86o/vHlc
1. Electronic densities of spherical symmetry
T h e title o f this c h a p t e r " T h e o r e t i c a l C h e m i s t r y " s h o u l d n o t b e e x t r a p o l a t e d p r e c o c i o u s l y to a s i t u a t i o n w h e r e fhe t h e r m o d y n a m i c e q u i l i b r i a d e t e r m i n e d b y f r e e e n e r g y d i f f e r e n c e s , a n d t h e k i n e t i c r e a c t i o n r a t e s , c a n b e c a l c u l a t e d w i t h sufficient p r e c i s i o n . O n e o f s e v e r a l m a j o r difficulties is t h a t a ( n o n - l i n e a r ) m o l e c u l e o r p o l y a t o m i c ion c o n t a i n i n g N n u c l e i h a s its e n e r g y d e t e r m i n e d b y p o t e n t i a l s u r f a c e s h a v i n g ( 3 N - 6 ) s p a t i a l v a r i a b l e s , a s s u m i n g t h e v a l i d i t y o f t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n ( w h e r e t h r e e r o t a t i o n a l a n d t h r e e t r a n s - l a t i o n a l d e g r e e s o f f r e e d o m c a n be n e g l e c t e d ) . T h e t y p i c a l q u a n t u m - c h e m i c a l c a l c u l a t i o n a s s u m e s a f i x e d s e t o f i n t e r n u c l e a r d i s t a n c e s ( b y t h e w a y , this m a n i f o l d o f d i s t a n c e s is sufficient to c h a r a c t e r i z e the n u c l e a r s k e l e t o n w i t h o u t a n y e x p l i c i t r e f e r e n c e to the p o i n t - g r o u p , if t h e d i s t i n c t i o n b e t w e e n o p t i c a l l y a c t i v e e n a n t i o m e r s is n e g l e c t e d ) . H e n c e , Such a c a l c u l a t i o n is i n t r i n s i c a l l y s u i t a b l e f o r s p e c t r o s c o p i c t r a n s i t i o n s a n d f o r i o n i z a t i o n p r o c e s s e s s t u d i e d in p h o t o - e l e c t r o n s p e c t r a , in b o t h c a s e s o b e y i n g t h e F r a n c k - C o n d o n p r i n c i p l e , w h e r e a d e f i n i t e d i s t r i b u t i o n o f n u c l e a r p o s i t i o n s o c c u r s . O n t h e o t h e r h a n d , t h e g e n e r a l c h e m i c a l r e a c t i o n is a d i a b a t i c in t h e s e n s e t h a t t h e i n t e r n u c l e a r d i s t a n c e s are a l l o w e d to v a r y .
O b v i o u s l y , t h e e v a l u a t i o n o f p o t e n t i a l s u r f a c e s in ( 3 N - 5 ) - d i m e n s i o n a l s p a c e s b e c o m e s f a r m o r e c o m p l i c a t e d a n d t i m e - c o n s u m i n g , w h e n t h e n u m b e r 2( o f n u c l e i i n c r e a s e s . T h i s is o n e r e a s o n w h y t h e o r e t i c a l c h e m i s t r y in t h e s t r o n g s e n s e is g e n e r a l l y r e s t r i c t e d to s m a l l m o l e c u l e s in t h e g a s e o u s s t a t e . H o w e v e r , it is n o t p o s s i b l e to c o n c l u d e f r o m t h e p o s i t i v e d i s s o c i a t i o n e n e r g y o f a g i v e n m o l e c u l e (to i s o l a t e d , g a s e o u s a t o m s ) t h a t t h e r e is s t a b i l i t y in t h e c o n d e n s e d s t a t e s ( l i q u i d s , a m o r p h o u s a n d c r y s t a l l i n e solids). F o r i n s t a n c e , t h e d i a t o m i c m o l e c u l e s B H a n d S i O r e a r r a n g e b y c o n d e n s a t i o n , a n d N O is t h e r m o d y n a m i c -
THEORETICAL CHEMISTRY OF RARE EARTHS 113 ally highly unstable relative to N2 and Oz. T h e s a m e is true (at m o d e r a t e t e m p e r a t u r e ) for CO in spite of this molecule having one of the highest k n o w n dissociation energies. T h e g a s e o u s state is particularly f a v o u r a b l e t o w a r d uni- positive c o m p l e x ions if the criterion of dissociation energies to m o n a t o m i c species is accepted. It is confirmed b y m a s s - s p e c t r a that nearly all c o m b i n a t i o n s MX + (including noble gases being M) are stable t o w a r d dissociation, w h e r e a s the calculations for neutral H L i H e (containing six electrons) show marginal stability, as the only p r o b a b l e case of a neutral helium c o m p o u n d . On the other hand, q u a n t u m - c h e m i c a l studies of H~, C H ; ~, H e l l +, H e O +, C~ (having a ground state with a total spin quantum number S = ~) and A r F + h a v e b e e n t h o r o u g h and convincing. All of these species are f a r too oxidizing or too acidic to be c o m p a t i b l e with any k n o w n solvents or anions to f o r m solid salts.
