86o/vHlc
2. Electronic configurations and multiplet structure
The 5d group can m o s t conveniently be said to start with lutetium, followed by hafnium, tantalum, tungsten . . . T h e r e is no particular r e a s o n t o place the 4f group as an inclusion in the 5d group f r o m cerium to lutetium though it is true that the g r o u n d ' state of the g a s e o u s l a n t h a n u m a t o m belongs to [Xe]5d6s z.
H o w e v e r , this has no m o r e influence on the c h e m i s t r y of the invariantly trivalent
T H E O R E T I C A L C H E M I S T R Y OF R A R E E A R T H S 139
lanthanum than [Ar]3d4s 2 has on the chemistry of scandium. As e m p h a s i z e d by Connick (1949) it is not plausible that the lanthanides are trivalent simply because t h e i r neutral atoms contain ( Z - 5 7 ) 4f electrons, since the actual fact is that eleven of the gaseous elements are barium-like with q = ( Z - 56) and two 6s electrons. T h e r e is no striking difference b e t w e e n the chemistry of the four lanthanum-like ground states (La, Ce, Gd and Lu) and the 11 others. As we have seen in the previous sections, the r e a s o n for almost invariant trivalence is that 13 is so small, c o m p a r e d with 5r, that the M(II) aqua ions have negative E ° and are able to evolve h y d r o g e n gas, and that 14 is so large, c o m p a r e d with 7K, that the M(IV) species in acidic solution h a v e E ° sufficiently positive to evolve oxygen. Our primordial question is: w h y is (•4-13) so overwhelmingly large compared with 2K ? Obviously, we do not have the simple situation of crossing a closed shell, as when aluminum has 14 = 120.0eV (depleting the neon shells) much higher than 13 = 28.45 eV (losing only a 3s electron) or when scandium has 14 = 73.47 eV (3p) considerably higher than 13 = 24.76 eV (3d). Table 23.1 shows the lanthanides to h a v e / 4 some 20 eV higher than 13 (disregarding the effects of q = 7) though a 4f electron is r e m o v e d in both cases (excepting La +2 and Gd +2, and even there it hardly makes any difference). In other words, the electron affinity of a partly filled 4f shell is far smaller than the corresponding L
Curiously enough, the linear trend of I as a function of Z (and h e n c e of the number q of 4f electrons) and the almost constant K serve as an empirical background with only qualitative theoretical justification, whereas many fine details of the small deviations from linearity as a function of q are well understood. Judd (1970) once r e m a r k e d in a book review that the writer "regards atoms and other comparatively simple systems as analog computers for the very properties they exhibit and h e n c e in need of no special analysis". This is a p r o f o u n d observation (though it m a y be a c c o m p a n i e d by a few e x p l a n a t o r y comments given below) but it is perfectly true that the monatomic entities of which the discrete energy levels are classified by Moore (1949, 1952, 1958) are the best (and only) quality available, and that the solution of the Schr6dinger equation would yield the o b s e r v e d energy differences, if it is valid. F u r t h e r m o r e , transparent lanthanide c o m p o u n d s are entirely exceptional by having recogniz- able J-levels corresponding to approximate spherical s y m m e t r y (JCrgensen, 1955b; Carnall et al., 1968) analogous to atomic spectra.
2.1. Huge differences between ionization energy and electron a1~nity, and the phenomenological baricenter polynomial
One of the great difficulties for good approximations (such as H a r t r e e - F o c k calculations) to the Schr6dinger equation for atoms containing many electrons is that the eigen-values E are negative, out of all proportion with chemical experience. A remarkably good approximation to the total binding e n e r g y of Z electrons to their nucleus with Z protonic charges has been given by Gombas and Gaspar as - Z 24 rydberg (where 1 rydberg = 13.60 eV). This expression only deviates seriously (for relativistic reasons) for Z above 90. One way of alleviat- ing this difficulty is to consider the atomic core consisting of strongly bound,
140 C.K. J O R G E N S E N
closed shells as the zero-point of energy, and to consider a single electron bound to this core. If the ionic charge is +z, R y d b e r g s u g g e s t e d the binding energy
- e , t = - ( z + 1) 2 rydberg/(n - d ) 2, (23.13) where the Rydberg defect d is a strongly decreasing function of l (being larger for s than for p electrons, and so on) but generally varying v e r y little with the principal quantum n u m b e r n. By the way, the success of eq. (23.13) in line spectra showing long series (of increasing n) was the reason w h y (negative) term values were generally given (reiative to an ionization limit) the first half of this century, whereas more complicated spectra without clear-cut ionization limits were tabulated relative to the ground state of the m o n a t o m i c entity as zero- point. F o r the sake of uniformity, the latter practice has now been generalized (Moore, 1949). Though this is a minor detail in most atomic spectra, we consider the baricenter of the relativistic effect spin-orbit coupling in eq. (23.13), the two j-values (l + l) and, at lower energy, ( l - ~) possible for positive l corresponding to (2l + 2) and (2l) states (with i~entical energy in spherical s y m m e t r y ) separated by (1 + l) times the Land~paramet~r~,~. H e n c e , the baricenter is situated ½l~',~ below the higher of the two j-levels. It is an empirical fact (Land6, 1924; Jorgensen, 1955a) that ~',~ is roughly proportional to (z + 1)2Z 2 but, u n d e r equal circumstances, much larger for external p than f o r external d electrons, and usually negligible for external f electrons (this statement starts to break down from barium with Z = 56). In typical cases, ~',t is at most a few p e r c e n t of e,~.
