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Ligand field calculation of the lower electronic energy levels of the lanthanide monoxides
JOURNAL
OF MOLECULAR
SPECTROSCOPY
131,301-324 (1988)
Ligand Field Calculation of the Lower Electronic Energy Levels of the Lanthanide Monoxides P. CARETTEANDA.HOCQUET Laboratoire de Spectroscopic Molthlaire, Unit6 Associtfe au CNRS No. 779, Universitt? de Lille Flandres Artois. 59655 Villeneuve d’Ascq Cedex, France
The ligand field theory applied to the lanthanide monoxides predicts that the lower electronic states of these molecules are due to the configuration 4fN-‘6s (4fN for EuO and YbO) of the divalent lanthanide ion Ln’+. We present here the results of a ligand field calculation of the energies of the lower electronic states for all the stable rare-earth monoxides. o 1988 Academic Press, Inc.
I. INTRODUCTION
The lanthanides are the elements pertaining to the group of rare earths which are characterized by the progressive filling of the 4fshell of their electronic configurations. They commence with the element cerium (2 = 58 ) and end with the element ytterbium (2 = 70). On the basis of the similarities of their chemical properties to those of the other lanthanides, one associates with this series the elements lanthanum (2 = 57) and lutetium (Z = 7 1) which possess an empty and a completely filled 4f shell in their normal configurations. The 4fN and 4fN-’ open shells give rise to numerous terms: as an example the configuration 4f8 leads to 295 atomic electronic states. The energies involved with the 4f, 5 d, 63, and even 6p electrons are of the same order of magnitude and the interactions which are usually taken as perturbations of the central field are very strong and lead to important configuration interactions which increase the complexity of the lanthanide spectra. The lanthanides possess the common feature of a 4feigenfunction essentially located inside the 5s25p6 closed shells of the xenon structure (Fig. 1). This effect known as the “lanthanide contraction” reinforces itself progressively with the filling of the 4f shell (Fig. 2). As a result of this contraction the surroundings of the lanthanide atoms are of little chemical significance. II. BASIS OF THE LIGAND FIELD THEORY APPLIED TO THE STUDY OF THE LANTHANIDE MONOXIDE SPECTRA
This theory was first developed for the study of the rare-earth salts and solutions of their compounds. At low temperatures the absorption spectra of these chemical species are usually found to consist of groups of sharp lines the positions of which are relatively unaffected by the environment of the lanthanide ion and may be correlated with the positions of the energy levels of the 4f N configuration of the free, tripositive ion. A 301
0022-2852/88 $3.00 Copyright 0
1988 by Academic Press, Inc.
AU rights of repmdwtion
in any form reserved.
302
-0.60
CARETTE AND HOCQUET
t
FIG. I. Double zeta slater type orbitals 4f, 5s, and 5p for NdI.
model was introduced for the study of the lanthanide monoxides (I, 2), model which enables a pseudoatomic treatment of a molecular problem. In this theory the electronic structure of lanthanide monoxide may be approximated by the ionic model Ln 2+02-, The molecular electronic levels result from the splitting of the atomic levels of the parent Ln 2+ free ion configuration by the axial electric field between the R, separated negative ligand of charge -2e and positive metal centers, neglecting metal-ligand orbital overlap (Fig. 3 ) . The perturbed Hamiltonian is usually written as, similar
H = Ho + HLF,
where Ho is the Hamiltonian iltonian.
(1)
for the free Ln2+ ion and HLF is the ligand field Ham-
FIG. 2. Double zeta slater type orbital 4Jfor LaI, NdI, GdI, ErI, and LuI.
LIGAND FIELD CALCULATION
303
OF LnO
q outer
,\\
r,=q
Re
\
inner
region
r<= Re
region
r,= Re r< = ri
FIG. 3. Coordinate system
for the ligandfieldpotential.
The interaction of a metal centered electron i with the scalar potential function V(r:,
oi9
Cpi)
=
F
z
(2)
may be expressed in terms of ri, the metal center origin coordinate, using the expansion ’ = [r? + Rz - 2riRAOS Bi]-“2 = 5 PIA ui)tF, r: k=O
(3)
where Vi = cos Bi, t; = Ti, ri> = max(R,, ri), ri< = min(R,, r;), and Pk( vi) are the Legendre polynomial functions. The total ligand field potential energy is obtained by summing eV( r:, Bi, vi) over all the indices of the electrons pertaining to the metal ion. Therefore the ligand field Hamiltonian HLF = $ 2e2 5 Pk( ui)tk i=l ri> kc0
(4)
is expressed as a sum of one-electron Hamiltonians hLF = 2” $ Pk(Ui)tfs ri> k=O
(5)
This formulation yields a semiempirical method of calculation of the electronic energy levels of the lanthanide monoxides from the known energy levels of the parent divalent free ions Ln*+.
304
CARETTE AND HOCQUET TABLE I Structure of the Hamiltonian Matrix Ho of the Free ion Ln2+
(4f Nvcl
(4fN-‘vcVcnp&pl
1
3
3
2
The Hamiltonian matrix of the free ions can symbolically be divided into four regions with regard to the atomic states arising from the configurations 4f Nand 4f N-‘nl (Table I). Whatever the encountered coupling case the problem is to express the three following types of matrix elements, type 1
(4~“4%I~LF14fNICI,)
type 2
(4fN-‘Vc,
type 3
(4fN~Ie_F14fN-‘*c,
M&l
(6)
&14fN%,
q&k,)
&l+p),
(7)
(8)
in terms of the one-electron matrix elements ( n ‘l’m’1hLF I n lm ) . Index c is related to core electrons and index p to peripheral electrons. The one-electron matrix elements can be written as
or (n’l’m’/h&&z)
= c
1%”
R,?~Ktf,‘hLFR~,Y~r2sin
B dr dp d8
(10)
by taking the wave function $ (I, (o, 0) as a product of R( r), the radial wave function, and Y,( cp, e), the angular wave function, written R and Ylmfor simplicity. Rodrigues’ formula yields Y,m = (-l)*
(21+ 1)(1- m)! “2 47r(l+ m)! [
f3)ezmp 1 &(cos
(11)
LIGAND FIELD CALCULATION OF LnO
305
and allows one to write:
(12) Therefore: * 2r Y,!,tPk(cos e)Y,_,sin 0 dp de ss0 0
= [(21’ + 1)(21+ 1)]“2 It follows from the properties of symbol 3 J that: (i) the one-electron operator hLF is diagonal with respect to m; (ii) the energy levels m and -m are degenerate; (iii) the matrix elements of hLF are nonzero for 1’ + k + 1 = 2g where g is an integer and for I’, k, 1 fulfilling the triangular relation A( l’kl), i.e., 11-1’1
nl)F(II,
1, m),
(14)
k=O
where (15) and Ck(lt, 1, m) = (&)I”
1
s,z’ &tmYYkOYl_,,,sin 8 dp de.
