Unconventional Look at the Diameters of Quantum Systems: Could the

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Unconventional Look at the Diameters of Quantum Systems: Could the Characteristic Atomic Radius Be Interpreted as a Reactivity Measure? Paweł Szarek,* Marcin Witkowski, and Aleksander P. Woźniak

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Centre of New Technologies, University of Warsaw, Krakowskie Przedmieście 26/28, 00-927 Warsaw, Poland S Supporting Information *

ABSTRACT: Decoding of atomic and ionic radii of transition metals in terms of energy response against changes in electron number and external potential variation has been considered. Energy as a functional of electron density by means of its derivatives is linked to response density and the electron detachment process. Employing charge sensitivity analysis and the electronegativity equalization principle, we interpret the electronic structure transformations (electron-following/preceding perspectives) into atomic diameters. Additionally, qualitative associations described within hard/soft acid/base theory and the approximate correlations of respective conceptual density functional theory reactivity descriptors were considered to meet postulates of the correspondence principle, giving the characteristic radius the attribute of latent variable related to quantum mechanical observables. By means of local density approximation, an insight from statistical analysis of frontier electron density complements the picture of classical (electrostatic) radii formulations as well as provides a view on the electron correlation effect on the atomic size. The presented radius identifies ground and excited states, as well as spin configurations through measurable properties. Its correspondence with empirical radii is illustrated. The provided mathematical interpretation associated with energy evolution contrasts with the classically understood physical boundary.

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INTRODUCTION Despite its deficiency, the atomic radius has become a commonly employed phenomenological approach, with applications in many areas of chemistry, physics, and engineering. However, it remains purely a classical and conceptual tool since it lacks quantum mechanical associations via expected values of observables.1 Historically, the idea originates from the Bohr model of atom,2 where finite atomic dimensions were implied by rigid electron orbits, thus a wellknown fundamental physical constant, the Bohr radius, a0 = ℏ2/mee2. Moreover, the elements had been suggested to have a definite size in connection with early X-ray crystallography experiments.3,4 Additionally, present-day techniques such as scanning tunneling microscopy5 or scanning transmission electron microscopy6 allow imaging individual atoms, although with insufficient resolution compared to atomic scale. Multiple theoretical approaches presenting various definitions7−37 of atomic diameters are a consequence of atomic architecture, where the electron wave−particle duality and uncertainty principle make problematic setting down sharp outlines. The main purpose of atomic radii was to reproduce experimental bond lengths. According to quantum mechanics, the equilibrium bond distances emerge solely from electrostatic interactions, by virtue of the Hellmann−Feynman theorem.38−42 Therefore, in some measure, the classical aspects of characteristic radius are justified. In fact, apart from empirical approaches, based on crystallographic and in silico results,3,8,10,13,14 the majority of procedures use electrostatic © XXXX American Chemical Society

criteria to define atomic diameters. These methods interpret electrostatic potential strength, including comparisons of local electric potential with selected atomic properties such as electronegativity,15 chemical potential43 or ionization potential,20 the positions of electrostatic potential extrema,17−19,25 the frontier orbitals’ Coulomb potential,16,36 or classical turning point formulations.20,27,28,37 In practice, the atom or ion size denote the distance atomic nuclei can come near to another, which is governed by atomic interactions. Displacement, sharing or exchanging electrons, affects approachable distance through both the repulsive and attractive parts of a potential. These electronic processes depend on the ability of atoms to attract electrons with different strengths, measured with electronegativity. Coincidentally, the same processes are involved in reactivity. Nevertheless, atomic dimensions should reflect atom predispositions to adopt hard or soft attributes,44−48 which are referred to by electronic response functions and conceptual density functional theory reactivity descriptors.49−51 Regardless of classical roots and its presumed role (of the closest distance to another atom, under equilibrium conditions with respect to distinct types of strong or weak interactions), the arguments raised imply that characteristic radius could be defined as a latent variable in regard to measurable parameters or quantum Received: January 8, 2019 Revised: March 16, 2019

