The Noncommutative Quadratic Stark Effect For The H-Atom

Using both the second order correction of perturbation theory and the exact computation due to Dalgarno-Lewis, we compute the second order noncommutative Stark effect,i.e., shifts in the ground state energy of the hydrogen atom in the noncommutative space in an external electric field. As a side result we also obtain a sum rule for the mean oscillator strength. The energy shift at the lowest order is quadratic in both the electric field and the noncommutative parameter $\theta$. As a result of noncommutative effects the total polarizability of the ground state is no longer diagonal.Comment: 7 pages, no figure

The Electronic Origin of Geometrical Deformations in Cyclohexadienyl and Cyclobutenyl Transition Metal Complexes

(40) (a) Davies4' has stated the opinion that ". . . the comparatlve success of London's method for aromatic hydrocarbons may be attributed to the dependence of the theoretical anisotropy on the square of the area of the rings in a molecule'' and that ". . . any method that takes this into account is likely to give reasonable results for the ratio . . ." of a given calculated anisotropy to that calculated, by the same method, for benzene. We might also add that, at least for paramagnetic systems of the type considered here, another important requirement for obtaining 'reasonable' ratios is the use of a a-electron wave function which is Iteratively self-consistent with respect to atomic charges and bond orders. (see subsection 4 of this section). The Electronic Origin of Geometrical Deformations in Cyclohexadienyl and Cyclobutenyl Transition Metal Complexes Roald Hoffmann* and Peter Hofmann Contribution from the Department of Chemistry. Cornell University, Ithaca, New York 14850. Received May 19,1975 Abstract: A case is presented for an electronic factor in the out-of-plane bending of the saturated carbon in cyclohexadienyl-M(CO)3 complexes, M = Fe+, Mn, Cr-. In the cyclohexadienyl ligand hyperconjugation extends the nonbonding MO wave function to the methylene hydrogens. The phase of the CH2 hydrogen contributions to that MO is such that when the C6H7 ligand is bound to the M(C0)3 group there arises a secondary M-CHz interaction which is destabilizing. In cyclobutenyl and cyclooctatrienyl complexes this interaction is lacking and thus these should be less bent than cyclohexadienyl complexes. A similar analysis rationalizes the bending away from the metal in cyclopentadiene-Fe(C0)3 complexes, its lessening in cyclopentadienone complexes, and the bending toward the metal in fulvene or cyclopentadienyl-carbonium ion complexes. The charge distribution and substituent effects in C6H,M(CO)3 complexes are examined, as well as a case of hypothetical isomerism in benzyl-M(C0)3. There exists a substantial chemistry of transition metal complexes of cyclohexadienyl and cyclobutenyl ligands, exemplified by structure 1. Examples exist for M = Fe+, Mn, Cr-, and their lower transition series analogues. The assign- 3 . 4 1 ment of a formal charge to the metal is, of course, arbitrary. Nevertheless it focuses on the basic electronic similarity of these complexes, an aspect that might be obscured by an argument over the cationic or anionic nature of the coordinated cyclohexadienyl ligand. In all known structures of type 1 the six-membered organic ring is highly nonplanar, and distorted in the same way-atoms 1 through 5 remain in an approximate plane, but the saturated carbon 6 moves out of that plane and away from the metal. It should be noted that the free organic ligand is either planar or only moderately distorted. In the crystal structure of the tetrachloroaluminate salt of the heptamethylbenzenonium cation, 2, the six-membered ring is essentially planar.' However, in three recent structures of stabilized u complexes, 3,1° dihedral angles up to 17' have been found.'' Stabilized anionic u complexes, that is Meisenheimer complexes, have been known for some time.l* Several crystal structures of such highly substituted cyclohexadienyl anions are a~a i l a b l e , '~ and in all the six-membered ring is approximately planar. The problem of potential nonplanarity of cyclohexadienyl radicals has been discussed re~e n t 1 y . I~ At any rate it is clear that upon formation of a transition metal complex there is a significant enhancement of th

An exactly solvable model for the Fermi contact interaction

A model for the Fermi contact interaction is proposed in which the nuclear moment is represented as a magnetized spherical shell of radius r 0 . For a hydrogen-like system thus perturbed, the Schrödinger equation is solvable without perturbation theory by use of the Coulomb Green's function. Approximation formulas are derived in terms of a quantum defect in the Coulombic energy formula. It is shown that the usual Fermi potential cannot be applied beyond first-order perturbation theory.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46454/1/214_2004_Article_BF00548828.pd

Fine and hyperfine structure of the muonic ^3He ion

On the basis of quasipotential approach to the bound state problem in QED we calculate the vacuum polarization, relativistic, recoil, structure corrections of orders $\alpha^5$ and $\alpha^6$ to the fine structure interval $\Delta E^{fs}=E(2P_{3/2})-E(2P_{1/2})$ and to the hyperfine structure of the energy levels $2P_{1/2}$ and $2P_{3/2}$ in muonic $^3_2He$ ion. The resulting values $\Delta E^{fs}= 144803.15 \mu eV$, $\Delta \tilde E^{hfs}(2P_{1/2})=-58712.90 \mu eV$, $\Delta \tilde E^{hfs}(2P_{3/2})=-24290.69 \mu eV$ provide reliable guidelines in performing a comparison with the relevant experimental data.Comment: 15 pages, 4 figures, 3 table

PROTON MAGNETIC SHIELDING IN MOLECULES

Author Institution: Department of Chemistry, Carnegie Institute of Technology“A simplified method is described for calculation of proton shielding in molecules, in which molecular wave functions are constructed from gauge invariant atomic orbitals. The method is applied to the hydrogen molecule in simple LCAO MO description, and the total nuclear shielding constant is computed to be $3.2 \times 10^{-5}$ Applications of the method to the hydrogen halides are discussed.