebra. © 1996 John Wiley & Sons, Inc.     

Particles and holes in the unitary group method

Based on the definition for complementary Gel'fand states, we proved the simple relationship between the matrix elements of particle states and those of hole states by unitary calculus.     

Many‐particle Dirac identities for arbitrary elementary spins

The eigenvalues of arbitrary conjugacy class‐sums of the symmetric group, within subspaces that contain irreducible representations with at most k rows, are considered. Explicit expressions for these eigenvalues in terms of the eigenvalues of single‐cycle class‐sums with cycle lengths up to k are obtained. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 407–411, 2000     

The freeon theory of magnetism. I. The Heisenberg interaction

The freeon theory of magnetism is a viable alternative to the well-known Heisenberg theory. In particular, the freeon-exchange theory provides a deeper insight into the nature of the magnetic interaction than does the spin-exchange theory. In addition, it avoids the superfluous M_S quantum number in zero-field splitting, the overcounting of spin states, the spin frustration, and the spin paradigm. The freeon-exchange theory employs the algebra of the symmetric group and/or the unitary group in place of the spin algebra. The basis vectors of freeon theory are the Gel'fand states which are uniquely labeled by Gel'fand diagrams; the latter provide both the electron configuration and the spin quantum number. Both the spin and the freeon formulations support the Landé interval rule. In this article, we apply freeon theory to transition-metal dimers, trimers, and tetramers; these are examples of molecular magnets which have applications to microcircuitry. The freeon theory follows the permutation-group principle laid down by Herman Weyl over one-half a century ago. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 287–297, 1997     

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.     

Application of the group function theory to infinite systems

We consider application of the group function theory to an arbitrary infinite system consisting of weakly overlapping structural elements which may be atoms, ions, molecules, bonds, etc. We demonstrate that the arrow diagram (AD) expansion developed previously is ill‐defined for such a system resulting in divergences in any physical quantity associated with the entire system such as, for example, the energy and charge density. A “linked‐AD” theorem is then formulated and proven, which results in a diagrammatic expansion that scales correctly with the system size. Coulomb systems with long‐range interactions between structure elements are also considered and the diagrammatic expansion is rearranged in such a way as to also give the correct (linear) scaling. We give an explicit expression for the total energy up to the third order with respect to overlap. Finally, we discuss the problem of choosing structure elements (SE) in a general insulating system and propose a variational method based on a configuration interaction (CI) type expansion within the Fock subspace associated with every SE. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 511–534, 2000     

On the class of attainable multidimensional NMR spectra

It is shown that a large class of two‐dimensional NMR spectra is characterized by a matrix algebra and an invariant subspace. Both the matrix algebra and the invariant subspace are determined by the system matrices of the bilinear system which describes the NMR experiments.     

On the computation of large-scale self-consistent-field iterations

The computation of the subspace spanned by the eigenvectors associated to the N lowest eigenvalues of a large symmetric matrix (or, equivalently, the projection matrix onto that subspace) is a difficult numerical linear algebra problem when the dimensions involved are very large. These problems appear when one employs the self-consistent-field fixed-point algorithm or its variations for electronic structure calculations, which requires repeated solutions of the problem for different data, in an iterative context. The naive use of consolidated packages as Arpack does not lead to practical solutions in large-scale cases. In this paper we combine and enhance well-known purification iterative schemes with a specialized use of Arpack (or any other eigen-package) to address these large-scale challenging problems.     

Symmetric group graphical approach to the direct configuration interaction method

An approach to the configuration interaction method based on symmetric groups (SGA ) is developed. The formalism is an alternative of the unitary group approach (UGA ). In many aspects the present formulation seems to be superior to UGA . In particular, in SGA the orbital and the spin parts of the configuration state functions may be processed separately. In consequence its graphical formulation is much simpler and the coupling constant expressions are more compact than the UGA analogs. A special emphasis is put on direct CI implementations. In addition to formulas for coupling constants, explicit expressions allowing for separation of external and internal space contributions are also presented.     

The combinatorics of graphical unitary group approach to many electron correlation

The combinatorics of Gel'fand states which are useful in the graphical unitary group approach to many electron correlation problem and spin free quantum chemistry is considered. Using operator theoretic methods it is shown that the generators of Gel'fand states are S-functions.     

Algebraic approach to the asymmetric rigid rotor

A pure algebraic treatment of the eigenvalue equation corresponding to the asymmetric top is presented. The algebraic method employs the Holstein–Primakoff bosonic realization of the angular momentum algebra. Explicit determination of the linear boson transformation coefficients of the eigenstates is carried out by means of the coherent states formalism. No reference to special functions is needed and a completely algebraic approach is achieved. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 704–709, 2000     

Symmetrizer: Algorithmic determination of point groups in nearly symmetric molecules

Symmetry is an extremely useful and powerful tool in computational chemistry, both for predicting the properties of molecules and for simplifying calculations. Although methods for determining the point groups of perfectly symmetric molecules are well‐known, finding the closest point group for a “nearly” symmetric molecule is far less studied, although it presents many useful applications. For this reason, we introduce Symmetrizer, an algorithm designed to determine a molecule's symmetry elements and closest matching point groups based on a user‐adjustable tolerance, and then to symmetrize that molecule to a given point group geometry. In contrast to conventional methods, Symmetrizer takes a bottom‐up approach to symmetry detection by locating all possible symmetry elements and uses this set to deduce the most probable point groups. We explain this approach in detail, and assess the flexibility, robustness, and efficiency of the algorithm with respect to various input parameters on several test molecules. We also demonstrate an application of Symmetrizer by interfacing it with the WebMO web‐based interface to computational chemistry packages as a showcase of its ease of integration. © 2012 Wiley Periodicals, Inc.