Most Probable Distance between the Nucleus and HOMO Electron

Article pubs.acs.org/JPCA

Most Probable Distance between the Nucleus and HOMO Electron: The Latent Meaning of Atomic Radius from the Product of Chemical Hardness and Polarizability Paweł Szarek* and Wojciech Grochala Center for New Technologies, University of Warsaw, Ż wirki i Wigury 93, 02089 Warsaw, Poland S Supporting Information *

ABSTRACT: The simple relationship between size of an atom, the Pearson hardness, and electronic polarizability is described. The estimated atomic radius correlates well with experimental as well as theoretical covalent radii reported in the literature. Furthermore, the direct connection of atomic radius to HOMO electron density and important notions of conceptual DFT (such as frontier molecular orbitals and Fukui function) has been shown and interpreted. The radial maximum of HOMO density distribution at (αη)1/2 minimizes the system energy. Eventually, the knowledge of the Fukui function of an atom is sufficient to estimate its electronic polarizability, chemical potential, and hardness.

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the nucleus at this fixed distance. The situation becomes more complex for heavier atoms, for which the size of an atom is determined by collective arrangement of all electronic shells. The theoretical estimates of empirical characteristic distance, called covalent radii, as defined for neutral atoms, have been derived in the past in density functional theory (DFT) using chemical and classical electrostatic potential,16,17 or hardness18 and Fukui function.19,20 The characteristic radius falls under categories21 of distances at which atoms start to measurably interact with other atoms the effective atomic radius (usually referred to as the van der Waals radius22,23). However, as pointed out in a recent study,21 the van der Waals radius determined from solids or molecules may actually be equal to or smaller than the mean value of r of the outermost shell (as proposed by Slater24) for some elements, which seems to be incorrect. The second type, the valence radius, corresponds to the valence region where most of the valence charge is enclosed (as those by Bragg25 and Slater24). The last one defines the “heart” of an atom, which may be called a core radius.26 The atomic radii have been correlated with various atomic properties, associated with the atomic charge distribution, and therefore may be determined with different origin: empirical data, electrostatic potential or electron density related, DFT Kohn−Sham approaches, or classical turning point formulations. The sets of experimental data, such as crystallographic as well as theoretical in silico results were used to define empirical

INTRODUCTION The maximum hardness principle (MHP) formulated by Pearson1 and rigorously proved by Parr and co-workers,2−4 states that the chemical system at a given temperature should evolve to a configuration of maximum absolute hardness, under a constraint of constant nuclear, external, and chemical potentials. One useful reformulation of the MHP is the minimum polarizability principle (MPP) proposed by Ghanty and Ghosh5 and later advocated by Chattaraj et al.6 Neglecting the knowledge on electron wave function and quantum mechanics, the intuitive realization of MHP, treated as a fundamental law of nature (and despite examples of its violation7−10), should lead to electron falling onto nucleus and, in the case of moleculeseven as simple as dinuclearto nuclear fusion. Of course, the single atoms have their electrons away and dispersed, and a pair of atoms in a molecule settles at some finite bond length. It is thus tempting to address the question: how keeping a distance (between nucleus and electron or between two nuclei) harmonizes with MHP? Here we will focus on the first of these problems, namely that of the most probable distance of an electron from a nucleus not being null. This problem is obviously understood well on the grounds of quantum mechanics, but we aim at getting additional insight using the hardness/softness formalism (the whole conceptual DFT field is introduced through review articles in refs 11−15). For a hydrogen atom, in its ground state, the most probable distance between the proton and the electron is known as the Bohr radius, a0 and, from quantum mechanics, the average distance is about 1.5a0. Using a0, the electronic energy of a hydrogen atom might be determined (even within classical mechanics), although the electron is not rigidly kept away from © 2014 American Chemical Society

Received: July 24, 2014 Revised: September 27, 2014 Published: October 6, 2014 10281

dx.doi.org/10.1021/jp507423p | J. Phys. Chem. A 2014, 118, 10281−10287

The Journal of Physical Chemistry A

Article

core and outer shells, correlating very well with ionic and covalent radii.

