
Successive Ionization Energies of Atoms: Theoretical Interpretation
John R. Strikwerda and Roger L. DeKock, Department of Chemistry and Biochemistry
Calvin College, Grand Rapids, Michigan
Results and Discussion
Introduction
Experimental studies show that successive ionization energies among most
elements roughly follow an arithmetic progression (Figure 1). There have been
many attempts to explain this observation using classical mechanics.1,2
However, few have accounted for this behavior while adhering to quantum
mechanical principles,3 and none have done so in a clear and transparent
manner. The present work re-examines the theoretical interpretation of the
trends in successive ionization energies of atoms, using modern electronic
structure theory, and investigates afresh what atomic properties may be
extracted from these successive ionization energies.
r0  na0
(2)
2. Vee remains relatively constant regardless the ionization state
150
100
50
0
0
1
2
3
4
5
6
7
8
Ionization Event
Theory and Method
Under Hartree-Fock theory, the energy of a valence electron can be expressed
in terms of the following elements
• T, the electron’s kinetic energy
• Ven, the electron-nuclear coulombic attractive interaction energy
• Vcv, the core electron-valence electron repulsive interaction energy
• Vee, the valence electron-valence electron repulsive interaction energy
The first concept (eq 2) we term relaxation compensation. It states
that although valence orbitals relax significantly upon ionization,
changes in orbital energies do not result from this relaxation. Instead,
differences in valence orbital energies from one ionization state to
the next (and therefore differences in ionization energies) result from
the loss of one Vee interaction, which remains relatively constant.
Hence, the magnitude of Vee takes on a crucial role in atomic
electronic structure and can itself be interpreted as the slope of the
arithmetic progression of successive ionization energies.
300
0
T
Vne + Vcv
Vee 2s,2s
ε = T + Ven + ΣVcv + ΣVee
(1)
Employing General Atomic and Molecular Electronic Structure System
(GAMESS)4 software and the recently developed5 spin constrained
unrestricted Hartree-Fock (CUHF) theory, we calculated the specific T, Ven, Vee,
Vcv Hartree-Fock energy integrals for valence electrons in the elements neon
and argon. In monitoring these values across the valence ionization series,
insight into electronic behavior upon ionization is obtained.
References
(1) Benson, S. W. The Journal of Physical Chemistry 1989, 93, 4457-4462.
(2) DeKock, R. L.; Gray, H. B. Chemical Structure and Bonding; University Science Books, 1989.
(3) Pyper, N. C.; Grant, I. P. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1978, 359, 525543.
(4) Schmidt, M. W. Baldridge, K. K. Boatz, J. A. Elbert, S. T. Gordon, M. S. Jensen, J. H. Koseki, S. Matsunaga, N. Nguyen, K. A. Su, S.
Windus, T. L. Dupuis, M.; Montgomery Jr, J. A. Journal of Computational Chemistry 1993, 14, 1347-1363.
(5) Tsuchimochi, T.; Henderson, T. M.; Scuseria, G. E. ; Savin, G. E. Journal of Chemical Physics 2010, 33, 134108-134108-10.
(6) DeKock, R.L.; Strikwerda, J.R.; Yu, E.X. Chemical Physics Letters 2012, 547, 120-126.
(7) Koopmans, T. Physica 1934, 1, 104-113.
Vee 2s,2p
-500
2a
In this formula, n is the principal quantum number of the valence
electrons in the atom; I is the ionization energy of the hydrogen
H
atom, and a0 is the bohr radius. If an atom obeys the arithmetic
progression, then it can be easily shown that the ratio of the squares
of the neutral atom’s radius with N valence electrons to that of its ion
with charge  is given by the following equation.
2
 r0 
N ( N  1)   (  1)
  
( N   )( N  1)
 r 
(4)
Representative ratios computed by equation 4 are given in Table 1.
These values agree well with our Hartree-Fock data, reflecting a
shrinking cationic radius. In contrast, evoking the frozen orbital
approximation demands these radius ratios to be one. The concept of
relaxation compensation, however, is consistent with a shrinking
cationic radius like that predicted from equation 4.
Table 1: Neutral to cation radius ratios, as
obtained from Hartree-Fock calculations (column
2) and computed from equation 4 (column 3),
for neon cations Ne+, Ne2+, Ne3+.
100
100
-600
(3)
150
-400
The orbital energy is then defined as below, where the summation of Vee and
Vcv is done over all interactions with that particular valence electron.
IH
Iv
200
200
Energy (eV)
Energy (eV)
∆𝑇 + ∆𝑉en + ∆Σ𝑉cv ≅ 0
y = 27.572x
R² = 0.9646
200
Central to an understanding of electronic behavior upon ionization in
atoms is the qualitative concept of atomic and ionic size. Previously,6
we related the effective radius, r0 , of an atom to the average valence
ionization energy, I v .
1. For a given valence electron, upon ionization, the terms T, Ven,
ΣVcv change (relax) in such a way that their sum remains roughly
constant, i.e.,
300
250
Atomic and Ionic Size
Figures 2a and 2b show data representative of our work. To
summarize:
Energy (eV)
Figure 1: Valence ionization series of
neon. A best-fit arithmetic progression
is drawn over top the data.
Relaxation Compensation and the
Constancy of Vee
50
0
T
Vne + Vcv
Vee 2s, 2s
Vee 2s,2p
-250
-300
0
1
2
3
4
5
Ionization State
6
7
-350
0
1
2
3
4
5
Ionization State
6
7
2b
Figures 2a and 2b: Individual energy integrals plotted as a function of ionization state. Figure 2a
shows data associated with a 2s electron in the neon ionization series. Figure 2b does similarly for a
3s electron in the argon series.
Neon
Radius Ratio
HartreeFock
Equation
4
2
1.10
1.11
2
1.17
1.22
2
1.34
1.33
 r0 
 
 r 
 r0 
 
 r2 
 r0 
 
 r3 
Conclusion
This work represents a new theoretical framework to understand the electronic behavior of an atom upon ionization. The “old” idea of frozen
orbitals (Koopmans’ Theorem7) is replaced by the dynamic model of ionization outlined above. Critical to this model is the constancy of the
valence electron-valence electron repulsion interaction as well as the concept of relaxation compensation. The proposed model is capable of
explaining the general arithmetic behavior of successive atomic ionization energies and, unlike frozen orbitals, is consistent with shrinking cationic
radii. A manuscript outlining this project is in preparation.
Acknowledgements
•
•
•
•
Dr. Michael Schmidt, Iowa State University
Jared Weidman, Calvin College
Jansma Family Research Fund in the Sciences and Business Fellowship
Calvin College