(16)
In order to include these matrix elements in more complex configurations one must express the one-electron ligand field Hamiltonian operator in tensor operator form. The formula of composition of spherical harmonics pk(cos 8) =
&
4
(17)
allows us to write
(18)
306
CARETTE AND HOCQUET
where Ct is the irreductible angular tensor operator defined by Racah (3) and where
is the radial tensor operator. In the case of diatomic molecules the symmetry of the group C,, restricts the problem to contributions of the components along the molecular axis, so that the ligand field Hamiltonian operator takes the following form: HLF
=
(19)
c (BOkCbi. ki
The sole nonvanishing matrix elements of this operator are those which couple electronic states with the same value of the projection of the total angular momentum along the molecular axis. The molecular energy level calculation requires, therefore, the diagonalization of a series of Hamiltonian matrices for every possible value of this projection. III. MATRIX ELEMENTS OF THE LIGAND FIELD HAMILTONIAN
OPERATOR
In order to perform the diagonalization of the ligand field Hamiltonian matrices one must first survey the vector coupling cases occurring in the free ion Ln2+ configurations and choose the most appropriate basis sets for the description of the corresponding monoxide molecules. One can then express the matrix elements of the ligand field Hamiltonian for these basis sets. Only after the completion of these two steps can one proceed by calculating the angular and radial parts of these matrix elements.
Basis Sets of the Lanthanide Free Ions A study of the coupling occurring in the configurations of the lanthanides as depicted by Z. R. Goldschmidt (4), B. G. Wyboume (5), and W. C. Martin et al. (6) allows us to define three basis types of eigenvectors for the description of the spectra of the free ions LnIII. They are type 1
14fN~&LJa~J,),
type 2
14fN-‘r&LJc,
type 3
I4f N%%L,
(20) nlJp, JaWa),
nl, &L&G,),
(21) (22)
where y symbolizes the set of quantum numbers which is necessary for the complete description of the atomic states as required by the group theory. This symbol will be dropped later for the sake of brevity. The type 1 eigenvectors correspond to the Russell-Saunders coupling case. All the configurations 4 f N are studied in this coupling scheme since the electrostatic parameters Ek (4 f, 4f ) which are responsible for the breaking of the degeneracy of an electronic term and which split it in (S,, L,) terms, have a much more important effect than the spin-orbit parameter r( 4 f) which couples them. The choice of this type of eigenvector for the 4 f N configurations is more and more justified as N becomes small. The type 2 basis set is used in a Hund’s coupling case (c). The angular momenta of the core with the quantum number J, and of the peripheral electron with the
LIGAND
FIELD CALCULATION
OF LnO
307
quantum number J,, couple to give the total atomic angular momentum with the quantum number J,. This basis set is particularly suitable for the description of configurations with a weak electrostatic exchange interaction between the 4fN-’ core and the peripheral electron. This case occurs when the peripheral electron is an outer electron such as the 6s and 6p electron. The overlap of the eigenfunctions 4fwith the eigenfunctions 6s and 6p is very small owing to the small spatial extension of the eigenfunctions 4 f. In the case of the configurations 4 fN-‘6s, the electrostatic interaction of the 4 f N-l core characterized by the Ek( 4f, 4f) parameter is more important than the spinorbit interaction with the {( 4f) parameter which is itself much stronger than the exchange electrostatic interaction with the G3( 4 f, 6s) parameter between the 4fN-’ core and the peripheral electron 6s. In the case of the configurations 4 f N-‘6p one must take into account, in addition, the spin-orbit interaction described by the parameter <(6p) which is more important than the exchange electrostatic interactions with the parameters FZ( 4f, 6p), G2( 4 f, 6p), and G4( 4f, 6p) which are very weak. The type 3 basis set is used when the spin-orbit interaction in the 4 f N-’ core is very weak or zero. This case occurs, for instance, when the 4f shell is half filled, i.e., for the I4f’, “s> core of GdIII ( 7). A proper description of some configurations sometimes requires a change of basis set when the number of core electrons changes. Thus the configurations 4 f N-15 d which contain a semiouter peripheral electron, change, when N increases, from a coupling case (a) (SC, f) S,, (L,, 2)La to a coupling case (c)(S,, L,)J,, (f, 2).&. For small values of N the electrostatic interaction between the 4 f and 5 d electrons is weak and the coupling is a Russell-Saunders coupling. For values of N near the maximum value 14, the spin-orbit parameter p( 4 f) increases in a noticeable way. The spinorbit parameter r( 5 d) increases also and the exchange electrostatic interaction 4 f 5 d decreases slightly. This evolution through the Lanthanide series results in a coupling case J-J. This trend is enhanced by the fact that for N - 1 > 7 the main parameter G, (4f, 5 d) has a nonzero contribution for the sole states arising from the configuration 4f’35d(‘P,). Matrix Elements of the Ligand Field Hamiltonian
The ligand field calculations require the use of basis sets with separated wavefunctions for each configuration. For the configurations 4f Nand 4 f N-‘n 1the basis sets practically used can be written as follows, I4f “A)
(23)
and 14fN-‘A)
@ In&).
(24)
When the configurations 4 f “, 4 f N-‘5 d, 4 f N-‘6p, and 4 f N-‘6s are well described in a Russell-Saunders coupling scheme, as is the case for La111and GdIII, the order of magnitude of the various interactions allows us to neglect the spin-orbit interaction, prior to the ligand field calculation. We can therefore deal with a basis set in which
CARE-X-I-E AND HOCQUET
308
the coupling between S and L is not taken into account. The eigenvectors then take the form I ‘V- Ns,L&.M~)
(25)
and 14fN-‘ScLc, nl, SaLaMs.M&
(26)
In this case the matrix elements of the ligand field Hamiltonian for the configurations 4fNand 4fN-‘&can be written as
