Kinetic Energy Matrices Research Articles

Dynamical effects of QCD in q2 $ \bar{{q}}^{{2}}_{}$ systems

We study the coupling of a tetraquark system to an exchanged meson-meson channel, using a pure gluonic theory based four-quark potential {\em matrix} model which is known to fit well a large number of data points for lattice simulations of different geometries of a four-quark system. We find that if this minimal-area-based potential matrix replaces the earlier used simple Gaussian form for the gluon field overlap factor $f$ in its off-diagonal terms, the resulting $T$-matrix and phase shifts develop an angle dependence whose partial wave analysis reveals $D$ wave and higher angular momentum components in it. In addition to the obvious implications of this result for the meson-meson scattering, this new feature indicates the possibility of orbital excitations influencing properties of meson-meson molecules through a polarization potential. We have used a formalism of the resonating group method, treated kinetic energy and overlap matrices on model of the potential matrix, but decoupled the resulting complicated integral equations through the Born approximation. In this exploratory study we have used a quadratic confinement and not included the spin-dependence; we also used the approximation of equal constituent quark masses.

Spectral properties of linear gyroscopic systems

We study how the natural frequencies of a linear gyroscopic system behave under variations in the kinetic and potential energy matrices as well as in the gyroscopic force matrix.

CH-stretching overtone spectra of a fast rotating methyl group. I. Toluene C6D5CH3

The CH-stretching overtone spectra of the methyl group in gaseous toluene C6D5CH3 are recorded with conventional Fourier transform near infrared spectroscopy in the ΔvCH=1–4 regions and by intracavity laser photoacoustic spectroscopy in the ΔvCH=5 and 6 regions. All spectra exhibit a complex structure. They are analyzed with a theoretical model which takes into account, within the adiabatic approximation, the coupling of the anharmonic CH stretch vibrations, described by Morse potentials, with the quasifree internal rotation of the methyl group and with isoenergetic combination states involving the six angle deformation modes of the methyl group. Using uniquely determined canonical potential-energy and kinetic-energy matrices allows us to solve the problem of indeterminacy caused by the description of molecular vibrations in such redundant coordinates. A set of Fermi resonance parameters and their variation with the internal rotation coordinate are determined from the fitting of the ΔvCH=1–6 overtone spectra. At Δv=1 and 2, the vibrational energy is expressed in a normal mode basis. Fermi resonance phenomena involving mainly the HCH bending modes lead to strong intramolecular vibrational energy redistribution. At this energy, a Coriolis coupling through internal rotation, which gives rise to a characteristic asymmetric top vibration-rotation profile, further perturbs the vibrational modes perpendicular to the rotation axis. From the second overtone (Δv=3), the vibrational energy is described in a local mode basis and perturbations due to Fermi resonance vanish progressively up to Δv=5. At Δv=6, a strong spectral perturbation is again observed. But, the efficient Fermi resonance phenomena are now essentially related to combination states involving HCH and HCC bending combination modes. This simple calculation successfully describes the relative intensity and frequency of each peak within a given overtone.

Unambiguous formalism of molecular vibrations: Use of redundant coordinates and canonical matrices

This paper introduces a new approach to the problem of describing molecular vibrations in redundant valence coordinates. A consistent definition for canonical force field, which is different of that by Kuczera and Pupyshev et al., is outlined. This new definition is also generalized to other matrices like kinetic-energy matrices and transformation matrices between different sets of vibrational coordinates. The importance of such canonical matrices follows from the fact that they are uniquely determined and allow us to consider the redundant coordinates as though they were independent.

Vector parametrization of the N-atom problem in quantum mechanics with non-orthogonal coordinates

This article aims to present a general method that enables one to build kinetic energy matrices in getting rid, for the angular coordinates (internal and Eulerian), of the heaviness of differential calculus (for expressing kinetic energy operators) and numerical integration (for calculating matrix elements). Therefore, instead of 3N-3 coordinates, only N-1 radial distances are to be treated as coordinates. In the present formulation, the system is described by any set of n vectors {Ri, i=1, ..., n} and the kinetic energy operator is expressed in term of (n-1) angular momenta {Li, i=1, ..., n-1} and the total angular momentum J. The formalism proposed is general and gives a remarkably compact expression of the kinetic energy in terms of the angular momenta. This expression allows one to circumvent the seeming angular singularities.