N e v e r t h e l e s s , there are a quite specific set of properties of lanthanide c o m p o u n d s closely related to the b e h a v i o u r of gaseous a t o m s and positive ions.
For instance, a s y s t e m containing q electrons can show the values of S = 0 (singlet), 1 (triplet), 2 (quintet), 3 (septet), 4 (nonet) . . . . when the n u m b e r q is even, and
S = ½ (doublet), 3 (quartet), ~ (sextet), 7 (octet) . . . .
when q is odd. T h e w o r d s in p a r e n t h e s e s are the old names, c u s t o m a r y in atomic s p e c t r o s c o p y , w h e n R u s s e l l - S a u n d e r s coupling is a good a p p r o x i m a t i o n . In actual practice, it is of little i m p o r t a n c e that the highest possible value of S is ½q.
W h e r e a s the t w o - e l e c t r o n a t o m s H e , Li +, Be +z .. . . h a v e sharply defined singlet and triplet energy levels, the lowest q u a r t e t levels of the t h r e e - e l e c t r o n a t o m s Li, Be + . . . . h a v e far higher energies than the first ionization e n e r g y c o r r e s p o n d - ing to f o r m a t i o n of the closed-shell electron configuration ls z (containing only one state, a singlet) by belonging to the highly excited configuration ls2s2p. The strong bonding of two electrons in the inner shell ls p r e v e n t s S values higher than 2 f r o m being readily o b s e r v e d in the six-electron a t o m s C , N +, O +2 . . . . and in the eight-electron a t o m s O, F +, N e +z . . .
T h e general c o n s e n s u s a m o n g organic c h e m i s t s is that normal m o l e c u l e s are diamagnetic and h a v e S = 0. Species having positive S are called " f r e e r a d i c a l s "
and are e x p e c t e d rapidly to dimerize or to p e r f o r m other reactions, with S vanishing in all end products. This does not p r e v e n t a few molecules (such as N O and CIO2) containing an odd n u m b e r of electrons and having ground states with S = ½. It is m u c h m o r e striking that certain molecules h a v e positive S in spite of an e v e n n u m b e r of electrons. Oz w a s s h o w n by F a r a d a y to be p a r a m a g - netic, and L e n n a r d - J o n e s d e m o n s t r a t e d that the six states of the l o w e s t M.O.
(molecular orbital) configuration constitute three e n e r g y levels, the lowest with S = 1 and the other t w o with S = 0. This case of a triplet ground state is a c o n s e q u e n c e of a general result of two orbitals having e x a c t l y the s a m e e n e r g y for group-theoretical reasons, containing only two of the four possible electrons.
Such b e h a v i o u r is far m o r e f r e q u e n t in the transition groups. Thus, the large m a j o r i t y of m a n g a n e s e ( I I ) and iron(III) c o m p o u n d s contain five 3d-like electrons
114 C.K. JORGENSEN
and show S = 5 in their ground states. Actually, in the 3d group, the c o m p o u n d s s e e m to be particularly stable w h e n S is high.
H o w e v e r m u c h the positive S values m a y seem u n e x p e c t e d to chemists working mainly with e l e m e n t s outside the transition groups, the high S values of ground states belonging to an electron configuration with one or s e v e r a l partly filled shells are well k n o w n in m o n a t o m i c entities. Charlotte M o o r e has compiled all the e n e r g y levels e x c e p t f o r the lanthanides and the e l e m e n t s f r o m thorium onwards. It is to be h o p e d that the f o u r t h v o l u m e of " A t o m i c E n e r g y L e v e l s "
will be published soon. E v e n b e f o r e 1927, H u n d had established rules for the various q u a n t u m n u m b e r s of the ground state of a m o n a t o m i c species containing one partly filled /-shell. F o r historical reasons, these n o n - n e g a t i v e integers are called:
I = 0 1 2 3 4 5 . . . s p d f g h . . .
and a given shell c h a r a c t e r i z e d by the c o m b i n a t i o n of the positive integer n (larger than l) and l is able to a c c o m m o d a t e a m a x i m u m of ( 4 / + 2) electrons.