Hw14fNU&.M~) = (4fNIY,I
(4fNSXM$Mil
Hd4fNIC/,)
= ~(~fNtih=~~fN~c),
(27)
where & = SCL, MscML,, and (4fN-‘S:L:,
n’l’, ShLgMlr,Mi.I HLFI~~~-‘&L~, nl, SaLaMsaM~)
= (n’r$I)I @ (4fN-‘ly,)Hp)4fN-‘&)
@ In&)
= (n’~~~~hLFIn~J/p)(4fN-‘lC/,4fN-‘~~) + (N-
l)(n‘r,~~Inrltb)(4fN-‘ry,lhLF14fN-’~~),
(28)
where & = MSpMLp, for the diagonal elements. As the operator HLFdoes not act on spins and has the z’z axis as a symmetry axis, it does not change the projections MJ of the angular momenta J. Therefore Eq. (28) can be rewritten as (4fN-‘SbL&
n’l’, S:L:MkI
HLF14fN-‘ScLc, nl, S,L,MS,M~)
= &Sa, S’,bXSc, SWMs,,
Mlr,)Wf~,,
MB)
(L~‘~ML,M~IL%L~)
C
ML, M% X (LcLpM~M~IL,ML.){(n’l’rtl,lhLFlnlrCb)G(~c, + w-
VW
1)(4fN-‘lVcIhLF14fN-‘~,)8(n,
n’)6(6
l’)wP,
1c/p)}-
(29)
The off-diagonal elements can be written in the form (4s
NW&M$MiaI
=
HLFIJ~
(4fN’b!l Hwi4fN%)
= 6(M&, MS,)&%
N-‘ScL,
nl, %LMs,ML)
@ b@p)
Sal
C
(L~LPML,MLPIL,ML~)(L,L)PMLEML~I
L:WJ
ML,&
x
{
fi(4fN-‘&Lc
I}4fNS~L:,)(4f~pIh~~Inplp~p)},
(30)
where ( 4 f N-1ScL, I>4 f NSLLi ) is the coefficient of fractional parentage according to Fano and Racah (3) and Judd (8). In the J-J coupling case the type 1 and 2 matrix elements can be easily written as
(4fNSbLbJhM;gIHLF14fNScLcJaMJ,)=
(4fNtiIHLFi4fN$c)
= ~(4fNtihd4fN’h),
(31)
LKiAND
FIELD CALOJ’LATION
309
OF LnO
where J/c = ScLcJeMJ,, and (4fN-‘SLLLJ’,, = (n’l’$;l
n’l’Jb, JeM>,l HLF14fN-‘&L,J,, 0 (4f”‘J/cl
= &SC, WNMJ~,
H~~l4_f~-l$,)
0
nlJp, JaMJB) lnlrC/p)
M;.> C (J~JPMJ,MJPI JBW~)(JCJPMJ,MJPI JaNa)
MJc X ~~n’rf~b,hLFlnl~~~~~~, $3 + (N-
l)(4fN-‘ry,IhLF14fN-‘~,) x 0,
n~(f,
r~(+,,
lyp)),
(32)
where tip = JpMJp. As the calculation of the coefficient of fractional parentage ( 4fN-‘S, L, I } 4fNSL LL) is tedious it is valuable to use it in the same form for the Russell-Saunders and J-J coupling cases. For this purpose the eigenvectors of the J-J coupling case are expanded in terms of the eigenvectors of the Russell-Saunders coupling case and the type 3 matrix elements can be written as (4fNs:L’,JgM;,lHLF14fN-‘S,L,Jc, = (4fNIY,l H~d4fN-‘~c)
= 6(M;s,
@
nl, JAGa)
IWP)
MJ.) C (JcJ~M~,M~pI J3Ga)(JcJ~WcWPI
JaWa)
MJ. MJP =
712 z (ZL’,l J&=5/2
J,J;)‘J;‘{
\TN(4fN-‘S,L,l}4fNS:L’,) x
(4f~PlkFbP~P$P)),
(33)
where (SLLkl J,J’p)(J;) IS ’ an abbreviation for the fourfold coupling coefficient (G’SP)&,
(LLPL,
=
Jai (ScLdJc,
(SPLPIJP,
Ja)
[(2La + 1)(2&a + 1)(2J, + 1)(2Jp + 1)]“2
The problem remains to calculate the (4 f Nv: I hLF14f “Itc) and (4 f N-‘v: I hLF X l4f N-‘Ic/,) one-electron matrix elements using the one-electron wavefunctions. One must express them in tensor operator form derived from formula (19) and calculate their angular parts. Calculation
of the Angular Parts of Matrix Elements
By using the Wigner-Eckart theorem one can calculate the angular matrix elements ( 4fN$L IC,kI4fN\t,) for each coupling case, say (4f NS:LbM$ML, = A(&, S:)G(Ms,,
I Ct I4f NS,L,MscMLc) MhJ(-l)L+‘k
L:.
_M,
k L, 0
L, ML,
x (4f Ns:L:llckllv-NscL,)
(35)
CARETTE
310
AND
HOCQUET
and
The expression of the scalar product of two tensor operators operating on different systems leads to the following expression:
(4fNs~L~J~IICkI14fNS,L,J,)
= (-l)sc+4+J;+kd(2Jc + 1)(2J; + 1)
The reduced matrix element ( 4fNS::L~llCkl14fNS,L,) (4fNs~L~llCk114fNs,L)
can be expressed as follows,
= (4fllCkl14f)(4fNWGllUkl14f”‘&L),
(38)
where Uk is a unit tensor operator such as
(n’l’IIUkllnl) = 6,,&
(39)
which was calculated by Nielson and Koster ( 9).
Calculation of the Radial Coeficients of Matrix Elements Usually in ligand field calculations the radial coefficients are taken as adjustable coefficients to fit the experimental results. It is therefore necessary to have previous knowledge of a large amount of experimental data. When there is a lack of experimental data one must calculate those radial coefficients. Otherwise this method is a powerful tool in guiding the experimental work. The radial coefficients can be written as sums of two integrals corresponding to the two regions delimited by the imaginary spherical boundary of radius R, (Fig. 3 ), i.e., the inner region with r < R, and the outer region with r > R,,
R”~prk’2R$,dr + Rk e
*R .tlp&Rldr], s&
(40)
where R,,/ are the radial parts of the wavefunctions which are not accurately known. One must therefore use approximate forms to describe them. The best results have been obtained in the ligand field calculations by using McLean linear combinations of Slater type orbitals P
(41) where p is the number of Slater type orbitals used for each value of n’ in the HartreeFock expansion and where
LIGAND FIELD CALCULATION
311
OF LnO
TABLE II Multiplicative Coefficients Used for the Evaluation of the &, Coefficients of Double Zeta Slater Type Orbitals of the LnIII Ions
s7s
IL’=7
0.51
0.55
%s
FL’=6
1.21
1.44
‘6~
El’=6
1.04
1.17
%d
n’=6
0.86
0.85
‘5d
II’=5
1.08
1.17
55f
IL’=5
0.97
0.84
S4f
n’=4
1.00
1.00
with
Unfortunately these expansions have not been calculated yet for the Ln 2+ions. The tables of McLean (10) yield the radial wavefunctions for elements with 2 = 55 to 2 = 92. For some of those elements the wavefunctions are calculated for both the neutral and the singly ionized atoms. By comparing the calculated wavefunctions for the same orbital of the neutral and of the singly ionized atoms of these particular elements we have deduced empirical rules which allow us to estimate the double zeta functions of the LnIII ions for the orbitals 6s, 73, 6p, 5d, 6d, 4f, and 5f (Table II). One can notice that the {nfcoefficients for the inner 4for 5d orbitals are not much modified through the neutral atom to the singly ionized atom, contrary to those of the outer 6p and 6s orbit& which are more changed because they are more affected by the screening effect of the removed orbital. The wavefunctions of the orbitals ns, (n - l)d, and (n - 2)f have been evaluated by extrapolating the observed change through the neutral atom to the singly ionized atom, neglecting contributions of weight less than 0.05. Orbitals (n + l)s, np, nd, and (n - 1)fwhich are involved in excited configurations have been calculated using the orthogonalization formula Rsl,,(r)
= AR:?(r)
+ RRz+i,l(r).