characteristic radii of atoms. Those include the classical radii by Pauling22,23 and Slater24 or more recently by Cordero et al.27 and revised radii by Pyykkö et al.28 Next to the empirical, the electrostatic criteria have been the most popular and explored among theoretical definitions of characteristic radii. Gordy29 has recognized covalent radius of a single bond as a distance, at which the electrostatic potential felt by an electron matches electronegativity. Later, Politzer et al.16 have provided a theoretical justification and calculated the distance, where constant chemical potential of a system at equilibrium equals classical electrostatic potential. The generalization19 of the previous method shows also that half of the electrostatic potential due to the Fukui function, at distance close to the covalent radius, equals hardness. Ghanty and Ghosh30 suggested a method for finding the neutral atom radius using the effective LUMO charge density Coulomb potential and electrostatic potential of the atom. Sen and Politzer31,32 have investigated the electrostatic potential curve and estimated the negative ion radii from the positions of the curve minima. An analogous approach has been utilized by Ghanty and Ghosh,33 who applied it to predict the atomic as well as ionic radii within spin-polarized Kohn−Sham theory calculations of neutral atom. Yang and Davidson proposed radii, for which the Slater average potential felt by electron equals ionization potential.34 Apart from the empirical atomic radii, in his paper, Slater24 has also pointed out a good correlation of the set with the distance of the maximum radial charge density of outermost shells of atoms. These so-called Bragg−Slater atomic radii have been found to characterize inner cores of electrons of atoms,24,25,35,36 hence determining the nearing between atoms. Using a small electron density criterion, Boyd defined37 the size of an atom correlating with Pauling values. A similar attempt to define radii of atoms and ions has been made by Deb et al.38 They proposed the universal density value (0.008 714), derived from the ratio of the Dirac exchange constant to the Thomas Fermi kinetic energy constant, as a beacon for finding a characteristic radius of an atom. In general, the electrostatic approaches may be considered as simple approximations to the problem but justified by solutions of the Euler−Lagrange equation. The numerical solutions of Kohn−Sham equations lead to radial distances associated with extrema of electrostatic potential.39 In other work, the radii evolve from atomic density profiles obtained by the selfconsistent solution of quadratic Euler−Lagrange equations.40 At the classical turning points, the electronic state energy coincides with the potential energy. In the classical harmonic oscillator the particle spends most of its time at the turning points, where it has the lowest kinetic energy. Thus, the atomic radii, correlating with van der Waals results, were evaluated from the classical turning point equation, at the position where the Slater average potential felt by an electron equals the negative of the first ionization potential of an atom.34 The previous definition might be rewritten as a radius based on the atomic hardness.18 A scale of orbital radii,41 based on the classical turning point and calculated using self-interaction corrected Kohn−Sham DFT with local spin-density approximation for exchange and correlation, has also been proposed. The size of atoms and ions has also been defined on the basis of not positive-definite kinetic energy density.42 Using the interface region (zero kinetic energy density) separating the negative regions (classically forbidden) and enclosing positive electronic drop regions, one may divide the atom into inner

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CHARACTERISTIC RADIUS ROLE BEYOND AN INTERATOMIC SEPARATION The atomic radii were defined initially to reproduce experimental bond lengths. They were also found useful in modeling diffusion, defects, ion conductivity, site preferences, surface tension, and close packing models in crystallography, among other things, all applications where size matters. Moreover, within the conceptual density functional theory formalism, one can attempt to relate electronic properties of atoms and/or molecules to their geometrical parameters. As a result, the classical “macroscopic” theories are used to explain atomic behavior, loosely based on the correspondence principle. For example, when an atom is treated as a metallic conducting sphere of radius r, its electric capacity has been identified18,43−47 with the softness48 (i.e., inverse of hardness) η = (4πε0r )−1

(1)

Likewise, using a classical expression for the energy needed to charge a conducting sphere, Hati and Datta49 proposed

1 ⎛⎜ C ⎞⎟ 2⎝α⎠

1/3

η=

(2)

In fact, a year earlier, Ghanty and Ghosh had already suggested a correlation between cube root of polarizability and softness, demonstrating it numerically for atoms, molecules, and clusters.50 The cubic relationship between polarizability and softness has been derived in ref 18 as well. Besides size, Ghanty and Ghosh with collaborators have also investigated correlations of polarizability with other quantities, such as activation energy of proton transfer, aggregation number for molecular clusters, or ionization potential and some dependencies for excited state polarizabilities.51−54 Combining (1) and (2) we arrive at

α = C(2πε0r )3

(3)

where C is some empirical constant and α is the electronic polarizability. Other simplified equations30 yielda α = kα

