Exchange Stabilization and the Variation of Ionization Energy in the p and dn Series Antony B. Blake University of Hull, Hull. England HU6 7RX

The way in which ionization energies vary within a series of consecutive elements helps us to understand a number of trends in the stabilities of compounds and the oxidation- reduction behavior of ions. This article is concerned with two types of ionizations that are of special importance to chemists, namelv. the Drocesses D" - pm-I and dn - dn-l. The first

.. .

is pertinent to the chemistry of the main-group elements from Group I11 onward, and the second is pertinent to that of the transition elements in oxidation states +2 and above (pro- vided n > Oi. Our main DurDose is to clarify current textbook

~~~ . .

interpretations of the peculiar decrease in ionization energy followine comnletion of a half-filled D or d shell.'

The & r e Shows the first ionizatibn energies of the atoms B to Ne (electron configurations ls22s22pn, n = 1 to 6), and the energies required to remove an electron from the gaseous ions Sc2+ to Zn2+ ([Ar13dn, n = 1 to lo), plotted against atomic number. In hoth cases we observe a tendency for the energy to increase with increasing atomic numher, but with a distinct drop in ionization energy on going from the atom or ion with a half-filled shell (p5d5) to the next one in the series. Thus, the energy of the process M2+@) - M3+(g) + e - ( g ) increases fairly steadily from Sc to Mn, and again from Fe to Zn, but the second approximately linear portion is shifted downward by about 500 kJ mol-' from an extrapolation of the first. This shift to lower energies has some familiar chemical conse- quences, for example, the instability of MnC13 compared with FeCln and the fact that CoF? is not much more oxidizing than M~F;.

I t is not surprising that most textbooks of inorganic chemistry give some discussion of these effects, and of their origins in terms of atomic structure. Although the overall rising tendency is easily explained hy the progressive increase in the effective nuclear charge experienced hy the electrons, it is not so easv to give a simple exulanation of the decrease following the half-Filled sheli. ~ w d a ~ p a r e n t l y different ex- nlanations are commonlv offered. Many authors point out that In a more than half-filleh shell, a t least one orhital is occupied by two electrons, and these electrons should experience a greater net repulsion than those that have an orbital to themselves, since sharing an orhital implies that the two electrons in it are constrained, to some extent, to occupy the same region of space. This explanation is simple and obvious and is accepted readily by most students. On the other hand, some more advanced books discuss the problem in terms of the seemingly different and more mysterious concept of "ex- change energy": electrons with parallel spins tend to keep anart. in conseauence of the Pauli principle; thus, the average repul&m herwc~n tw , r k t r t ~ n . in ~ I I I ' I V W ~ ~ d , i ta i . ia Ivy. ii their wi~i; ;ir(, w~rdllel t hm ii thc,v :ire op~,u+~l, the Liierence arising from the presence of a so:called"exchange integral." The first five electrons to enter the d shell go in with parallel spins, thereby taking maximum advantage of the exchange stabilization; but the sixth and subsequent electrons must have the opposite spin direction, and although they still achieve minimum repulsion among themselves, their repulsion with the first five is not reduced hy the exchange effect, as it would have been had they been able to go in with the same spin direction as the first five. The two interpretations-extra repulsion from double occupancy of orbitals, and extra re-

"

Fe" {~r l3d"- l~r l3d"~ '

First ionization energies of the atoms B to Ne (electron configurations ls22s22p") ( 18) and of the divaient ions ScZ+ to Zn2+ (electron configurationr [ArISd") (19).

pulsion due to loss of exchange stabilization with the first half-shell-are distinct, a t least superficially, and hoth seem physically rea~onable .~ The question is "Which effect is the more important, and which should we teach?"

In 1111; article we qhall l c ~ ~ k . a t tht v:ir~,itic,n~ i n ionimti~m r~terr\ . oredicted in rht. 11" iu~d r ln series, anrl lr\. t o inl i t ~ I I C I I I intothk "classical" or ~ou lombic repulsion energy and the quantnm-mechanical exchange correction. We shall see that the drop a t the half-filled shell is due mainly to the exchange effect, though with quite asuhstantial contribution from the douhle-occupancy effect. Before embarking on the analysis, however, we first outline a simple method by which ground- state electron repulsion energies can he calculated, so as to make plain the significance of some concepts that we use later.