(43)
Expansion of formula (40) using a Slater type radial wavefunction leads to 1 1 R k (nl, n’I’) = 2e2 \l(2n)l(2n,)! (c + #‘+‘Rp’ xy[n+n’+k+
l,(1.+T’)R,]+(1.+~)~R~r[n+n’-k,(r+T’)R,l},
(44)
312
CARETTE AND HOCQUET TABLE III
Lower Energy Levels (cm-‘) for Each Configuration through the LnlII Series According to W. C. Martin et al. (6) for Observed Values and according to L. Brewer (I I) for Calculated Values (in the Last Case Estimated Uncertainties are Specified) 4fN
4?‘5d
4fN-‘6s
4fQ
4fN-‘7s
4fN-‘6d
4fN-*5d2
4fN-*Sdbs
0
3 277 I* 84, 15 262
26 283
33 856
Cl
8 972 1, 500 22 ml 18 039
16 976
22 89,
33 385
5 708
where y and I’ are the incomplete gamma functions x e-V’-’ dt r(a, x) = J0 and co qa,
x) =
sx
e-V-’
The spherical coefficient B$( n ‘I’,n I) is predominant can notice the following ordering, B:(4f,
4f) > B:(5d,
dt .
in all the configurations and one
5d) > B$(6s, 6s) > B:(6p, 6p),
(45)
which is directly related to the more or less inner character of the orbitals and controls the reordering of the energy levels. The nonspherical coefficients am very weak for the 5f orbital but much more important for the 5 d and 6p orbitals. This has the result that the splitting of the levels arising from the configurations 4 f Nor 4 f N-‘6s is much weaker than that of the levels arising from the configurations 4 f N-‘5 d or 4 f N-‘6p.
LIGAND FIELD CALCULATION
313
OF LnO
Large off-diagonal radial coefficients are mixing states pertaining to different configurations; these coefficients are responsible for strong configuration interactions. IV. LOWER ELECTRONIC ENERGY LEVELS OF THE LANTHANIDE
MONOXIDES
Before performing the ligand field calculation, weighted mean energies of the ion states have been evaluated according to the formula
E(&L)
=
c
Ja
(2;a +
1)
C (2Ja + l)E(Sd,Ja).
(46)
J.
Crude calculations performed using the 4fN, 4fN-‘5d, 4fN-‘6s, and 4fN-‘6p configurations, neglecting spin-orbit and electrostatic interactions, show that the lower energy levels for most of the lanthanide monoxides arise from the 4fN-‘6s configuration, except for EuO and YbO the ground states of which arise from the configuration 4fN. This particular behavior is due to the large energy separation between the 4f N configuration and the others in EuIII and YbIII (Table III). Except for the two special cases La0 ’ and GdO ( 7), with respectively an empty 4f shell and a half-filled 4fshell, the lower complex of states arising from the configuration 4fN-‘6s must be studied in a J-J coupling scheme. A multiconfiguration calculation performed in the case of CeO has shown that the configurations other than 4fN-‘6s are of little weight on the composition of the eigenvectors as well as on the values of the energies of the lower states, so that the problem can be restricted to a one configuration calculation. On the other hand one can notice that the B!j coefficients have, for a 4forbital, a less important effect than that of the electrostatic coefficients and even than that of the spin-orbit coefficient (41, as a result of which the complex of states arising from the configuration 4fN-‘6s can be ascribed to the ground term of 4fN-‘. The Hamiltonian matrix of the free ion depends only on the two parameters sbrand G3(4f, 6s) to the exclusion of terms depending on the inner 4f electrostatic interaction. In this case, the matrix elements of the electrostatic interaction 4f6s and of the core spin-orbit interaction are given by the following formulas (5):
(4fN-‘ScLcJc, 63, JaIK,14fN-1ScLcJc, 65, Ja) = + G3(4L -
6s)
(2J, + 1)
tL,(L,
+ 1) - S(&
+ 1) - Jc(Jc +
111, (47)
for diagonal matrix elements, with the plus sign if J, = J, + $ and minus if J, = J, - I.
27
(4fN-‘S&J,
= Ja T :, 6s, JaIH,,14fN-‘S,L,J,
= J, f 4, 6s, Ja) = T1’J”f;; a
X i(/csc + L, + J, + 1.5)($
+ L, - J. + 0.5)(L,
+ J, - SC + OS)(S,
+ J, - L, + 0.5) (48)
’ The ligand field calculation of the energy level scheme of La0 will be the subject of a separate paper.
”
(Cm- ’ )
Energy
Eiganfunctions
2
2
1
1
1
n (cm’ ’ )
Energy
0.2115.5
5 > + 0.2214.5 4 > -
5 > -
0.,614.5
6 > + 0.3915.5
6 > + 0.1911.5
ll.1015.9
0.1116.5
0.9014.5
0.4415.5
5 > -
0.8815.5
7 > +
0.8715.5
0.i217.5
7 > -
0.9816.5
6 > -
0.0817.5
6 > + 0.1915.9
> -
0.9716.5
, 7 > + 0.06!6.5
8 > + 0.0717.5
7 > + 0.1116.5
0.9917.5
0.9917.5
EigcnfunctionP
Calculated Lower Electronic Levels and Eigenfunctions of the Lanthanide Monoxides
TABLE IV
5 z
5 >
5 >
6 >
7 >
7 >
8 >
8 >
8 > -
0.2017.5
7 > -
1.*cl,,.5
+
0.9816.5
>
7
8 > +
0.3517.5 0.33b7.5
+
7
>
* > -
0.9217.5 0.9417.5 0.0717.5
0.1316.5
0.1616.5 7
7
7
>
,
,
TABLE IV-Conlinued
3
3
3
3
ul
w
n
E=nlY (cm- ’ ,
Eiqenfunctions
TABLE IV-Continued
903 3 399 1 155 428 0
5.5 6.5
812
634
5.5
1
5.5
3
4.5
4.5
Energy (Cm-‘)
4.5
n
5 > 6 > +
0.9415.5 0.9615.5
Eigenfunctions
0.2616.5
0.3114.5
>
-
5 a + 6
0.0914.5
0.1013.5
3 4
,
>
3 > +
-0.6112.5
6 a 6 > +
0.7215.5
0.5016.5
-
5 > -
0.5915.5
>
5
0.5511.5
+
3 > +
0.7513.5
-
I
0.7213.5 >
8 > +
0.9617.5
>
7
0.9117.5
TABLE IV-Continued
4.5 6.5
0.9315 0.8216
> + Cl.1415 > - 0.5716
5.5
5.5
> - 0.1914 > + 0.0515
> >
3.5 4.5
1 273
4.5
1 269
835
685 757
6.5
6.5
7.5
131
0
7.5 7.5
8.5
I
I
6.5
975
5 208
6.5
1 598
5.5
1 090
5 320
5.5
8 715
5 597
5.5
6.5
8 847
5.5
5.5
9 111
5.5
189
1 821
4.5
12
5 606
4.5
5.5
5 902
4.5
TABLE
IV-Continued
7 > + 9 > 9 >
0.9917.5 1.0017.5
8 > 1.0016.5 0.9917.5
7 > + 7 >
0.99t7.5 0.9917.5
0.1517.5
0.1617.5
0.1617.5
0.1617.5
7 > +
8 b -
7 > +
8 > +
0.0516.5
0.0516.5
0.0416.5
0.0516.5
6 >
f
>
7 >
7 >
TABLE IV-Continued
320
CARETTE AND HOCQUET TABLE V Spin-Orbit Coefficients sb/( cm-‘) Used for the Ligand Field Calculation &III
bf
hII
NdIII
757
640
780
SmIII
TbIII
1671
DyIII
I600
ErIII
HOI11
1900
2170
2300
TrnIII
2500
for off-diagonal matrix elements; and 6s, JaIHso14fN-‘UcJc, 6s, Ja)
(4fN-‘ScLcJc,
= +[z,+
[Jc(J, + 1) - L,(Lc + 1) - S,(S, + l)]
(49)
with the plus sign if N - 1 d 7 and minus if N - 1 3 7. These matrix elements are necessary to build up the Hamiltonian matrix of the free ion LnIII when the configuration 4fN-‘6s has not been experimentally studied. For EuIII the configurations 4f65 d, 4f66s, and 4f’ have been analyzed in a RussellSaunders coupling scheme, contrary to the configuration 4 f 66p which has been studied in a J-J coupling scheme. The observed states arise from the configuration 4 f ‘( *S)
TABLE VI Calculated and Observed Lower Energy Levels of CeO
Energy
state
Xl
Calculated r&=0.5
R
z,=2
(Cm-')
Observed
X2
2 3
0 197
0 191
Wl
1
1 026
1 003
%
2
1 160
1 133
912
"1 "2
01
1 756 1 919
1 657 1 823
1 679 1 870
"1 X3 XL % Wb
0+ 4 3 3 2
1 2 2 3 2
1 2 2 2 3
1 2 2 2 2
"3 "4
2 1
3 944 3 724
3 726 3 733
3 463 3 642
Tl "2
01
4 109 4 391
3 980 4 202
3 821 4 133
"3
0+
4 476
4 352
4 458
978 495 146 268 994
886 111 410 991 146
