Research: Science and Education
Electronic Structure Principles and Atomic Shell Structure P. K. Chattaraj* and B. Maiti Department of Chemistry, Indian Institute of Technology, Kharagpur 721302, India; *
[email protected] The purpose of this article is to familiarize the reader with the relationship between shell structure and stability in atoms on the basis of quantities such as hardness and electronegativity, which are based on density functional theory, and polarizability. It is shown that the periodicity of these properties is framed on the principles of maximum hardness and minimum polarizability. Important chemical reactivity parameters such as electronegativity (χ) (1) and hardness (η) (2, 3) are defined within density functional theory (DFT) (4 ). Originally the concept of electronegativity was introduced by Pauling (5) as the power of an atom in a molecule to attract electrons to itself. The concept of hardness was given by Pearson (6 ) in his famous hard–soft acid–base (HSAB) principle (6, 7), which states that “hard likes hard and soft likes soft”. Acids, the acceptors of electrons, are termed hard if they do not have easily excitable outer electrons and have small sizes and high positive charges; bases, the donors of electrons, are called hard if they have low polarizabilities and high electronegativity values, are hard to oxidize, and are associated with empty orbitals of high energy. Otherwise they are termed soft. Electronegativity (8) and hardness (9) are respectively defined as the following first-order and second-order derivatives:
χ = µ = ∂E ∂n
v r
≈ I +A 2
(1)
and
∂2E η=1 2 ∂n 2
v r
∂µ =1 2 ∂n
≈ v r
I –A 2
(2)
for an n-electron system with energy E and chemical and external potentials µ and v(r), respectively. In eqs 1 and 2, I is ionization potential and A is electron affinity. Electronic Structure Principles These reactivity indices are better appreciated in terms of the associated electronic structure principles. Electronegativity difference is the major driving force behind the electron-transfer process in chemical reactions. Electrons are transferred from a species of lower electronegativity to one with higher electronegativity until the electronegativity values of the two species become equal. Within a molecule, all the constituent atoms have the same electronegativity value, which is roughly equal to the geometric mean of the electronegativities of the isolated atoms (10). Another important hardness-related principle is the maximum hardness principle (11, 12) (MHP), which states that “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible”. Formal proofs of electronegativity equalization (8), HSAB (7, 9), and maximum hardness (12) principles are provided within the purview of the DFT. It has been demonstrated that the validity of the
HSAB principle demands that of the MHP (7 ). A many-particle quantum system is completely characterized by n and v(r). While χ and η measure the response of the system when n changes at fixed v(r), the polarizability (α) measures the response of the system for the variation of (v(r)) at fixed n when a weak electric field is the source of v(r) in addition to that arising out of a set of nuclei. Based on the inverse relationship (13) between α and η, a minimum polarizability principle (MPP) has been proposed as “the natural direction of evolution of any system is towards a state of minimum polarizability” (14 ). Validity of both MHP and MPP has been tested in various physicochemical process (3, 12, 14–18). The maximum hardness and minimum polarizability criteria complement the minimum energy criterion for stability. In general, a stable state (minimum energy configuration) or a favorable process is associated with the maximum hardness and minimum polarizability and a transition state with the minimum hardness and maximum polarizability. A molecule at equilibrium geometry possesses maximum hardness and minimum polarizability values when compared with the corresponding values for any other geometry obtained through a non-totally symmetric distortion. In the internal rotation process the most stable isomer is associated with the maximum η and the minimum α value, and the least stable is associated with the minimum η and the maximum α value. For several chemical reactions it has been observed that the reaction proceeds in the direction that produces the hardest and least polarizable species (3, 12, 15). Since a system (atom, ion, molecule, etc.) is generally more reactive in its excited electronic state than in the corresponding ground state, it was anticipated and confirmed through ab initio calculations for atoms, ions, two-state ensembles, and molecules (14, 16 ) that the system is hardest and least polarizable in its ground state. These results hold good in a dynamic situation as well (14 ). It has also been demonstrated (17 ) that η is a reliable diagnostic in analyzing the chaotic dynamics of Rydberg atoms. Ab initio and DFT calculations show that the symmetryallowed conrotatory transition state is of lower energy, harder, and less polarizable than the corresponding symmetryforbidden disrotatory stationary point associated with the electrocyclic isomerization of cyclobutene to cis-butadiene (18). Both stationary points possess higher energy and α values and smaller η values than the reactant and the product of this concerted stereospecific reaction. Chemical bond making, bond breaking, and electron transfer in solution have been analyzed in the light of the MHP (19). As concrete examples, the making and breaking of C–C, C–N, C–O, and C–S bonds in various organic compounds vis-à-vis the validity of the MHP have been considered. Electron transfer free energy for the creation of the cation and anion from their conjugate radical is given by the absolute hardness of the radical .