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DOI: 10.1021/acs.jpcc.9b00221 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C mechanical expectation values. Such attempts have been made, resulting in loose correlations.19,23,24,35,43,48,52,53 Among measurable physical quantities, those related to atom size and reactivity as well include electron density,16,19,23,24,35 electric dipole polarizability,54−57 or ionization potential.15−20,25,42,43 Identifying how tendencies toward certain reactivity affect an atom size, by quantifying the electronic structure evolvement upon interaction, might be pragmatic, especially in reference to transition metals acting as catalysts (represented by “bare” metal ions in gas-phase reactions or metal−ion complexes in solution). The general connection between reactivity and atomic dimensions has been established through Pearson’s hard and soft acids and bases (HSAB) theory.44 The harder systems are considered smaller than the softer ones. However, these relations between classification based on equilibrium constants and the characteristic radii are qualitative at most. The picture could be complemented with a quantitative interpretation based on fundamental electronic response functions.46,56−58 However, a single-parameter description of acid−base interactions has been shown to be insufficient.59−62 From another standpoint, chemical reactions involve bondbreaking and -forming processes, which implicate changes in the electronic structure described by ground-state electron density (or overall atomic electron population), and distortion in the geometrical structure characterized by positions of the nuclei. The electron density is determined via nuclear potential, which exemplifies “electron-following” mapping of chemical activity.63−65 In the complementary “electronpreceding” view,66 the displacements of electron distribution are aimed at adjusting nuclear positions via the preceding step to reduce regional electronic strain due to external potential.67 Therefore, characteristic radius with impartial definition could be used to recognize the chemical propensity of atoms, delimiting interatomic nearing and resulting from electron density reorganization. Moreover, the mapping transformations can be derived within charge sensitivity analysis68,69 and are related to electronegativity equalization principle.70,71 Thus, chemical processes, as seen from these two perspectives, associate with deformation of electron distribution around the nucleus and with electron-attracting power, linking the atomic size with a particular electronic state. Although valence or more precisely frontier density has been identified to have a major impact on reactivity,72−74 yet direct use of electron density to determine characteristic radii has its weaknesses. The atomic outer limits are inconclusive because electrons are subject to Heisenberg’s uncertainty principle and their distribution around the nucleus is given as probability density function. Moreover, the behavior of electron density around the nucleus changes with distance because of screening effects and electron correlation. The character of electron density close to the nucleus (mostly corresponding to core electrons) is rigid, in contrast to flexible distribution in the tail region, generally associated with the valence orbitals (yet, penetration effects should be appreciated). Therefore, further from the nucleus, it might be considered an “electron atmosphere”,75 which becomes more condensed “fluid” at short distance. The electron density is a common physical property among observables used to assess atomic diameters.10,11,21−24,26 However, depending on the approach employed, it might be interpreted arbitrarily and deliver different radii (Figure 1). Dilemma arises because various descriptors might be used to characterize the head and tail of

Figure 1. Size of hydrogen atom might be represented by the position of the maximum of electron radial probability density located at a0 or the expected radial distance, ⟨r⟩ ≅ 1.5a0, as well as the radius delimited by electron density cutoff, ρ = 0.001, equal to ∼3a0. Mathematically, a0 approximately corresponds to the most probable distance between the nucleus and the electron in a ground-state hydrogen atom (head of probability distribution). The two other measures are a consequence of a long tail of the radial wave function.

probability density function, including mode, expected value, median, variance, or higher standardized moments or their combinations (Figure 2). Further issues concern including

Figure 2. Selected estimators for probability density function. The amount of enclosed electron number given in percentage.

whole or part of density (a more tightly bound core, or the valence, specifically related to frontier orbitals) to appoint characteristic distance, eventually how to consider screening and penetration effects? The answer to these complications might be going back to the source since the electron density is defined as a derivative of an energy functional. Therefore, the atomic diameters should be possible to determine as information encoded directly in an energy functional. Typically, atoms are non-neutral electrically and occur in various redox forms (formal oxidation states), which in the case of, e.g., transition metals are multiplied by spin configurations. The general problem is that diameters of atoms are not fixed but conform to the surrounding environment.70 The assessments are influenced, for instance, by ligand effects, such as ligand−ligand repulsion76,77 increasing the central ion size or ligand field78 affecting its spin state. Besides, the characteristic radii determined from molecular data are distinct from those based on condensed phase systems on account of crystal field effects. For example, the Shannon characteristic radii to comply with molecular, i.e., covalent, systems provide a parallel set with values shifted by a B


The Journal of Physical Chemistry C

2 y i i ∂μ yz i ∂ρ(r ) zy zz = jjj ∂ E zzz zz = jjjj f (r ) = jjjj z j z z k ∂N {v k ∂v(r ) { N k ∂N ∂v(r ) {

constant, ΔR ≅ 0.14 Å.33,34 Ultimately, the Shannon radii, representing the most common (average) separations, are used for swift detection of unexpected experimental distances. However, the total energy of a system is a function of its state, which should translate into appropriate atomic radii.

