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
2n
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
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•
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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).