Article Cite This: J. Am. Chem. Soc. 2017, 139, 15068-15073
pubs.acs.org/JACS
Isoelectronic Theory for Cationic Radii Noam Agmon* The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel S Supporting Information *
ABSTRACT: Ionic radii play a central role in all branches of chemistry, in geochemistry, solid-state physics, and biophysics. While authoritative compilations of experimental radii are available, their theoretical basis is unclear, and no quantitative derivation exists. Here we show how a quantitative calculation of ionic radii for cations with spherically symmetric charge distribution is obtained by charge-weighted averaging of outer and inner radii. The outer radius is the atomic (covalent) radius, and the inner is that of the underlying closed-shell orbital. The first is available from recent experimental compilations, whereas the second is calculated from a “modified Slater theory”, in which the screening (S) and effective principal quantum number (n*) were previously obtained by fitting experimental ionization energies in isoelectronic series. This reproduces the experimental Shannon-Prewitt “effective ionic radii” (for coordination number 6) with mean absolute deviation of 0.025 Å, approximately the accuracy of the experimental data itself. The remarkable agreement suggests that the calculation of other cationic attributes might be based on similar principles.
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INTRODUCTION Covalent (rA) and ionic (rI) radii are an integral part of every general chemistry course, demonstrating the periodic properties of the elements. Starting with the works of Bragg1 and Wasastjerna,2 ionic radii have found many important applications in solid-state chemistry:1−5 The ratio of the cation/anion radii determines feasible crystal structures3,4 and probable coordination numbers (CNs); similarity of cationic radii indicates that one cation can be replaced by the other, opening the road to preparation of novel mixed crystals; and the charge to radius ratio is found to correlate with ionic binding energies. These principles transcend beyond binary crystals to more complex materials, such as hybrid organic− inorganic perovskites,6 which hold promise for solar-cell technology. A more detailed history of the ionic radii concept can be found in Gibbs et al.7,8 The authoritative compilation by Shannon and Prewitt (SP)9,10 also introduced a dependence of rI on CNs and spin states (high spin, HS, vs low spin, LS, in some transition metals). The SP “effective ionic radius” for a given metal M was obtained by subtracting the Pauling4 oxide radius, rO2− = 1.40 Å, from M−O bond distances, and averaging over all Ms with the same CN, oxidation, and spin states. Setting instead rO2− = 1.26 Å produced their “crystal radii”.10 The Shannon compilation has received over 35 000 citations, attesting to the importance and relevance of rI to present day research in a variety of fields beyond crystallography and mineralogy (e.g., ion mobility and permeability in solution chemistry and biophysical systems). Yet, in spite of the fundamental significance, there is no satisfactory quantitative theory for rI, which is a serious impedance on both intuitive understanding and practical applications. Some qualitative trends have been noticed in the © 2017 American Chemical Society
past. For example, Slater pointed out that cationic/anionic radii are approximately 0.85 Å smaller/larger than the rA of the neutral atom.11 Therefore, covalent bond distances can also be obtained by summing the ionic radii. Note that when we compare rI and rA, the ion (I) and atom (A) have the same nuclear charge (Z) but a different number of electrons (Q): Q < Z and rI < rA for a cation while Q > Z and rI > rA for an anion. Slater has also suggested12 that in each shell the ratio rI/rmax (where rmax is the radius of maximal radial electron density in the outer shell) is a constant >1. He