0 80 812
932 040 141 617 772
TABLE VII
179
77
(X,3.5
(2,2.5 (3,4.5
(3)5.5
(1,l.S (2,3.5
(2)4.5
(1)2.5
(2,S.S
(1,3.5
(1,4.5
(1,5.5
X(6.5,
Calculated and Observed Lower Energy Levels of CeO, PrO, SmO, GdO, TbO, DyO, HoO, and ybo
322
CARETTE AND HOCQUET
and from the core states 4f6( ‘F) for the configurations 4 f ‘nl. The states of the configuration 4f 66p lie in a range extending from 78 982 cm-’ to 90 155 cm-’ above the ground state 4f’( ‘S) . As this configuration acts as a secondary component among the states in competition for the ground state, the mean energy 80 000 cm-’ has been assigned to all the states arising from the 4f N-‘6p configuration. The calculated ground state of EuO is then a *I; state arising from the configuration 4f’ ( ’ S) with a complex of states arising from the configuration 4f6( ‘F) 6s lying 8 000 cm-’ above it. Finally, for YbIII the four configurations 4f14, 4f136s, 4f135d, and 4f136p have been analyzed in a J-J coupling scheme. A ligand field calculation performed on the 37 states arising from the configurations 4f14( ’ S) and 4f13( ‘F)nl yields a ground state fl = 0 with a complex of states arising from the configuration 4 f I3(*F)nIlocated 900 cm -’ above it. Results of the ligand field calculation through the lanthanide series are collected in Table IV. The eigenfunctions displayed are the contributions of the three eigenvectors of more important weights, I J,, Ja) in the J-J coupling scheme and I L&, n,l,; La&) in the LS coupling scheme (i.e., for EuO and GdO). The values of the spinorbit coefficient used in this calculation are given in Table V. A common value was used through the lanthanide series for the electrostatic interaction parameter G3( 4f, 6s) fixed to 150 cm-’ in conformity with the value obtained by Dulick (I) for this parameter for CeO. V. DISCUSSION
A number of experimental results on lanthanide monoxide spectra are now available, especially from laser spectroscopy, which allow us to check the predictions of the ligand field theory. The first molecule of this series which has been extensively studied was CeO (12) for which a complete low-energy level diagram has been derived from laser induced fluorescence experiments. Table VI gives the experimental values of the energies of the 16 low energy levels of CeO as well as the corresponding theoretical values with an effective ligand charge Z/e equal to 2e and OSe. Despite the factor four variation of the radial coefficients B$ n ‘I’, nl), the change of the energies of the levels with respect to 2, is minor. This result justifies the fact that we have taken 2, = 2 in Eq. (2). This choice is not connected with any hypothesis on the actual ionicity of the lanthanide monoxide molecules. In the same way a comparison of experimental results with our ligand field calculationscanbeachievedforPrO(13),SmO(14),Gd0(7, 15),Tb0(16),Dy0(17), Ho0 (18)) and YbO (19) (see Table VII). This table also gives the corresponding values as resulting from ligand field calculations performed by M. Dulick et al. (20) on all the lanthanide monoxides. A nonambiguous assignment of the observed low energy levels of these molecules is possible. The labelling of the states used here for CeO, PrO, and Ho0 is one which has been developed for CeO (12). States with tl - 4 are labelled Xi, R’i, Vi, Ui, and Ti, respectively, = J,,J,l,J.-2,J.-3,andJ, where the subscript i identifies each state from the lowest in each stack of levels with constant J, - Q. The J, value taken into account for this labelling is that of the eigenvector 1J,, J.) with the more important weight entering in the composition of the molecular eigenvector. This method of labelling the states in the J-J coupling
LIGAND FIELD CALCULATIONOF LnO
323
scheme breaks down when there is sufficient mixing between eigenstates with different J, such that it is no longer possible to assign a well-defined J, to the state. This is the case in SmO, TbO, and DyO where the labelling is changed to reflect the order of states of a given Q, (i)fl being the ith state with Q from the lowest. The ground state is labelled X ( Q) . Dulick has used direct radial functions generated by the Hartree-Fock method in the numerical evaluation of the radial integrals, except in the case of CeO for which he has used empirical estimates obtained from a least-squares fit of the 16 low-lying electronic states, assigned in the CeO laser induced fluorescence spectra, to the 4f6s levels of the ligand field model. Apart from this last case for which a comparison is meaningless, Table VII shows that Dulick’s values are closer to those observed for PrO, DyO, and Ho0 whereas ours are closer for SmO, TbO, and YbO. These discrepancies, in the absence of any fundamental differences, must be ascribed to the choice of the radial eigenfunctions on the one hand and of the G3 parameter on the other hand. The largest discrepancy between Dulick’s values and ours occurs for EuO where our value for the separation of the configurations 4f66s and 4 f' is 7 937 cm-’ as compared to Dulick’s value of 3 299 cm-‘. The separation between the 4f N-‘6s and 4 f N configurations is very sensitive to the value of the 6s eigenfunction. An increase of 3% of the { coefficients of the 6s double zeta Slater type orbital of Yb2+ results in a change of the separation between the first O- state and the ground state 0’ from 921 cm-’ to 2 024 cm-‘. Only the ground state has been observed in the EuO molecule ( 19) and it is therefore impossible to know what is the more realistic value. The fact that, in most cases, the separations X2 - X1, W2 - W,, etc. . . are much smaller in our calculations than in Dulick’s is a consequence of fixing the G3 parameter at 150 cm-’ instead of 300 cm-‘. VI.
CONCLUSION
The ligand field theory was able to predict the lower energy level scheme of the lanthanide monoxides. Further developments of this method can be envisaged now in three main directions, say, (i) testing the ligand field calculated molecular wavefunctions by evaluating molecular quantities which are sensitive to the wavefunction compositions, such as electric dipolar moments, LandC factors, etc. . . . ; (ii) generalizing the multiconfigurational calculations in order to assign the excited states of the lanthanide monoxides, and CeO seems to be a good candidate for checking the method on account of the large amount of available experimental data; (iii) widening this type of calculation to the actinide series for which the inner 5f shell plays a part analogous to the 4 f shell in the lanthanide series. RECEIVED: February 18, 1988 REFERENCES 1. M. DULICK, Ph.D. thesis, M.I.T., 1982. 2. R. W. FIELD, Ber. Bunsenges. Phys. Chem. 86,771-779
(1982).
