Showing papers on "Antisymmetric relation published in 1975"

A Quantitative Assessment of „Through-space”︁ and „Through-bond”︁ Interactions. Application to Semi-empirical SCF Models

Abstract: The scheme of ‘through-space’ and ‘through-bond’ interaction of (semi)localized orbitals, originally proposed by Hoffmann, is reexamined in terms of SCF many-electron treatments. It is shown that the two types of interaction can be characterized by examining the corresponding off-diagonal matrix elements of the Hartree-Fock matrices of the localized or the symmetry adapted localized orbitals and of the partially diagonalized Hartree-Fock matrices referring to ‘precanonical orbitals’. The procedure outlined is applied to three practical examples using the semiempirical many-electron treatments SPINDO, MINDO/2 and CNDO/2: a A reassessment of ‘through-space’ and ‘through-bond’ interaction in norbornadiene indicates, that the latter type of interaction is also of importance for the orbital based mainly on the antisymmetric combination of the localized x-orbitals. The differences in the predictions derived from the three models are critically examined. b The competition between ‘through-space’ and ‘through-bond’ interaction in the series of bicyclic dienes from norbornadiene to bicyclo[4.2.2]-dcca-7,9-diene and in cyclohexa-1,4-diene, i. e. their dependence on the dihedral angle UI is reexamined. It is found that the rationalization for the orbital crossing near ω = 130° deduccd from PE. spectroscopic data can not be as simple as originally suggested and that the relay’ orbitals responsible for ‘through-bond interaction affecting both the symmetric and the antisymmetric combination of the π-orbitals extend over the whole CC-σ-system of the six membered ring. c ‘Through-bond’ interaction of the two lone pair orbitals in 1,4-diazabicyclo[2.2.2]octane is found to be large for their symmetric and the antisymmetric linear combination. The analysis quoted, draws attention to some of the dangers involved in using semiempirical treatments for the interpretation of PE. data in terms of Koopmans′ theorem, without due caution.

Interaction of waves with two-dimensional obstacles: a relation between the radiation and scattering problems

Abstract: A relation connecting the reflexion and transmission coefficients for scattering of water waves by a fixed body with the far-field radiated waves due to forced motions of the same body is derived Two alternative derivations are given, including a simple argument based on the analysis of an appropriate linear superposition of the two problems, and a more formal application of Green's theorem to the two potentials For bodies with horizontal symmetry, the transmission and reflexion coefficients are related to the phase angles of the far-field radiated waves associated with symmetric and antisymmetric forced motions of the body Some general conclusions follow for arbitrary symmetric bodies, and these are verified in specific cases by comparison with existing solutions The applicability of these relations to other types of wave problem is noted

Spin recoupling and n-electron matrix elements

Abstract: For n-electron systems with well defined total spin antisymmetric states are constructed by successively coupling the spins associated with each orbital. A second quantized scheme is used and the matrix elements of these states are expressed both for spin-independent and spin-dependent interactions in terms of recoupling coefficients of SU(2). The latter are evaluated to give very simple expressions. As a particular case a simple formula for the matrix elements of generators of U(N) for two-column partitions is obtained.

A simple class of U( N) Racah coefficients and their application

Abstract: Using permutation group techniques, a general expression is derived for the special class ofU(N) Racah coefficients for which the representations [f1] and [f3] in the recoupling matrix for [f1]×[f2]×[f3]→[f] are either both totally antisymmetric or both totally symmetric. For the totally antisymmetric case further specialization gives a simple expression for aU(N) Racah coefficient which is needed in taking the average of the product of operators over the states of an irreducible representation ofU(N), where this result can be useful in the study of identical fermion systems by spectral distribution methods.

The intrinsic properties of rank and nullity of the Lagrange bracket in the one dimensional calculus of variations

Abstract: This paper establishes the existence of symplectic structure in degenerate variational problems, i.e. problems whose full development involves a hierarchy of equations of constraint as well as various equations of motion. Any variational problem, degenerate or otherwise, may be called regular if the equations of the second variation provide a complete description of the infinitesimal relationships subsisting between any orbit and all its infinitesimal neighbour orbits. It is proved that Poincare’s conserved antisymmetric derived bilinear differential form in the orbit manifold of any regular degenerate problem admits no null vectors other than those which represent infinitesimal deviations due to indeterminacy in the evolution of the orbit. Conversely, it is shown how, given any continuous system of orbits endowed with a conserved antisymmetric closed bilinear differential form having this unique property of rank and nullity, one can construct at least one regular variational

The canonical form of an antisymmetric tensor and its application to the theory of superconductivity

Abstract: It is shown that a real or complex antisymmetric second order tensor transforming according to either the real orthogonal or the unitary group can always be put into a canonical form in which it appears as the direct sum of a zero matrix and of 2×2 antisymmetric matrices. This theorem is used to establish the necessary and sufficient relations satisfied by the contractions in the general Bogoliubov-Valatin transformation, and to put these contractions into a canonical form. The later result shows that the general Bogoliubov-Valatin transformation is always equivalent to a pairing in which each paired state is paired with only one other state.