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Research: Science and Education
Atomic Shell Structure One of the most brilliant discoveries in chemistry is the concept of chemical periodicity and atomic shell structure, originally proposed by Mendeleev and later modified after Moseley’s work to state the periodic law as “The properties of chemical elements and their compounds are periodic functions of the atomic numbers of the elements” (20). Atoms with completely filled shells and subshells possess extra stability in comparison to their open-shell neighbors. This idea prompted Lewis (21) and Langmuir (22) to postulate the octet rule and Sidgwick (23) to put forward his 18-electron rule for the transition elements. Similar physicochemical behavior of atoms present in the same group of the periodic table is the hallmark of chemical periodicity, which can be understood by applying a simple aufbau principle to generate the necessary electronic configurations in terms of the orbitals calculated even at the SCF level (24 ). Properties such as ionization energies, electron affinities, and atomic radii exhibit discernible periodicity (24 ). From the MHP and the MPP it is expected that the atoms with closed-shell or closedsubshell structures will have large hardness (25) and small polarizability values. Therefore, hardness and polarizability can be used as indices for stability and reactivity (instability), respectively. To verify this prognosis we analyze the periodic behavior of these reactivity parameters below. For this purpose χ and η values are taken from Goycoolea et al. (26 ) except for H, He, and Rb, which are taken from Robles and Bartolotti (27). The values from Goycoolea et al. were obtained from a selfinteraction-corrected DFT calculation including a correlation functional. The Robles and Bartolotti values were generated through a spin-polarized DFT with the Gunnarson– Lundqvist exchange-correlation functional. In ref 26, H and He values are not available and Rb values appear to be erroneous. The polarizability values are from ref 28.
Figure 1. Plot of hardness vs atomic number (1 ≤ Z ≤ 56).
Figure 2. Plot of electrophilicity index vs atomic number (1 ≤ Z ≤ 56).
Results, Discussion, and Summary Figure 1 depicts the periodicity in atomic hardness values plotted against the atomic number (Z ). In any period, the alkali metal is the softest and the noble gas atom is the hardest. In general, hardness increases along a period and decreases along a group. Elements with completely filled subshells also exhibit local maxima in their hardness in most cases. Parr et al. (29) defined the electrophilicity index, ω, as µ2/2η. The plot of the electrophilicity index (ω = µ2/2η = χ2/2η) versus Z is presented in Figure 2. Required χ and η values are taken from refs 26 and 27 . Whereas in Figure 1 the maximum hardness values imply maximum stability, in Figure 2 the maximum values of χ2/2η imply maximum reactivity toward an electrophilic attack. As expected, the most electronegative element in a group possesses maximum electrophilicity, and conversely. The periodic behavior of this quantity is easily discernible. The reciprocal of the electrophilicity index can be used as an index of the propensity of the nucleophilic attack. It was shown earlier (30) that for the main group elements belonging to the same group, the ratio of electronegativity to hardness (χ/η) is more or less constant but for the lightest ones. This implies that electronegativity can be considered a good measure of relative electrophilicity
812
Figure 3. Plot of polarizability vs atomic number (1 ≤ Z ≤ 56).
for elements within the same group. Figure 2 corroborates this. It may also be noted that ω F > ωCl and χF > χ Cl, but AF < ACl. Atomic polarizability values are plotted versus Z in Figure 3. It is heartening to note that in any period the alkali metal is the most polarizable and the noble gas is the least polarizable, in conformity with the MPP. In general polarizability decreases along a period and increases along a group, a signature of the inverse relation between η and α. However, except for in a few elements such as Pd, subshell structure is not very prominent in this figure, unlike the corresponding hardness behavior shown in Figure 1. It is interesting to note that η decreases and α increases in the order He, Ne, Ar, Kr, Xe. Electron
Journal of Chemical Education • Vol. 78 No. 6 June 2001 • JChemEd.chem.wisc.edu
Research: Science and Education
affinities of these atoms, which are otherwise very difficult to obtain, can be estimated from their η values. In summary, an atom with a closed-shell structure is the most stable, hardest, and least polarizable among all the atoms in a given period, as expected from the principles of maximum hardness and minimum polarizability. Hardness, electrophilicity index, and polarizability show beautiful periodic behavior across the periodic table. Acknowledgment
13.
14.
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