324
CARETTE
AND
HQCQUET
3. U. FANO AND G. RACAH, “Irreducible Tensorial Sets,” Academic Press, San Diego, CA, 1959. 4. Z. R. GOLDSCHMIDT, “Spectroscopic and Group Theorical Methods,” Racab Memorial Volume, NorthHolland, Amsterdam, 1968. 5. B. G. WYBOURNE, ‘Spectroscopic Properties of Rare Earths,” Interscience, New York, 1965. 6. W. C. MARTIN, R. ZALUBAS, AND L. HAGAN, “Atomic Energy Levels-The Rare Earth Elements,” NSRDSNBS60, 1978. 7. P. CARETTE, A. HOCQUET,M. DOUAY, AND B. PINCHEMEL,J. Mol. Spectrosc. 124,243-271 (1987). 8. B. R. JUDD, “Operator Techniques in Atomic Spectroscopy,” McGraw-Hill, New York, 1963. 9. C. W. NIELSON AND G. F. KOSTER, “Spectroscopic Coefficients for the pN, dN, fN Configurations,” MIT Press, Cambridge, MA, 1963. 10. A. D. MCLEAN AND R. S. MCLEAN, At. Data Nucl. Data Tables 26, 197-381 (1981). II. L. BREWER,J. Opt. Sot. Amer. 61, 1666-1682 (1971). 12. C. LINTON, M. DULICK, R. W. FIELD, P. CAREITE, P. C. LEYLAND, AND R. F. BARROW, J. Mol. Spectrosc. 102,441-497 (1983). 13. M. DLJLICKAND R. W. FIELD, J. Mol. Spectrosc. 113, 105-141 (1985). 14. C. LINTON, Guo BUJIN, R. S. RANA, AND J. A. GRAY, J. Mol. Spectrosc. 126, 370-392 (1987). IS. Yu. N. DMITRIEV, L. A. KALEDIN, E. A. SHENYAVSKAYA,AND L. V. GURVICH, Acta Phys. Hung. 55,
467-479 (1984). 16. A. N. KULIKOV, L. A. KALEDIN, A. I. KOBYLIANSKY, AND L. V. GURVICH, Canad. J. Phys. 62, 18551870 (1984). 17. C. LINTON, D. M. GAUDET, AND H. SCHALL, J. Mol. Spectrosc. 115, 58-73 (1986). 18. Y. C. LILJAND C. LINTON, J. Mol. Spectrosc. 104,72-88 (1984). 19. S. MCDONALD, Ph.D. thesis, M.I.T., 1985. 20. M. DULICK, E. MURAD, AND R. F. BARROW, J. Chem. Phys. 85, 385-390 (1986).
OF MOLECULAR
SPECTROSCOPY
131,301-324 (1988)
Ligand Field Calculation of the Lower Electronic Energy Levels of the Lanthanide Monoxides P. CARETTEANDA.HOCQUET Laboratoire de Spectroscopic Molthlaire, Unit6 Associtfe au CNRS No. 779, Universitt? de Lille Flandres Artois. 59655 Villeneuve d’Ascq Cedex, France
The ligand field theory applied to the lanthanide monoxides predicts that the lower electronic states of these molecules are due to the configuration 4fN-‘6s (4fN for EuO and YbO) of the divalent lanthanide ion Ln’+. We present here the results of a ligand field calculation of the energies of the lower electronic states for all the stable rare-earth monoxides. o 1988 Academic Press, Inc.
I. INTRODUCTION
The lanthanides are the elements pertaining to the group of rare earths which are characterized by the progressive filling of the 4fshell of their electronic configurations. They commence with the element cerium (2 = 58 ) and end with the element ytterbium (2 = 70). On the basis of the similarities of their chemical properties to those of the other lanthanides, one associates with this series the elements lanthanum (2 = 57) and lutetium (Z = 7 1) which possess an empty and a completely filled 4f shell in their normal configurations. The 4fN and 4fN-’ open shells give rise to numerous terms: as an example the configuration 4f8 leads to 295 atomic electronic states. The energies involved with the 4f, 5 d, 63, and even 6p electrons are of the same order of magnitude and the interactions which are usually taken as perturbations of the central field are very strong and lead to important configuration interactions which increase the complexity of the lanthanide spectra. The lanthanides possess the common feature of a 4feigenfunction essentially located inside the 5s25p6 closed shells of the xenon structure (Fig. 1). This effect known as the “lanthanide contraction” reinforces itself progressively with the filling of the 4f shell (Fig. 2). As a result of this contraction the surroundings of the lanthanide atoms are of little chemical significance. II. BASIS OF THE LIGAND FIELD THEORY APPLIED TO THE STUDY OF THE LANTHANIDE MONOXIDE SPECTRA
This theory was first developed for the study of the rare-earth salts and solutions of their compounds. At low temperatures the absorption spectra of these chemical species are usually found to consist of groups of sharp lines the positions of which are relatively unaffected by the environment of the lanthanide ion and may be correlated with the positions of the energy levels of the 4f N configuration of the free, tripositive ion. A 301
0022-2852/88 $3.00 Copyright 0
1988 by Academic Press, Inc.
AU rights of repmdwtion
in any form reserved.
302
-0.60
CARETTE AND HOCQUET
t
FIG. I. Double zeta slater type orbitals 4f, 5s, and 5p for NdI.
model was introduced for the study of the lanthanide monoxides (I, 2), model which enables a pseudoatomic treatment of a molecular problem. In this theory the electronic structure of lanthanide monoxide may be approximated by the ionic model Ln 2+02-, The molecular electronic levels result from the splitting of the atomic levels of the parent Ln 2+ free ion configuration by the axial electric field between the R, separated negative ligand of charge -2e and positive metal centers, neglecting metal-ligand orbital overlap (Fig. 3 ) . The perturbed Hamiltonian is usually written as, similar
H = Ho + HLF,
where Ho is the Hamiltonian iltonian.
(1)
for the free Ln2+ ion and HLF is the ligand field Ham-
FIG. 2. Double zeta slater type orbital 4Jfor LaI, NdI, GdI, ErI, and LuI.
LIGAND FIELD CALCULATION
303
OF LnO
q outer
,\\
r,=q
Re
\
inner
region
r<= Re
region
r,= Re r< = ri
FIG. 3. Coordinate system
for the ligandfieldpotential.
The interaction of a metal centered electron i with the scalar potential function V(r:,
oi9
Cpi)
=
F
z
(2)
may be expressed in terms of ri, the metal center origin coordinate, using the expansion ’ = [r? + Rz - 2riRAOS Bi]-“2 = 5 PIA ui)tF, r: k=O
(3)
where Vi = cos Bi, t; = Ti, ri> = max(R,, ri), ri< = min(R,, r;), and Pk( vi) are the Legendre polynomial functions. The total ligand field potential energy is obtained by summing eV( r:, Bi, vi) over all the indices of the electrons pertaining to the metal ion. Therefore the ligand field Hamiltonian HLF = $ 2e2 5 Pk( ui)tk i=l ri> kc0
(4)
is expressed as a sum of one-electron Hamiltonians hLF = 2” $ Pk(Ui)tfs ri> k=O
(5)
This formulation yields a semiempirical method of calculation of the electronic energy levels of the lanthanide monoxides from the known energy levels of the parent divalent free ions Ln*+.