calculated rmax from an effective one-electron model, where the effect of the other electrons is manifested by a screening parameter (S) and an effective principal quantum number (n*). These depend only on Q and were calculated from the so-called Slater rules.12 This makes rmax a function of only Q (electronic charge) and Z (nuclear charge). In spite of this, Ghosh and Biswas suggested equating rI = rmax for ions,13 and rA = rmax for neutral atoms.14 We have recently shown15 that the latter works well, provided that the empirical Slater rules are replaced by a procedure for calculating S and n* from experimental ionization energies (IEs) in isoelectronic series (IS).16 (And then we denote rmax by rIS.) In contrast, rI ≈ rmax holds only for cations of high positive charge (q ≡ Z − Q), but otherwise is a poor approximation to experiment (exp), with rmax < rexp I as noted by Slater. Considering the relation between cationic and atomic (metallic) radii prompted Johnson to suggest that rI = krA,17 where k ≈ 0.64 is a universal factor. A similar relation was recently advocated by Heyrovska for ionic radii in water, but Received: July 27, 2017 Published: October 3, 2017 15068
DOI: 10.1021/jacs.7b07882 J. Am. Chem. Soc. 2017, 139, 15068−15073
Article
Journal of the American Chemical Society with an assortment of different k values which are all related to the Golden ratio.18 Testing these relations depends, of course, on the sources selected for the rI and rA values. Modern compilations of ionic9,10 and covalent radii19,20 do not exhibit such a fixed ratio. For alkali metals, k varies systematically from 0.57 (Li) to 0.725 (Rb), averaging to 0.66 (close to the Johnson ratio). For IS of cations with noble-gas electronic configurations (Figure 1), k decreases from around the Johnson ratio to a rather small value with increasing q. According to Fajans’ rules,21 large low-charged cations have a tendency of forming purely ionic bonds, whereas small and highly charged cations (lower right corner in Figure 1) will tend to form more covalent bonds. Thus, k ≡ rI/rA is an approximate measure of the varying ionic character.
lattices show close agreement with the SP radii for the more electropositive cations, but substantial upward deviations for the electronegative ones.8 Although this may be a deficiency of the SP assumption of constant rO2−, there is no doubt concerning the wide applicability of the SP radii that consequently beseech theoretical foundation. Below we show how to solve the Slater problem of relating rI to rmax, and the Johnson problem of relating rI to rA. The three radii are connected such that rI is a charge-weighted average of rmax and rA. Furthermore, upon replacing rmax by rIS (Modified Slater Theory) we obtain a remarkable quantitative agreement with the SP radii of all cations with spherical charge distributions (Results), without any new adjustable parameter. We discuss possible physical interpretations of these results (Conclusions).
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THEORY Slater Theory. The Slater theory for covalent radii is based on the analytic solution of hydrogen-like atoms.12 The radial wave function, Rn,l(r), depends on two quantum numbers: The principal quantum number, n = 1, 2, ..., and the azimuthal quantum number, l, ..., n − 1, where r is the distance of the electron from the nucleus. Setting l = n − 1 (its maximally allowed value), one obtains the following: R n , n − 1(r ) = A(αr )n − 1exp( −αr /2)
(1)
which is known as a “Slater-type orbital” (STO). Here A is a normalization constant, α ≡ 2Z/(na0), and a0 is Bohr’s radius (a0 ≈ 0.529 Å). Slater considered the radial electron density, ρ(r ) = 4π[rR n , n − 1(r )]2
Figure 1. Ratio of experimental ionic radii10 (for cations of CN = 6) to covalent radii20 in isoelectronic series of Q electrons (data in Table 1 below). For example, the Q = 10 series (green) involves the cations Na+, ···, Cl7+.