THEORY In reference to fundamental response functions, i.e., electric dipole polarizability,48,79,80 α (approximating linear response function, χ(r,r′) = [δ2E/δv(r)δv(r′)]N), and chemical hardness,46,47 η (coupled with ionization potential, I, and electron affinity, A), we have proposed a definition of atomic radius81,82 (1)

where e is the electric charge, acting as a normalization constant given in the SI standard (instead of capacitive energy, hardness, the equivalent elastance of the system, η/e2, is used). Both α and η have numerous correlations with atomic size.36,48 Those relations establish links between the presented definition and chemical principles or conceptual density functional theory.49−51 Such radius upholds aspects of the dimensional extent of hard and soft systems,59−62 described within HSAB theory. Simultaneously, it measures the tendency of a system to induce charge deformation, i.e., gradual polarization and electronic transitions. Reference to Energy Response. Formulating eq 1, in terms of energy derivatives, dE, with respect to electric field, dF, and electron number, dN, we obtain R=

ij d2E yz ij d2E yz jj z −jjj 2 zzz j 2 zz k dF { N , F → 0 k dN { v

Following both the adopted hardness definition in eq 1, i.e., (I − A)/2, and the one used in LDA, which does not include factor 1/2, as well as the slope of linear dependence between experimental and LDA-estimated polarizability (antisymmetric function with respect to change in Cartesian axis sign, against the symmetric Fukui function-based approach), the LDA model gives about 2 times greater radii than eq 1, thus αη ∝

(6)

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(2)

RESULTS AND DISCUSSION Unfolding the Sizes of Ground-State TransitionMetal Ions. Transition metals exhibit a rich selection of available oxidation states multiplied by relevant spin configurations. They have up to 12 electrons in their valence shell orbitals, i.e., (n − 1)dns, which hypothetically could be involved in chemical interactions.94−99 The respective groundstate electronic configurations of cations from 3d-block elements, in term symbols, are listed in Table S1 in the Supporting Information, SI. The radii estimated using computational methods and evaluated using experimental polarizabilities and hardness (ionization potentials) are compared in Table 1. Although only limited experimental results are available, the correspondence between theoretically Table 1. Radii (in Bohr) Calculated from Theoretically Evaluated Response Functions vs Experimental Data-Based Onesa

2

∫ r 2f (r)dr−(∫ rf (r)dr)

1 {R f (r) ≡ R Ml} 2

Computational Details. The electric dipole polarizabilities and chemical hardness (ionization potentials) for transition-metal ions were calculated with coupled cluster method, CCSD(T),90,91 and ANO-RCC92 basis set in Gaussian93 software.

which might be interpreted as a mathematical limit of system energy evolution. Therefore, radius refers to the internal energy barrier coupled with the charge delocalization (i.e., transitions between highest occupied and lowest unoccupied orbitals56,57,83) acting in opposition to the ability to induce dipole moment or energy level transformation through external potential (e.g., Stark effect or photon-related transitions). Hence, fluctuation in the potential (in response to electronic charge flow, unscreening, or caused by external potential variation) competes with the relaxation of charge distribution (electron-following/preceding views). This entangled relation between charge distribution and potential fields, both of which minimize the system energy, determines the size of an atom. The relation between size and changes in electron number or external potential is implied by both atomic architecture (wave−particle duality under Born−Oppenheimer approximation) and energy representation through a functional of electron density,84 thus designating the radial limits of atomic influence. The relation in eq 1 expressed within local density approximation, LDA,85 takes the form81,86 R f (r) =

(4)

Therefore, local density approximation introduces a key modification, in the form of a mixed energy derivative, to the radius defined in eq 2, which yields a product of two secondorder energy derivatives with respect to electron number, and to the external potential gradient. The Fukui function integrals in eq 3 correspond to multiple moments of Frontier orbital electron density (the quadruple and dipole moment) derivatives with respect to electron number;89 thus, the equivalent notation is ÄÅ É 2Ñ ÅÅ dQ ij dDi yz ÑÑÑÑ 1 ÅÅ ii zz ÑÑ R Ml = − jj ∑ Å 3 i = x , y , z ÅÅÅÅÇ dN k dN { ÑÑÑÖ (5)

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R = e−1 αη

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(3)

In free isolated atoms, eq 3 simplifies to the square root of the first term only. The Fukui function74,87,88 in the above notation might be represented as a second-order partial derivative of energy with respect to electron number and external potential.87

ion

Rcalc.