When two electrons o c c u r in two different shells outside the atomic core, several energy levels (characterized b y the quantum n u m b e r J ) are observed, each corresponding to ( 2 J + 1) states (independent, mutually orthogonal wave functions) and which in (the usually satisfactory approximation of) Russell- Saunders Coupling can be assembled in terms each characterized by the quantum numbers S and L, and each containing (2S + 1)(2L + 1) states. If one of two electrons has l = 0, the two terms with S = 0 (singlet) and S = 1 (triplet) and L equal to l of the other electron is f o r m e d . Since the triplet (at lower energy) contains 3 times as m a n y states as the singlet, the baricenter of such a configuration is situated a quarter of the singlet-triplet distance a b o v e the triplet energy. In the general case of two different shells (nl) and (n'l') there o c c u r many terms, L = (1 + 1'), (l + I ' - 1), (1 + l ' - 2) . . . (It - I' I + 1), [l - l'[ combined both with S = 0 and S = 1. In all of these cases, the n u m b e r of mutually orthogonal states is (4l + 2)(4/' + 2). T h e situation is entirely different in the case of two electrons in the same nl shell, where only (2l + 1)(4i+ 1) states occur, distributed on the series of terms
1S, 3p, 1D, 3F, 1G, 3H, l I . . . a ( 2 l - 1), 1(2l) (23.14) not given in order of decreasing energy, though ~S is always the highest and (according to H u n d ' s rules) the lowest is the triplet with L = ( 2 l - 1 ) . The number of states for one partly filled shell I q is the permutational expression for q indiscernible objects distributed on (4l + 2 ) distinct sites, and the general recursion formula for the n u m b e r of terms has been given by Karayianis (1965).
T h u s f3 and f " present the terms
THEORETICAL CHEMISTRY OF RARE EARTHS 141 48, 4D, 4F, 4G, 41, 2p, aZD, bZD, a2F, b2F, aZG, b2G, a2H, b2H, 21, ZK, 2 L (23.15) again not in energetic order (the H u n d ground term is 4I). It is noted that the same combination of S and L m a y be present twice, and for the more complicated configurations (such as f6, f7 or f8) such a combination m a y be present m a n y times. T h e numbers of states, J-levels and terms are:
q = 0.14 1.13 2.12 3.11 4.10 5.9 6.8 7 States 1 14 91 364 1001 2002 3003 3432
J-levels 1 2 13 41 107 198 295 327 (23.16)
S,L-terms 1 1 7 17 47 73 119 119
H u n d 1S 2F 3H 4I 5I 6H 7F 88
Highest L iS 2F 1I z L 1N 20 iQ 2Q
We return below to the interesting c o n s e q u e n c e s for the properties of lanthanide c o m p o u n d s containing a partly filled 4f shell, that the n u m b e r of different energy levels of 4f q is as high as given in eq. (23.16). It is also possible to study the energy differences b e t w e e n various configurations of the same monatomic entity M +" or b e t w e e n configurations of M +"+1 and M +" representing ionization energies. A closer analysis (JOrgensen, 1969a, 1973a) shows that it is interesting to evaluate the baricenter of each configuration by weighting the energies of each J-level with their n u m b e r (2J + 1) of states, and w h e n neces- sary, supplementing the o b s e r v e d J-levels with calculated positions of missing J-levels. It is noted that this baricenter refers to the actual J-levels belonging (in a classificatory way) to a given configuration, and hence includes conceivable differences of correlation energy in the monatomic entities. A parameter A . ( n l , n'l') of interelectronic repulsion is defined for the energy of the baricenter of (nl)l(n'l') 1 (or alternatively (nl) z giving A . ( n l , nl)) of M +n-2 outside the closed shells of the ground state of M +" by using the latter ground state as the zero-point of energy
-e,,t - e,,v ~+ A . ( n l , n'l') (23.17)
with the two one-electron energies taken f r o m eq. (23.13). Since the baricenters of the ( 4 / + 2)(4/'+ 2) or ( 2 / + 1)(4/+ 1) states are usually known with a great precision, table 23.2 gives such values in the unit 1000 cm -1 = 0.12398 eV for the systems Cs, Ba +, L a +2, Ce +3 and Pr +4 containing one electron and Ba, La ÷, Ce +2 and P r +3 containing two electrons (4f, 5d or 6s) outside the x e n o n closed shells.
