Use Of Normal Coordinates Research Articles

Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator.

We present the first use of curvilinear vibrational coordinates, specifically polyspherical coordinates, in combination with vibrational coupled cluster theory. The polyspherical coordinates are used in the context of both the adaptive density-guided approach to potential energy surface construction and in the subsequent vibrational coupled cluster calculations of anharmonic vibrational states. Results obtained based on the polyspherical coordinate parametrization are compared to results obtained with the use of rectilinear vibrational coordinates, namely, normal coordinates and hybrid optimized and localized coordinates for the formaldehyde molecule. This comparison is carried out with the full vibrational configuration interaction model, using the respective fully coupled potential energy surfaces and untruncated kinetic energy operators. The polyspherical coordinates are shown to facilitate an acceleration of convergence for truncated methods when compared to the use of normal coordinates. We furthermore report on calculations on the hydrogen peroxide molecule in the polyspherical coordinate parametrization. The polyspherical vibrational coordinates are shown to perform very well, even for truncated methods, especially when considering the difficulty that rectilinear vibrational coordinates can exhibit in treating complicated internal molecular motion.

Multidimensional anharmonic calculation of the vibrational frequencies and intensities for the trans and cis isomers of HONO with the use of normal coordinates

The equilibrium geometry and the potential energy and dipole moment surfaces have been determined for the cis and trans isomers of the HONO molecule by an ab initio Moller–Plesset (MP2) calculation with a wide set of atomic orbitals. The multidimensional anharmonic vibrational Schrodinger equations are solved using the variational method with the Hamiltonian and wave functions written in the normal coordinates of cis and trans isomers. All one- and two-dimensional and a number of three-dimensional vibrational problems are solved to obtain the energy levels and vibrational eigenfunctions. The frequencies and intensities for the fundamental, overtone and some combination bands are determined in good agreement with the available experimental results. The calculation shows the strength of coupling between different vibrational modes and reveals the presence of strong resonances between the ( v 1, v 3, v 6) and ( v 1, v 3−1, v 6+2) states of cis-HONO. This fact may be important for understanding the energy redistribution between the intermolecular degrees of freedom. The magnitude and direction of vibrationally averaged ground-state dipole moment of both isomers, as well as the direction of transition dipole moments, are in good agreement with the experimental findings. The changes in the values of dipole moment and some geometrical parameters of cis- and trans-HONO on vibrational excitation are also computed.

Radial gauge in Poincaré gauge field theories

The generalization of the Fock–Schwinger or ‘‘radial’’ gauge condition xiAi =0 to the gauge theories of the Poincaré group, which describe the gravitational field, is treated. It is shown that the choice of a radial gauge is equivalent to the use of normal coordinates and of tetrads parallel transported along autoparallel lines starting at the origin. The formulas that give the fields in the radial gauge starting from an arbitrary gauge and the formulas that give the gauge potentials in terms of the gauge field strengths are derived. The residual gauge freedom, which consists of the arbitrariness in the choice of the origin of the coordinate system and a tetrad of orthonormal vectors at this point is discussed in detail. It is the analog of the usual Poincaré invariance in flat space-time theories. The whole treatment can be extended to gauge theories of the affine and Euclidean groups. As an application, some properties of the homogeneous and isotropic states with random geometric fields are found.

Use of Normal Coordinates in Nuclear Physics

Use of Normal Coordinates in Nuclear Physics

On the Use of Normal Coordinates for the Solution of Lumped Parameter Transient Heat-Transfer Problems

T USE of normal coordinates in solving lumped parameter forced vibration problems has been a valuable asset to the vibration analyst. The same method may be applied to the lumped parameter transient heat-transfer problem. Standard digital computer programs are available which will solve the matrix eigenvalue problem. Using a method such as the serial Jacobi method,' 2 the eigenvalues and eigenvectors of a 45 X 45 matrix can be obtained in about 6 minutes on an IBM 704 computer. This report shows how the matrix eigenvalue problem for heat transfer is set up and how the eigenvalues and eigenvectors are used to construct the transient solution.

Further Developments of New Methods in Heat-Flow Analysis

SUMMARY Lagrangian methods in heat-flow problems and transport phenomena were introduced by the writer in some previous work. The present paper develops further one particular aspect of the method,-i.e., the elimination of “ignorable coordinates.” This is accomplished by a special choice of generalized coordinates, each of which is constituted by an arbitrary temperature distribution and an “associated flow field.” The latter is a vector field which is derived from the corresponding scalar field by a variational method. The procedure is valid for a certain class of nonlinear problems, provided we replace the temperature by the heat content as the unknown. It is shown that for normal coordinates derivation of the associated flow field is immediate. The use of normal coordinates and their associated flow fields is illustrated by an example. Introduction of Dirac functions and associated flow fields yields a procedure which constitutes a generalization of the classical formulation by Green’s functions and integral equations. This is illustrated by application to onedimensional problems of heating of a homogeneous or composite slab and directly verified by classical methods in the Appendix. N A PREVIOUS PUBLICATION~~~ haveintroducednew methods in heat-flow analysis. These methods have a two-fold basis; first, a new formulation of the thermodynamics of irreversible processes, and, second, the application of Lagrangian techniques to the mathematical analysis itself. The earlier developments were carried out in references 2, 3, and 4. We have recently reviewed in more detail in reference 5 the basic thermodynamic concepts and principles. The formulation of dissipative phenomena into a variational language can be achieved in many ways.$ The particular method which we have chosen is different from the classical variational approaches. It appears to be the most general and fits into a unified thermodynamic theory embracing a large category of physical phenomena, which leads to equations of the same type as in Lagrangian mechanics. A particularly useful concept which has been introduced is the generalized thermodynamic force by a @inciple of virtual work. In the case of thermal problems this results in a representation of the physical system by means of a vectorial flow field and to the use of a generalized thermal force to represent the externally applied tem