movable cursor. The nuclear charge can be modified and the relationship between effective nuclear charge and depth and internuclear distance of the energy minimum can be demonstrated. The calculus capabilities of DERIVE have been u s e l l in evaluating a number of different integrals. Average values in a simple harmonic oscillator present no problem. My students have calculated kinetic and potential energies in different quantum states. Even transition moment integrals can be evaluated and the results used to calculate oscillator strengths that can be compared to experimentally derived results from the lab.
Computer Assisted Molecular Modeling Exercises for Undergraduates 11. Aromaticity In Heterocyclic Molecules Vernon G. S. Box City College of The City University of New York New Yoh, NY 10031 Huckel's rules are necessary conditions for any molecule to show aromaticity. However, there are many molecules, particularly uncharged oxygen-containing heterocyclic molecules, t h a t do fulfill the requirements of Huckel's rules but are nevertheless nonaromatic. Recently some criteria for ammaticity were stated (46) which were based on the lengths of the bonds in the molecule. These criteria were shown to complement Hiickel's rules and to clearly identify molecules which are truly aromatic, and those which obey Huckel's rules but are nevertheless nonaromatic. These new criteria were based on data gathered from an extensive study of the X-ray crystallographic structural data of several hundred organic molecules (naturally occumng and synthetic) and can be distilled into two main points: 1. Bond types must be integers, no bond can be a fractional R-bondand all bonds must be either single, double, or tri-
ple bonds. 2. Bonds in aromatic molecules, and all delocalized R-bonds, have lengths which fall within the normal ranges of
lengths for R-bonds,and in fact towards the longer end of this range, see Table 2. The values of the bond lengths in Table 2 have been rounded. The numeric value of bond type (usually single, double, or triple) is equal to the integral value of the Table 2. Bond TypeBond Length-Bond Order Relationships ( 1 ) Bond Type
Average Bond Length
Bond UpperILower Numeric Order Bond Length Bond Type Limit (pm) Range (ex VESCFHMO)
