Hermitian operators for two-centre wavefunctions

Semi-analytical solutions of the Schrödinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, are derived in such a way that the resulting matrix eigenvalue equations contain real symmetric band matrices. Numerical techniques appropriate to the solution of the two simultaneous matrix eigenvalue equations are described; in particular the bisection method is used to determine precisely the significant truncation order of the matrices for a given numerical precision.     

A method of calculating eigenvectors of real symmetric tridiagonal matrices in a hyperspherical space

A novel direct method of calculating eigenvectors for real symmetric tridiagonal matrices has been proposed. An eigenvector is represented as a unit vector in a hyperspherical space set by hyperpolar angle coordinates. An algorithm has been examined for a series of tridiagonal matrices. Numerical calculations as compared with a conventional Householder algorithm have shown a good agreement within 8-byte double precision for most eigenvectors computed.     

Simple derivation of conditions for instability in the Hartree–Fock and projected Hartree–Fock schemes

The conditions for instability of solutions of Hartree–Fock and projected Hartree–Fock equations are derived in a form involving finite real symmetric matrices. These conditions are also expressed in terms of the Fock–Dirac density matrix, both at the spin–orbital and at the orbital level. The particular variations which give rise to the so-called singlet and triplet instabilities are described.     

Algebraic reductions of the real symmetric 4 × 4 secular problem

Procedures for the construction of the eigenvector matrix and the spectrum of 4 × 4 real and symmetric matrices are given. The Lie algebra of the group O(4)? is used as well as the relation to O(3)?. Perturbations are analyzed in terms of the group parameters.     

A new algorithm for the symmetric (Löwdin) orthonormalization

A new method for the numerical calculation of (inverse) square roots of real symmetric matrices is proposed. Simplicity of the algorithm, good numerical accuracy, short computational time and the possibility to solve the problem in an (n, n) array are some of its features; in addition a new theoretical view on this problem is presented.     

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.     

Efficient linear algebra routines for symmetric matrices stored in packed form

Quantum chemistry methods require various linear algebra routines for symmetric matrices, for example, diagonalization or Cholesky decomposition for positive matrices. We present a small set of these basic routines that are efficient and minimize memory requirements.     

对称群标准表示矩阵计算新方法

在多体问题的对称群方法中,群表示矩阵的计算是关键性问题。Young-Yamanouchi规则给出了标准正交表示的计算方法,然而该法相当繁琐,颇难使用,本文将2列Young表标准表示的计算方法[1]推广到任意不可约表示,给出对称群标准表示矩阵计算新方法.     

Graph generators

We consider the construction of highly symmetrical vertex transitive graphs. Some such graphs represent the degenerate rearrangements in which a molecule or an ion is formed by breaking and making bonds so that the final and the initial skeleton is identical. The approach is closely related to Cayley's graphs for selected groups. We restrict the choice of generators to symmetric matrices. Successive multiplications of such matrices generate other permutation matrices of the same dimension, each new matrix representing a new vertex for a transitive graph under the construction. In particular we restrict our discussion to matrices of dimension 3 and 4 and proceed to construct systematically all transitive graphs using 4 × 4 symmetric matrices as generators.     

The symmetric groups and the Waller–Hartree spin-free method

The relations between the Waller–Hartree spin-free method and the symmetric group theory are given. It is shown that the Gallup method is a special case of ours with S = M. Furthermore, all the irreducible representation matrices and other matrices needed are written explicitly in terms of Sanibel coefficients which makes the method more useful. However, it was shown that the cases with S ≠ M for the spin-free pure spin states might be beyond the power of the symmetric group theory.     

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     

Contraction algorithms for third-order reduced density matrices: Symmetric group approach

In this work, we present the mapping formulae for the contraction of the third-order reduced density matrices represented in the basis of the irreducible representations (IR) of the symmetric group S3 into the second-order ones which are represented in the basis of the IR of group S2. These algorithms, which can be useful in several fields, have been applied for the approximation of reduced density matrices within the spin-adapted reduced Hamiltonian theory. Some results obtained with this procedure are also presented.     

Equations-of-motion method: Calculation of the k lowest or highest solutions

The method of optimal relaxation to determine the eigenvalues of symmetric matrices, as proposed by Shavitt, has been adapted to solve the equation-of-motion problem. Matrices Z and Y are obtained by one diagonalization, while matrices A and B remain unchanged. This procedure is particularly useful for high-dimensional or nonorthogonal bases, if one needs only the lowest transition energies.     

The n-electron problem and matrices representing the symmetric groups

The application of the representation theory of symmetric groups to the solution of quantum chemical problems is discussed. This is done using matrices which do not represent S_n in the normal sense, but which are much simpler to calculate than the Yamanouchi-Kotani representation. A complete discussion of the calculation of these matrices is given.     

The Fix–Heiberger procedure for solving the generalized Ill-conditioned symmetric eigenvalue problem

We review a formalism introduced by Fix and Heiberger in 1972 for solving the generalized (or nonorthogonal) eigenvalue problem for ill-conditioned symmetric matrices and we discuss its application in quantum chemistry. A few examples dealing with the calculation of high Rydberg orbitals are presented.     

Matrix isolation infrared spectroscopic study of the interaction of CH3ReO3 with NH3

The matrix isolation technique has been combined with infrared spectroscopy to identify and characterize the product of the codeposition of CH3ReO3 with NH3 into inert matrices at 14 K. This codeposition led to the formation of the isolated 1:1 complex between these two reagents and its isolation in argon and nitrogen matrices. The complex is characterized by perturbations to all of the vibrational modes of the NH3 subunit in the complex, including a large, 185 cm(-1) blue shift of v2, the symmetric deformation mode. In addition, shifts of the -ReO3 antisymmetric stretch and -ReO3 symmetric bending of the CH3ReO3 subunit in the complex were observed. This complex, while predicted theoretically, has not been reported previously.     

OnPLS—a novel multiblock method for the modelling of predictive and orthogonal variation

This paper presents a new multiblock analysis method called OnPLS, a general extension of O2PLS to the multiblock case. The proposed method is equivalent to O2PLS in cases involving only two matrices, but generalises to cases involving more than two matrices without giving preference to any particular matrix: the method is fully symmetric. OnPLS extracts a minimal number of globally predictive components that exhibit maximal covariance and correlation. Furthermore, the method can be used to study orthogonal variation, i.e. local phenomena captured in the data that are specific to individual combinations of matrices or to individual matrices. The method's utility was demonstrated by its application to three synthetic data sets. It was shown that OnPLS affords a reduced number of globally predictive components and increased intercorrelations of scores, and that it greatly facilitates interpretation of the predictive model. Copyright © 2011 John Wiley & Sons, Ltd.     

A computational method for symmetry factorization without using symmetry operators

A computational method is suggested for symmetry factorization of symmetric matrices according to the symmetry properties of their basis. The method avoids the use of symmetry operators and is convenient for computer programming.     

Decomposition of the first-order reduced density matrix: an isopycnic localization treatment

In this work, we propose a partitioning of the first-order reduced density matrix corresponding to an N-electron system into first-order reduced density matrices associated with regions defined in the real space (regional matrices). The treatment is based on an isopycnic orbital localization transformation that provides regional matrices that are diagonalized by identical localized orbitals, having many attributes associated with chemical concepts (appropriate localization in space, high transferability, etc.). Although the obtained numerical values are similar to those arising from previous studies, their interpretation is more rigorous and the computational cost is much lower.