(2)
defining the covalent radius as the r where ρ(r) obtains its maximum (denoted rmax). By differentiation,
rmax = n2rH/Z
(3)
Here rH = a0, Bohr’s radius for the hydrogen atom (H). The energy levels of hydrogen-like atoms also depend only on n and Z:
Use of IS is prevalent in discussing atomic and ionic radii (and other periodic properties, such as IEs).4,5 Consider, for example, the series F−, Ne, and Na+. All have Q = 10 electrons, whereas the Z increases from 9 to 11. With increasing Z, one expects the radius to decrease monotonically. Indeed, textbooks triumphantly point out that the sodium cation is smaller than fluoride, failing to mention that the radius of neon is even smaller. Perhaps an isolated sodium cation (in the gas-phase) is indeed even smaller than neon (if radii of isolated species could have been measured), but this ideal radius increases when Na+ interacts with anions in the crystal. The dependence of ionic radii on the ion’s CN (in solids or liquids) is missing in the classical theoretical treatments of isolated ions by Pauling4 and Slater,12 but it was often tacitly assumed that the CN = 6.1−5 Later, a CN dependence was deduced experimentally for the SP radii.9,10 A more recent empirical power-law relation (see below), while showing promising agreement with some of the SP radii, depicts also a dependence of rI on CN.22 A theoretically sound choice for rI is provided by the distance from the nucleus to the point of minimal total electron density along the cation−anion bond. This point is known as the “bond critical point” (BCP) in Bader’s quantum theory of atoms in molecules (QTAIM).23 When the quantum chemistry calculations of the electron density are performed for an ion with its solvation shell, the dependence of the ensuing “bonded radii” on the CN can be obtained. Extensive calculations for crystal
En = −RH(Z /n)2
(4)
where RH ≡ e2/(8πϵ0a0) = 13.598 eV is Rydberg’s constant (energy of the ground state H atom). Eq 4 can thus be written in the form of a Coulomb law: 2En = −e2Z/(4πϵ0rmax), where e is the electronic charge and ϵ0 the vacuum permittivity. The radius and lowest ionization energy of a Q-electron atom/ion are determined by the outermost electron. The remaining Q − 1 electrons screen the nuclear charge Z by an amount S, resulting in a smaller “effective” nuclear charge Zeff = Z − S. Slater devised empirical rules12 to determine S so that it is only a function of Q. Additionally, for n > 3 he has suggested to replace n by a smaller, “effective” value, n*. Thus, eqs 4 and 3 become: IE = RH(Z − S)2 /n*2
(5)
rmax = rHn*2 /(Z − S)
(6)
Modified Slater Theory. While eqs 5 and 6 have their theoretical basis in the theory of hydrogen-like atoms, the Slater rules are completely empirical. Now, both the IE and rmax depend on S and n*, assumed to be functions only of Q, but IEs are experimental observables. Hence, in lieu of using Slater’s rules, one can determine S and n* by fitting IEs in IS to eq 5.16 15069
DOI: 10.1021/jacs.7b07882 J. Am. Chem. Soc. 2017, 139, 15068−15073
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Journal of the American Chemical Society For example, the Q = 10 IS (mentioned in the Introduction) involves Ne, Na+, Mg2+, Al3+, and so forth. Their IEs are the first IE of Ne, the second IE of Na, the third of Mg, and so forth. Fits performed for the then available IEs (n ≤ 5) were very good, excepting the first 1−3 members of each IS.16 When the ensuing S(Q) and n*(Q) are used in eq 6 for rmax one obtains rIS. The case q = 0 (namely, Q = Z) corresponds to a neutral atom. Then rIS is identified with rA. Good agreement with experiment19 was obtained with rH = 0.42 Å (instead of rH = a0 used by Slater).15 The question now is how to extend this to ions. Theory of Cationic Radii. The case q > 0 (i.e., Z > Q) corresponds to cations. Slater suggested12 that rI/rmax = c(n) namely, a constant depending only on n [e.g., c(n) = 3.38, 2.49, and 2.25 for n = 1, 2 and 3, respectively]. c(n)rmax is supposedly where ρ(r) of eq 2 goes down to about 10% of its maximal value. Figure 2 shows that, for modern data, rI/rIS is not constant for fixed Q. A more elaborate relation connects these two radii.
