Superfine and hyperfine structures in the v3 band of 32SF6

We present a first detailed account of our theoretical approach to reproduce observed superfine and hyperfine structures in the ν₃ band of SF₆ and we display various observed and calculated patterns of superfine clusters exhibiting hyperfine effects. The main operators of the hamiltonian are derived and the associated constants are related to molecular parameters. We show that, owing to the off-diagonal terms in the hyperfine hamiltonian, a mixing occurs between vibration—rotation states with different point-group symmetry species. As a consequence, superfine and hyperfine structures have to be considered simultaneously and hyperfine hamiltonian matrices connecting several vibration—rotation states need to be diagonalized to reproduce the spectra. We analyse in greater detail a few typical examples from which several molecular constants have been determined (e.g. t₀₄₄, c_d). For the first time, the sign c_d is obtained. Also an effective change, Δc_d, is found between upper and lower levels which can be readily interpreted as a manifestation of the tensor spin—vibration interaction.     

Some iterative techniques for large CI matrices. I. Excited eigenvalues

Iterative methods for computing eigenvalues and eigenvectors of large symmetric matrices are discussed using the language of multidimensional partitioning technique. New variants are proposed and found useful particularly for the computation of highly excited states of CI matrices.     

Simplified diagrammatic expansion for effective operators

For a quantum many-body problem, effective Hamiltonians that give exact eigenvalues in reduced model space usually have different expressions, diagrams, and evaluation rules from effective transition operators that give exact transition matrix elements between effective eigenvectors in reduced model space. By modifying these diagrams slightly and considering the linked diagrams for all the terms of the same order, we find that the evaluation rules can be made the same for both effective Hamiltonian and effective transition operator diagrams, and in many cases it is possible to combine many diagrams into one modified diagram. We give the rules to evaluate these modified diagrams and show their validity.     

Eigenvector and eigenvalues of some special graphs. IV. Multilevel circulants

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.     

Topological analysis of eigenvectors of the adjacency matrices in graph theory: The concept of internal connectivity

The topological properties of eigenvectors of adjacency matrices of a graph have been analyzed. Model systems studied are n-vertex-m-edge (n-V-m-E) graphs where n = 2–4, m = 1–6. The topological information contained in these eigenvectors is described using vertex-signed and edge-signed graphs. Relative ordering of net signs of edge-signed graphs is similar to that of eigenvalues of the adjacency matrix. This simple analysis has also been applied to naphthalene, anthracene and pyrene. It provides a sound basis for the application of graph theory to molecular orbital theory.     

A block variational procedure for the iterative diagonalization of non-Hermitian random-phase approximation matrices

We present a technique for the iterative diagonalization of random-phase approximation (RPA) matrices, which are encountered in the framework of time-dependent density-functional theory (TDDFT) and the Bethe-Salpeter equation. The non-Hermitian character of these matrices does not permit a straightforward application of standard iterative techniques used, i.e., for the diagonalization of ground state Hamiltonians. We first introduce a new block variational principle for RPA matrices. We then develop an algorithm for the simultaneous calculation of multiple eigenvalues and eigenvectors, with convergence and stability properties similar to techniques used to iteratively diagonalize Hermitian matrices. The algorithm is validated for simple systems (Na(2) and Na(4)) and then used to compute multiple low-lying TDDFT excitation energies of the benzene molecule.     

Fock space perturbation theory

Diagonal and non-diagonal operators in Fock space are defined. With a universal Fock space wave operator W the Fock space hamiltonian H can be transformed to a diagonal operator L containing all relevant information about eigenvalues of H for arbitrary particle number in a simply coded form. W and L are constructed by perturbation theory, even in a spinfree form, and illustrated diagrammatically.     

An iterative method to solve the algebraic eigenvalue problem

An iterative method based on perturbation theory in matrix form is described as a procedure to obtain the eigenvalues and eigenvectors of square matrices. Practical vector notation and elementary schematic algorithm codes are given. The particular programming characteristics of the present computational scheme are based upon eigenvector corrections, obtained through a simple Rayleigh–Schrödinger perturbation theory algorithm. The proposed methodological processes can be used to evaluate the eigensystem of large matrices.     

Improvement of Weyl’s Inequality

Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized.     

The relationship between the eigenvalues and eigenvectors of a similarity matrix and its associated Carbó index matrix

The eigenvalues and eigenvectors of a quantum similarity matrix are also generalized eigenvalues and eigenvectors of the associated matrix of Carbó indices. This establishes bounds on the spectrum of the Carbó index matrix; for example, a quantum similarity matrix is positive semidefinite if and only if the associated Carbó index matrix is also positive semidefinite. The generalized eigenvalue problem for the Carbó index matrix has a diagonal metric matrix on the right-hand-side. Every generalized eigenvalue problem can be written in this diagonal form (i.e., this form is not special to this application). This diagonally structure generalized eigenvalue problem is especially convenient because it can be converted to a conventional eigenvalue problem by a particularly simple partial Löwdin transformation.     