'The discussion that follows is based on certain simplifying as- sumptions, of which the most important is the neglect of spin-orbit coupling. Although this approximation is valid for our purposes far the lighter elements, including those of the 3d and 4d series, the conclusions reached may not be correct far heavier elements where spin-orbit coupling is strong. Charge correlation effects are also ne- glected, since these are expected to be relatively unimportant in the outer shells of many-electron atoms.

21t should perhaps be mentioned that anumber of textbooks offer neither of the above explanations but refer instead to a supposed "special stability" of half-filled shells. This mythical concept has been admirably demolished (along with the "special stability" sometimes attributed to filled shells) in a recent article by B. J. Duke ( I ) .

Volume 58 Number 5 May 1981 393

Electron-Repulsion Energies We are interested in atoms or ions in which the outer elec-

trons are in a partly or completely filled p or d shell, and all the other electrons are in filled shells (s2, p% etc.) which, to- eether with the nucleus. constitute the oositivelv charged "ore" of the atom and remain essentially Lndisturbed in'the ionization process. Each of the outer electrons can be assigned a "private" energy, 11, which is the sum of its average kinetic energy and the average (negative) energy of its electrostatic interaction with the core, &d on top of this, the outer elec- trons have a collective repulsion energy, E,,,, due to all the electrostatic revulsions between them: When an electron is removed from an atom with outer shell p n , the energy input is given by

and if the repulsion energies are those for the ground states, El is the ionbatinm enerq. lporinp; (1 for the present,~ur first task i.. to calrulat~ EICP iur 11 = 2,3,. . .. 6. \\.esh:lll then do the same for d".

T o illustrate how this isdone, we take the configurationp2. An electron in a p orhital can he in any one of six states (mi = 1 , O , or -1, with m. = +%or -%), and there are therefore 15 possible ways of arranging two electrons in the samep shell. (Remember that the two electrons cannot simultaneouslv occupy the same state. The numher of ways of choosing twb states from 6 is 15.) If we use boxes to represent the orbitals and an arrow for thespin direction (up for rn, = +%,down for -%), one such arrangement is as follows:

In using thequnlitum nun~hers I and rn, for a single electron, we rwall that I wlls us the maenitude of the electron's orbital

-

angular momentum (1 = 0 for an s orhital, 1 for a p , 2 for a d , etc.) while m, gives the value of its vrojection (angular mo-

. ..

mentum being a vector quantity) aldngthe z axis. Similarly, for an atom in a particular state there will be a total orbital angular momentum due to all the electrons, to which we assign the quantum number I,, and a definite projection of this in the r direction, given by ML, which can take any of the 2L + 1 integer values from -L to +L. MI. is equal to the sum of the rn, values of the individual electrons. We shall also have atotal spin quantum number S , the z-component, M s , of which can take any of the 2 s + 1 values from -S to +S, M s being equal to the sum of the rn, values. (S may be an integer or half-in- teger.) Filled shells make zero contribution to L and S .

I t can he proved that as long as the only forces acting on the electrons are the electrostatic attractions and repulsions within the atom, each many-electron wavef~~nction that satisfies the Schrodinyer equation for the atom and is a m - sistmt with the indistinguishability of electrons must corre- spond todefinite valuesd'L and 3 t'urther,all of the r?L + 1 1 ( 2 + 1) staws (wavefunctionc) with a particular value of I, and S will hnve the samc energy ( i e . will he "degeneratt:"). Thr energy level correspmd~ng to such a set of degenerate state. is called a "term" td the configuration, and is given a svml~ol in the form 2sv11.. where the value of L is indicated cbnventionally by a letter rather than a number, using S.P.D.F.G.. . . forL =0.1.2.3.4.. . . . T h e t e r m s o f ~ ~ a r e V . . . . . . . . . .

ID, and LS, and their derivation is a standard eiercise fo; students of atomic structure. The lowest-energy term is 3P,

-.

as predicted hy Hund's rules. If we write out some of the possible "arrow-in-hox" di-

agrams for p2 with their MI, and M s values and list them against the terms to which they could belong, as in Table 1, we see that certain combinations of MI. and M s can arise from only one diagram, whereas others can arise from two or three.