Rempr.

Sc+3 Ti+4 V+5 Cr+6 Ni+ Cu+ Zn+2

1.42 1.23 1.11 1.02 1.23 1.25 1.02

1.32a, 1.45b 1.13a, 1.28b 0.97a 0.85a 1.24c,d 1.58e 0.875a,f

a

Ionization potentials to estimate the hardness of empirical radii are from ref 100, and the polarizabilities are taken from (a) ref 101, (b) ref 102, (c) ref 103, (d) ref 104, (e) ref 105, (f) ref 106.

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specified in Table S1 in the SI. The shifted values of ionization potentials, as in eq 7, marked with a broken line, approximate the respective “electron affinities” of a metal. It is striking that in the N-electron system, these electron affinities of metal atoms with atomic number Z align (almost parallel) ionization potentials of atoms with Z − 1 atomic number, eq 8 [note that η = (dμ/dN)Z = −(dμ/dZ)N,48,107 and benchmark values of hardness obtained from isoelectronic series108].

and empirically determined radii is notable. The hardness defined as a second derivative of energy with respect to electron number46 in cations might be approximated as a difference in successive ionization potentials (by making an assumption that electron affinity for the N-electron state is ionization potential of the N + 1 electron state, eq 7). ΔE (7) ΔN The radii estimated according to eq 1, using computed properties, are presented in Figure 3 (numerical values of AN ∝ IN + 1 =

(AN )Z ∝ (IN + 1)Z ≈ (IN )Z − 1

(8)

Deviation from this trend is observed in the case of core cations on the edge of 3d and 3p orbital occupation, where hardness significantly increases (about 3 times higher than its value in [Ar]3d1 or [Ne]3s23p5 ionization states, illustrated in Figure 5). Moreover, ion polarizabilities, with respect to atomic

Figure 3. Theoretically determined radii, as defined in eq 1 of valence states of transition-metal cations (according to Table S1 in the SI), from neutral to the core. Ni0 has two possible configurations: [Ar] 3d94s1 shown by a cross, and [Ar] 3d84s2 represented by a circle. Figure 5. Hardness (line with data points) and ionization potentials (line) with respect to ionic charge of Zn. Empty points are average values of hardness taken over neutral up to the particular ionization state.

hardness, polarizabilities, and radii can be found in SI Tables S2−S4). The results in Figure 3 show that ionic radii drop considerably at the lowest oxidation states (0, +1, or +2), whereas at higher oxidation states (>+2), further contraction is particularly slow. The exception from this monotonic behavior occurs in the case of the core radii (the largest inner noble gas configuration, hence 3d04s0); the radius notably increases compared to oxidation states with single or few electrons occupying d orbitals. This effect is directly related to significant elevation in hardness consequent to crossing between subshells while extracting electrons. Periodic Properties vs Ionization States. Figure 4 renders ionization potentials for valence electrons in 3d transition metals calculated for electronic configurations

charge, behave monotonically, much like power series and do not exhibit significant steps or jumps, thus cannot correct for sudden hardness inflation. Analogies with the Electronic Structure. Neglecting the many-electron interactions, the anomaly in the core radius reflects hydrogen-like ion orbital behavior (Figure 6). Although the 3p orbital penetrates deeper than 3d, the mean distances are ⟨r⟩3d < ⟨r⟩3p (normalized values are listed in Table 2). Therefore, the majority of 3p electron density (∼0.26e) is further from the nucleus than density in the 3d

Figure 4. Ionization potentials, I, solid line, and assigned electron affinities, A, broken line, of 3d transition metals.