Basis set dependence of the locality of the kinetic energy operator

We have analyzed the locality of matrices of the kinetic energy operator in even-tempered Gaussian and Slater-type (STO) bases, as well as in several three-dimensional, one-dimensional, and pseudo-one-dimensional basis sets of orthogonal polynomials and trigonometric functions. We find that the locality of the kinetic energy matrix in Gaussian bases is dependent upon the basis function parameters, while STO bases are found to always produce kinetic energy matrices which are very nearly equal to the matrices of local operators. In addition, it is observed that in the limit of completeness, the locality measure for all of the one-dimensional basis sets appears to converge to a common value. In the case of a basis set of particle on a ring eigenfunctions, an exact value for this limit has been determined analytically. Three-dimensional and l=0 pseudo-one-dimensional basis sets of orthogonal polynomials give kinetic energy matrices that do not behave similarly in the limit of completeness. It is found that all of these basis sets result in kinetic energy matrices which clearly exhibit nonlocal behavior, except for those involving exponentials. Such bases, like the STO bases, appear always to yield kinetic energy matrices that remain nearly local regardless of the basis function parameters and the number of basis functions used.

Kinetic energy matrices in a basis of equidensity orbitals

The matrix of the kinetic energy operator can be divided into two components, one of which is equivalent to the matrix of a function so that it is effectively local. This decomposition is basis-set dependent and is particularly simple when equidensity orbitals are used. It is shown that for a one-dimensional problem the norm of the local component is √5/3 times the norm of the whole kinetic energy matrix, independent of the density used to define the orbitals. In the three-dimensional case this ratio depends on the density but reasonably simple expressions are obtained.

Band structure of holes in p-type delta -doping quantum wells and superlattices.

p-type \ensuremath{\delta}-doping quantum wells and superlattices are semiconductor systems of considerable interest for basic research and device applications. In this paper, a method for calculating potentials and band structures of such systems is developed. The method relies on a plane-wave expansion of the multiband effective-mass equation, uses kinetic energy matrices of any size, and takes exchange correlation into account in a more rigorous way than this was done before. The method is used to calculate potential profiles, subband and miniband structures, as well as Fermi level positions for a series of p-type \ensuremath{\delta}-doping quantum wells and superlattices. Exchange-correlation effects turn out to be rather large. Only if they are properly taken into account reasonable agreement with experimental photoluminescence data can be achieved. For comparison, potentials and energy levels are also calculated for electrons of n-type \ensuremath{\delta}-doping systems. The potential wells for electrons are considerably deeper and wider, and exchange-correlation effects are less pronounced than for holes. The physical reasons for these differences and their implications on luminescence spectra from n- and p-type \ensuremath{\delta}-doping structures are discussed. \textcopyright{} 1996 The American Physical Society.

Linear dependencies among basis set products and near locality of some nonlocal operators

Linear dependencies among basis-set products and the decomposition of the matrices of some one-electron operators into local and nonlocal components have been investigated for several common quantum chemical basis sets and for a harmonic oscillator basis. For the first ten atoms and some simple diatomics, the kinetic energy, Fock, and density matrices were investigated. It was found that, for the basis sets used, these operators are all nearly local, in the sense that their matrices could be reproduced as the matrices of simple multiplicative-function operators, even though there are significant numbers of linear dependencies among the products of the basis functions. SCF eigenfunctions for these systems were found to have no linear dependencies among products. Basis sets of one-dimensional harmonic oscillator eigenfunctions were found to give a kinetic energy matrix that is not nearly local.