∫ ρHOMO(r)r 3 dr = kα⟨r 3⟩HOMO

(4)

and η = kη

⟨r 2⟩HOMO ⟨r 3⟩HOMO

(5)

where, in the case of the hydrogen atom, the integrals, which correspond to the exact density of the 1s orbital, are 3 3 ⎧ ⎪⟨r ⟩HOMO = 7.5a 0 ⎨ ⎪ 2 2 ⎩⟨r ⟩HOMO = 3a0

(6)

The empirical constants kα and kη (eqs 4 and 5) are 1.48 and 0.50, respectively, for a large set of elements; however, for the hydrogen atom the best estimate yields the values of kα ≅ 0.57, kη ≅ 0.59 (eq 17 below). Thus, according to eqs 5 and 6 the atomic hardness scales as the inverse of the atomic radius and according to eq 3 the polarizability is proportional to the cubic radius (a “volume”), Figure 1. 10282



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α=S

∫

r 2 cos2 θf (r ) dr

(9)

(for the more general forms, which are applicable to molecules, see refs 55 and 56) where S is the global softness, θ is the angular dependence of the Fukui function, and f(r) is the Fukui function (see the comment57 on the conversion to SI units). Combining (7) and (9) and putting s(r) = Sf(r) for local softness, we rewrite

∫

r ⃗·α(⃗ r ) dr = r2

∫ s (r ) d r = S ∫ f (r ) d r = S

(10)

58−61

The Fukui function, for simplicity, particularly in the case of the hydrogen atom might be approximated by the molecular shape function:62−64 Figure 1. Hardness and polarizability scaling with radial distance.

⎡⎛ ⎛ ∂μ ⎞ ⎤ ∂ρ(r ) ⎞ ρ (r ) = σ (r ) f (r ) = ⎢⎜ ⎟ =⎜ ⎟ ⎥≅ ⎢⎣⎝ ∂N ⎠v ⎝ ∂v(r ) ⎠ N ⎥⎦ N

The existence of a relationship between the stipulated “classical” radius and the hardness (connected to the energy “gap” between occupied and unoccupied orbitals), also suggests close links of the characteristic radius with some other properties of Fukui’s frontier molecular orbitals.

(11)

In the above, μ is the chemical potential, v(r) is the external potential, and σ(r) is the molecular shape function. It follows from relations 10 and 11 that

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RESULTS AND DISCUSSION We assume here a Bohr atom-like model, where electrons are represented by shells of ultrathin surface charge density (Figure 2). In general, this charge density would be associated not only

[ r ⃗·α(⃗ r )] = r 2s(r ) = r 2Sf (r ) ≅ r 2S

ρ (r ) N

(12)

For the sake of our shell model we introduce the radial distribution function, D, which gives the probability that the electron will be found anywhere, in the shell of radius r: D(r ) = 4πr 2ρ(r )

(13)

Because the whole density is uniformly distributed over the infinitesimally thin surface of radius r, around the nucleus, and yields the exact polarizability of an atom, Figure 2 (the polarizable charge on a surface of conducting sphere), we may write 2 ⎧ ⎪ N = 4πr ρ(r ) ⎨ ⎪ 2 ⎩ α = 4πr [ r ⃗·α(⃗ r )]

Figure 2. Hypothetical model radial distribution function of electron density and local polarization.

Using (12) and conditions in (14) we find the value of r, to be (see also comment in ref 44)

with global hardness of an atom in its ground state but also with its exact electronic polarizability. Of course, there would be no preference for any radius and intuitively r = 0 due to MHP realization might be anticipated. Let us introduce the following approximations to subsequently perform analysis of the problem. The electronic dipole polarization tensor, for atoms with constraints on frozen nuclear positions may be evaluated from55 α=

∫

r ⃗·α(⃗ r ) dr

α 1 2 =S r ⇒r= 2 4πr 4πr 2

αS − 1 =

αη

(15)

As a test of our result (15), we show three estimates of r for the hydrogen atom, from classical mechanics (16), conceptual DFT (17), and experiment (18): ⎧ 1 I − A I ≫A I ⎯⎯⎯⎯→ ≅ ⎪η = = S 2 2 ⎪ ⎪ in au 2 2 ⎨I = − 1 Z e =⇒ r ≥ ⎪ n2 2a0 ⎪ ⎪ α ≅ 4πε a 3 ⎩ 0 0

(7)

where α is the electronic polarizability, α⃗ (r) is the local polarization vector, and r ⃗ is the pointing vector. The local polarization vector55 is defined as ⎛ ∂ρ(r ) ⎞ αi(r ) = −⎜ ⎟ ⎝ ∂i ⎠ N