304
CARETTE AND HOCQUET TABLE I Structure of the Hamiltonian Matrix Ho of the Free ion Ln2+
(4f Nvcl
(4fN-‘vcVcnp&pl
1
3
3
2
The Hamiltonian matrix of the free ions can symbolically be divided into four regions with regard to the atomic states arising from the configurations 4f Nand 4f N-‘nl (Table I). Whatever the encountered coupling case the problem is to express the three following types of matrix elements, type 1
(4~“4%I~LF14fNICI,)
type 2
(4fN-‘Vc,
type 3
(4fN~Ie_F14fN-‘*c,
M&l
(6)
&14fN%,
q&k,)
&l+p),
(7)
(8)
in terms of the one-electron matrix elements ( n ‘l’m’1hLF I n lm ) . Index c is related to core electrons and index p to peripheral electrons. The one-electron matrix elements can be written as
or (n’l’m’/h&&z)
= c
1%”
R,?~Ktf,‘hLFR~,Y~r2sin
B dr dp d8
(10)
by taking the wave function $ (I, (o, 0) as a product of R( r), the radial wave function, and Y,( cp, e), the angular wave function, written R and Ylmfor simplicity. Rodrigues’ formula yields Y,m = (-l)*
(21+ 1)(1- m)! “2 47r(l+ m)! [
f3)ezmp 1 &(cos
(11)
LIGAND FIELD CALCULATION OF LnO
305
and allows one to write:
(12) Therefore: * 2r Y,!,tPk(cos e)Y,_,sin 0 dp de ss0 0
= [(21’ + 1)(21+ 1)]“2 It follows from the properties of symbol 3 J that: (i) the one-electron operator hLF is diagonal with respect to m; (ii) the energy levels m and -m are degenerate; (iii) the matrix elements of hLF are nonzero for 1’ + k + 1 = 2g where g is an integer and for I’, k, 1 fulfilling the triangular relation A( l’kl), i.e., 11-1’1
nl)F(II,
1, m),
(14)
k=O
where (15) and Ck(lt, 1, m) = (&)I”
1
s,z’ &tmYYkOYl_,,,sin 8 dp de.
(16)
In order to include these matrix elements in more complex configurations one must express the one-electron ligand field Hamiltonian operator in tensor operator form. The formula of composition of spherical harmonics pk(cos 8) =
&
4
(17)
allows us to write
(18)
306
CARETTE AND HOCQUET
where Ct is the irreductible angular tensor operator defined by Racah (3) and where
is the radial tensor operator. In the case of diatomic molecules the symmetry of the group C,, restricts the problem to contributions of the components along the molecular axis, so that the ligand field Hamiltonian operator takes the following form: HLF
=
(19)
c (BOkCbi. ki
The sole nonvanishing matrix elements of this operator are those which couple electronic states with the same value of the projection of the total angular momentum along the molecular axis. The molecular energy level calculation requires, therefore, the diagonalization of a series of Hamiltonian matrices for every possible value of this projection. III. MATRIX ELEMENTS OF THE LIGAND FIELD HAMILTONIAN
OPERATOR
In order to perform the diagonalization of the ligand field Hamiltonian matrices one must first survey the vector coupling cases occurring in the free ion Ln2+ configurations and choose the most appropriate basis sets for the description of the corresponding monoxide molecules. One can then express the matrix elements of the ligand field Hamiltonian for these basis sets. Only after the completion of these two steps can one proceed by calculating the angular and radial parts of these matrix elements.
Basis Sets of the Lanthanide Free Ions A study of the coupling occurring in the configurations of the lanthanides as depicted by Z. R. Goldschmidt (4), B. G. Wyboume (5), and W. C. Martin et al. (6) allows us to define three basis types of eigenvectors for the description of the spectra of the free ions LnIII. They are type 1
14fN~&LJa~J,),
type 2
14fN-‘r&LJc,
type 3
I4f N%%L,
(20) nlJp, JaWa),
nl, &L&G,),
(21) (22)
where y symbolizes the set of quantum numbers which is necessary for the complete description of the atomic states as required by the group theory. This symbol will be dropped later for the sake of brevity. The type 1 eigenvectors correspond to the Russell-Saunders coupling case. All the configurations 4 f N are studied in this coupling scheme since the electrostatic parameters Ek (4 f, 4f ) which are responsible for the breaking of the degeneracy of an electronic term and which split it in (S,, L,) terms, have a much more important effect than the spin-orbit parameter r( 4 f) which couples them. The choice of this type of eigenvector for the 4 f N configurations is more and more justified as N becomes small. The type 2 basis set is used in a Hund’s coupling case (c). The angular momenta of the core with the quantum number J, and of the peripheral electron with the
LIGAND
FIELD CALCULATION
OF LnO
307
quantum number J,, couple to give the total atomic angular momentum with the quantum number J,. This basis set is particularly suitable for the description of configurations with a weak electrostatic exchange interaction between the 4fN-’ core and the peripheral electron. This case occurs when the peripheral electron is an outer electron such as the 6s and 6p electron. The overlap of the eigenfunctions 4fwith the eigenfunctions 6s and 6p is very small owing to the small spatial extension of the eigenfunctions 4 f. In the case of the configurations 4 fN-‘6s, the electrostatic interaction of the 4 f N-l core characterized by the Ek( 4f, 4f) parameter is more important than the spinorbit interaction with the {( 4f) parameter which is itself much stronger than the exchange electrostatic interaction with the G3( 4 f, 6s) parameter between the 4fN-’ core and the peripheral electron 6s. In the case of the configurations 4 f N-‘6p one must take into account, in addition, the spin-orbit interaction described by the parameter <(6p) which is more important than the exchange electrostatic interactions with the parameters FZ( 4f, 6p), G2( 4 f, 6p), and G4( 4f, 6p) which are very weak. The type 3 basis set is used when the spin-orbit interaction in the 4 f N-’ core is very weak or zero. This case occurs, for instance, when the 4f shell is half filled, i.e., for the I4f’, “s> core of GdIII ( 7). A proper description of some configurations sometimes requires a change of basis set when the number of core electrons changes. Thus the configurations 4 f N-15 d which contain a semiouter peripheral electron, change, when N increases, from a coupling case (a) (SC, f) S,, (L,, 2)La to a coupling case (c)(S,, L,)J,, (f, 2).&. For small values of N the electrostatic interaction between the 4 f and 5 d electrons is weak and the coupling is a Russell-Saunders coupling. For values of N near the maximum value 14, the spin-orbit parameter p( 4 f) increases in a noticeable way. The spinorbit parameter r( 5 d) increases also and the exchange electrostatic interaction 4 f 5 d decreases slightly. This evolution through the Lanthanide series results in a coupling case J-J. This trend is enhanced by the fact that for N - 1 > 7 the main parameter G, (4f, 5 d) has a nonzero contribution for the sole states arising from the configuration 4f’35d(‘P,). Matrix Elements of the Ligand Field Hamiltonian
The ligand field calculations require the use of basis sets with separated wavefunctions for each configuration. For the configurations 4f Nand 4 f N-‘n 1the basis sets practically used can be written as follows, I4f “A)
(23)
and 14fN-‘A)
@ In&).