Table 1. Some of the Fineen Possible Ways of Assigning m, and m. Values to Two Electrons in the p Shell, with the Resulting ML and M, Values, the Terms to Which They Can Belong, and the

ReDuision Eneraies of the Sinale-Determinant States Possible Repulsion

M & terms enerav'

1 3P 41, 0) - N1. 0) = Fo - 5F2

1 'P dl,-1)- 41,-1) = FO - 5F1

1 'P @,-I)- NO,-1) = Fo - 5F2

0 3P, ' D , ' S

The box diamams actuallv reoresent wavefunctions of a rather special type',namely, funitions ill which we can say definitely that a oarticular ~rbit i l l IS wcuvic~d and that the electron in it has a particular spin directi'n. For example, the first di- agram in Table 1 (with ML = 1, M s = 1) represents the wavefunction

" -

Here, the numbers 1 and 2 in parentheses label the two elec- trons, po stands for the orhital with rnr = 0, and the + sign indicates that rn, = +%. Notice that * obviously describes a state of the atom in which the orbitals po and pl each contain an electron: however, because of the form of 'P. we cannot say whirh electrun occupies which orbital, and this inahility fi& in with the tact that an individual electron rannor bc disrin- guished from any other electron. Notice also that if we inter- change the electrons, by switching the labels 1 and 2, the sign of *-is reversed. This property is also one that all ~ o m p l ~ t e many-electron wavefunctions must possess: the known vrooerties of svstems of electrons can he satisfactorilv ex- . . plained only if it is assumed that their wavefunctions (in- cludine snin) must alwavs be antisvmmetric. i.e. change sien

.. . .. -

whenever two electronc are interchanged. The antisymmetr). reuuiremenr is actuallv a generalized form of the Pauli ~ r i n - --r---

An interestina orovertv of anv antisvmmetric wavefunction that can be represented by a single box diagram is that it can he written as a determinant of one-electron states, in which electron labels vary along the rows and orbital-plns-spin designations vary down the columns (or uice-uersa). Thus * is the determinant"

3Similarly, an antisymmetric three-electron function would have the form

(where a, b, c include spin designations), and this is the determi- nant

abbreviated as label.

394 Journal of Chemical Education

". determinants ensure that they are always antisymmetric and satisfv the Pauli exclusion principle.

NU;, the combination Mi, = ~ , - M s = 1 must belong to the 3P term of p2 (L = 1, S = I ) , and as we have seen, this com- bination can arise from only one box diagram. Hence the corresponding wavefunction 'U3P;l,l) can be written as a single Slater determinant, eqn. (3). On the other hand, two different determinantal wavefunctions with ML = 1, MS = O can be constructed (corresponding to the two box diagrams shown in Table l ) , and neither of these determinants by itself is a proper wavefunction of p2. Therefore, the wavefunction 'P("P;1,0) cannot be written as a single determinant but will be some combination of these two determinantal functions, and the same is true of 'P(lD;l,O). Similarly, 'P(T;O,O), 'P(lD;O,O), and 'P('S;O,O) are combinations of three Slater determinants. I t is often quite hard to determine the coeffi- cients in such combinations.

We shall now see that the repulsion energy in a state that can be written as a single Slater determinant is very easy to calculate ( 2 4 ) . For a state of the atom with wavefunction T, the repulsion energy is given by the integral

pP*H,*,$d7 in which

H,,, = ~(e214arorp,) where r,, is the distance between the electrons labelledp and o. and the sum is over all nairs of electrons. Although this is . , -

a multiple integral over all space and spin for all of the elec- trons, it can be shown to reduce to a combination of two- electron integrals. For most states the result is still quite comnlicated. However, for asingle-determinant state (i.e. one corrksponding to a single box diagram) it is given by simply adding a term Ji, for each pair of electrons in the outer shell, and substracting a term Kj, for each pair that have parallel snins, the integrals J and K beina defined as follows: . . -

J;, = S$i*(l)dj*(2)(eV4a~~r12)$,(1)$j~2)dild (q) Kij = S$i*(I)$,*(2)(e2/4aror~%)9,(2)9,~l)d~~d~~

where d; and d; are the orbital wavefunctions of the two , . , , electrons. The J s , which are known as Coulomb integrals, have a simnle nhvsical internretation: J;; is the average enerm . . . - ... of repulsion hetween two electrons whose motions conform to the nrobabilitv densitv functions Id; 1 and Id; 1 2 , or (looking

, . , , . , , .