Until the half-filled shell containing ( 2 / + 1) electrons is r e a c h e d , the ground state of a s y s t e m containing q electrons (written as right, h a n d superscripts 3d q or 4f q) has S = ½q. In the second half of the shell, S = ½(4/+ 2 - q) in a g r e e m e n t with Pauli's hole-equivalence principle (actually, the n u m b e r of states and the distribution of other q u a n t u m n u m b e r s such as L and J are also the s a m e in the two cases). It is i m p o r t a n t for c h e m i s t s not to c o n f u s e the spin c o m p o n e n t Ms along a selected linear axis with S. T h o u g h the highest Ms value of a given s y s t e m c o r r e s p o n d s to the highest S, it is not valid to conclude that a lower Ms necessarily belongs to a lower S. T h e psychological p r o b l e m is that b o x e s with a r r o w s pointing in one of two o p p o s i t e directions h a v e b e e n v e r y p o p u l a r a m o n g chemists as a r e p r e s e n t a t i o n of Ms being the arithmetic sum of +½ or -½ of e a c h electron. H o w e v e r , quintet, triplet a n d singlet states can all h a v e Ms zero.
T h e c o n c e p t s derived f r o m a t o m i c s p e c t r a h a v e b e e n v e r y i m p o r t a n t in the r e c e n t progress of understanding s p e c t r o s c o p i c properties and such chemical questions as the deviations of the oxidation state f r o m M(III) and the conditions for metallic c h a r a c t e r of the c o m p o u n d s . We return to these individual p r o p e r - ties specifically d e p e n d e n t on 4f q in section 2, and we start with the s m o o t h l y v a r y i n g p r o p e r t i e s which can b e described as if the lanthanide M(III) is a sphere of electronic density gradually decreasing its radius f r o m l a n t h a n u m to lutetium.
The contributions of q u a n t u m c h e m i s t r y to this, a p p a r e n t l y simpler p r o b l e m , h a v e been m u c h m o r e qualitative than the specifically s p e c t r o s c o p i c statements.
1.1 Ionic radii and coordination n u m b e r N
Seen f r o m the point of view of q u a n t u m m e c h a n i c s , a m o n a t o m i c entity has no distinct surface. T h e structure of Schr6dinger's e q u a t i o n indicates an exponential d e c r e a s e of the w a v e function (and its square) as a function of large , i u ~ t n u c l e a r distance r. Actually, the w a v e function of a o n e - e l e c t r o n a t o m J e ~Jr rc~a.~
THEORETICAL CHEMISTRY OF RARE EARTHS 115 d e c r e a s e s a s y m p t o t i c a l l y as e x p (-(2E)l/2r) in atomic units
length: 1 b o h r = 0.52917 ,~ = 5.2917 × 10-" m,
energy: 1 hartree = 2 r y d b e r g = 27.2116 eV = 219474.6 cm -t, velocity of light in vacuo: 137.036 . . . b o h r / a t o m i c unit of time,
where - E is the negative e n e r g y of a stationary state. T h o u g h one might c h o o s e a limit of negligible size of the exponential function as a m e a s u r e of the radius there is no d o u b t that such a choice would p r o d u c e atomic radii of the type k n o w n f r o m solidified noble gases, m u c h larger than the radii k n o w n f r o m ordinary c o m p o u n d s . It is general e x p e r i e n c e f r o m c r y s t a l l o g r a p h y of solids that the closest interatomic c o n t a c t s fall into t w o categories: the f o r m e r t y p e of Van der Waals radii well-known f r o m internuclear distances b e t w e e n two a d j a c e n t neutral molecules in a crystal, and the chemical bonds. T h e f e w e x c e p t i o n s of internuclear distances falling in the o p e n interval b e t w e e n the t w o categories are quite interesting; one type is due to short c o n t a c t s b e t w e e n metalloid a t o m s due to strong h y d r o g e n bonds ( S p e a k m a n , 1972) such as F H F - and H 2 O H O H ~ and a n o t h e r t y p e is due to residual chemical interactions which are quite f r e q u e n t b e t w e e n a d j a c e n t sulfur a t o m s or b e t w e e n iodine a t o m s ( M u r r a y - R u s t et al., 1975).
