Orthonormality Constraints Research Articles

Polyhedra Identification Using the Orthonormal Constraints of Rotation Matrix

AbstractA method to identify polyhedric objects from single perspective line drawings is proposed. The method searches correct associations from the model database in four stages: 1) topological matching of vertex and edge connection; 2) the computation of the transformation matrix from the model coordinate system to the camera coordinate system; 3) the validity test of transformation matrix; and 4) the determination of position and orientation of object. Stage 2) computes the optimal solution of the matrix using nonlinear optimization. Thus, if the initial value is improper, the solution does not always converge to a correct solution as pointed out in previous research. Orthonormal constraints of transformation matrix are introduced to improve the convergence property of numerical computation and to reduce computation cost. Experimental results show the validity of the method.

Multiconfiguration-self-consistent field (MC-SCF) method for excited states

A method is derived to calculate the MC-SCF wavefunctions for the various excited states of a given symmetry. The appropriate orthonormality constraints are taken into account in the construction of the generalized Fock operators. A simplification of the two by two rotation method is suggested for deriving the SCF orbitals. The method is applied to the calculation of the wavefunctions for the first three states of the hydrogen molecule with 1Σu + symmetry.

A quadratically convergent multiconfiguration–self-consistent field method with simultaneous optimization of orbitals and CI coefficients

A quadratically convergent MC–SCF procedure is described which is based on the direct minimization of the energy. In comparison to the Newton–Raphson technique, which has previously been applied by several authors for orbital optimization, the convergence radius is much improved by taking into account in the energy expansion those parts of third and higher order terms which account exactly for the orthonormality constraints imposed on the orbitals. The nonlinear equations which define the improved orbitals are solved iteratively by a simple adaption of the Gauss–Seidel method. The coefficients of the configuration expansion can be optimized simultaneously with the orbitals, a necessary requirement for over-all quadratic convergence. The removal of redundant variables as well as useful approximations for the optimization of core orbitals are discussed. The convergence of the method is demonstrated to be much superior to classical Fock operator techniques and MC–SCF methods which are based on the generalized Brillouin theorem. The formalism is carried down to matrix operations and shows a simple structure.

Variational equations for the solution of the Hartree-Fock problem with angular momentum projection before the variation

A variational principle is used to determine the optimal angular momentum projected one determinant approach to theN-nucleon yrast-wave function for a given total spin value. The solution is given in terms of a set of coupled nonlinear equations. Besides an orthonormality constraint for the occupied orbits and a normalization condition for the total wave function, this set consists out of a matrix equation taking care of the fact that the spin-projected wave function does not depend on the orientation of the intrinsic determinant it is based on, and a second subset of equations, which can be considered as a Thouless theorem for the spin-projectedN-nucleon state, and describes the diagonalization of the total Hamiltonian in the subspace of linear independentN-nucleon shell model configurations contained in the test-determinant. Furthermore, a numerical method for the solution of these equations is proposed and an extension of the theory for the description of excited bands is given. Finally, the consistency of the equations is checked by solving them for a simple example analytically.

Quantum dynamics of relativistic massive strings

The string is defined here as a relativistic generalization of the classical massive inhomogeneous string. A covariant Hamiltonian method is used for quantization. It is shown that the orthonormal parametrization constraint can be used only when the string is homogeneous, while a special transverse gauge type of covariant constraint can be used always and it gives ${a}_{n}^{0}|m,\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0,s〉=0$ in the rest frame of reference. This condition ensures the positivity of norms of states with ${m}^{2}>0$ and ${p}^{0}>0$. The origins of difficulties with present relativistic quantum theories of the string are pointed out and also resolved. In addition, we study quantum electrodynamics of the string.

The convergence properties of direct energy minimisation with respect to linear coefficients in the MC-LCAO-MO-SCF approach

It is shown that in the LCAO-MO-MC-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner suggested by Kari and Sutcliffe or indeed by any similar method then the Hessian of the problem with respect to the linear coefficients is singular. The nature of this singularity is analysed and it is shown that in general it is possible to remove it in a level-shifting-like scheme, but that only in certain special cases is this procedure likely to be quickly convergent.

The convergence properties of direct methods of energy minimization with respect to linear coefficients in the LCAO-MO-SCF approach

It is shown that in the LCAO-MO-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner first suggested by Fletcher, then the Hessian of the problem is singular. It is suggested that this singularity may well account for the slow convergence observed using direct energy minimization methods to solve the SCF problem. Ways of avoiding the consequences of this singularity are discussed.