The spin optimized separated pair method

Abstract: The separated pair method, for systems containing even and odd numbers of electrons, is generalized by allowing for contributions from all available spin coupled states of the same total spin quantum numbers S and M. The total wavefunction is expressed as a linear combination of products of a given spin eigenfunction, which is constructed out of singlet and triplet coupled pairs, and a suitable spatial function built out of symmetric (for singlet) and antisymmetric (for triplet) spatial pair functions. The electron and spin density functions are then presented in a ‘ sum of pair ’ densities form, so enabling the effects of spin optimization on the chemical bonding to be interpreted in a simple and obvious way. Finally, a discussion is given of the practical application of the spin optimized separated pair method.

Exchange Symmetry of the Effective Interaction in the Landau-Migdal Theory of Finite Fermi Systems

Abstract: We investigate the particle exchange symmetry of the phenomenological effective particle-hole interaction used in the theory of Finite Fermi Systems. We find that experimental data and consistency relations support an interaction which is antisymmetric at the surface and in the exterior region of the nucleus and nonsymmetric in the interior. Besides, we show that the number of adjustable parameters of the effective interaction can be reduced substantially by the use of generalized Ward identities without any loss of agreement with experimental data.

Aeroelastic Loads Predictions using Finite Element Aerodynamics

Abstract: Aerodynamic and structural influence coefficients are utilized to determine the load distributions, deflections, and trim parameters for a vehicle in quasi-static aeroelastic equilibrium. A matrix formulation is used to solve the various quasi-static aeroelastic problems. Nonlinearities in the aeroelastic trim equations are accounted for by an iteration of the classical closed form solution. Aerodynamic and structural idealizations are related by a surface spline transformation. Solutions are developed for symmetric, antisymmetric, and asymmetric load conditions on symmetric vehicles of general geometric shapes which may include both lifting surfaces and lifting bodies. INTRODUCTION With the advent of large order matrix solutions for the analysis of complex structures a need has arisen for a complementary approach to the external loads problem. Finite element structural analysis techniques demand that the external loads be distributed over the structure at discrete points. Therefore, shear, moment, and torque distributions along a psuedoelastic axis are no longer sufficient to define the external load distributions required by the stress analysis. A general finite element approach to the problem of the determination of aeroelastic loads on a flexible vehicle flying in a state of quasi-static equilibrium is presented here. The vector point loads available from this solution are directly applicable to matrix structural analyses. Structural and aerodynamic influence coefficients obtained from finite element idealizations of the aircraft are utilized as a basis for the method. The technique is primarily an extension of the method first suggested by ~ e d m a n ( l ) * and later generalized by odde en.(^) This work extends the efforts of the above-mentioned authors by * Numbers in parentheses designate References at end of paper.

Spin-coupled wavefunctions

Abstract: A new method of formulating the construction of spin-coupled wave-functions, in which each electron is ascribed to a different spatial function, is described. The direct use of the matrix representatives, ∪(P), of the permutation operators P, normally required in constructing antisymmetric wavefunctions, is avoided. Instead, antisymmetrizing operators are introduced as matrices, in such a way as to considerably reduce the amount of data required to characterize the spin-state, thereby increasing the computational efficiency of the method. The theory is applied to the three-electron systems Li, HeH and LiH+ using a simple one-configuration model, in which the orbitals are written in elliptical coordinates, and the non-linear parameters are optimized numerically.

Erratum: Resonating-group study of N+ 16 O scattering with a generator-coordinate technique

Abstract: The scattering of nucleons by $sup 16$O is studied with the resonating- group method. The $sup 16$O wave function used is a translationally invariant antisymmetrized product of single-particle wave functions of (1s)$sup 4$(1p)$sup 12$ configuration in a harmonic-oscillator well having an appropriately chosen width parameter. A generator-coordinate technique is employed to facilitate the computation of the nonlocal interaction between the nucleon and the $sup 16$O nucleus. This technique is discussed in some detail in order to demonstrate its utility in a nuclear problem involving a relatively large number of nucleons. Calculated results are compared with experimental data over a wide range of energies, and it is found that the agreement is in general quite satisfactory. Also, the result shows that in this particular problem the heavy-particle pickup process, which is properly taken into consideration by the use of a totally antisymmetric wave function, has a relatively small contribution. (AIP)

Boundary properties of functions harmonic in a strip

Abstract: We study in the Lp-norm, 1≤p≤∞, the boundary properties of the solution to the Dirichlet problem for the stripA ={(x, y):−∞ 0} and its dependence on the structural properties of the given boundary values (symmetric, antisymmetric). In particular, for the case of symmetric boundary values we obtain direct and inverse theorems on approximation in terms of the general modulus of continuity of second order.