Figure 6. Radial probability density distribution of hydrogenic orbitals. D


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averages the frontier density, yet it does not take into account the electronic configuration of the N-electron state, only N ± 1 states. It might be illustrated with the example of a neutral Ni atom, having two possible electronic configurations: [Ar]3d94s1 and [Ar]3d84s2. According to eq 1, two radii for Ni0 are obtained, one for each electronic configuration. However, the LDA-based model yields two radii for Ni+ because Ni0 densities contribute to Ni+ Fukui function (an analogous condition appears for Ni−). Furthermore, we emphasize here the classical electrostatic nature of the LDA-based approach, which appears consistent with the classical atomic radius concept, yet with its consequences, as will be explained later. We recognize that fluctuations in radii, such as core radius expanding, as shown in Figure 1, depend on hardness (ionization potential) variation, and not on the polarizabilities. To demonstrate this, two hardness approximations based on the Fukui function might be employed. First is the Coulomb integral of frontier density110,111

Table 2. Normalized Expectation Values of Radial Density Distribution, ⟨r⟩ = ∫ r(rRnl(r))2dr, for Hydrogenic Orbitals atomic orbital

1s

2p

2s

3d

3p

3s

4s

⟨ri⟩/⟨r⟩1s

1.00

3.33

4.00

7.00

8.33

9.00

16.01

orbital and only about 0.07e is closer. The “valley” around node region of the 3p orbital resides up to 0.33e more on 3d than on the 3p orbital. Correspondingly, the mean distance, ⟨r⟩, from the nucleus is about 1.2 times greater for 3p compared to that for the 3d orbital. The radius, eq 1, reflects the properties of electrons in the highest occupied orbitals; hence, described features would rationalize to some extent the expansion of core ion diameters consequent to last 3d valence electron detachment. Yet obviously, the hydrogen-like orbitals indicate the role of shielding effects in multielectron systems on a range of characteristic radius. The more shielded the outer electron from the nucleus, the less energy is required to remove an electron from an atom. However, taking into account many-electron interactions by employing quantum mechanical methods with Gaussian-type basis set complicates the situation and the relation with expected values for orbital densities becomes not so evident. The radial extent of 3d orbitals is greater than that of 3p in early transition metals, whereas in late ones, i.e., Zn, the mean distances of 3d and 3p densities are similar to a hydrogen-like orbital picture. Yet, this feature varies with the ion charge and spin state and many exceptions appear; thus, it is difficult to claim any rule with a direct reference to electron density and the effect is principally energy-related. Role of Exchange Correlation Interactions. The association with frontier orbital density might be examined in more detail using analogous definitions of the presented radius, the one based on local density approximation, eqs 3 or 5. The radii estimated by the corresponding method are shown in Figure 7 (Table S5 in SI). The results by eq 1 and eqs 3 or 5

ηel =

∫∫

f (r )f (r′) dr dr ′ |r − r′|

(9)

representing respective electron−electron repulsions. The second one might be estimated from the Fukui potential107 ηv = −

dV = dN

∫

f (r ) dr r

(10)

describing the electron−nuclear attraction of frontier electron density in the effective external potential of unit charge. Note that also the half of the electrostatic potential due to the Fukui function measured at covalent radius amounts to hardness.36 Both approximations are classical Coulomb contribution to hardness. The radius formulated by LDA also represents the classical electrostatic approach interpreted as a difference between multipole moments (quadrupole and dipole). The comparison of radii estimated by αηi using (I − A)/2, ηel, and ηv is presented in Figure 8. Note that hardness kernel-

Figure 7. Radii of valence states of transition-metal cations (according to Table S1 in SI), from neutral to core, by the LDA approach. The density of Ni0 affects the Fukui function of Ni+ ion. Two radii for a Ni+ ion result from two possible configurations of Ni0: [Ar] 3d94s1 shown by a cross, and [Ar] 3d84s2 represented by a circle.

Figure 8. Ionic radii of Znn+, n = 0,...,12, with respect to different hardness approximations.

based and LDA approaches give identical results, eq 11, for spherically averaged density

are very similar, with average bivariate correlation coefficient109 of 0.964 ± 0.02. The agreement is better, 0.982 ± 0.018, if core radii are omitted, since results according to LDA are strictly monotonic with respect to all charge states. Corresponding monotonic behavior is related to the standard deviation aspect of eq 3, measuring the dispersion of the frontier density. Additionally, the Fukui function finite difference definition

αηel ≡

∫ r 2f (r)dr − (∫ rf (r)dr)2

(11)

From these examples, it is clear that fluctuations in radii, calculated according to eq 1, are caused by nonclassical terms in hardness. However, not all electronic configurations diverge depending on the selected approach, and the radii have E


