-
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
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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
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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