at it in another way) the energy of interaction between two clouds of negative charge 14; I2e and I@j1 2e. (If i = j , this is the classical "self-energy" of the charge cloud.) The Ks are called exchange integrals, and if 4j and @, are real functions, Kij can be identified with the energy of repulsion between two su- perimposed "overlap" probability densities 4i4, (the "self- energy" of the charge distribution @i@,e). Knowing the orbital wavefunctions mi, we can in principle calculate these integrals once and for all. Since J and K are always positive (at least, for single atoms or ions), it follows that the more pairs of electrons have parallel spins, the lower will be the energy. This is "exchange stabilization." The reason that it arises only for parallel-spin electron pairs is closely linked with the an- tisymmetry requirement, and a very clear explanation can be found, for example, in Linnett's little book "Wave Mechanics and Valency" (5).

The repulsion energies of some states of p2 that can be written as single Slater determinants are given in Table 1 in terms of the integrals J(mi,mr') and K(mi,ml') for p or- bitals.

We now proceed to calculate the Coulomb and exchange contrihutions to the repulsion energies of all the p" and d" ground terms. This determination is quite straightforward because, in any configuration, the wavefunction with the

largest possible number of unpaired spins, and the highest ML consistent with this, will always he a single-determinant function (since there is only one hox diagram that satisfies these conditions), and will also belong to the ground term (Hund's rules). Energies of p" Ground Terms

Using the ml Orbitals The "ml" orbitals usually used in calculations on free atoms

or ions are products of a radial function, R ( r ) , and an angular function of the polar coordinates (H,@), the latter generally being a complex function. The two-electron integrals J(ml,mr') and K(mi,ml') in eqns. (4) can be reduced, by in- tegrating over the H and @ coordinates of each electron, to linear combinations of integrals of the radial function alone, known as Slater-Condon integrals, of which for p electrons there are two. denoted F o and F?. Table 2 gives the values of the J s and KS in terms of Fo and F2(6), and in Table 3 we have listed the ground term of each confiauration P", with a box diagram for-one of its component states and the Coulomb and exchange contributions to its energy. [For example, for the p5 state shown, we have 2J(-1,O) + 2J(-1,l) + 4J(0,1) + J(0,O) + J ( l , l ) = 1OFo - 5Fa, and -K(-1,O) - K(-1,l) - 2K(0,1) = -15F2.1 Using Equivalent Orbitals

A disadvantage of the orbitals D-,. DO. and D T is that they . .. . .. . . implicitly s ing lek t a particular direction (the z axis), so that D" has a different soatial form from D-I and D I . (Its nodal . -

surface is the ny plane, whereas fur p l l and p i t h e z axis is a node.) However, any set of three independent linear combi- nations of these wavefunctions is equally acceptable from the quantum-mechanical point of view, and for pictorial purposes the orbitals p,, p,, and p,, defined by eqns. (5), are more convenient, because they are real functions and are spatially

Table 2. Coulomb and Exchange Integrals for the Orbitals p,, PO, and p-,

Table 3. Ground-State Electron-Repulsion Energies of p" Configurations, Calculated by the Use of the Orbitals p,,po, and

D-. and Analvsis of Eneraies for r," - 0"-'

p" ground Coulomb Exchange term. and energy energy

typical slate ZJ - 2 K Em,

Change Change in ZJ in - 2 K Change in Em,

Volume 58 Number 5 May 1981 395

PZ = (PI + P - J I J ~ P, = (PI - p-d l i J2

P. = PO ( 5 ) equivalent. Students are familiar with these orbitals, and Howald and Muharak (7) have pointed out that they can be used for calculating electron repulsion energies just as easily as p-I, po, and p l can. Their spatial equivalence also makes them very convenient for our purpose, because it means that they give only two distinct repulsions: the one between two electrons in the same orhital, and the one between electrons in any two different orbitals (Tahle 4). We shall use J,,,, to

Table 4. Coulomb and Exchange Integrals tor the Orbitals p., p,, and PZ

