Robin M. Hochstrasser University
of
Pennsylvania Philodelphio
I
I
The Ener~ies of the Electronic Configurations of Tramition Metals
The concept of the electronic configuration of an atom has a central role inchemicalperiodicity, for the obvious reason that atoms with similar configurations enter into valence theories in a similar way and are therefore expected to have similar chemical properties. The usual simplified versions of atomic structure characteristically avoid the details of the relationship between electronic configurations and spectroscopic observable~. Furthermore, it appears to this author that there is an apparent hiatus between atomic spectroscopy and theories of chemical bonding in the manner that these are often introduced in elementary courses. The purpose of this article is therefore twofold: first, to provide a precise yet non-mathematical definition of an electronic configuration and to discuss its physical structure in terms of the vector model of the atom (1, 2 ) ; second, to show how the energies of configurations are estimated from experimental spectroscopy. The motivation for this presentation arose when, a t a recent gathering of college teachers, the following question was posed: "Why do the 4s orbitals fill b e fore the 3d orbitals, yet the 4s electrons ionize before the 3d electrons in the transition elements?" The subsequent discussion lasted some hours-notwithstanding the supposed "freshman" character of the question. The wide variety of explanations that were given emphasized the need for an article dealing with the substance of this and similar questions. Accordingly, this article is focused on configurations involving delectrons. The Total Electronic Energy of a n Atom
The electronic energy derives from kinetic energy and the numerous forces on the electrons in the atom. For a system of spinning electrons in the neighborhood of a positively charged nucleus the following interactions must be considered (in order of importance of the energy contribution): (a) electron-nucleus attractive interaction, (b) electron-electron repulsive interaction, (c) interaction between spin motion and orbital motion for each electron, and (d) interaction between the spin motions of the electrons. I n addition to the above interactions for orbiting and spinning charged particles the effect of external perturbations, such as that obtained by placing the atom in a charged environment or an external electric or magnetic field, is also to change the energy. For our purposes the influence of the magnetic field will have importance; we therefore label this interaction (e). For hydrogen-like atoms, interaction (a) is by far the principal factor in determining the energy. The electron may take up any one of an infinite number of wave 154
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characteristics, the energy of each being known rigorously from theory. As is well known, the sets of angular functions having the same type (a) interactions are identified by their principal quantum number n (n = 1, 2, 3, . . . etc.), and the electrostatic energy is proportional to n P . The radial parts of these characteristics have (n - 1) nodes along any line radiating from the nucleus. Each angular function (a spherical harmonic) is characterized by integers C and mc which are identifiable with the orbital angular momentum and its projection along the z-axis, respectively (as determined, for example, by a magnetic field). The possible values for these quantum numbers are G = 0,1, . . . (n-1) ; mc = 0, + 1, *2, . . . +d. The spin angular momentum is 1) (h/2a), and the quantumnumber is simply I/%. The spin projections on a z-axis are m, $ = +1/2. Fundamental theory indicates that the square of the orbital angular momentum it(d 1)l. and any one component [say, mC]are simulta~eousobi servables: in other words these quantities or their quantum mechanical operators are constants of the motion of one electron in a central field of force. To develop an accurate description of the spectroscopic observable we start with the notion of a con$gurution for which the quantum numbers (n, G) are known for each electron. I n the ns, np, nd, . . . etc. configurations of the hydrogen atom the necessity of spatial quantization of spin and orbital angular momentum in~posesa restriction on the possible number of spectroscopic levels. The spin quantum number is so the total angular momentum auantum number j (= L 8) (we could be using lower case j, 1, and s for the one electron problem) is restricted to only two values [(d '/z)and (G '/z - I)] when G is different from zero, and to one value when Gis zero. The notion of spatial quantization is easily visualized with the aid of the vector model of the atom. Spectroscopic levels are normally denoted by a hold face term symbol 5, P, D, . . . etc., corresponding to the &value of the confignration, with the spin multiplicity as a presuperscript, and the J-value as a subscript. The spin multiplicity is 25 1 where S is the total spin quantum number which is always for hydrogen-like atoms. Usually the shell number (n) of the configuration precedes the term symbol. For example the configuration 3d provides the levels 32D6,2 and 32D3,2; these are the observable levels which are a t slightly different energies due to interaction (c). Each level is degenerate and the spatial quantization of the J value permits only 2 J 1 values of m ~ .The degeneracy can be lifted by a magnetic field in which one may ob1 spectroscopic states for each level. Since serve 2 J each G term is (24 1) fold degenerate (mCvalues),
+
+
+ +
-
+
+
+
+
+
+
and the spin multiplicity is (25 l ) , the total number of states (m,) arising from a configuration is (2d 1) (28 1) which is easily shown to equal (2J 1).