Slater (1964) e m p h a s i z e d that B r a g g ' s original suggestion was to assign atomic radii to e a c h e l e m e n t i n d e p e n d e n t of the nature of the c h e m i c a l bonding. This p r o p o s a l is r e m a r k a b l y a c c u r a t e but has certain w e a k n e s s e s . Thus, the c e s i u m - cesium distances in colorless N a C l - t y p e C s F are shorter than in metallic cesium, though the f o r m e r c o m p o u n d shows no sign of not being a l m o s t exclusively electrovalent. Students are often i m p r e s s e d by the highly different ionic and covalent radii given in t e x t - b o o k s f o r a given element. This distinction is to a large extent an illusion. Roughly speaking, the Cation M +z in a c o m p o u n d MX~ is said to h a v e an ionic radius 0.8.~ smaller t h a n the covalent radius of the M atoms, w h e r e a s the anion X is said to h a v e an ionic radius 0.8/~ larger than the covalent radius of X. Since the o b s e r v a b l e quantity is the M - X distance, the d i s c r e p a n c y bdtween the two descriptions usually falls within the s a m e interval of a b o u t 0.1 A as defined f r o m the dispersion of a p p a r e n t ionic radii of the same M +~ in different c o m p o u n d s . I t m u s t be added, in all fairness, that a certain boundar~y condition exists f o r this p r o b l e m of dividing the M - X internuclear distance in two ionic radii so that the choice of crystal t y p e in highly elec- trovalent c o m p o u n d s can be rationalized b y the packing of hard spheres of two distinct radii. Thus, the iodide-iodide c o n t a c t s in N a C l - t y p e LiI and the o x i d e - oxide c o n t a c t s in CaFz-type CeO2 give higher limits to the ionic radii of iodide and oxide.
It m a y be worthwhile considering explicitly the m e t h o d o l o g y of determining internuclear distances. Diffraction techniques of crystalline solids p r o v i d e a t i m e - a v e r a g e picture of the a v e r a g e c o n t e n t of the distinct unit cell, which is able exactly to fill the crystal without gaps or overlap b y translations in three directions. S o m e internuclear distances are particularly well known, w h e n they are g e o m e t r i c a l c o n s t a n t s multiplying the unit cell lengths (in which case the
116 C.K. J O R G E N S E N
atoms considered are said to o c c u p y special positions). Thus, the cubic (having three equivalent cartesian axes) crystal types CsC1, NaC1, CuCI (frequently called zincblende, because it is one of the modifications k n o w n of ZnS) and CaF2 have all their atoms at special positions. The distances to atoms at general positions (e.g. the Pt-C1 distances in the regular o c t a h e d r o n f o u n d in the cubic type K2PtC16) are much more difficult to evaluate with great precision, and the experimental data n e e d e d are relative intensities of spots on a Weissenberg diagram, or less preferably, lines on a D e b y e p o w d e r diagram. Actually, one of the early arguments for almost exclusive electrovalent bonding was the vanish- ing intensity of reflections d e p e n d e n t only on the difference b e t w e e n the two constituents in NaCl-type NaF, KC1 and RbBr having both the cation and the anion isoelectronic with neon, argon and krypton, respectively.
T h e r e is no doubt that crystallographers tend to overestimate the s y m m e t r y of their crystals, either by neglecting w e a k reflections, by studying twin samples, or simply by putting atoms on esthetically satisfactory positions and obtaining reasonable agreement with the calculated intensities. In the chemistry of the rare earths, one f r e q u e n t l y meets statistically disordered, non-stoichiometric compounds, even of cubic symmetry. It is quite obvious that m a n y inorganic compounds do not contain distinct molecules. Thus, in NaCI, each Na ÷ is surrounded by a regular o c t a h e d r o n of six C1 , and each C1 is surrounded by a regular o c t a h e d r o n of six Na +. H e n c e , in both cases, the coordination number N = 6 for both the cation and the anion. In CsC1, they both have N = 8. In a way, CaF2 is a super-structure of CsC1, lacking half the cations in a systematic manner (and retaining cubic symmetry). Ca +z has N = 8 with eight F - forming a cube, whereas F - has N = 4 with f o u r Ca +2 forming a regular tetrahedron.