Spline collocation methods for calculating orbital energies

The accuracy of the spline collocation method for solving molecular orbital equations is tested for the hydrogen atom. Two different types of grids are used that concentrate points near the nucleus. A 40 point grid defined by a polynomial transformation, which is suitable for molecular calculations, is found to produce an accuracy within a mHartree of the exact value. For many electron systems, the authors have previously employed an algorithm for solving Poisson's equation which uses the eigenvectors of the kinetic energy matrices as a basis. In order to ensure the accuracy of the eigenvalues and eigenvectors of the kinetic energy, a method is implemented using scaled splines that leads to kinetic energy matrices having a simple block structure. The eigenvalues of the kinetic energy matrices in this scheme are quite accurate.

Summation of the Born series for the normal modes of the Earth

SUMMARY We derive an explicit formula for the response of a laterally heterogeneous, self-gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long-period spectrum at a single station is given by u(ω) =RH0 [V(ω) +ωW−ω2T]−1·S0(ω) where R0 and S0(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω2T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.

Path-integral simulation of positronium in a hard sphere.

We employ the path-integral Monte Carlo method to study the binding energy of positronium trapped within a hard sphere. The two-body density matrix and the total- and kinetic-energy matrices are obtained by matrix squaring. The pair-product approximation is then used to obtain the entire density and estimator matrices for the Monte Carlo simulation. Ground-state binding energies are obtained as a function of sphere radius and the system is found to become unbound near a radius of 4 bohrs. Annihilation times are calculated which vary from 125 ps for free-space positronium down to 76 ps at a sphere radius of 4 bohrs.

Coupled free oscillations of an aspherical, dissipative, rotating Earth: Galerkin theory

Variational theory based on self‐adjoint equations of motion cannot fully represent the interaction of the earth's seismic free oscillations in the presence of lateral structure, attenuation, and rotation. The more general Galerkin procedure can model correctly the frequencies and attenuation rates of hybrid oscillations. Implementation of either algorithm leads to a generalized matrix eigenvalue problem in which the potential and kinetic energy interactions are separated into distinct matrices. The interaction of the earth's seismic free oscillations due to aspherical structure, attenuation, and rotation is best treated as a matrix eigenvalue problem. The presence of attenuation causes the matrices to be non‐Hermitian and requires the use of a general Galerkin procedure. Physical dispersion, represented as a logarithmic function in frequency, must be represented by a truncated Taylor series about a fiducial frequency in order to be incorporated in the Galerkin formalism in a numerically tractable manner. The earth's rotation introduces an interaction matrix distinct from the potential and kinetic energy matrices, leading to a quadratic eigenvalue problem. A simple approximation leads to an eigenvalue problem linear in squared frequency. Tests show that this approximation is accurate for calculations using modes of frequencies f ≳ 1 mHz, unless interaction across a wide frequency band is modeled. Hybrid oscillation particle motions are represented by matrix eigenvectors that can be significantly nonorthogonal. The degrees of freedom in the low‐frequency seismic system remain distinct, since source excitation is calculated by using dual eigenvectors. Synthetic seismograms that are constructed from Galerkin coupling calculations without reference to this eigenvector nonorthogonality can be disastrously noncausal.

Structure of spontaneously broken local gauge supersymmetry

The structure of gauge supersymmetry (with field equations R AB = λg AB ) before and after spontaneous breakdown is analysed. The gauge group of the vacuum metric, after the breakdown is examined. For real gravitational interactions, a metric is constructed which is covariant with respect to Einstein and local vierbein transformations. For the case λ ≠ 0 cancellation of all cosmological terms at the tree level in the gravitational equations that arise from the Bose-Bose sector, R μν = λg μν , and the Bose-Fermi sector, R μα = λg μα , of the superspace equations is exhibited. The field equations arising from the two sectors are shown to be consistent. For the general case the relationship of the broken gauge and internal symmetry, fictitious Goldstone particles (Goldstone bosons as well as germions) and mass growth for high-spin particles (such as spin 3 2 , spin 2 etc.) is discussed. An expansion of the Lagrangian quadratic in fields (e.g. kinetic energy and mass matrices) needed for diagonalization of the theory is given. Details of superspace Riemannian geometry are discussed.

Generalization of the rayleigh theorem to gyroscopic systems: PMM vol. 40, n≗ 4, 1976, pp. 606–610

The Rayleigh theorem on the properties of the spectrum of a linear conservative mechanical system is generalized to embrace the gyroscopic systems, i.e. to the case in which the equations of motion contain, in addition to the kinetic and potential energy matrices, an arbitrary skew-symmetric matrix of gyroscopic forces.