+
+
+
Many Electron Atoms
The normal symbolism is the same as for hydrogenic atoms except that in the simplest coupling scheme both L and S take a number of values according to the orbital quantum number of each electron. The major change in the description involves the energies of the terms arising from a particular configuration. Since the electron-electron interaction is not the same for each L value, the terms of a given principal quantum number are no longer a t the same energy as is the case with hydrogen. The term splittings may amount to a few electron volts which can be comparable with the inter-shell energies for higher principal quantum numbers. The observable levels (characterized by J ) arising from these terms (characterized by L and S) are relatively close in energy for lighter atoms. We therefore have the general situation of overlapping of levels from different configurations. The extent to which these overlap, and also the extent to which levels from different configurations may interact, are the factors which will either permit or forbid the unambiguous assignment of a particular configuration to an atom in a certain chemical situation. Term Splitting
For elements in the first row of the periodic table the lowest energy terms are clearly separated energetically from the terms arising from less stable "aufbau" configurations. For example in carbon, nitrogen, and oxygen the normal configurations 2p2, 2pa, and 2p4 give rise to levels which span 2.9 ev, 3.7 ev, and 4.4 ev, respectively; the lowest levels stemming from the promoted configurations 2p3s, 2p23s, and 2p83s are a t 7.5 ev, 10.4 ev, and 9.1 ev, respectively (3). It is therefore very reasonable to claim that these isolated atoms have the former set of ground configurations. Each observable level corresponds to some linear combination of detailed configurations with ml and m, specified for each electron. Although the valence s, p, d, f ... etc., orbitals are not eigenfunctions of the angular momentum it is possible to express the charge diitribution for every state in a given level as a combination Table 1 Configuration Term State (n, l ) (L, S ) (L,M L ; 8 , M a )
(IS
I
(0, 0 ; 0 , O )
Level (J) lSa
of these well-known valence functions. For example the 'S, =P,and 'D terms from the p2 configuration of carbon provide a total of [l 9 51 fifteen states for which the charge distributions are given in Table 1 in terms of the p,, p,, and p, wavefunctions. The situation for transition elements is very different in that, for nearly every case, the terms of one configuration overlap with those of another. At normal temperatures (kT * 0.025 ev) we assume that an atom will be in states arising from its lowest energy level if the energy required for electronic excitation to a level from a different configuration is much greater than kT. Throughout the transition series the configurations 3dn4s2, 3dn+'4s, and 3dn+2 are in extremely close competition for the extent that they might contribute to the normal state of the atom. Figure 1 shows the trend of the energies of the levels arising from the competing configurations. The diagram is intended to emphasize the overlapping of terms from different configurations. For Ti, V, Cr, Fe, Co, and Ni the lowest levels are separated by less than 1 ev fromlevels arising from a different configuration. I n nickel the separation is roughly equal to kT a t room temperature. I n the case of scandium (Fig. I), the terms arising from the competing configurations 3d4s2 and 3d24s are:
+ +
. . . .(a)
. . . (b)
We will not specify the levels of each term since in most cases the spin-orbital interaction is much smaller than the electron-electron repulsions that split the terms. This is true for lighter elements so spin-orbital coupling can play a significant role in a similar descrip tion of the second and thud transition series. The circled point a t scandium represents the energy of the single term %D. The two dots enclosing the shaded r e gion represent the energies of the lowest and uppermost terms from the set (b). The lowest energy terms are the ones with the largest spin multiplicity and among these, the term with largest orbital angular momentum is lowest (Hund Rule). The shaded region therefore encompasses all terms and levels from (b) between the extremes 4P (lower dot) and (uppermost dot). The figure shows that the levels from configurations (a) and (b) do not overlap and the assignment of the ground configuration of Sc to 3d4s2is more or less unambiguous. The competing configurations for Ni are: IF. . . . . . . . . . (c) 3d84sP : IS, ID, 'GI 3dP48 : 'D, aD... . . . . . . . . . . . . . . . .( d )
Cbrg? distnbutlon
' h (P.' n.2)
+ pVa+
Figure 1. Energy bracketed b y terms from the conflguratlons 3d"4ra and 3d"+'4r. The rhoded area portrays the region spanned b y tho 3d"+14s ~ ~ ~ f l g ~ which ~ o t contributes i o ~ most to the normal .tom in Cr and Cu. This illustrates the Hund Rule.
