(Conformal) Killing Vectors in the Newman-Penrose Formalism

Finding (conformal) Killing vectors of a given metric can be a difficult task. This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators, applied to the four coordinates.It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives.If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods.Several explicit examples are given to illustrate the method.     

Kinetics of the complexation of manganese(II), cobalt(II) and zinc(II) with DL-isocitric acid

The kinetics of complexation of Mn^II, Co^II and Zn^II by isocitric acid have been studied by the stopped flow method at 15,25 and 35°C, ionic strength 0.20 M (NaClO₄) and pH range 4.50–6.35. Under these experimental conditions, one process is observed for each system within a few seconds. A mechanism is proposed to account for the observed behaviour, which is associated with participation of the OH group in complex formation. Activation energies are also reported. TMC 2593     

Hetero-diels-alder reaction of enaminecarbaldehydes an entry to branched aminosugars

N-acyl-enaminecarbaldehydes 6a - g with an electron accepting group in the α- position react in a hetero-Diels-Alder cycloaddition with enolethers 7a - g to 4-amino- dihydropyrans 8a - g, 9a - g and 10a - g. This reaction represents a convenient entry to branched aminosugars of the garosamine-type. The rate of the cycloaddition depends strongly on the N-acyl group in 6. However, the phthalimide 11 does not react because of deconjugation of the electron accepting function in the α-opposition.     

Interactions between Oxygen Lone-Pair Orbitals and Double-Bond π-Orbitals in β,γ-Unsaturated,Bicyclic Ketones; a PE-Spectroscopic Investigation

The HE(Iα) photoelectron (PE) spectra of 2,3,5,6-tetramethylidene-2-bicyclo[2.2.1]heptanone ( 12 ), 5,6-dimethylidene-2-bicyclo[2.2.1]heptanone ( 14 ), 5,6-dimethylidene-2-bicyclo[2.2.2]octanone ( 16 ), and 5,6,7,8-tetramethylidene-2-bicyclo[2.2.2]octanone ( 17 ) have been recorded, Comparison with the PE data of other β,γ-unsaturated ketones and parent alkenes, and with the result of ab initio STO-3G calculations, confirm the existence of significant interactions between the oxygen lone-pair orbital n_o and the double-bond π orbital(s). It is argued that the major contributions to the basis energy shifts and to the cross term between the n_o and π orbitals are due to a ‘through-bond’ mechanism.