For comparison, E,l for Rb, Sr +, y+Z and Zr +3 and A . ( n l , n'l') for Sr, Y+ and Zr +2 are also given for 4d, 5s and 5p electrons outside the k r y p t o n closed shells. More extensive tables of this type are given in a r e c e n t chapter in Gmelin (J~rgensen,
1976b).
One of the conclusions f r o m table 23.2 is that 4f electrons are distinctly external b e f o r e barium and distinctly inside the closed shells after cerium, as first discussed theoretically by Maria G o e p p e r t M a y e r (1941). T h e R y d b e r g d e f e c t d increases from 0.31 in Ba + to 1.40 in L a +2 and A . ( 4 f , 4f) is already 13 eV in L a + and 19.4eV in P r +2 to be c o m p a r e d with more usual values such as A.(5s, 5 s ) = 6.3 eV in Y+. Although parameters such as A . are derived from
142 C.K. J O R G E N S E N
TABLE 23.2.
One- a n d t w o - e l e c t r o n p a r a m e t e r s (in t h e unit 1 000 cm - 1 = 0.12398 eV) f r o m eq.(23.17) for s e l e c t e d m o n a t o m i c entities
e ~ ess esp A,(4d,4d) A,(4d, 5s) A,(4d, 5p) A,(5s, 5s) A,(5s, 5p)
Rb 14 34 21
Sr + 74 89 65 Sr 58 47 39 45 35
y+2 165 158 123 Y* 78 60 52 51 46
Zr +3 276 238 194 Zr +2 95 75 65 52? 52
e4f esd e6~ A,(4f,4f) A,(4f,5d) A , ( 4 f , 6 s ) A,(5d,5d) A,(5d,6s)
Cs 7 17 31
Ba ÷ 32 75 81 Ba - - - - 25 51 43
L a ÷2 146 153 141 L a ÷ 104 75 56 65 52
Ce +~ 295 245 210 Ce +2 139 90 67 77 63
Pr +4 461 346 284 Pr +3 156 98 71 64? - -
relative positions of o b s e r v e d configuration baricenters, it is clear that there is a close relation to the reciprocal a v e r a g e radius of the orbital considered, as would also be true f o r the calculated integrals of interelectronic repulsion f o r H a r t r e e - F o c k radial functions ( F r e e m a n and W a t s o n , 1962). W h e r e a s eq. (23.17) is a definition of A , it is possible to define a p h e n o m e n o l o g i c a l b a r i c e n t e r p o l y n o m i a l using the s a m e set of p a r a m e t e r s for various ionic charges of a given e l e m e n t containing a electrons in the nl shell and b electrons in the n'l' shell:
- a ~ n t - b~n.t, + ½a(a - 1 ) A , ( n l , nl) + a b A , ( n l , n'l') + ½b(b - 1 ) A , ( n ' l ' , n'l') (23.18) which can be easily generalized to three or m o r e partly filled shells, the coefficients to A , being ½q(q - 1) for q electrons in the same shell and qlq2 for ql electrons in one shell repelling q2 electrons in a n o t h e r shell. It turns out that eq.
(23.18) is not a p e r f e c t a p p r o x i m a t i o n , but explains a v a r i e t y of facts s o m e w h a t surprising to chemists expecting a linear and not a parabolic (quadratic) depen- dence of the b a r i c e n t e r energies on the o c c u p a t i o n n u m b e r s of the two shells.
