The Relationship Between Atomic Size, Charge, and Polarizability
Jared D. Weidman and Roger L. DeKock, Department of Chemistry and Biochemistry, Calvin College
Results
Introduction
where 𝑛 is the principal quantum number of the valence
electrons in the atom, 𝑎0 is the bohr radius, 𝐼H is the
ionization energy of the hydrogen atom, and 𝐼v is the
atom’s average valence ionization energy [1]. The radii
obtained from this equation using experimental
ionization energy values correlate well with other
measures of atomic size, in particular the theoretical rmax
that we will discuss below. In our present work, we
examine other ways of calculating atomic size that
correlate with covalent or van der Waals radii [2] and
how these values change when comparing neutral atoms
to their ions. We also explore other uses of this effective
radius equation including ratios of size between neutral
atoms and their cations, atomic polarizability, and
polarizability ratios between neutral atoms and cations.
In doing this work, we seek to gain insight into the
electronic structure of atoms.
0.05
0.04
0.03
0.02
F
F-
F+
0.01
0.05
r max
0.04
<r>
0.03
√(<r^2>)
0.02
3
 re   N ( N  1)   (  1) 

     



 re   ( N   )( N  1) 
r 0.002
r 95%
0.01
0
Our work is also directly related to atomic polarizability,  . Polarizability
is a fundamental property of atoms that is important in van der Waals
interactions. It has been shown to correlate with the cube of atomic size. By
cubing our effective atomic radii, we obtained effective atomic
polarizability, e , and these results are shown compared to experiment in
Figure 2. Cubing Equation 2 results in an equation that can be used to
estimate the ratio of polarizabilities of a neutral atom compared to its cation:
b
r 98%
r 99%
r 0.001
0
0
0.5
1
Radius (angstroms)
1.5
2
0
0.5
1
Radius (angstroms)
1.5
2
Figure 1a: The radial probability density of the fluorine cation (grey), fluorine atom (blue), and fluorine anion (yellow) as
a function of the distance from the center of each species. Figure 1b: The radial probability density of the fluorine atom
labeled with several different measures of size.
Species
–
rmax
r0.001
F
0.40
1.86
F
0.39
1.63
F+
0.38
Table 1: rmax and r0.001 for the fluorine
atom and its singly-charged ions in
angstrom units.
We employed the quantum chemistry software
GAMESS (General Atomic and Molecular Electronic
Structure System) [3] to obtain theoretical results that
include the binding energy and radial size. Our studies
were done for elements 1-36 (H-Kr) minus the transition
metal elements in order to explore trends across the
periodic table. Results are shown in Figure 1.
a
300
250
1.44
200
150
100
50
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 49 50 51 52 53 54
Atomic Number
The effective size concept can be applied to cations as well to obtain neutral to cation radius ratios. To
obtain a simplified equation, we note that successive ionization energies of atoms roughly follow an
arithmetic progression. If this arithmetic progression is assumed, we can derive the ratio
N ( N  1)   (  1)
(2)
( N   )( N  1)
where N is the number of valence electrons in the neutral atom and  is the charge on the cation. Radius
re


re
ratios were calculated using this formula and compared to actual radius ratios from theoretical radii, and
results are given in Table 2. Note the excellent agreement between the two ratios depicted in Table 2.
Species
rmax ratio
Predicted ratio
Ne+
1.05
1.05
Ne2+
1.08
1.11
Ne3+
1.16
1.15
Ne4+
1.23
1.20
Ne5+
1.27
1.25
Table 2: Neutral to cation radius ratios for
the cations of Ne from rmax compared to
the predicted ratio from Equation 2.
Conclusion
When chemists refer to neutral and ionic radii, they are referring to very different ways of determining
atomic radius. Neutral atomic radii correlate best with the theoretical covalent radii, namely, the first
three types depicted in Figure 1b . Ionic radii, on the other hand, show better agreement with the
theoretical van der Waals radii, namely, the last five types depicted in Figure 1b. Our formula for neutral
to cation radius ratios correlates well with actual ratios from theoretical radii. Effective polarizabilities
obtained from cubing our effective radii were shown to agree well with experimental atomic
polarizabilities.
Rb 733.87 a.u.
350
b
300
αe (a.u.)
There are many different ways to measure the “radius”
of an atom. We were interested in two different groups
of radii, those that align closely with either
experimentally-determined covalent radii or with van
der Waals radii. The covalent radii were computed as
weighted averages of the individual radius measures of
each species’ valence atomic orbitals. These radii
include rmax, the maximum of the radial probability
distribution function; <r>, the expectation value of r;
and √(<r2>), the root mean square radius. The van der
Waals radii comprise two different kinds of calculations
that are both based on the total electron density curve.
One kind of these radii includes r0.001 and r0.002,
corresponding to the points where the total electron
density is equal to 0.001 or 0.002 atomic units. The
other kind of van der Waals radii includes r95%, r98%, and
r99%, corresponding to the radius where 95, 98, or 99
percent of the integration of radial probability density is
contained, respectively.
(3)
350
Atomic and Ionic Size Ratios
Theory and Method
3
2
We have completed theoretical studies of polarizability ratios using this
equation, compared them to experimental polarizabilities, and found
reasonable agreement between the two methods. There has been recent
interest in polarizability [4,5], and our work can provide new insight into
this property and its relationship to atomic size and ionization energy.
α (a.u.)
(1)
0.06
a
Radial Probability Density (a.u.)
re  na0
IH
Iv
0.06
Radial Probability Density (a.u.)
DeKock, Strikwerda, and Yu showed in 2012 how an
atom’s “effective” atomic size, 𝑟𝑒 , relates to its average
valence ionization energy according to the formula
Polarizability
250
200
150
100
50
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 49 50 51 52 53 54
Atomic Number
Figure 2a: Experimental atomic polarizabilities,  , for 34 elements. Figure 2b:
Theoretical effective atomic polarizabilities,  e ,obtained by cubing effective atomic
radius
Acknowledgements
•
•
•
•
Luke VanLaar, Matt Genzink, John Strikwerda
Dr. Michael Schmidt, Iowa State University
Calvin College
Camille and Henry Dreyfus Foundation, Senior Scientist Mentor
Program
References
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Matsunaga, N. Nguyen, K. A. Su, S. Windus, T. L. Dupuis, M.; Montgomery Jr, J. A. Journal of
Computational Chemistry 1993, 14, 1347-1363.
[4] Hohm, U.; Thakkar, A. J. J. Phys. Chem. A 2012, 116 (1), 697–703.
[5] Ma, L.; Indergaard, J.; Zhang, B.; Larkin, I.; Moro, R.; de Heer, W. A. Phys. Rev. A 2015, 91 (1),
010501.