Complementary properties of binary relations

Abstract: After defining the complementary relation R of a binary relation R on a set X, this paper constructs the binary relation C (‘is a complementary property of’) on the set P of nine well known elementary properties that R might possess. It deduces some theorems about C; especially that symmetry is the only one of these possible properties of R on X that is possessed by C on P. The set P may be enlarged to contain other elementary properties of R on X without changing the truth of these theorems, when the symbols of sets are properly modified. Finally, the paper discusses the desirability of a general theory of elementary properties of binary relations for the further development of statistical decision theory.

Free surface effects on the side force and moment acting on a ship hull with a drift angle

Abstract: The purpose of this research is to study the free surface effects on the side force and the yaw moment acting on a ship hull advancing with a steady drift angle. First, on the assumptions of thin ship, small drift angle and low wave height, the problem was divided into a symmetric and an antisymmetric one. Since the symmetric term had no effects on this problem, for the antisymmetric problem, approximate formulas to estimate the free surface effects were obtained on the assumption of slenderness. Next, by means of model experiment, free surface effects were examined, and it was ascertained that the above approximate formulas could explain the experiment results on the whole.

Hydrodynamics in weak gravitational fields small oscillations of an ideal liquid in a cylindrical vessel

Abstract: The problem of determining the frequencies and forms of small natural oscillations of an ideal liquid in a cylindrical vessel under conditions close to weightlessness is examined. It is assumed that a weak homogeneous gravitational field acts parallel to the vertical generatrix forming the cylinder. In contrast to [1], where only the first antisymmetric oscillation frequency is found for a semiinfinite cylindrical vessel, the frequencies of several axiosymmetric, antisymmetric, etc. oscillations are obtained as functions of the gravitational-field intensity and other parameters of the problem. The Ritz method is employed for two different variations of the problem, equivalent to that of oscillations of an ideal liquid under conditions of weightlessness [1–5].

A quantitative assessment of ′through-space′ and ′through-bond′ interactions, application to semi-empirical scf models

Abstract: The scheme of ‘through-space’ and ‘through-bond’ interaction of (semi)localized orbitals, originally proposed by Hoffmann, is reexamined in terms of SCF many-electron treatments. It is shown that the two types of interaction can be characterized by examining the corresponding off-diagonal matrix elements of the Hartree-Fock matrices of the localized or the symmetry adapted localized orbitals and of the partially diagonalized Hartree-Fock matrices referring to ‘precanonical orbitals’. The procedure outlined is applied to three practical examples using the semiempirical many-electron treatments SPINDO, MINDO/2 and CNDO/2: a A reassessment of ‘through-space’ and ‘through-bond’ interaction in norbornadiene indicates, that the latter type of interaction is also of importance for the orbital based mainly on the antisymmetric combination of the localized x-orbitals. The differences in the predictions derived from the three models are critically examined. b The competition between ‘through-space’ and ‘through-bond’ interaction in the series of bicyclic dienes from norbornadiene to bicyclo[4.2.2]-dcca-7,9-diene and in cyclohexa-1,4-diene, i. e. their dependence on the dihedral angle UI is reexamined. It is found that the rationalization for the orbital crossing near ω = 130° deduccd from PE. spectroscopic data can not be as simple as originally suggested and that the relay’ orbitals responsible for ‘through-bond interaction affecting both the symmetric and the antisymmetric combination of the π-orbitals extend over the whole CC-σ-system of the six membered ring. c ‘Through-bond’ interaction of the two lone pair orbitals in 1,4-diazabicyclo[2.2.2]octane is found to be large for their symmetric and the antisymmetric linear combination. The analysis quoted, draws attention to some of the dangers involved in using semiempirical treatments for the interpretation of PE. data in terms of Koopmans′ theorem, without due caution.

On the use of a finite difference method for solving anisotropic scattering problems

Abstract: A new method of solving the radiative transfer equation is developed in which the scattering and absorption coefficients may have arbitrary variations with depth, and in which both internal (thermal) emission and incident radiation are allowed. Specular and diffuse reflection at both boundaries also is taken into account. The method begins by forming a paired set of coupled first-order differential equations for the symmetric and antisymmetric parts of the radiation field after writing the scattering integral as a numerical quadrature. These differential equations are broken into finite difference form, in which the symmetric and antisymmetric parts of the radiation field are found on alternate grid points. Numerical results for a number of test problems are shown, demonstrating that the method is very fast, that it returns specific intensities and fluxes that are accurate to at least a percent, and that it can be applied to optically thick problems.