The r e a s o n f o r the non-linearity is that A , ( n l , n'l') tends to be c o n s i d e r a b l y smaller than the arithmetic a v e r a g e value of A , ( n l , nl) and A , ( n ' l ' , n'l') w h e n the a v e r a g e radii of nl and n'l' are highly disparate, as is n o r m a l l y the case w h e n f or d orbitals are c o m p a r e d with s orbitals. Thus, Sc ÷ has [Ar]3d4s at a b o u t equal distances below both [Ar]3d 2 and [Ar]4s 2 and L a ÷ has [Xe]5d6s below both [Xe]5d 2 and [Xe]6s 2. T h e s e are special cases of the t e n d e n c y of n d group neutral a t o m s to contain two (or less f r e q u e n t l y one) (n + 1)s electrons in the ground state, as if nd w e r e less stable than (n + 1)s though the latter electron is lost at first by ionization. This situation is e v e n more striking in the beginning of the lanthanides, w h e r e [Xe]4f 2 and [Xe]4f5d almost coincide in Ce ÷2 but the baricen- ter of [Xe]5d 2 occurs 5 e V higher. It is almost certain that the u n k n o w n configuration [Xe]4f 4 of the neutral cerium a t o m would s p o n t a n e o u s l y ionize to a free electron and the ground state of Ce ÷ belonging to [Xe]4f5d ~. Starting with
THEORETICAL CHEMISTRY OF RARE EARTHS 143 p r a s e o d y m i u m , the neutral a t o m [Xe]4f36s z contains two 6s electrons in the ground state (and already [Xe]4f%s would be an excited configuration) w h e r e a s the ground state of Pr + belongs to [Xe]4f36s and of Pr +2 to [Xe]4f 3 with [Xe]4f25d startil~g at 12847cm -~ (1.6eV) higher energy. The b a r i c e n t e r of [Xe]4f 3 is situated 2.4 eV a b o v e the ground state of P r +2, of [Xe]4fZ5d at 3.2 eV and of [Xe]4f26s at 4.7 eV which can be c o m p a r e d with eq. (23.18) and the data f r o m table 23.2 giving m u c h closer coinciding b a r i c e n t e r s (in units of 1000 cm-% - 9 1 5 , - 9 1 6 and - 9 0 8 . This s o m e w h a t u n s a t i s f a c t o r y a g r e e m e n t m u s t be seen against the b a c k g r o u n d that the o b s e r v e d ground state e n e r g y of P r +2 (relative to P r +5) is - 9 5 2 . 0 and of the [Xe]4f 3 b a r i c e n t e r - 9 3 2 . 5 showing a r e a s o n a b l e p e r c e n t a g e agreement. A p p a r e n t l y , A , ( 4 f , 4f) does not stay p e r f e c t l y constant.
2.2. Spin-pairing energy and the double zig-zag curve
T h e r e is no d o u b t that the overall separation b e t w e e n [Xe]4f q of M +" and [Xe]4f q-I of M +"+1 mainly d e t e r m i n e d by A , ( 4 f , 4f) b e c a u s e its coefficient in eq.
(23.18) changes ( q - 1 ) units, actually shows a characteristic variation as a function of q, as evident in table 23.1. It is possible to describe the deviation f r o m the s m o o t h variation by the f a c t that the baricenter of all states having the m a x i m u m value Smax is situated b e l o w the b a r i c e n t e r of the configuration 4f q for q b e t w e e n 2 and 12. Actually, the fraction of all the states having Smax d e c r e a s e s sharply in the vicinity of the half-filled shell and is (7/13) for q = 2 and 12, (5/13) for q = 3 and 11, (25[143) for q = 4 and 10, (9[143) for q = 5 and 9, (7/429) for q = 6 and 8, and only (1/429) f o r q = 7. It is possible to consider the p a r a m e t e r s of interelectronic repulsion in the theories of Slater (Condon and S h o r t l e y , 1953) and R a c a h (1949) as entirely p h e n o m e n o l o g i c a l p a r a m e t e r s to be c o m p a r e d with the o b s e r v e d distribution of e n e r g y levels. T h e writer (JCrgensen, 1957b) pointed out that the baricenter of all the states of I q having a given value of S and the baricenter (at higher energy) of all the states having ( S - 1) are s e p a r a t e d to the extent 2DS where D is a definite linear c o m b i n a t i o n of S l a t e r - C o n d o n - S h o r t l e y or of R a c a h p a r a m e t e r s . W h e r e a s it is a tautological s t a t e m e n t to say that the triplet and singlet baricenters of a t w o - e l e c t r o n s y s t e m are s e p a r a t e d b y 2D, it is not trivial that the s a m e linear c o m b i n a t i o n of p a r a m e t e r s of interelectronic repulsion can be used for all the q (with the s a m e I) and that it can be used for configurations (such as f% fs, f6 and
f7)
presenting three or four alternatives of S.The separation 2DS is the difference quotient of - D S ( S + 1) and it is actually possible (JCrgensen, 1962d, 1969a) to write the deviation of the baricenter of all the states having a given S f r o m the baricenter of all the states of l 4 as
