Photoelectron Spektra of Azabenzenes and Azanaphthalenes: II. A Reinvestigation of Azanaphthalenes by High-Resolution Photoelectron Spectroscopy

The bands with I_v < 13 eV in the photoelectron spectra of quinoline (IX), isoquinoline (X), cinnoline (XI), quinazoline (XII), and quinoxaline (XIII) have been reassigned in a way consistent with the assignment proposed for pyridine (II), the diazines (III, IV, V), s-triazine (VI), and 1,2,4,5-tetrazine (VII). The bands corresponding to the ejection of an electron from a π-orbital have been identified by a regression calculation based on a HMO perturbation treatment. It has been found that the combined through-space and through-bond interaction of the lone pairs in III, IV, V and in their corresponding benzologues XI, XII, XIII are the same within experimental error ( ±, 0.2 eV). Our assignment is also supported by an empirical correlation of the pK_{a, 1?} values and the mean lone-pair ionization potentials of the azaderivatives I to XIII.     

(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.