Core Electron Ionization and the Periodic Table Jared D. Weidman and Roger L. DeKock, Department of Chemistry and Biochemistry, Calvin College • • • • Employs the concept of effective nuclear charge First developed by Slater in 1930 (ref. 1) Improved upon by Gould in 1991 (ref. 2) Example for Ne shown below 2 e Z E N , Ze is effective nuclear charge 2 2n N is number of electrons We employed two theoretical models in order to interpret the empirical results of Moseley: a simple spreadsheet screening constant model and a sophisticated computational chemistry model. Both models are based upon quantum mechanics. In the following material, the energy results are expressed in hartree units. 11 11 10 10 9 9 8 8 7 Screening electron 1s 2s 2p 3s 3p 0.31 0.00 0.00 0.00 0.00 0.90 0.25 0.34 0.00 0.00 0.95 0.25 0.34 0.00 0.00 1.00 0.90 0.90 0.25 0.34 1.00 0.95 0.95 0.25 0.34 The K transition illustrated for Ne: (Ne )* 1 2 1s 2s 2p 6 Ne 2 2 1s 2s 2p h + 5 1 Z (1s) 2 Z (2s) 6 Z (2p) E (Ne ) Er 2 2 2 2 1 2 2 2 2 2 2 2 2 Z (1s) 2 Z (2s) 5 Z ep ep ep (2p) E (Ne ) Ep 2 2 2 2 1 2 2 2 2 E Ep Er * 2 er 2 er 2 er E k(Z 1), empirically true, Moseley, 1913 Computational Chemistry Model • 6 5 5 √(ΔE)=.6342(Z–1.0032) 3 3 2 2 10 15 √(ΔE)=.6331(Z–.9796) 4 20 5 Z (atomic number) Gould Screening Constants, 1991 6 5 snl does not depend upon Z or the ionic state GAMESS software (ref. 3) • General Atomic and Molecular Electronic Structure System • CUHF calculations (ref. 4) • Constrained Unrestricted Hartree-Fock • First introduced in 2010 • Made available in GAMESS May, 2013 b 7 4 2s1s 2s2s 5s2 p for 2p electron of Ne The basis for atomic number of the elements was provided by the X-ray spectroscopy experiments of English physicist Henry Moseley in the early 20th century. Moseley measured the energy of X-rays emitted by the elements, and found the following relationship: √(ΔE)=k1(Z-k2), where ΔE is the energy, Z is the element’s atomic number, and k1 and k2 are empirical constants. This relationship is known as Moseley’s Law. Moseley found that the value of k2 is near 1 for the Kα transition. His results were purely empirical, that is, with no theoretical interpretation. In our work, we seek to provide a quantum mechanical interpretation of Moseley’s Law. No such direct interpretation exists in the current literature. Methods 12 Ze Z , is the screening constant Introduction CUHF Model 12 n is principal quantum number 1s 2s Screened 2p electron 3s 3p a Screening Constant Model √(ΔE) Moseley’s Law, √(ΔE)=k1(Z-k2), empirically describes the direct correlation between an element’s atomic number (Z) and the energy of its emitted X-ray (ΔE). We employed two quantum mechanical methods to calculate the electronic energies of the reactant and product species involved in the X-ray emission. This was done for elements B – Ar in order to model Moseley’s Law. In doing this, we seek to theoretically interpret the value of the constant k2, which Moseley empirically found to be nearly equal to 1 for the Kα emission. Our computational data shows excellent agreement with experiment. Interpretation of our theoretical results shows that the fact that the value of k2 is nearly equal to 1 is purely coincidental. Results √(ΔE) Abstract Screening Constant Model 10 15 20 Z (atomic number) Figure a: The results from the screening constant model for the Kα transition illustrate excellent agreement with the empirical result of Moseley; the value of k2 is 1.0032, near 1 as required by the experimental data. Figure b: We calculate energy differences between the relevant ions using CUHF in order to compare with Moseley‘s experimental data. Our computational results again show excellent agreement with the experimental data of Moseley in that we find the value of k2 to be 0.9796, that is, near 1. Conclusion We showed that both a screening constant model and CUHF model are accurate for modeling electronic transitions. It is clear by comparing the resultant equation (see methods) of the screening constant model to the empirical result of Moseley that there is no simple way to “derive” the (Z—1) term. That is, the empirical observation that k2 = 1 is purely coincidental. Attempts to state otherwise (ref. 5, ref. 6) are doomed to failure. Acknowledgements • • • • John Strikwerda Dr. Michael Schmidt, Iowa State University Calvin College Dr. Luke and Pauline Schaap Summer Research Fellowship References 1. “Atomic Shielding Constants”, J. C. Slater, Physical Review, Vol. 36, pp. 57–64 (1930). 2. “Energies and Wave Functions for Many-Electron Atoms”, Y.-D. Jung and R. J. Gould, Physical Review A, Vol. 44, pp. 111-120 (1991). 3. "General Atomic and Molecular Electronic Structure System“, M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis, J. A. Montgomery, Journal of Computational Chemistry, Vol. 14, pp. 1347-1363 (1993). 4. “Constrained-Pairing Mean-Field Theory. IV. Inclusion of Corresponding Pair Constraints and Connection to Unrestricted Hartree–Fock Theory”, T. Tsuchimochi, T. M. Henderson, G. E. Scuseria, and A. Savin, Journal of Chemical Physics, Vol. 133, pp. 134108-134108-10 (2010). 5. “Teaching Moseley’s Law. A Classroom Experiment”, E. Lazzarini and M. Bettoni, Journal of Chemical Education, Vol. 52, pp. 454-456 (1975). 6. “The Bohr-Moseley Synthesis and a Simple Model For Atomic X-ray Energies”, M. A. B. Whitaker, European Journal of Physics, Vol. 20, pp. 213-220 (1999).