D[(S(S + 1)) - S ( S + 1)] (23.19)
where the a v e r a g e value of the S ( S + 1) for the whole configuration can be written in various equivalent w a y s
2•+2 8 ~ + 2
( S ( S + l ) ) = ~ q ( q + Z ) - 4 ~ - - ~ ½ q ( q - l ) = ~ q - ½ q ( q - 1 ) = 3 q ( 4 1 + z - q ) 16l + 4 '
(23.20)
144 C.K. JORGENSEN
w h e r e the first w a y is related to R a c a h ' s theory, Sm,x f o r q <- ( 2 / + 1) being ½q, where the coefficient to ½q(q - 1) indicates w h a t multiple of D might be added to A . if one so desires, the s e c o n d w a y illustrates h o w the Pauli exclusion principle d e c r e a s e s ( S ( S + 1)) f r o m the value 3q it would h a v e had f o r non-equivalent electrons (it is actually multiplied by the f a c t o r 1 - (q - 1)/(4l + 1)) and the last w a y e m p h a s i z e s Pauli's h o l e - e q u i v a l e n c e principle by being s y m m e t r i c in (4l + 2 - q ) and q electrons. T h e coefficients in eq. (23.19) for various I and q h a v e been explicitly tabulated (J0rgensen, 1962a). In the case of a partly filled f shell, Smax c o r r e s p o n d s to a spin-pairing stabilization going d o w n to a sharp m i n i m u m -168/9/13 at q = 7 and generally having the e x p r e s s i o n -8/9/13 multiplied b y
½q(q - 1) for q b e l o w 7 as easily seen f r o m the first part of eq. (23.20). C h e m i s t s should be w a r n e d not to c o n f u s e the q u a n t u m n u m b e r S with the c o m p o n e n t Ms along a linear axis. Actually, it is possible (J0rgensen, 1969a, 1971a) to write e x p r e s s i o n s analogous to eq. (23.19) containing - ( M s ) 2 rather than - S ( S + 1), but is is not feasible to s p e a k in a significant w a y a b o u t spin-pairing energy related to r e v e r s a l of the "spin d i r e c t i o n " of a n u m b e r of electrons, b e c a u s e Ms lower than Sma x does not normally c o r r e s p o n d to a definite S in the eigenvalues.
T h o u g h D is (2l + 3)[(2l + 2) times the a v e r a g e value of the " e x c h a n g e integral of the t w o - e l e c t r o n o p e r a t o r " Kav (the strict definition being the a v e r a g e value of the (4/2 + 2l) integrals stabilizing a closed-shell H a r t r e e - F o c k function relative to its H a r t r e e product) and 9El f o r f electrons, it is not a p p r o p r i a t e to relate 2DS with the c o r r e s p o n d i n g multiple of Kay in r e p r e s e n t a t i v e Slater a n t i - s y m m e t r i z e d determinants. A classical case is p2 having the baricenter o f its 15 states containing no contribution of Kav if J(x, x) is written J(x, y) + 2Kav but shows the terms 3p at - K a y , 1D at +Kay and IS at +4K,v. A m o n g the 15 Slater determinants, 3 contain the contribution J(x, x) f o r m i n g IS and two of the five 1D states. T h e spin-pairing e n e r g y p a r a m e t e r D is }Kay. It is an o b v i o u s s t a t e m e n t that if several terms show Smax a n d differ in energy, the lowest has a lower e n e r g y than the baricenter of all the states having S . . . . F o r 4f q this h a p p e n s to be a distinct function of L of the ground term. Actually, the f u r t h e r stabilization is - 9 E 3 f o r 3H (q = 2 and 12) and 6H (q = 5 and 9) and is - 2 ~ E 3 for 4I (q = 3 and 11) and 5I
~ 4 . ( q = 4 and i 0 ) gr0urid t e r m s As a numerical q~estion, the R a c a h p a r a m e t e r E 3 turns out quite a c c u r a t e l y to be one tenth of E I.
The refined spin-pairing energy theory was originally introduced in an a t t e m p t (Jcrgensen, 1962c) to rationalize the variation of the w a v e n u m b e r s of electron t r a n s f e r bands, where an electron p r o v i d e d by one or m o r e reducing ligands r e d u c e s R(III) having 4f q ground state to R(II) with 4f q+~. T h e " r e f i n e m e n t s "
are a question of the coefficient to E 3 and of the first-order s p i n - o r b i t coupling stabilizing the lowest J - l e v e l to the e x t e n t -½(L + 1)sr4f for q below 7 and -~L~'4f for q a b o v e 7. F o r our p u r p o s e s , it is m o s t useful to consider the difference of spin-pairing energy r e m o v i n g an electron f r o m 4f q t o f o r m 4f" ~ and we intro- duce two p a r a m e t e r s of a linear variation of the o n e - e l e c t r o n e n e r g y differences, V as a zero-point and ( E - A ) as a p a r a m e t e r indicating the stabilization of 4f electrons f r o m one e l e m e n t to the next.