Goldschmidt pointed out that many CaFz-type minerals are strongly non-stoi- chiometric~ T h u s , y t t r o f l u o r i t e Ca~-xYxF2+x contains yttrium(Ill) to the extent of x up to 0.3, and the excess fluoride is situated on the e m p t y cation positions (compared with CsC1) like the excess o x i d e in the non-stoichiometric UOz+x. On t h e whole, it is more usual to have a deficit of anions in the statistically disordered fluorites, as is k n o w n from thorianite Thl-xRxOz-0.~x (in the following, R denotes a mixture of various lanthanides). It is possible to drive synthetic samples of Thl_~La~Oz-0.5~ up to x = 0.54. The N e r n s t lamp ( M f b i u s , 1962, 1964) is a mixed oxide such as CaF2-type Zrl-~YxO2-0.sx or Zrl_xMgxOz-x (with x around 0.1) conducting electricity a b o v e 500°C, not by transport of electrons (like in metals and semiconductors) but by diffusion of oxide anions among the vacant positions (more similar to hot AgI or to molten salts). Such mixed oxides are conveniently prepared by co-precipitating mixed aqueous solutions (contain- ing the quadrivalent and trivalent metals) with aqueous ammonia (not producing non-volatile cations). The hydroxide precipitate is washed, dried and calcined, say at 800°C, to an intimately mixed oxide, though the crystal type (J0rgensen and Rittershaus, 1967) identified with D e b y e p o w d e r diagrams does not always c o r r e s p o n d to t h e r m o d y n a m i c equilibrium. Thus, equal a m o u n t s of erbium(III) and zirconium(IV) p r o d u c e the disordered fluorite Er0.sZr0.5Ol.75 statistically lacking an eighth of the oxide. The same result is obtained with dysprosium(III)
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 117
when the mixed hydroxide is calcined at moderate temperatures, whereas DyzZr207 prepared at 1200°C is a cubic superstructure of fluorite, pyrochlore, where M(III) has N = 8 in a distorted cube and M(IV) has N = 6 in a distorted octahedron. The pyrochlore structure has been carefully investigated in Er2Ti207 where the smaller Ti(IV) pulls the oxide anions away from Er(III). Caro (1972) has discussed the interesting situation that certain disordered mixed oxides show epitaxial layer structures which also can be studied using electron micrographs.
One of the striking aspects of certain crystal structures containing rare earths is that small but systematic deviations occur from high symmetries. Thus, the cubic perovskite SrTiO3 has the large strontium(II) surrounded by twelve oxide anions forming a tetrakaidecahedron (usually called a cuboctahedron) and the smaller titanium(IV) surrounded by six oxides anions in a regular octahedron.
Related materials such as BaTiO3 (which should not be called titanates because they d o not contain discrete titanate anions like carbonate in BaCOa or phos- phate and orthovanadate in RPO4 and RVO4) have transitions between crystal- line modifications of slightly different symmetry. In particular, some of these modifications are ferroelectric, having quite unusually large dielectric constants.
This phenomenon has not been reported for the perovskites RMO3 formed by a large R(III) and a small M(III), such as aluminum. However, these perovskites have generally non-cubic symmetries. Their magnetic properties can be rather unusual in cases of M = chromium (having S = 3) and iron (S = 2) having a high spin quantum .number. Another interesting fact (Schneider and Roth, 1960) is that La(III) and the heaviest lanthanides have sufficiently different ionic radii that LaErO3, LaTmO3 and LaYbO3 are perovskites though some of these samples decompose by heating above 6500C, forming among other products a type of mixed oxide NdYO3 only characterized by its powder diagram (J~rgen- sen and Rittershaus, 1967). However, there is spectroscopic evidence (Faucher and Caro, 1975) that perovskites containing lanthanides such as La~_xEuxA103 have rather low local symmetry, and one should not expect N = 12 to cor- respond to roughly equal L a - O distances.