Table 5. Some Assignments of Orbltals and Spins to Two Electrons in the Orbitals p,, p,, and p., wlth the Resulting

Octahedral Symmetry Types and Mr Values, and the Repulslon Energies at the Single-Determinant States

Possible M1 terms Regulsion energy

Table 6. Ground-State Electron Repulsion Energies ot p" Configurations, Calculated by the Use of the Orbitals p., pr, and

p, and Analysis of Energies for p" - p"-' p" ground Coulomb Exchange term, and energy energy

typical state Z J -ZK Em D t 2 . r ,+ 0 0 0 . . I

9 3T7 ~ ~ . + p y t ~ Fa- 2F2 -3F2 Fo - 5 Fz d 4A, I P ~ + P ~ + P ~ + ~ 3F0- 6F2 -SF, 3Fa- 15Fz $ 3T7 / P ~ ~ P ~ ~ P ~ ~ ~ 6Fa- 6F2 -SFz 6Fo - 15F2 d T; 1 P , ~ P ~ ~ P ~ ~ 1 10F0- 8F2 -12F2 10F0 - 20F2 P' ' A , lpx2p?pZ21 15F0 - 12F2 1 8 F 2 15F0 - 30Fz

Change in ZJ Change in - 2 K Change in E,.,

Note: J = A,, = Fo - 2F2: K = 3F2

Table 7. Coulomb and Exchange Integrals for the "cubic" d nrhitals

so, cc. E . 77. rc A + 4 8 + 3 C I A + 4 B + C C

@. 07 A + 2 B + C B + C ft, 67. 7. 7 c E A-2B+ C 3B+ C flc flt A - 4 B + C 4B+ C

stand for J(xx), etc., Jd,ff fur J(xy), etc., and K for K(xy), etc.

Since these orbitals do not all have definite values of ml, the single-determinant wavefunctions constructed from them cannot usually be assigned ML values. However, if in imagi- nation we impose on the atom an octahedral "crystal field" (which can be as weak as we like), we can classify the free-atom terms having L = 0,1,2, etc. by the labels used in octahedral symmetry, according to the well-known correlation S - A1, P - TI, D - E + Tz, and so on. A little group theory (8) also enables us to find to which symmetry type each singIe.de. terminant wavefunction (or box diagram) belongs. As in the L, ML case, it turns out that at least one box diagram with the maximum numher of unpaired spins is of a unique symmetry type and thus can contribute to only one state ofp". For the p2 case, illustrated in Table 5, the components of the 3T1 term are single-determinant functions; therefore, we can calculate the repulsion energy of this term (which is really the 3P term) by the Z J - 2 K rule, using now the J a n d K integrals in Tahle 4. (The latter are derived readily from Tahle 2 by use of eqns. 5.) The results for all the p n ground terms are listed in Table 6. The Ionization Energies

Let us now look at the changes in the ionization energy, eqn. (I), as we go from one p n configuration to another. We must hear in mind, of course, that U, and also F o and F2, are not

nuclear charge increases. Indeed, the overall rising trend of the ionization energies is due mainly to the increase in -U with increasine nuclear charge. a n d F n and Fq also increase

nuclear chargk when an electron is removed. However, the variations seem to he smooth enoueh to he nedected in a semi-quantitative discussion of the "&upn after the half-filled shell. and we shall proceed as if Fo and F? were constants.

WL look first at the results obtained uBing p,, p,, and p, (Table 6). If there were an unlimited supply of vacant orbitals, then each time we added an electron tip" to givepn+', the number of electron pairs would increase by n and the repul- sion energy would increase by n(Jdiff - K ) , since all the elec- trons would he in separate orbitals with parallel spins. The effect of this would merelv be to make the steadv increase in ionization energy slightly Lss steep. In fact, this picture is true onlv for the first three electrons: once the shell is half filled. we must modify the contribution of each subsequent electron to Erep by an extra J,,,, - Jdiff = 6F2, because it has to share an orbital, and by an extra 3K = 9F2 representing loss of ex- change stahilization, because its spin is opposite to that of the first three electrons. The resulting 15Fz of "extra" repulsion energy, which will apply to each of the electrons in the second half shell, can be identified with the anomalous downward shift in ionization energy, and we see that 215 of it arises from the increased Coulomhic repulsion associated with sharing an orbital, and 315 from the loss of exchange stabilization due to the necessarv reversal of spin.

It is instructive to repeat this analysis using the orhitalsp-1, D". and D, (Table 3). The situation now is complicated . ".