T H E O R E T I C A L C H E M I S T R Y O F R A R E E A R T H S 145
q = 1 V + 2~'4f
q = 2 V + ( E - A ) + 8 D + 9 E 3 + ~ n f A ) + ~3D + 1 2 E 3 + ½~'4f q = 3 V + 2 ( E - 16
q = 4 V + 3 ( E - A ) + ~ D
q = 5 V + 4 ( E - A ) + ~ 3 D - 1 2 E 3 - 1 ~ 4 f q = 6 V + 5 ( E - A ) + ~ D - 9 E 3 - ~4f q = 7 V + 6 ( E - A ) + ~ D - 2 ~ 4 f
q = 8 V + 7 ( E - A ) - ~ D + 3~4 f q = 9 V + 8 ( E - A ) - ~ D + 9 E 3 + ~4f q = 10 V + 9 ( E - A ) - ~ D q- 1 2 E 3 32 + l~"4f q = l l V + 1 0 ( E - A ) - i 3 D 24
q = 12 V + l l ( E - A ) - ~3D - 1 2 E 3 16 - ½~4f q = 13 V + 12(E - A ) - 8 D - 9 E 3 - ~'4f q = 1 4 V + 1 3 ( E - A ) - 3 ~ 4 f
(23.21)
T h e s a m e e x p r e s s i o n s c a n b e u s e d w i t h o p p o s i t e sign w h e n ' a d d i n g a n e l e c t r o n to 4f q-I f o r m i n g 4f q. S i n c e V ( o b v i o u s l y ) is a n o t h e r , n e w z e r o - p o i n t , s u c h ex- p r e s s i o n s s t a r t V - (q - 1)(E - A). T h e e x p e r i m e n t a l l y k n o w n f a c t t h a t t h e i o n i z - a t i o n e n e r g y f o r q = 7 is m a r g i n a l l y larger i n c o n d e n s e d m a t t e r t h a n f o r q = 14 is a n i n d i c a t i o n t h a t 7 ( E - A ) a p p r o x i m a t e l y e q u a l s 48D/13. I n s u c h a c a s e , eq.
(23.21) p r e d i c t s c o m p a r a b l e i o n i z a t i o n e n e r g i e s f o r q a n d f o r (7 + q) e l e c t r o n s . It is a n i n t e r e s t i n g ( b u t difficult) q u e s t i o n w h e t h e r o n e s h o u l d a s s u m e i n v a r i a n t p a r a m e t e r s f o r a definite o x i d a t i o n state, or allow a s m o o t h v a r i a t i o n f r o m o n e e l e m e n t to the n e x t . It has b e e n t h e g e n e r a l a t t i t u d e to e v a l u a t e ( E - A ) as o n e p a r a m e t e r f r o m e x p e r i m e n t a l d a t a (a t y p i c a l v a l u e is 3000 cm-1), to a s s u m e a fixed v a l u e of D = 6500 c m -~ ( o n e r e a s o n f o r this c h o i c e is t h e d i v i s i b i l i t y w i t h 13) a n d e i t h e r a fixed v a l u e of E 3, s a y 500 c m -~, or allow E 3 to i n c r e a s e s m o o t h l y f r o m t h e k n o w n v a l u e s 4 6 0 c m -~ f o r q = 2 in P r ( I I I ) to 6 3 0 c m -1 f o r q = 12 in T m ( I I I ) . T h e m i n o r r e l a t i v i s t i c effect e x p r e s s e d as a s p i n - o r b i t c o u p l i n g is u s u a l l y t a k e n ,into a c c o u n t a c c e p t i n g the s m o o t h (but n o t l i n e a r ) i n c r e a s e of 6"4f f r o m 6 5 0 c m -~1 f o r q = 1 in C e ( I I I ) to 2 9 5 0 c m -1 for q = 13 in Y b ( I I I ) . A s a n u m e r i c a l e x a m p l e , w e m a y c o n s i d e r the e x p r e s s i o n s a d d e d to V w i t h ( E - A ) = 3.2, D = 6.5, E 3 = 0.5 a n d e m p i r i c a l l y v a r y i n g ff4f all in u n i t s of 1000 cm-~:
q = 1: 1.3 2 : 1 2 . 5 3 : 2 0 . 9 4 : 2 1 . 6 5 : 2 2 . 2 6 : 3 0 . 2 7 : 4 0 . 3
q = 8 : 0.8 9 : 1 1 . 9 1 0 : 1 9 . 8 1 1 : 2 0 . 0 1 2 : 2 0 . 0 1 3 : 2 7 . 2 1 4 : 3 7 . 2