which indeed has the expected limiting behavior as a function of q. When q = 0 it gives rI = rA, the atomic radius, whereas when q → ∞ it gives rI → rIS, which is a covalent radius by definition, in line with Fajans’ rules. It is amusing to consider the connection between eq 7 and the proportionality relations previously considered for rI.12,17,18 Suppose the Slater relation, rI = c(n)rmax were exact. Then by substituting rIS = c−1rI in eq 7 we obtain the Johnson relation, rI = krA, with k−1 − 1 = q(1−c−1). Choose, as an example, univalent cations (q = 1) with Slater’s c(2) = 2.49. This gives k = 0.63, almost precisely the Johnson constant. It appears, therefore, that the Slater and Johnson relations are connected via eq 7, which is therefore more general than both. In summary, to obtain the radius of a cation with nuclear charge Z and electronic charge Q, we simply insert Z, S(Q), n*(Q) and rH = 0.42 Å into eq 6, to obtain rIS. We then qaverage it with rA via eq 7. While rA could also be obtained from rIS (for q = 0),15 this may introduce additional errors. Hence, the most recent experimental compilation of atomic radii is used for rA.20
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RESULTS
The Slater theory is based on a spherically symmetric wave function, eq 1, and hence it could be expected to be most useful for cations with spherically symmetric charge distributions. This includes ions with closed-shell electronic configurations: noblegas or filled d-shell electronic configurations (for example, [Ar]3d10 for Cu+). In addition, HS d5 electronic configurations also possess spherically symmetric charge distribution, with one electron in each d orbital. Results for all these cations will be given below, whereas results for cations with open-shell electronic configurations are summarized in the Supporting Information (SI). Table 1 shows results for cations with noble-gas electronic configurations, up to n = 5. This includes the helium (Q = 2), neon (Q = 10), argon (Q = 18), and krypton (Q = 36) series. The noble gas starting each series (not shown) has rA = rIS, which is a special case of eq 7 for q = 0. rA were taken from the recent compilation by Pyykkö.20 S and n* were previously extracted from fits of eq 5 to IEs in IS (up to n = 5).16 Using rH = 0.42 Å, rIS is now calculated from eq 6, and averaged out with rA via eq 7. The calculated ionic radii, rcalc I , are compared with the SP “effective ionic radii”, rexp I , for CN = 6 (when available) and the HS electronic configuration (when relevant),10 as these choices give the best agreement and possess high symmetry, close to the spherical symmetry of the isolated ion treated by the modified Slater theory. With no adjustable parameters, the mean absolute deviation (MAD) between the calculated and measured values is only 0.025 Å. This is approximately the error in the SP database in predicting bond lengths in oxides and fluorides,7 and often smaller than the difference between rexp I for different CNs. In the compilation of Cordero et al.,19 transition elements tend to have larger radii, and the MAD increases (mildly) to 0.035 Å (Table S1). Table 1 (last column) also shows results from an empirical power-law relation,22 which in the present notation is written as follows:
Figure 2. Ratio of experimental ionic radii10 (for cations of CN = 6) to the Slater radius of maximal outer electron density, when S and n* are those obtained from fitting eq 5 to experimental IEs in ref 16.