slightly bythe fact that the repulsion between two eiectrons depends on which orbitals they occupy, but we can proceed by calculating the auerage J or K and examining the-energy incrementsin terms of these averages. Thus we haveJ,,, = Fo + 2Fz, Jdi if = FO - Fz, and K = 4Fz. (Note that these av- erage values differ from those for p,, p,, and p,.) We again find an "extra" repulsion energy 15F2 in the segnd half shell, but this is now made up, on average, of J,,, - Jm = 3Fz due to double occupancy and 3K = 12Fz due to loss of exchange stahilization. i.e. and % of the total, respectively. The dif- ference between this result and that ohtained withp,, p,, and p, is a useful reminder that the distinction between "Cou-

396 Journal of Chemical Education

lomhic" and "exchange" components of the net repulsion energy is to some exteGt an arbitrary one: the extent to which the exchange "correction" is necessary will vary with the or- bital wavef;nctions used in the calc&tion. Still, it seems reasonable to say that the anomaly in the ionization energies of the elements B to Ne is due mainly to the effects of quantum-mechanical exchange. In other words, it is a conse- quence of the Pauli principle.

Energies of d" Ground Terms The electron-repulsion energies of d orbitals depend on

three radial integrals, and it is customary to use the Racah parameters, defined as A = Fa - 49F4, B = F2 - 5F4, and C = 35F4. As in the pn case, the calculation is easier and more physically satisfying if real orhitals are used, and we shall use the well-known "cubic" orhitals d,~, d,n -y 2 , dxy, dyi, and d,,. To simplify the notation, we shall relabel them as 0, e , 5; f , and 7, respectively. Their J and K integrals are given in Table 7 191 \-,.

The reader will notice that even with these real orhitals we have to consider five types of interactions, compared with only two for the real p orhitals. (If the complex d orbitals with mi = 0, f 1, f 2 are used, there are nine distinct interactions.) I t appears to he impossible to find a set of d orhitals for which all int.ernct,ions hetween electrons in different orbitals are ------ ~ equal, as we can in the p case. (One can liken this circumstance to the fact that in three dimensions it is impossible to arrange five points so that all ten interconnections are equivalent.) I t is true that sets of spatially equivalent d orhitals exist, namely, the pentagonal-prismatic orbitals of Powell (10) and Pauling and co-workers ( I l ) , but even with these there are still three types of interactions, and moreover, these functions do not seem to give single-determinant ground states, which are es- sential for the simple approach we wish to use. So we have to content ourselves with the familiar "cubic" d orbitals.

The n-electron wavefunctions are again classified according to their symmetry properties in an imaginary octahedral en- vironment. For most configurations the free-ion ground term splits into two or three "crystal-field" terms, but in all cases a t least one component is a single-determinant function of unique symmetry type (see Appendix). The repulsion energies can thus be obtained as hefore, and in Table 8 we list the "Coulomhic" and "exchange" contributions, and the analysis of d" - d*-'.

From Table 7 , we have J,,,, = A + 4 8 t 3C,Jdiff = A - B

+ C, and K = 2.5B + C. On the average, therefore, we expect addition of each electron in the second half-shell to involve an extra repulsion of J,,,, _Jd l f f = 5 B + 2C due to double occupancy, plus an extra 5 K = 12.58 t 5C due to loss of exchange stahilization with the five electrons of the first half- shell. In other words, the downward shift in ionization energy amounts to 17.5B + 7C, of which 2/7 arises from the increase in Coulomhic repulsion associated with sharing an orbital, and 517 from loss of exchange stabilization. " Small deviations from the "average" behavior are inevitable

because of the inequivalence of the five orhitals. The d 2 , d3, d7, and d8 configurations can achieve arrangements with especially low repulsion energy; for example, the second

-~~~~ ~

than the~&erage'per eiectron pair. This + 2 8 + 2 c - 6 _ K t 1 1 8 + 5 C 6(J- M t A - 4.58 ds - d' 7J+ 58+ 2C -75+ 12.58 t 5 c 7 ( j - - + A d9 + ds G + ~ B + Z G -8K+ 14 8 C 5C a(?- K) + A + 4.58 8" -S 9 J t 58+ 2 6 -9K+ 12.5B+ 5C s ( J - % + A

N ~ ~ ~ : J = J ~ ~ ~ = A - ~ + c : K = ~ . ~ B + c ; A = ~ ~ s B + ~ c . or thedivalent ions of the first transition series. C-45- 30-50 kJ mol-' (14.