(23.22) T h o u g h it is o b v i o u s t h a t t h e d e c i m a l l a c k s p h y s i c a l s i g n i f i c a n c e , it is q u i t e c h a r a c t e r i s t i c to find t h e s a m e d e v e l o p m e n t (the d o u b l e zig-zag c u r v e ) r e p e a t e d b e t w e e n q = 1 a n d 7, a n d b e t w e e n q = 8 a n d 14. T h e a l m o s t h o r i z o n t a l p l a t e a u x at q = 3, 4, 5 a n d 10, 11, 12 are v e r y c h a r a c t e r i s t i c f e a t u r e s of t h e r e f i n e d s p i n - p a i r i n g e n e r g y t h e o r y . W h e r e a s eq. (23.22) is i n t e n d e d f o r a p p l i c a t i o n s to
1~ C.K. JORGENSEN
c o n d e n s e d matter, as discussed below, the ionization energies for g a s e o u s ions in table 23.1 suggest s o m e w h a t larger ( E - A ) . I F D - - 6 5 0 0 and E 3= 500 cm -~
are m a i n t a i n e d , / 3 of M +2 c o r r e s p o n d s to ( E - A) close to 4200 cm ~ and 14 to an e v e n larger value, unless D is c o n s i d e r a b l y larger. H o w e v e r , it is p r o b a b l y not worthwhile analyzing this situation in detail, b e c a u s e the m e a n value of the I of q = 7 and 8 according to eq. (23.21) should be close to the m e a n value of I for q = 1 and 14 at equal distance. Actually, the m e a n value of 14 for Gd +3 and Tb +3 is 5.14eV higher than 14 of C e +3 but only 3.29eV below 14 of L u +3, and by the s a m e token, 15 of Pr +4 is 6.74 eV below the m e a n value of 15 for Tb +4 and D y +4, which again is only 4.09 eV below 15 of H f +4. It is not e a s y to fix a definite value of these t w o distances, which should both be 1 3 ( E - A ) / 2 . O n e w a y out of this situation would be to a s s u m e that ( E - A) d e c r e a s e s as a function of increasing Z, as rationalized b y V a n d e r Sluis and N u g e n t (1972) b y ( E - A ) being a difference b e t w e e n two huge p a r a m e t e r s with A r e p r e s e n t i n g effects of inter- electronic repulsion, w h e r e it is highly p r o b a b l e that A . ( 4 f , 4f) = 156000 cm -~ for Pr +3 given in table 23.2 increases p e r c e n t a g e w i s e to the s a m e extent a s E 3 and hence s o m e 5000 cm -1 per unit of Z. This a r g u m e n t is far too strong to explain the effect o b s e r v e d in the gaseous ions, and o p e n s the fascinating discussion as to w h y the description using the p h e n o m e n o l o g i c a l b a r i c e n t e r polynomial (JCrgensen, 1969a, 1973a) is so constructive, w h e n one might h a v e e x p e c t e d e n o r m o u s effects of differential changes of the various p a r a m e t e r s (in particular of A , ) going f r o m one configuration to another. T h e whole c o n c e p t of one- electron energies is e x c e e d i n g l y difficult to t r a n s f e r f r o m o n e ionic charge to a n o t h e r of a m o n a t o m i c entity, or f r o m one oxidation state to another.