The motivation behind the newly proposed relationship is deceptively simple. Whereas a “perfect” covalent bond can materialize experimentally (e.g., in a homonuclear diatomic molecule), a “perfect” ionic bond does not exist, because it always contains some contribution from covalent bonding. As a result, cations increase their radii and conversely for anions.17 For a monovalent cation (q = 1), we suggest that rI is the average of rIS and rA, namely rI = (rIS + rA)/2. Thus, we get rIS ≤ rI ≤ rA. According to Slater,11 the radius of maximal radial density, here rIS, characterizes a purely covalent bond. Hence, rIS is the (hypothetical) radius the cation would have had were its bonding covalent. The radius rA is the covalent radius the cation would have had if a full electron were transferred to it (from the anion). While rI and rA have the same Z (but different Q), to calculate rIS one considers ions having the same Q (but different Z). Proceeding to q > 1, Fajans’ rules21 imply that a cation has an increased tendency to engage in purely covalent bonds as q → ∞ (cf. Figure 1), and then rI → rIS, see Figure 2. This suggests the charge-weighted average, qr + rA rI = IS q+1 (7)
rIpower 15070
⎛ (n − 1)m ⎞0.22 = 1.39⎜ − 1.38 ⎟ q ⎝ ⎠
(8)
DOI: 10.1021/jacs.7b07882 J. Am. Chem. Soc. 2017, 139, 15068−15073
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Journal of the American Chemical Society Table 1. Isoelectronic Cationic Radii of Closed-Shell Configurationsa np6 configurations Q 2 2 2 2 2 2 2 10 10 10 10 10 10 10 18 18 18 18 18 18 18 36 36 36 36 36 36 36 MAD =
isoelectronic series
atomic
ionic radii
ion
Z
n*
S
Zeff
rIS
rA
rcalc I
rexp I (VI)
rpower I
Li+ Be2+ B3+ C4+ N5+ O6+ F7+ Na+ Mg2+ Al3+ Si4+ P5+ S6+ Cl7+ K+ Ca2+ Sc3+ Ti4+ V5+ Cr6+ Mn7+ Rb+ Sr2+ Y3+ Zr4+ Nb5+ Mo6+ Tc7+
3 4 5 6 7 8 9 11 12 13 14 15 16 17 19 20 21 22 23 24 25 37 38 39 40 41 42 43
0.996 0.996 0.996 0.996 0.996 0.996 0.996 1.930 1.930 1.930 1.930 1.930 1.930 1.930 2.737 2.737 2.737 2.737 2.737 2.737 2.737 3.451 3.451 3.451 3.451 3.451 3.451 3.451
0.65 0.65 0.65 0.65 0.65 0.65 0.65 7.32 7.32 7.32 7.32 7.32 7.32 7.32 14.68 14.68 14.68 14.68 14.68 14.68 14.68 31.63 31.63 31.63 31.63 31.63 31.63 31.63
2.35 3.35 4.35 5.35 6.35 7.35 8.35 3.68 4.68 5.68 6.68 7.68 8.68 9.68 4.32 5.32 6.32 7.32 8.32 9.32 10.32 5.37 6.37 7.37 8.37 9.37 10.37 11.37
0.177 0.124 0.096 0.078 0.066 0.057 0.050 0.425 0.334 0.275 0.234 0.204 0.180 0.162 0.728 0.591 0.498 0.430 0.378 0.338 0.305 0.931 0.785 0.679 0.598 0.534 0.482 0.440
1.33 1.02 0.85 0.75 0.71 0.63 0.64 1.55 1.39 1.26 1.16 1.11 1.03 0.99 1.96 1.71 1.48 1.36 1.34 1.22 1.19 2.10 1.85 1.63 1.54 1.47 1.38 1.28
0.75 0.42 0.28 0.21 0.17 0.14 0.12 0.99 0.69 0.522 0.42 0.35 0.30 0.27 1.34 0.96 0.743 0.616 0.54 0.46 0.42 1.52 1.14 0.92 0.79 0.69 0.61 0.54 0.025
0.76 0.45 C 0.27 C 0.16 0.13 0.10 A 0.08 1.02 0.72 0.535 0.40 0.38 C 0.29 C 0.27 1.38 C 1.00 0.745 0.605 0.54 0.44 C 0.46 1.52 1.18 0.90 0.72 0.64 0.59 0.56 0.00
0.68 0.39 0.24 0.14 0.07 0.01 −0.04 1.02 0.68 0.505 0.39 0.31 0.24 0.18 1.25 0.87 0.68 0.56 0.46 0.39 0.33 1.42 1.02 0.82 0.68 0.58 0.51 0.44 0.07
a
All radii are in Å. n* and S from Agmon.16 rA from Pyykkö.20 Experimental ionic radii (CN= 6) from Shannon,10 where C = calculated, A = from Ahrens.5
Table 2. Isoelectronic Cationic Radii for Full and Half-Full d-Shell Configurationsa nd10 configurations
isoelectronic series
atomic
ionic radii
Q
ion
Z
n*
S
Zeff
rIS
rA
rcalc I
rexp I (VI)
rpower I
28 28 28 28 28 28 28 46 46 46 46 46 46 46
+
29 30 31 32 33 34 35 47 48 49 50 51 52 53
2.539 2.539 2.539 2.539 2.539 2.539 2.539 2.660 2.660 2.660 2.660 2.660 2.660 2.660
25.54 25.54 25.54 25.54 25.54 25.54 25.54 43.68 43.68 43.68 43.68 43.68 43.68 43.68
3.46 4.46 5.46 6.46 7.46 8.46 9.46 3.32 4.32 5.32 6.32 7.32 8.32 9.32
0.782 0.607 0.496 0.419 0.363 0.320 0.286 0.895 0.688 0.559 0.470 0.406 0.357 0.319
1.12 1.18 1.24 1.21 1.21 1.16 1.14 1.28 1.36 1.42 1.40 1.40 1.36 1.33
0.95 0.80 0.68 0.58 0.50 0.44 0.39 1.09 0.91 0.77 0.66 0.57 0.50 0.45
0.77 E 0.74 0.62 0.53 0.46 C 0.42 0.39 1.15 C 0.95 0.80 0.69 0.60 0.56 0.53
1.25 0.87 0.68 0.56 0.46 0.39 0.33 1.42 1.02 0.82 0.68 0.58 0.51 0.44
25 26 27
2.475 2.475 2.475
21.09 21.09 21.09
3.91 4.91 5.91
0.658 0.524 0.435
1.19 1.16 1.11
0.84 0.68 0.57 0.045
0.83 0.645 0.53 0.00
0.87 0.68 0.56 0.08