Volume 58 Number 5 May 1981 397

pulsion-energy calculations also serves to minimize the ex- change corrections, as well as making the physical effect easier to visualize. Nevertheless, as far as the trend in ionization energy in a series of elements is concerned, the exchange effect clearly emerges as dominant. Conclusions

(1) The decrease i n p n - pn-' or d n - dn-' ground-state ionization energy following completion of the first half-shell is due mainly to extra interelectronic repulsion resulting from the fact that the electrons of the second half-shell have op- posite spins to those of the first (i.e., t o the absence of "ex- change stabilization" with respect to repulsions between the first and second half shells). This effect, which can be regarded as a correction to the ~ u r e l v electrostatic model by incoruo-

t o extra repulsion arising directly from the fact tha t the electrons of the second half-shell have t o share orbitals with those of the first.

(2) The relative importance of these two factors depends on the choice of orbitals used to construct the many-electron wavefunctions: the use of orbitals tha t are, as far as possible, spatially equivalent (e.g. p,, p,, and pJ maximizes the con- tribution from orbital sharing, hut the exchange contribution remains dominant. Acknowledgment

This article has henefitted greatly from discussions with Dr. Peter G. Nelson, for whose perceptive criticism and many constructive suggestions I am most grateful. Dr. Nelson made a partial study of the d n - dn-' case several years ago (15) , the conclusions of which have been published by Johnson (16). I am also grateful t o Dr. David E. Webster for very helpful comments on the manuscript. Appendix

d12+2, ete. We wish to find a single-determinant component of 3F, and since the spin part will be symmetric when the number of parallel spins is a maximum, we need an antisymmetric spatial function. In cubic symmetry the orbitals transform according to the standard ir- reducible representations r z and e , and their antisymmetrized direct products span the following irreducible representations: e2 - A?; ts2 -TI; tze and etz - TI + T?. Since TI occurs twice, its components will in general be mixtures of tz2 and t2e products. Az arises only from the product Oc, and hence IB+c+l PA21 is a single-determinant com- ponent of 3F.

A more complete analysis can be made by inspecting the "Clebsch-Gordan" coefficients connecting products of particular one-electron functions with the two-electron states of appropriate symmetry that can be constructed from them. For real p and d ar- bitals in cubicsymmetry theseare given by Griffith (17),Table A20. For T? we find that the [ and q components are mixtures of products, but the [component arises only as the product Or, and lfl+Pl PT2) is therefore the only other single-determinant component of 3F that can be constructed from the real d orbitals. We can also extend these results to d3 by noting that the orbital symmetry behavior arising from two empty orbitals will be similar to that of d2. The rest of the results follow.

error or by s variational procedure, a determinantal function that gives the same energyas that obtained in the ML scheme Literature Cited

I l l Duk8.B. J.,Edur in C h m , IS, 186 (1978). (21 Skier. J. C.,Phys. Re".. 34,1293 (1929). (21 Slator, J. C.."QuantumTheoiy of AfornlcStructure: McGrar-Hill, New York, 1960,

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Pauling,L.,Isrod J. Chrm.. lO.211 (19721. ,,*? n-0 ,"\ " "9, ,.-, .~ v, , ,* , ,p . . , , , . (131 J0hnson.D. A.,Ado. infnow. Chem ondRodiochem., 20. L(1977). Finding a Component Of the Ground Term (14, Lennad~,Iones, , I , and F'ople, J. A , P m c . Roy . So,, A 2 0 , 166 (1950).

When mi orbitals are used, it is easy to find a single-determinant {::i ~ ~ , " h ~ 5 ~ i , ~ . h s ~ k ~ ~ ~ ~ r ~ , " ~ ~ ~ ~ ~ c b o f ,noiganieChemistry,.. Cambridge, component of the ground term. From the states with the maximum 1968. section fi.6. number of unpaired spins, we select the one with the highest ML. 117) Rec u), pp. 168,396-298. When we use that do not have definite values, a different 118) NBs clreulsr 270~1, National Bureau of Standards, Washington, DC. 1968. approach is necessary, and as usual, group theory provides the (") RerU1,p'37y' quickest solution. Let us take d2 as an example, using the orbitalsd,%,

398 Journal of Chemical Education