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In accordance, Shannon radii for the octahedral environment are proportional to a high degree to the standard deviation from the mean distance of frontier electron density with respect to the nucleus position (eq 3). On the contrary, the discrepancy between Shannon’s collection and the results by eq 1 is mostly related to the influence of crystal potential. In addition, Shannon’s set employs statistical analysis of atomic properties in solid-state systems in contrast to isolated species in gas phase presented here. Interestingly, the predicted disparity in sizes of high- and low-spin configurations occurs with insignificant variation in ion polarizability (the ratio is very close to 1.0) with respect to spin transition, whereas hardness substantially increases for high-spin states (about 1.5 to more than 2.0 times). Hence, the spin-related divergence in particular ion radius is predominantly driven by occupied− unoccupied frontier orbital energy difference in high- or lowspin states and effect of electron pairing energy (or degree of electron repulsion). The extent predicted using hardness definition with 1/2 factor, i.e., η = (I − A)/2, correlates with radial density maximum of the highest occupied orbital, HOAO, in atoms.81,82 A correlation with the position of the maximum of radial density of the outermost atomic orbital has also been pointed out by Slater.10,11 The so-called Bragg− Slater atomic radii determine nearing between atoms. Yet, the correspondence in transition metals is mostly qualitative in contrast to main-group elements. Moreover, the relaxed radial density for the HOAO (calculated as the difference between the N-electron state in a particular spin configuration and the N − 1 ground charge state) often cannot discriminate between spin cases or may give smaller radius for high spin. Hence, electron density becomes an unreliable property to estimate transition-metal diameters. Within 3d transition-metal ions considered as a whole, only the neutral and singly oxidized atoms exhibit meaningful differences in radii. Higher oxidation states might be grouped into collections of similar size, yet without sharp partition. Assuming a very small deviation in diameter, of few percent or less (with an order of magnitude of about 0.1 Bohr), the batch of ions with almost uniform size might be selected. For instance, high-spin Cr2+ has comparable size (∼1.01 au) to Sc2+ and Zn2+, while its low-spin state matches high-spin Fe3+ (∼0.86 au). The ions discussed next should be assumed to have high spin unless specified differently. On the contrary, the low-spin Fe3+, 0.60 au, might be compared to Cr5+, Mn5+, and Ni4+, 0.62 au. Again, Fe2+ (0.95 au) is placed next to Cu2+, 0.94 au, also very close to Ni2+ and Co2+ ions estimated to 0.97 au. Yet, in low spin, it falls among Cu4+, Ni5+, Fe4+, and low-spin Mn3+, with radii 0.65 ± 0.01 au. An interesting pair is Co2+ and Ni2+, which have similar size in high-spin configuration, 0.97 au, and then again in low-spin configuration (0.80 au, which is also similar to Cr3+, V3+, and low-spin Mn2+). Although the ions might have comparable size, the related polarizabilities and hardness are usually fairly different. Despite that these measures are motivated by HSAB theory, the radii reflect Fajans’ rules102,112,113 in terms of polarizing power (ion chargeto-ion radius ratio), which is a measure of the ion charge density. Such relationship is supported by electric permittivity.114 While it would reflect the most effective overlap distance with another atom, the Pauli repulsion might affect the nearing in chemical compounds.115 Hence, ions may present different chemical properties. The size of ions might be relevant in the case of fixed site preferences in crystals, biomolecules, or

comparable range from all models. Either this might be the result of canceling between different effects or the interactions other than London dispersion are negligible at the system state in question. It appears that significant exchange correlation (Fermi correlation) effects usually occur for radii with electronic configurations with closed or half-filled subshells (because the electron repulsion is decreased). Spin State-Determined Ionic Radius. Table 3 compares the radii of high- and low-spin configurations of selected metal Table 3. Radii of High-Spin, H, and Low-Spin, L, Configurations of 3d 4 to 3d 8 Transition-Metal Ionsa ion

spin

Cr2+

L H L H L H L H L H L H L H L H L H L H

Mn3+ Mn2+ Fe3+ Fe2+ Co3+ Ni4+ Co2+ Ni3+ Ni2+

term symbol 3

H4 D0 3 H4 5 D0 2 I11/2 6 S5/2 2 I11/2 6 S5/2 1 I6 5 D4 1 I6 5 D4 1 I6 5 D4 2 H11/2 4 F9/2 2 H11/2 4 F9/2 1 G4 3 F4 5

α (au)