Cu Zn2+ Ga3+ Ge4+ As5+ Se6+ Br7+ Ag+ Cd2+ In3+ Sn4+ Sb5+ Te6+ I+7 5 3d HS configurations 23 Mn2+ HS 23 Fe3+ HS 23 Co4+ HS overall MAD =
a All radii are in Å. n* and S from Agmon.16 Covalent radii from Pyykkö.20 Experimental ionic radii (CN= 6) from Shannon (where E = estimated; C = calculated).10
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DOI: 10.1021/jacs.7b07882 J. Am. Chem. Soc. 2017, 139, 15068−15073
Journal of the American Chemical Society Here m is the CN (we set m = 6), and rI is in Å. According to this expression, the cationic radius increases (mildly) with principal shell and CN, and decreases with increasing cationic charge. While this expression is one of the most useful ones published thus far, its MAD is about 3 times that of the present work, which in comparison has unprecedented accuracy. Results for closed d-shell configurations (3d10 and 4d10) are shown in Table 2. Agreement with experiment is less good here, the MAD increasing to 0.05 (or 0.06 with rA from is poorer, Cordero et al.,19 see Table S2). Agreement with rpower I with a nearly doubled MAD. Note, however, that most of the error is in the first two members of each series. For q ≥ 3 the errors in the two methods become comparable. The HS d5 cations also possess spherical symmetry, so those that appear in the SP compilation were added to Table 2. Indeed, these show very good agreement with the theory, with MAD of only 0.028 Å. An outlier is Cu+, for which the theory gives an ionic radius of 0.95 Å, compared to 0.77 Å in the SP compilation. We note the symbol E there, which stands for an estimated value, and that the older value by Pauling is actually 0.96 Å.4 Shannon explained the difference in stating that Pauling’s value is more suitable for ionic compounds whereas his value suits the more covalent oxides.10 It is amusing to compare Cu+ with K+. Being well-down the same row in the periodic table as K, Cu clearly has a much smaller radius than K, and similarly Cu+ is much smaller than K+. Yet their rIS values are quite similar (0.79 and 0.73 Å, respectively), slightly smaller than the atomic radius of argon (0.96 Å).20 Indeed, both cations relate to the same Ar core. A similar behavior is noted for Ag+ and Rb+. Tables S3 and S4 present a similar comparison for ions with open-shell electronic configurations. These generally show worse agreement with the SP values than for the closed-shell cations, as might be expected from a spherically symmetric theory when applied to non-spherically symmetric ions. However, theory tends to systematically overestimate the experimental values, suggesting that there might be a suitable correction for this. The two radii, rIS and rA, are now shown to be approximately the radii of two orbitals of the neutral atom. Orbital radii (maxima of r2Ψ2i ) for atoms were calculated from the relativistic Dirac equation by Waber and Cromer.24 For each atom with principal quantum number, n, Table 2 there lists the radius of the filled n − 1 shell (denoted herein by r1) and the radii of the valence orbitals in shell n, the largest of which we term r2. (Similar results just appeared in Table 4 of Guerra et al.).25 Table S5 shows that, for cations with noble-gas electronic configurations, r1 ≈ rIS and r2 ≈ rA. Moreover, this substitution orbital in eq 7 gives a reasonable approximation for rexp I , denoted rI (MAD = 0.055 Å). For the cation, orbital 1 (with radius r1) is the Highest Occupied Molecular (actually, atomic) Orbital (HOMO), whereas orbital 2 (with radius r2) is the Lowest Unoccupied Molecular (atomic) Orbital (LUMO). The (closed-shell) cationic radius is thus the q-averaged HOMO and LUMO radii of the isolated ion. This suggests that when an isolated cation is introduced into an ionic material, partial electron transfer takes place from the neighboring anion(s) into its LUMO orbital, and this increases its radius beyond the Slater theory value of rIS.