T h e refined spin-pairing energy t h e o r y is also used for a p u r p o s e where one would not h a v e e x p e c t e d it to w o r k v e r y well. If one c o m p a r e s the lowest energy level of the configuration 4fq-~5d with the 4f q ground state, M c C l u r e and Kiss (1963) pointed out that it followed eq. (23.21) for small a m o u n t s of R(II) i n c o r p o r a t e d in CaF2 in spite of the large p a r a m e t e r s of interelectronic repulsion involving 4f and 5d orbitals simultaneously, and in spite of the cubic rather than spherical s y m m e t r y separating the two 5d sub-shells by 1 to 2 e V . Similar o b s e r v a t i o n s were m a d e by L o h (1966) for small a m o u n t s of R01I) in CaF2 crystals, w h e r e the transitions w e r e f o u n d below 10 eV in eleven elements, to be c o m p a r e d with aqua ions and some other c o m p l e x e s of Ce(llI), Pr(III), T b ( I I I ) previously discussed (J~rgensen, 1956a, 1962c; J 0 r g e n s e n and Brinen, 1963;
R y a n and J¢rgensen, 1966). Martin (1971) called the c o r r e s p o n d i n g energy difference b e t w e e n the lowest J - l e v e l of [Xe]4fq-~5d and the lowest J level of [Xe]4f q in m o n a t o m i c ions the system difference and pointed out, almost at the same time as B r e w e r (1971) and N u g e n t and Vander Sluis (1971) that the variation with q is a l m o s t e x a c t l y described by the refined spin-pairing energy theory. It is also possible to define this s y s t e m difference b e t w e e n the lowest J - l e v e l of [Xe]4fq-15d6s and of [Xe]4f%s in g a s e o u s R +, and b e t w e e n the lowest J - l e v e l of [Xe]4fq-~5d6s 2 and of [Xe]4fq6s 2 in the neutral a t o m R °. T h e curves (as a function of q) run parallel with a f e w t h o u s a n d cm ~ b e t w e e n R +2 and R ÷, and b e t w e e n R + and R ° in striking c o n t r a s t to the large distance 50000 cm -~
THEORETICAL CHEMISTRY OF RARE EARTHS 147 between R +3 and R +2. It may be noted that the conditional oxidation state M[III]
or M[II] determined f r o m the n u m b e r of 4f electrons (JOrgensen, 1969a) is the ionic charge in the latter ions, but is M[II] for M + and M ° considered here. In eight cases (La °, L a +, L a +2, Ce °, Ce +, Gd °, Gd + and Gd +2) the system differences are negative, the configuration with one 5d electron being the more stable. The results for R(II) in CaF2 (McClure and Kiss, 1963) are similar to the system differences f o u n d in gaseous R +, whereas those for R(III) in CaF2 (Loh, 1966) are much closer to R +3 than to R +z, and actually about 17000 cm -1 below R +3 for q below 7, and about 10000 c m -1 below the gaseous ion for q at least 8.
The minute irregularities in the variation of complex formation constants (relative to R(III) aqua ions) as a function of q discovered by Fidelis and Siekierski (1966), cf. Mioduski and Siekierski (1975), now called the tetrad effect can be explained (J~rgensen, 1970a; Nugent, 1970) as covalent bonding decreas- ing E 3 to a slightly larger extent than E 1.
2.3. Electron transfer spectra and redox processes.
The clear-cut analogy between spectral lines of monatomic entities and the narrow absorption bands due to transitions b e t w e e n the energy levels belonging to the partly filled 4f shell of lanthanide(III) compounds convinced most physicists about the essentially monatomic character of all transitions to be expected. H e n c e , it was not surprising that inter-shell transitions to the excited configuration 4fq-~5d could be identified, but it was not generally felt that electron transfer bands would be prominent. This statement has a necessary exception: the orange or yellow colours of a large majority of cerium(IV) c o m p o u n d s , corresponding to v e r y strong, broad absorption bands in the violet or the near ultra-violet, c a n n o t be c o n n e c t e d with the absent 4f electrons, and the comparison with iron(III) suggests that an electron is transferred f r o m the surrounding reducing anions to the oxidizing central atom. It might be argued that the e m p t y orbital accepting the electron is 5d (rather than 4f) since the absorption band is so strong. This is in contrast to the first (and f r e q u e n t l y luminescent) excited level of the linear uranyl ion UO2 ~2 which is about 1000 times less intense (J~rgensen, 1957 a; J~rgensen and Reisfeld, 1975). T h e r e is no d o u b t t o d a y that it is due to electron transfer from two MO consisting essentially of oxygen 2p orbitals, to the e m p t y 5f shell. Because of strong spin-orbit coupling (large ~sf) and unusual conditions (almost identical one- electron energies of at least four of the seven 5f orbitals) it is not meaningful to ask w h e t h e r the excited level is a singlet or a triplet. C o n c o m i t a n t with its long half-life (10 -4 to l0 3 S) is its strongly oxidizing character as a photo-chemical reactant, for instance, h y d r o g e n atom abstraction from organic molecules (Rabinowitch and Linn Belford, 1964; Burrows and K e m p , 1974) and it has been established that Mn(H20)~ 2 yields a quasi-stationary concentration of manganese (III) by flash photolysis (Burrows et al., 1976).
The first authors to mention the possibility of electron transfer bands in trivalent lanthanides were Banks et al. (1961) noting an absorption edge in the