The scaled Hermite–Weber basis in the spectral and pseudospectral pictures

Computational efficiencies of the discrete (pseudospectral, collocation) and continuous (spectral, Rayleigh–Ritz, Galerkin) variable representations of the scaled Hermite–Weber basis in finding the energy eigenvalues of Schrödinger operators with several potential functions have been compared. It is well known that the so-called differentiation matrices are neither skew-symmetric nor symmetric in a pseudospectral formulation of a differential equation, unlike their Rayleigh–Ritz counterparts. In spite of this fact, it is shown here that the spectra of matrix Hamiltonians generated by Hermite collocation method may be determined by way of diagonalizing symmetric matrices. Furthermore, the symmetric matrix elements do not require the evaluation of Hermite polynomials at the grid points. Surprisingly, the present numerical results suggest that the convergence rates of collocation and Rayleigh–Ritz methods are entirely the same.AMS subject classification: 65L60, 81Q05, 65L15, 34L40, 42C10     

Atom and Bond Fukui Functions and Matrices: A Hirshfeld‐I Atoms‐in‐Molecule Approach

The Fukui function is often used in its atom‐condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms‐in‐molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld‐I atoms‐in‐molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld‐I approach to obtain atom‐ and bond‐condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms‐in‐molecule approach shows low correlation between the two partitioning schemes.     

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.     

Calculation of Tanabe-Sugano Diagrams by Matrix Diagonalization

The chemistry and spectroscopy of transition elements depend on complex interactions found for partially filled d subshells. There are complex electron-electron repulsion effects within a partially filled subshell and additional complications due to participation of some of the d orbitals in chemical bonding. The number of states involved can be quite large, and the mathematical treatment involves matrices as large as 28 by 28. The direct solution for eigenvalues and eigenvectors of matrices this large was not even attempted in the 1930s, but is well within the capabilities of current undergraduates and computers. Full matrices in the form of spreadsheets are provided in this paper for octahedral and tetrahedral symmetry for all cases from d2 through d8. Diagonalization of such matrices with MACSYMA is illustrated. Construction of full Tanabe-Sugano diagrams is possible for students for any choice of input parameters.     

Iterative methods for the calculation of a few of the lowest eigenvalues and corresponding eigenvectors of the AX = λBX equation with real symmetric matrices of large dimension

New methods for the iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of a generalized eigenvalue problem are proposed. These methods use only multiplication of the A and B matrices on a vector. © 1994 by John Wiley & Sons, Inc.     

A new approach to the molecular rotational hamiltonian

The present paper is devoted to a new approach of the rotational hamiltonian of a non degenerate vibronic state of the semi-rigid molecules. It is based upon two points. Firstly the hermitian operators on a finiten dimensional vector space belong to an ² dimensional euclidian vector space. Secondly, the vector space of the rotational states is a direct sum of irreducible representations of the rotation group. Accordingly in each one of those representations the rotational hamiltonian can be represented by its set of real components on a orthonormal basis of hermitian operators. The components of the reduced hamiltonian of Watson limited to its quartic terms are determined.     

A concurrent algorithm for parallel calculation of eigenvalues and eigenvectors of real symmetric matrices

Elementary Jacobi Rotations are used as the basic tools to obtain eigenvalues and eigenvectors of arbitrary real symmetric matrices. The proposed algorithm has a complete concurrent structure, that is: every eigenvalue-eigenvector pair can be obtained in any order and in an independent way from the rest. Examples based on diagonally dominant real symmetric matrices are given.     

A note on the partitioning technique

A new derivation of the wave operator of the partitioning technique is given. Furthermore this approach is applied to derive wave operators for the inverse hamiltonian and in general for functions of the hamiltonian.     

Investigation of some soluble lattice spin models

The diagonalization of a class of lattice spin models of a particular structure is first reviewed and secular polynomials for these models are calculated explicitly from the corresponding secular matrices. The structure of the eigenvectors of the given secular matrices is investigated and used to determine the eigenvalues theoretically, and proofs which have not appeared are presented. These results can be compared to the results obtained from the full secular polynomials. © 1996 John Wiley & Sons, Inc.     

Two new graph-theoretical methods for generation of eigenvectors of chemical graphs

Two new graph-theoretical methods, (A) and (B), have been devised for generation of eigenvectors of weighted and unweighted chemical graphs. Both the methods show that not only eigenvalues but also eigenvectors have full combinatorial (graph-theoretical) content. Method (A) expresses eigenvector components in terms of Ulam’s subgraphs of the graph. For degenerate eigenvalues this method fails, but still the expressions developed yield a method for predicting the multiplicities of degenerate eigenvalues in the graph-spectrum. Some well-known results about complete graphs (K _n) and annulenes (C _n ), viz. (i)K _n has an eigenvalue −1 with (n−1)-fold degeneracy and (ii) C _n cannot show more than two-fold degeneracy, can be proved very easily by employing the eigenvector expression developed in method (A). Method (B) expresses the eigenvectors as analytic functions of the eigenvalues using the cofactor approach. This method also fails in the case of degenerate eigenvalues but can be utilised successfully in case of accidental degeneracies by using symmetry-adapted linear combinations. Method (B) has been applied to analyse the trend in charge-transfer absorption maxima of the some molecular complexes and the hyperconjugative HMO parameters of the methyl group have been obtained from this trend.