Derek W. Smnh I The 'Anomalous' Ionization Potential of Univenih, of Waikato 1

Hamilton. New Zealand I Bismuth

Periodic variations in the ionization potentials (I.P!s) of atoms are of great importance in understanding the com- parative chemistry of the elements, and these variations are discussed a t some length in all texts on inorganic and general chemistry. The general tendency towards lower I.P.'s as we descend a group is readily explained, and devia- tions from this hehavior (e.g., the higher LP. of gallium compared with aluminium) can he understood in terms of screening effects and effective nuclear charges. Similarly, the tendency towards higher LP.'s as we pass horizontally along a period is well understood, and discontinuities (such as the higher I.P. of nitrogen compared with oxygen) can he readily explained. These points are adequately dealt with in most standard texts.

The purpose of this paper is to draw attention to a less well known anomaly in atomic I.P.'s, and to offer an expla- nation. The first LP.'s of lead, hismuth, and polonium are, respectively,' 7.416,7.289, and 8.48 eV. The student is enti- tled to ask why hismuth should have a lower I.P. than lead, in conflict with predictions based on effective nuclear charges. Had no experimental value for the I.P. of hismuth been available, we might have offered estimates in the range 8.0-8.5 eV; many hooks actually quote such values.

Before we can attempt to explain this anomaly, we have to make quite clear what the I.P. measures. To a fair ap- moximation (Koo~man's theorem), the I.P. is the negative bf the orbital energy of the relevant electron, and I.PT'S are commonlv used as a measure of the bind in^ energies of the outermost electrons. This is not strictly true; the experi- mental I.P. is equal to the difference between the total en- ergy of the atom A and the total energy of the ion A+, hoth in their ground states. If it were possible to obtain accurate solutions to the Schrodinger equation for large atoms and ions, the only rigorous way of calculating an LP. would he to determine the total enereies of A and A+. and take the difference. This must be borne in mind when making quali- tative rationalizations. The conventional arguments which we apply to the discussion of I.P!s imply a Hamiltonian consisting of a one-electron operator (taking account of ki- netic energy and electron-nucleus attraction) and a two- electron operator (which deals with coulomb and exchange interactions between electrons). But an accurate calcula- tion of the total enerw of an atom or ion requires another component in the ~ G i l t o n i a n ; the spin-orb& coupling op- erator. If this he applied as a perturbation to the Russell- Saunders terms, which are eigenfunctions of the combined one- and two-electron operators, its effect (to first-order) is to split the degeneracy of terms having non-zero values of hoth L and S. For example, the ground state term 3P of an ns2np2 atom (such as carbon) is split as shown in Figure 1, into states labelled 3Po, 3P1, and 3Pz, the suhscript indicat- ing the total quantum number J. Figure 1 shows only the first-order effect of spin-orbit coupling; the relative ener- gies of the J states will he further affected by second-order interactions. In the case of an ns2np4 atom (such as oxy- gen) the splitting of the ground state 3P term is inverted relative to Figure 1; X (the spin-orbit coupling constant) is negative for species having a more than half-filled suhshell. Thus the ground state of the C atom is 3Po while that of the 0 atom is 3Pz; in either case, the ground state energy is lower than would have been the case in the ahsence of spin-

Fgure 1. Spliiing by spin-wbn coupling of ? term of an nJZnp2 atom.

Figure 2. Effect Of spin-orbn wupling on the ionization energies of Pb, Bi, and PO.

orbit coupling (the 'center of gravity' in Figure 1). For light atoms such as C and 0, spin-orbit coupling is relatively un- important; the total spread of the 3 P ~ states in carbon is only 0.005 eV. But for heavy atoms, the spin-orbit coupling constant is very large; the spread of the 3 P ~ states for the lead atom is 1.32 eV. Thus the effects of spin-orbit coupling in heavy atoms and their ions are likely to affect their LP.'s significantly.

In the Ph atom, we estimate that the ground state 3Po is stabilized to the extent of 1.05 eV relative to the 3P term in the ahsence of spin-orbit coupling. This figure is obtained from the observed relative energies of the 3 P ~ states: and

'Liebman, J. F., J. CHEM. EDUC., 50,831 (1973). %These, and all other atomic spectral data, are taken from

Moore, C. E., "Tables of Atomic Energy Levels," National Bureau of Standards Circular 467, Vol. 3, Washington, 1956.

578 / Journal of Chemhl Education

the 'center of gravity rule,' that the energy of the 3P term is the weiehted mean of the enereies of the 3P.r states. each state having a weight of W + 7, its total degenerac;. The Pb+ ion (6s26u) has the mound Russell-Saunders te rn %P. which is split into 2 ~ 1 1 2 i d 2P312 by spin-orbit coupling: The around state 2P1/2 is stabilized bv 1.16 eV relative to the 3~ term at the center of gravity. ~ h u s the ground states of both atom and ion are about eauallv stabilized bv spin- . - . . orbit coupling, and the LP. is not anomalous.

The ground state Hussell-Saunders term for the Bi atom is 4S; having zero orbital angular momentum this is unaf- fected (at least to first-order) by spin-orbit coupling. But the Bi+ ion (isoelectronic with Pb) has a 3P ground term, split by spin-orbit coupling. The ground state 3Po is stabi- lized by 1.72 eV relative to the center of gravity which rep- resents the energy of the 3P term in the absence of spin- orbit coupling. Thus the I.P. of bismuth is much lower than

would have been expected if spin-orbit coupling were not important.

The ground state Russell-Saunders term 3P of the Po atom is split by spin-orbit coupling; the ground state 3Pz is stabilized by 0.80 eV relative to the center of gravity. Since the giound state 4S of the Po+ ion is not split by spin-orbit coupling, the I.P. of Po is larger than might have been ex- pected. These points are illustrated in Figure 2.

The second I.P. of bismuth is 16.68 eV, more or less, aa might have been expected without introducing apin-orbit coupling. The relatively large increase between the first and second I.P.'s, arising from the spin-orbit stabilization of the atomic ground state, is of some chemical significance since it may help to stabilize the curious Bi+ ion which ap- pears to exist3 in Bi10Hf3Clls.

"Friedman, R. M., and Corbett, J. D., Chem. Comm., 422, (1971); Inorg. Chem., 12,1134 (1973).

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