The three 61assical types of rare earths are the hexagonal A-type R203 where R has N = 7 (two triangles in planes perpendicular on the axis containing the seventh neighbor atom), the complicated and low-symmetry B-type, and finally the cubic C-type (also known for Sc203, In203 and T1203) where N = 6. A quarter of R in the C-type has a surrounding similar to Ti in pyrochlore ErzTi207 whereas 75% of the R have six oxide neighbors very far from forming a regular octahedron. Bergerhoff (1964) and Caro (1968) have pointed out that these highly irregular structures are much more regular when seen from the point of view of the oxygen atom. Actually, these and many other oxides have to a good approximation four atoms M in a circumscribing regular tetrahedron, as is known from the so-called basic beryllium acetate OBe4(OzCCH3)6 whereas N is also 4 in BeO and ZnO, but the structure is hexagonal (like in the wurtzite modification of ZnS). The corundum type (a-A1203) is also known from V 2 O 3 (which is a physical metal above the Mott transition temperature), Cr203, Fe203 and Ga203 and has N -- 4 for oxide and N = 6 for M(III) being much closer to a
118 C.K. JORGENSEN
regular o c t a h e d r o n than is the case f o r C - t y p e rare earths. It m a y be noted that in a binary c o m p o u n d MaXb w h e r e all the M a t o m s are equivalent, and where all the X a t o m s are equivalent, it is o b v i o u s that N (the n u m b e r of M - X contacts) for M is (b/a) times N of X.
T h e whole question of N f o r trivalent lanthanides does not h a v e the s a m e c o n t e x t as in the iron(3d), palladium(4d) and platinum(5d) transition groups. The n u m b e r q of 4f electrons does not h a v e a specific influence on N going f r o m one lanthanide to the next, w h e r e a s the d groups are v e r y sensitive in this respect. It m a y be instructive to c o m p a r e with one of the least transition-group-like elements, nickel. It- is b e y o n d discussion that certain c o m p o u n d s exist of Ni(IV), Ni(III) and Ni(0). T h e s e oxidation states are defined (J~rgensen, 1969a) f r o m the p r e s e n c e of six, s e v e n and ten 3d-like electrons in the p r e p o n d e r a n t electron configuration. H o w e v e r , the large m a j o r i t y is nickel(II) containing eight 3d-like electrons. A m o n g these c o m p o u n d s , b o t h the aqua ion Ni(H20)~ 2, the N a C I - t y p e NiO, the highest a m m o n i a c o m p l e x Ni(NH3)~ 2 and a large n u m b e r of other e x a m p l e s h a v e N -- 6 with the neighbor a t o m s in a regular o c t a h e d r o n , and all the k n o w n Ni(NH3),(H20)~_2, are a p p r o x i m a t e l y octahedral. Such cases, also k n o w n f r o m rutile-type (a tetragonal modification of TiO2) NiF2 and cubic p e r o v s k i t e s such as KNiF3 all have N = 6 and are p a r a m a g n e t i c , c o r r e s p o n d i n g to S - - 1 of t h e ground state. The diamagnetic (S = 0) nickel(II) c o m p l e x e s usually h a v e N = 4 with the f o u r neighbor a t o m s in a square, such as Ni(CN)42 with all nine nuclei situated on a G r e e k cross, or in a rectangle, such as m a n y sulfur-containing c o m p ! e x e s like d i t h i o c a r b a m a t e s Ni(S2CNX2)2 and dithiophos- phates Ni(S2P(OX)2)2 where X is CH3, C2H5 . . . . (J0rgensen, 1968b). H o w e v e r , other diamagnetic nickel(II) c o m p l e x e s h a v e N = 5 like the r e d - o r a n g e Ni(CN)~ 3 ( f o r m e d in strong cyanide solutions) which is t e t r a g o n a l - p y r a m i d a l like Cu(NH3)~ z f o r m e d in c o n c e n t r a t e d a m m o n i a ( R o m a n o and B j e r r u m , 1970). A few sporadic e x a m p l e s of unusual s t e r e o c h e m i s t r y a m o n g p a r a m a g n e t i c (S = 1) c o m p l e x e s are now known, such as N = 4 in regular tetrahedral NiC142 and NiBr42 occurring in certain n o n - a q u e o u s solvents, e.g. CH3CN, and in salts of large cations, e.g. N(C2Hs)LThis s u m m a r y of nickel(II) c h e m i s t r y shows that N has the m o d e r a t e values 4, 5 and 6 also k n o w n f r o m e l e m e n t s outside the transition g r o u p s such as p h o s p h o r u s ( V ) , indium(III) and tin(IV) and that spec- troscopic properties (color and a b s o r p t i o n spectra) and the choice b e t w e e n the alternatives S = 1 and 0 are intimately c o n n e c t e d with the s t e r e o c h e m i s t r y . T h e octahedral cases with S = 1 h a v e one anti-bonding electron in e a c h of the two d-orbitals (pointing t o w a r d s the neighbor atoms) having angular functions pro- portional to (x 2 - y2) and (3z 2 - r2), w h e r e a s the three other d-orbitals with angular functions proportional to (xz), (yz) and (xy) are filled with six electrons.