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CONCLUSIONS
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ASSOCIATED CONTENT
This work showed how cationic radii (for cations with spherically symmetric electronic configurations and CN = 6) can be calculated with unprecedented accuracy and no adjustable parameters, based on the radii of two atomic Slater orbitals (rA and rIS). For the isolated neutral atom, rA is the radius of the covalent shell, whereas rIS is the radius of the closed-shell below it. For the isolated cation (obtained by removing all the electrons from the atomic covalent shell), rA is the radius of the LUMO whereas rIS is the radius of the HOMO orbital. Both these limiting radii are obtained here from experimental data: rA from tabulated atomic radii, whereas rIS is calculated from S and n* previously obtained from fitting experimental IEs to eq 5. rIS can be interpreted as the hypothetical radius of the bare cation, which one cannot measure. It is not the experimental ionic radius in condensed phases. When the isolated cation is brought into contact with other atoms/anions, partial electron transfer occurs into its LUMO orbital, increasing its radius beyond that of the HOMO orbital. Less charge transfer occurs for highly charged cations, for which the HOMO−LUMO gap is larger, and then rI → rIS. All this is depicted by eq 7, which is therefore the main new result of the present work. We have shown that eq 7 has the expected behavior with q, and it generalizes the more approximate proportionality relations of Slater and Johnson. Yet, it presently has no theoretical derivation. Possibly, for a closed-shell cation of charge q, the linear combination of isolated cationic orbitals that approximates its frontier orbital is also given by eq 7, namely [qΨHOMO + ΨLUMO]/(q + 1). Such an approximation, if applicable, could allow evaluating other ionic properties besides its radius. The theory presented herein for ionic radii is not yet complete. Due to the availability of the S and n* parameters only up to n = 5,16 the analysis did not cover the whole periodic table. More IE values may now be available that will allow to complete the analysis for the heavier atoms, although this may require a relativistic correction.25 It is also desirable to extend the theory to ions with openshell configurations, as well as to anions. Perhaps use could be made of electron affinities to get the relevant S and n* values for anions. Additionally, the Slater model deals with an isolated atom/ion, hence has no CN dependence. It is traditional to compare with experimental data for CN = 6, and this indeed produced here the best agreement. Perhaps one could test eq 7, and elucidate the theoretical CN dependence, by more detailed QTAIM studies, starting with diatomic molecules26 and progressing e.g., to gas-phase clusters of hydrated ions.
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jacs.7b07882.
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Tables with similarly calculated ionic radii: using another database for rA; for open-shell cations; from orbital radii (PDF)
AUTHOR INFORMATION
Corresponding Author
*
[email protected] 15072
DOI: 10.1021/jacs.7b07882 J. Am. Chem. Soc. 2017, 139, 15068−15073
Article
Journal of the American Chemical Society ORCID
Noam Agmon: 0000-0003-4339-8664 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS The Fritz Haber Research Center is supported by the Minerva Gesellschaft für die Forschung, München, FRG. REFERENCES
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DOI: 10.1021/jacs.7b07882 J. Am. Chem. Soc. 2017, 139, 15068−15073
