Point nondegenerate

Let us recall that in elementary catastrophe theory critical points of potential functions are examined. A potential function can have noncritical points, nondegenerate critical points and degenerate critical points. To degenerate critical points correspond sensitive states lying in the state variable and control parameter space in the catastrophe manifold M their... 

Consider the function at a certain set of R s.Rfl. where c()(r, R) is nondegenerate. When the nuclei move away from that point, and approach it back via a different route, the wave function must return to its original value. However, in... 

The second difference is the appearance of a doubly degenerate E symmetry species whose characters are not always either the - - 1 or — 1 that we have encountered in nondegenerate point groups. 

From the point of view of the present state of computational possibilities, an extension of open-shell methods to a and a + rr electronic systems is rather tempting. This extension is easy for the method of Longuet-Higgins and Pople e.g., the CNDO/2 method is amenable to radicals having one nondegenerate open shell if in the original terms for F matrix elements (66),... 

Returning to the example, the optimal point x = (4, 3) is a nondegenerate vertex because... 

In the case of k = C and Kx = 0, Xl" has a further nice structure. Suppose X has a holomorphic symplectic form u, i.e. u is an element in H X, which is nondegenerate at every point x [Pg.8]

A theorem due to Hohenberg and Kohn points to the central role of the electron density in representing the properties of a system. In 1964, Hohenberg and Kohn (1964) proved that the properties of a system with a nondegenerate ground state are unique functionals of the electron density. 

The first system to be discussed is ethene. Ethene is a closed-shell molecule with a point group that only includes nondegenerate irreducible representations. Its MCD spectrum therefore can only include terms. Despite this restriction, the ethene MCD spectrum is a useful testing ground and many of the insights obtained can be usefully apphed to other MCD calculations. 

Let (A, A) be a principally polarized abelian variety over an algebraically closed field k. If the characteristic of k is not equal to the prime p, then the kernel of multiplication by p on A(k) is a finite group isomorphic to (Z/pZ)29. The polarization A induces a nondegenerate alternating pairing A k)[p] x A( )[p] -+ pp(k). Hence, we can try to classify principally polarized abelian varieties with a symplectic basis for the group of points of order p. However, this no longer works in characteristic p. 

A chemist isolated an unknown transition metal complex with a formula of ABe. Five potential structures were considered, belonging to point groups C, Dih, D6h, Dlh, and Djj. Spectroscopic studies led to the conclusion that the p orbitals originating on A in the complex were completely nondegenerate. Sketch a structural formula that is consistent with each of the five point group assignments and decide which structures can be eliminated on the basis of the experimental results. 

A nondegenerate irrep that is symmetric with respect to the principl axis is denoted A, while B indicates antisymmetry with respect to this axis. In point groups with a horizontal plane of reflection, primes and " respectively indicate symmetry and antisymmetry with respect to the plane, while g and u indicate symmetry and antisymmetry with respect to inversion. For doubly degenerate irreps a subscript m indicates which spherical harmonics VJ, m form basis functions for that irrep. Numerical subscripts are used on nondegenerate irreps to distinguish them where necessary the numbers indicate the first of the vertical planes or perpendicular twofold axes (in the order specified in the character table) with respect to which the irrep is antisymmetric. 

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