η (au)

4.104 3.782 1.908 1.840 4.758 3.390 1.744 1.693 3.533 3.380 1.710 1.647 1.001 0.986 3.246 3.178 1.495 1.509 2.874 2.932

0.1821 0.2688 0.2186 0.3246 0.1330 0.3383 0.2038 0.4511 0.1164 0.2688 0.1546 0.3341 0.1839 0.3878 0.2033 0.2952 0.2532 0.3599 0.2242 0.3188

R = αη (Bohr) 0.86 1.01 0.65 0.77 0.80 1.07 0.60 0.87 0.64 0.95 0.51 0.74 0.43 0.62 0.81 0.97 0.62 0.74 0.80 0.97

Shannon (Bohr) 1.22 1.10

1.04 1.22 1.17 1.47 1.03 1.15 0.91 1.23 1.41 1.06 1.13 0.93 1.30

a Shannon’s34 estimates are about √2 higher and thus correlate better with full gap, i.e., (I − A). All Shannon radii are for octahedral coordination, except square planar low-spin Ni2+.

cations. The results show very good qualitative agreement with Shannon33,34 estimates. All low-spin configurations display shorter radii than their high-spin counterparts. This complies with HSAB tendencies. The Shannon estimates fit the radius calculated with respect to full gap, i.e., 2η = I − A (Figure 9).

Figure 9. Ionic radii of high- and low-spin configurations, according to Shannon34 (octahedral coordination, except low-spin Ni2+, which is square planar) and eq 1, using 2η = I − A. F



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organometallics. However, for the occupation of flexible coordination sites, the electrostatic factors are mainly involved.

AUTHOR INFORMATION

Corresponding Author

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*E-mail: [email protected]. Tel.: +48 22 5543 600.

CONCLUSIONS The purpose of characteristic atomic radius is to represent nearing distance between atomic nuclei. Despite its ongoing popularity for applications in chemistry, physics, and engineering, yet as an outmoded model, incompatible with the modern theoretical description of the atomic structure, it usually seeks to justify a sharp physical boundary given to probabilistic electron density distribution. Owing to its classical character, as well as to the Hellmann−Feynman theorem implications on equilibrium bond lengths, it is most often linked to electrostatic or electric properties of atoms and correlated with electronic response functions such as electronegativity or ionization potential. Numerous definitions producing miscellaneous diameters primarily depend on the interaction nature in the equilibrium state. The wave−particle duality and uncertainty of electron position resulting in probability distribution make difficult setting down definite atomic boundaries; thus, such measures retain the label of an abstract concept. The characteristic atomic radius determined through energy response to changes in electron number and external potential comes as a natural consequence of atomic architecture and contrasts with the classically formulated physical atomic boundary. The spatial dependence of electron density is reinterpreted via an energy functional and through energy derivatives linked to response density and electron detachment process designating the radial limits of atomic influence. The unyielding character of near-nuclear (core) electronic charge contrasts with the behavior of long tails of monotonically decaying electron density, which is related to system ionization potential (electron detachment) as well as through response density to energy derivatives. Such features imply direct involvement of frontier orbitals’ density in affecting the internuclear nearing distance. The influence of nonclassical effects (electron correlation) on atomic size with respect to preconditioned electronic configuration of the system can be obtained from comparison of radius determined through measurable atomic properties with the local density approximation approach. Moreover, it implies how ionic radius is affected by rigid and flexible coordination sites, ligands, or crystal field, explaining the correspondence with empirical radii and chemical activity. Furthermore it evaluates a system charge density (or current density in the case of moving ions) being a principal source term of the electromagnetic field, which might affect physicochemical processes.116

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ORCID

Paweł Szarek: 0000-0002-5706-0173 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project was funded by the National Science Centre Poland, grant No UMO-2015/19/B/ST4/02718. The computations have been realized with resources of Interdisciplinary Centre for Mathematical and Computational Modelling, ICM, and Wrocław Centre for Networking and Supercomputing, WCSS.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b00221. Term symbols for ground-state electronic configurations of metal ions (Table S1); hardness for transition-metal ions (Table S2); electric dipole polarizabilities for transition-metal ions (Table S3); radii determined through the square root of polarizability and hardness product (Table S4); radii obtained from the approach based on local density approximation (Table S5) (PDF) G


Article

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