(14)

a03 = 0.5a0 4a0

(16)

i ∈ {x , y , z} (8)

r=

with i the electric field in the direction of the ith tensor, ρ(r) the electron density, and N the electron number. Alternatively, with the local density approximation56 the electronic polarizability is

r= 10283

kα⟨r 3⟩k η

αexp

⟨r 2⟩ = ⟨r 3⟩

(Iexp − Aexp) 2

kαk η3a0 2 = 1.004a0

= 1.03a0

(17)

(18)



Article

where η is the chemical hardness, I is the ionization potential, A is the electron affinity, Z is the nuclear charge, e is the elementary charge, a0 is the Bohr radius, n is the principal quantum number, ε0 is the vacuum permittivity, and αexp, Iexp, and Aexp are experimental values of respective quantities. The justification of approximation for α in (16) is given in the Supporting Information. The agreement of the result from eq 15 with the Bohr model of H atom (i.e., r = a0) is quite good for both conceptual DFT and experimental estimates of r. Encouraged by these results, we have tested applicability of eq 15 to other elements (and despite the large differences of the Fukui functions and other properties for diverse atoms). Figure 3 illustrates how the atomic radius correlates with the latest

electrons, of the universal functional without the classical Coulomb repulsion energy term, are both null:20 δG[ρ] δρ(r )

=0 (19)

r ≈ αη

⎛ ∂ ⎡ δG[ρ] ⎤⎞ ⎜ ⎢ ⎥⎟ ⎝ ∂N ⎣ δρ(r ) ⎦⎠v

=0 (20)

r ≈ αη

Although the conditions of fixed μ and v, associated with the MHP, are never met, at fixed N, μ, and v, the second-order expression for E[N,v] shows that the larger the hardness, the lower the energy:67 E[N ,v] = Nμ − −

1 2

1 2 Nη+ 2

∫ v(r)[ρ(r) − Nf (r)] dr

∬ ρ(r) ρ(r′) ω(r ,r′) dr dr′

(21)

From eqs 19−21 it follows that adopting the electronic distribution of the HOMO with the radial maximum of density at (αη)1/2 minimizes the system energy.68,69 Importantly, it has been proposed before that the density of the HOMO can be used as a good approximation of the Fukui function.70 The knowledge of the Fukui function, HOMO density, and electrostatic potential of an atom is sufficient to estimate its electronic polarizability, chemical potential, and hardness, because - the chemical potential is equal to the electrostatic potential at the characteristic radius μ = Φ(r ≈ (αη)1/2),16,17 - the hardness is equal to the electrostatic potential of the Fukui function (electrostatic potential due to a distribution of charge equal to the Fukui function) at the same point,20 - and the “covalent” radius might be approximated by the distance to the maximum radial probability of the HOMO density (or square root of polarizability and hardness product). Moreover, polarizability and hardness may be estimated by knowing solely the Fukui function, because eq 15 may be rewritten as

Figure 3. Comparison of atomic radii by eq 15 with Cordero et al.,27 in Å.

experimental estimates of the covalent radius from Cordero et al.27 (i.e., the corrected Slater radii). It turns out that eq 15 performs quite well for a broad set of atoms (from H to Cs) with the linear coefficient of 1.07 and the coefficient of determination of 0.86.65 Hence, one may derive the size of an atom while knowing its Pearson hardness and electronic polarizability using eq 15. Indeed, it makes sense that the atomic radius should be related in some way to its polarizability and hardness. As pointed out before by Ghanty and Ghosh,33 “It is thus of interest to investigate the prediction of atomic and ionic radii as well through the same calculation as used for obtaining hardness and polarizability”. We notice that Vela and Gázquez56 and later Ghanty and Ghosh33 have come up with similar, but not identical, relationships66 to that shown in our eq 15. The r (=(αη)1/2) in the hydrogen atom is associated with the most probable distance between the proton and the electron. Similarly, in heavier atoms r points close (see Supporting Information for details) to the maximum of radial probability of HOMO density. In other words, the size of an atom, as defined by r from eq 15, corresponds to the most probable distance between the HOMO electron and the nucleus. Furthermore, at the characteristic distance r the derivatives with respect to the local electron density distribution and to the number of

r=

αη =

∫ r 2f (r) dr

(22)

2

where r comes as an expectation value for the Fukui function. The hardness itself may be estimated from the Fukui function alone, with a formula resembling the Coulomb repulsion term of the universal functional,71,14 namely η=

