Quadratic Coordination

The existence of MX more stable in the point-group than in T has been consider- 

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Within the frame of crystal field theory it is possible to derive additional information on the coordination number x of CuLx polyhedra from the quotient 4g,i/4igj. which attains the values 4 for octahedral. 5.4 for quadratic pyramidal and 6.9 for quadratic coordinations [5]. Further discrimination between the ligands H2O and OH- is based on the reasonable assumption that g [Cu( "Si-O ) aH20 ] > g, [Cu( sSl-0-) aOH]. ... 

We suppose that the planar quadratic coordination of the palladium(II) chelates allow CO2 molecules or other small molecules like traces of the synthesis solvent to be coordinated to the Pd(II) centre and therefore change the polarity /solubility of the chelate. Voluminous and flexibel residual ligand substitutes protect the central palladium ion against interaction of small molecules and we can observe a drastic increase of solubility. Otherwise, if the coordination sphere of the chelate centre ion is octahedral or tetrahedral, other molecules (e.g. the solvent) cannot interact with the metal centre easily. 

For a quadratic surface, the gradient vector is a linear function of the coordinates. An alternative way of using... 

Baker J, KInghorn D and Pulay P 1999 Geometry optimization In delocalized Internal coordinates An efficient quadratically scaling algorithm for large molecules J. Chem. Phys. 110 4986... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

An example that is closely related to organic photochemishy is the x e case [70]. A doubly degenerate E term is the ground or excited state of any polyatomic system that has at least one axis of symmetry of not less than third order. It may be shown [70] that if the quadratic tenn in Eq, (17) is neglected, the potential surface becomes a moat around the degeneracy, sometimes called Mexican hat, The polar coordinates p and <(>, shown in Figure 20, can be used to write an expression for the energy ... 

On the basis of tbe above analyses, it follows that them is no need to compute multidimensional potential surfaces if one wishes to handle the R-T effect in the framework of the model proposed. In spite of that, such computations were carried out in [152] in order to demonstrate the reliability of the model for handling the R-T effect and to estimate the range in which it can safely be applied in its lowest order (quadratic) approximation. The 3D potential surfaces involving the variation of the bending coordinates pi, p2 and the relative azimuth angle 7 = 4 2 1 were computed for both component of the state. 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

SHAPE. Gives the shape functions in terms of local coordinates for bi-linear or bi-quadratic quadrilateral elements. 

We have found the principal axes from the equation of motion in an arbitrary coordinate system by means of a similarity transformation S KS (Chapter 2) on the coefficient matrix for the quadratic containing the mixed terms... 

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