ChemInform Abstract: MOLECULAR DYNAMICS, ROOT‐MEAN‐SQUARE AMPLITUDES, STATISTICAL THERMODYNAMICS, AND MOLECULAR POLARIZABILITY FOR THE ISOTOPIC SPECIES OF DIOXYGEN MONOFLUORIDE

A brief survey of vibrational spectral studies for the four isotopic species of dioxygen monofluoride has been made. On the basis of group theoretical considerations, symmetry coordinates have been constructed and kinetic energy matrices (orG matrix elements), potential energy matrices, and secular equations have been derived to calculate the valence force constants. The mean-square amplitudes and root-mean-square amplitudes for both the bonded and nonbonded atom pairs have been calculated at the room temperature. On the basis of a rigid rotator and harmonic oscillator model, enthalpy function, free enthalpy function, entropy, and heat capacity have been calculated from 200 to 2000 °K for all the four isotopic species. On the basis of a delta-function potential model based on the variational method and delta-function electronic wave functions, the bond parallel components, the bond perpendicular components, the contribution by the nonbonding electrons, and the average molecular polarizability have been calculated. The results obtained from these studies clearly confirm a double bond character for the oxygen—oxygen distance and a bond order of less than one-half for the oxygen—fluorine distance. The results have been discussed in relation to the nature of the two characteristic bonds involved in this molecular system.

Molecular dynamics, root-mean-square amplitudes, statistical thermodynamics, and molecular polarizability for the isotopic species of dioxygen monofluoride

A brief survey of vibrational spectral studies for the four isotopic species of dioxygen monofluoride has been made. On the basis of group theoretical considerations, symmetry coordinates have been constructed and kinetic energy matrices (orG matrix elements), potential energy matrices, and secular equations have been derived to calculate the valence force constants. The mean-square amplitudes and root-mean-square amplitudes for both the bonded and nonbonded atom pairs have been calculated at the room temperature. On the basis of a rigid rotator and harmonic oscillator model, enthalpy function, free enthalpy function, entropy, and heat capacity have been calculated from 200 to 2000 °K for all the four isotopic species. On the basis of a delta-function potential model based on the variational method and delta-function electronic wave functions, the bond parallel components, the bond perpendicular components, the contribution by the nonbonding electrons, and the average molecular polarizability have been calculated. The results obtained from these studies clearly confirm a double bond character for the oxygen—oxygen distance and a bond order of less than one-half for the oxygen—fluorine distance. The results have been discussed in relation to the nature of the two characteristic bonds involved in this molecular system.

On a new method for the calculation of dipole moment derivatives from infrared intensities

The recent method of Girljavallabhan, Joseph and Venkateswarlu for the calculation of dipole moment derivatives from infrared intensities is unsatisfactory because their matrix of intensity constants is not generally diagonalized by the normal coordinate transformation that diagonalizes the potential and kinetic energy matrices.

The normal coordinates of nitromethane-like molecules

The form of the vibrational kinetic and potential energy matrices for molecules with structures similar to nitromethane is discussed, and the kinetic energy matrix elements tabulated. Symmetry aspects of the vibrational problem are discussed in terms of the group defined by Longuet-Higgins. The possible forms of the normal coordinates are derived, and their relevance to the associated internal rotation problem indicated.

Normal co-ordinate analysis of 1–4 trans polybutadiene

The frequencies of the active fundamental vibrations of 1–4 trans polybutadiene and of its symmetrical deutero-derivatives, viz. (CH 2CDCDCH 2) m , (CD 2CHCHCD 2) n and (CD 2CDCDCD 2) m , have been calculated by means of valence force constants derived from cyclo-olefins. The method for obtaining the kinetic and potential energy matrices for an isolated polymer chain is briefly discussed, and compared with a procedure previously presented in the literature. A detailed discussion of the i.r. spectrum of 1–4 trans polybutadiene, based on normal co-ordinate analysis, is given.