∬

f (r ) f (r′) dr dr ′ |r − r′|

(23)

In a more general case, following the derivation by Vela and Gázquez,56 we have r=

αη =

∫∫

∫

r 2f (r ) dr − ( rf (r ) dr )2

(24)

which might be interpreted as the standard deviation of the Fukui function spread out around the mean. Especially from the simpler formula (22), which defines the radius through the expectation value of the radial FMO density distribution, we 10284



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might conclude the atomic radius to be the center of mass of (radial) frontier electron density. Taking into account the importance of the incomplete electron-cloud after and before determining the shape of molecules,72−75 it implies atomic f lexibility (i.e., off-center ions) through the Hellmann− Feynman force concept, because the Fukui function describes the preference for rearrangements of the electron density (after variation in electron number and in light of the sensitivity of a system’s chemical potential to an external perturbation at a particular point). The evolution of the radius from the neutral atom to the ion might help to interpret structural changes associated with normal modes of lowest frequency in the molecules or second-order Jahn−Teller distortions. In principle, the use of exact Fukui function should work well with the formulas by eq 22 or eq 24. However, the characteristic radii obtained by eq 22 or eq 24, using the simplest approximation to the Fukui function, namely 1/2(ρN+1 − ρN−1), though reproducing trends of (αη)1/2, are not in good quantitative agreement with theoretical or experimental values, especially for heavier atoms. This might be attributed to the oversimplified shape of the Fukui function assumed above. Despite some ambiguity with indicating the precise shape of the Fukui function, relations 22 and 24 give a simple and promising way for determining atomic and ion polarizability in at least a qualitative manner. They also provide good tests for the Fukui function when experimental data are available. As a referee correctly reminded us, a systematic study of correlations between the hardness, polarizability, and radius for a large set of organic molecules by Blair and Thakkar has recently appeared.76 Among many relationships, the authors present correlation between polarizability, volume, and modified hardness: α ≈ 0.031201V/η. The formula may be rewritten r = 1.97 × (αη)1/3 to compare with our eq 15. The cubic-root formula, when applied to atoms gives more than 1.5 larger radii as compared with results of this work or the work of Slater and Cordero et al. The differences are greater for small atoms, and although they decrease in groups, they increase in periods. The radii from the cube-root expression are substantially more than twice as large as radii for H and He and around 2 times for the second period. The cube-root radii oscillate between 0.5 and 1.5 times the van der Waals radii. It thus seems that the simple formula derived by us here on the basis of a small number of assumptions yields a better agreement with experiment.

experimentally and theoretically, given that two of the three parameters are known.

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ASSOCIATED CONTENT

S Supporting Information *

Polarizability of uniformly charged sphere. Correlation of (αη)1/2 with the maximum of radial probability of HOMO density in atoms. Comparison of atomic radii with Slater and Pyykkö and Atsumi values. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*P. Szarek. Phone: +48 22 554 0813. E-mail: pawel.szarek@ cent.uw.edu.pl. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors express their gratitude to Dr. Tapan K. Ghanty and to Dr. hab. Leszek Stolarczyk for their valuable comments and suggestions. The financial aid from Polish National Science Center through FUGA postdoctoral internship (UMO-2013/ 08/S/ST3/00554) is acknowledged. We dedicate this work to Prof. Andrzej W. Sokalski on the occasion of his birthday in recognition of his contribution to theoretical chemistry of catalysts.

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ADDITIONAL NOTES The formulas derived in this work in eqs 4−24 are in atomic units. b During the proof review, a new work on atomic radii by Pekka Pyykkö appeared: dx.doi.org/10.1021/jp5065819. a

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REFERENCES

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CONCLUSIONS The new definition for the classical covalent radius of an atom has been proposed here from the conceptual density functional theory. The radius is expressed as a square root of the product of electronic polarizability and hardness. The definition may be applied universally to neutral atoms as well as to extended to ions. Furthermore, it has been shown that such a value of the covalent radius nearly coincides with the maximum radial probability density of the HOMO electrons. The presented relations for characteristic radii further support the role of the HOMO density as a good approximation to the Fukui function. Moreover, we argue that the system energy is minimized for electronic distribution by maximizing this characteristic radius. Hence it is one possible manifestation of MHP with respect to the electron density distribution. The new concept of the atomic radius opens an alternative way for crude estimation of the electronic polarizability and Pearson hardness, both 10285



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