The f o r m e r f e a t u r e (of two electrons in two orbitals having the s a m e energy) provides a ground state with S = 1 as in 02. T h e quadratic ( " s q u a r e - p l a n a r " is rather redundant) cases with S = 0 h a v e the strongly anti-bonding orbital (x 2 - y 2 ) directed t o w a r d the f o u r ligating a t o m s situated on the x- and y - a x e s e m p t y , and the four other d-orbitals filled. T h e t e t r a g o n a l - p y r a m i d a l cases of N = 5 and S = 0 have essentially the s a m e distribution of electrons on the f o u r lower
THEORETICAL CHEMISTRY OF RARE EARTHS 119 orbitals, but an additional complication is the absence of a centre of inversion allowing the mixture of /-values of the nickel atom with opposite parity. Such weak intermixing has observable effects on the intensities of the absorption bands due to internal transitions in the partly filled shell (J~rgensen, 1975b).
For reasons to be discussed below, it is quite certain that the 4f orbitals hardly have any influence on the chemical bonding in trivalent lanthanide c o m p o u n d s . This by no means excludes a considerable a m o u n t of covalent bonding. Un- fortunately, it is very difficult to obtain reliable evidence for such bonding, because in the L.C.A.O. (linear combination of atomic orbitals) model, it is due to the five e m p t y 5d orbitals and the e m p t y 6s orbital. The same difficulty occurs at the end of the d groups, where the covalent bonding in copper(I), zinc(II) and gallium(III) c o m p o u n d s is highly probably, and due to the e m p t y 4s orbital and, perhaps to a much lower extent, to the three e m p t y 4p orbitals. We return below to the more qualitative reasoning behind the comparison of c o m p o u n d s with metalloids of varying electronegativity, but it may be mentioned that strongly calcined CeO2 and ThO2 dissolve r e m a r k a b l y slowly in acids, even in boiling sulfuric acid. Though it cannot be excluded that the exceedingly large Madelung potential in these fluorite-type crystals provides a high Arrhenius activation energy for dissolution, it w o u l d seem likely to be an expression of partly covalent bonding. Actually, the X-ray emission spectra (Bonnelle, 1976) of these oxides suggest a certain covalence involving a mixture of o x y g e n 2p orbitals with Ce4f and Th5f orbitals, e.g. having the angular function proportional to (xyz) pointing toward all eight neighbor atoms. A n y b o d y dissolving sesquioxides in aqueous acids will also notice that A-type oxides react rapidly, if not violently, like MgO, w h e r e a s calcined C-type containing no carbonate is un- reactive. F u r t h e r m o r e , it is striking that one may heat the suspension for some twenty minutes, and then, within a few minutes, a kind of autocatalytic reaction seems to occur, producing a limpid solution.
For m a n y years the general model used for M.O. calculations in molecules was L.C.A.O.; which might be justified by the very weak effects of chemical bonding (except hydrogen) c o m p a r e d with the binding energy (roughly Z 24 rydberg) of the atoms consisting of a nucleus with the charge + Z surrounded by Z electrons. H o w e v e r , this is a rather vague argument, because the loosest bound electrons have ionization energies I between 0.4 and 1.2rydberg, not much more than the bond dissociation energies. Whereas attempts to solve the Schr6dinger equation directly with a self-consistent potential for electrons were previously made for infinite crystals (in the model of augmented plane waves) there has recently been a certain interest in the X a model for oligoatomic systems (Johnson, 1973) providing I in good agreement with photo-electron spectra. One of the qualitative descriptions related to such direct techniques disregarding the L C A O hypothesis is the valence-shell electron-pair repulsion (VSEPR) elaborated by Gillespie (1972). This description has certain virtues for molecules f o r m e d by elements outside the transition groups, where a given spherical region of the valence shells is divided between bulky lone-pairs, and bonding electron pairs having smaller angular requirements when the other atom