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The term splitting for configuration (c) is very large (-3 ev) and the energy bracketed by the uppermost ('S) and lowest (3F)terms is the gap between the circled dots above nickel. The term splitting between SD (lowest dot) and 'D is rather small. However, the two sets of terms will interact considerably in the region of overlap, and both configurations undoubtedly contribute to the normal state. The assignment of a particular configuration to the normal atom is further complicated by the fact that levels arising from different configurations involving d and s orbitals can interact strongly with one another. Accordingly a given spectroscopic level cannot be uniquely assigned to a particular configuration; instead, the wavefunction is a superposition of functions from both configurations. The degree of mixing is largest when the energy gap is least. Therefore the question of which is the normal configuration of a transition metal can hardly be answered without considering the interactions within and between the closely lying configurations. The best known example of this is the 'S3level of chromium (d5s1)which because of intraconfigurational interactions (minimum electron repulsion between unpaired spins; ef. Hund Rule above) lies below the lowest level from the d4s2 configuration. So the role of the central field is apparently secondary in that case.'
The result of increasing the effective nuclear charge is to partially un-screen the outer electron and hence to decrease the relative importance of penetration. When the screening is large (non-coulomb effective field) the relative cont,ribution of penetration is greatest. The effective nuclear charge can be increased by removing one electron from the atom. I n many cases this ionization can cause a reversal to the hydrogenic order of orbitals. Figure 2 shows the energies bracketed by the levels from the 3dn4s and 3dn-'4s2 configurations of monopositive ions in the transition series. Clearly in the case of V+, Cr+, Co+, Ni+, and Cu+ the 3d subshell is more stable than the 4s orbitals. Increasing the effective nuclear charge has therefore revived the hydrogenic order of orbitals. Of course, there are strong interactions between the two configurations, and the term splitting is larger than the energy separation between the configurations as was the case for the atoms. Accordingly the reverse order of 3d and 4 s is maintained in Sc+, Ti+, Mn+, and Fe+, although for Ti+ and Fe+ the configurational energy gap is very slight indeed.
Relative Stability of 3d and 4s Orbitals
The detailed examination of spectroscopic levels which led to the above conclusions also shows that the 3d orbitals lie a t higher energy than the 4 s orbitals. This reversal is not peculiar to the transition metals. I n fact the nitrogen levels from 2p24s definitely occur a t lower energies than the levels from 2p23d. I n neon the reversal is already sufficient to provide for no overlapping of levels from 2p64s and 2p63d. The reversal probably occurs a t carbon for the first time although in that case the configurations will overlap considerably. A reversal of the above type occurs because the effective field in which the electrons are moving is no longer of the simple coulomb form which is epitomized by bydrogen: the electron-electron interactions have become comparable with the intershell energies for higher principle quantum numbers. One way of overcoming the deviations from hydrogenic effective field is to increase the effective nuclear charge. The argument may proceed as follows: the effect of electron-electron interactions on orbital energies is to stabilize the most penetrating orbitals. For example 4s is stabilized relative to 3d, as can be rationalized from their radial distribution curves (4).
Figure 2. Energy brmcketed by terms from the configurmtionr 3d-9 and 3d"+' of the mono-positive ions. The shaded area shows the region spanned by the 3d"+'conflgurotion. In this case the competition for the normal state is keener than in that shown b y Figure 1.
For doubly positive ions the return to hydrogen order is complete. The ground terms all stem from the configurations 3dn. The author would like to thank Dr. D. A. Hutchison for some helpful discussion. Literature Cited (1) H E ~ ~ B E RG., Q , "Atomic Spectra and Atomic Structure," Dover Publications, Ino., New York, 1948, pp. 71-119. ROBIN M . , "Behavior of Electrons in ( 2 ) HOCHSTEASSER, Atom," W. A. Benjamin, Ino., New York, 1964, pp.
78-98.
G . H. "The Theory of U., AND SHORTLEY, Atomic Spectra," Cambridge University Press, 1963, pp. 327-77 (an authoritative account of the basic theorv. especially chapter 14). (4) HOCHSTRASSER, ROBINM., op. eit., pp. 6676. (3) CONDON, E.
Some of the term from the d-s configurationsof MnandMn+ are not definitely assigned. Accordingly Figures 1 and 2 contain only the lowest energy terns for this element.
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