(24)
When the configurations 4 f “, 4 f N-‘5 d, 4 f N-‘6p, and 4 f N-‘6s are well described in a Russell-Saunders coupling scheme, as is the case for La111and GdIII, the order of magnitude of the various interactions allows us to neglect the spin-orbit interaction, prior to the ligand field calculation. We can therefore deal with a basis set in which
CARE-X-I-E AND HOCQUET
308
the coupling between S and L is not taken into account. The eigenvectors then take the form I ‘V- Ns,L&.M~)
(25)
and 14fN-‘ScLc, nl, SaLaMs.M&
(26)
In this case the matrix elements of the ligand field Hamiltonian for the configurations 4fNand 4fN-‘&can be written as
Hw14fNU&.M~) = (4fNIY,I
(4fNSXM$Mil
Hd4fNIC/,)
= ~(~fNtih=~~fN~c),
(27)
where & = SCL, MscML,, and (4fN-‘S:L:,
n’l’, ShLgMlr,Mi.I HLFI~~~-‘&L~, nl, SaLaMsaM~)
= (n’r$I)I @ (4fN-‘ly,)Hp)4fN-‘&)
@ In&)
= (n’~~~~hLFIn~J/p)(4fN-‘lC/,4fN-‘~~) + (N-
l)(n‘r,~~Inrltb)(4fN-‘ry,lhLF14fN-’~~),
(28)
where & = MSpMLp, for the diagonal elements. As the operator HLFdoes not act on spins and has the z’z axis as a symmetry axis, it does not change the projections MJ of the angular momenta J. Therefore Eq. (28) can be rewritten as (4fN-‘SbL&
n’l’, S:L:MkI
HLF14fN-‘ScLc, nl, S,L,MS,M~)
= &Sa, S’,bXSc, SWMs,,
Mlr,)Wf~,,
MB)
(L~‘~ML,M~IL%L~)
C
ML, M% X (LcLpM~M~IL,ML.){(n’l’rtl,lhLFlnlrCb)G(~c, + w-
VW
1)(4fN-‘lVcIhLF14fN-‘~,)8(n,
n’)6(6
l’)wP,
1c/p)}-
(29)
The off-diagonal elements can be written in the form (4s
NW&M$MiaI
=
HLFIJ~
(4fN’b!l Hwi4fN%)
= 6(M&, MS,)&%
N-‘ScL,
nl, %LMs,ML)
@ b@p)
Sal
C
(L~LPML,MLPIL,ML~)(L,L)PMLEML~I
L:WJ
ML,&
x
{
fi(4fN-‘&Lc
I}4fNS~L:,)(4f~pIh~~Inplp~p)},
(30)
where ( 4 f N-1ScL, I>4 f NSLLi ) is the coefficient of fractional parentage according to Fano and Racah (3) and Judd (8). In the J-J coupling case the type 1 and 2 matrix elements can be easily written as
(4fNSbLbJhM;gIHLF14fNScLcJaMJ,)=
(4fNtiIHLFi4fN$c)
= ~(4fNtihd4fN’h),
(31)
LKiAND
FIELD CALOJ’LATION
309
OF LnO
where J/c = ScLcJeMJ,, and (4fN-‘SLLLJ’,, = (n’l’$;l
n’l’Jb, JeM>,l HLF14fN-‘&L,J,, 0 (4f”‘J/cl
= &SC, WNMJ~,
H~~l4_f~-l$,)
0
nlJp, JaMJB) lnlrC/p)
M;.> C (J~JPMJ,MJPI JBW~)(JCJPMJ,MJPI JaNa)
MJc X ~~n’rf~b,hLFlnl~~~~~~, $3 + (N-
l)(4fN-‘ry,IhLF14fN-‘~,) x 0,
n~(f,
r~(+,,
lyp)),
(32)
where tip = JpMJp. As the calculation of the coefficient of fractional parentage ( 4fN-‘S, L, I } 4fNSL LL) is tedious it is valuable to use it in the same form for the Russell-Saunders and J-J coupling cases. For this purpose the eigenvectors of the J-J coupling case are expanded in terms of the eigenvectors of the Russell-Saunders coupling case and the type 3 matrix elements can be written as (4fNs:L’,JgM;,lHLF14fN-‘S,L,Jc, = (4fNIY,l H~d4fN-‘~c)
= 6(M;s,
@
nl, JAGa)
IWP)
MJ.) C (JcJ~M~,M~pI J3Ga)(JcJ~WcWPI
JaWa)
MJ. MJP =
712 z (ZL’,l J&=5/2
J,J;)‘J;‘{
\TN(4fN-‘S,L,l}4fNS:L’,) x
(4f~PlkFbP~P$P)),
(33)
where (SLLkl J,J’p)(J;) IS ’ an abbreviation for the fourfold coupling coefficient (G’SP)&,
(LLPL,
=
Jai (ScLdJc,
(SPLPIJP,
Ja)
[(2La + 1)(2&a + 1)(2J, + 1)(2Jp + 1)]“2
The problem remains to calculate the (4 f Nv: I hLF14f “Itc) and (4 f N-‘v: I hLF X l4f N-‘Ic/,) one-electron matrix elements using the one-electron wavefunctions. One must express them in tensor operator form derived from formula (19) and calculate their angular parts. Calculation
of the Angular Parts of Matrix Elements
By using the Wigner-Eckart theorem one can calculate the angular matrix elements ( 4fN$L IC,kI4fN\t,) for each coupling case, say (4f NS:LbM$ML, = A(&, S:)G(Ms,,
I Ct I4f NS,L,MscMLc) MhJ(-l)L+‘k
L:.
_M,
k L, 0
L, ML,
x (4f Ns:L:llckllv-NscL,)
(35)
CARETTE
310
AND
HOCQUET
and
The expression of the scalar product of two tensor operators operating on different systems leads to the following expression:
(4fNs~L~J~IICkI14fNS,L,J,)
= (-l)sc+4+J;+kd(2Jc + 1)(2J; + 1)
The reduced matrix element ( 4fNS::L~llCkl14fNS,L,) (4fNs~L~llCk114fNs,L)
can be expressed as follows,
= (4fllCkl14f)(4fNWGllUkl14f”‘&L),
(38)
where Uk is a unit tensor operator such as
(n’l’IIUkllnl) = 6,,&
(39)
which was calculated by Nielson and Koster ( 9).
Calculation of the Radial Coeficients of Matrix Elements Usually in ligand field calculations the radial coefficients are taken as adjustable coefficients to fit the experimental results. It is therefore necessary to have previous knowledge of a large amount of experimental data. When there is a lack of experimental data one must calculate those radial coefficients. Otherwise this method is a powerful tool in guiding the experimental work. The radial coefficients can be written as sums of two integrals corresponding to the two regions delimited by the imaginary spherical boundary of radius R, (Fig. 3 ), i.e., the inner region with r < R, and the outer region with r > R,,
R”~prk’2R$,dr + Rk e
*R .tlp&Rldr], s&
(40)
where R,,/ are the radial parts of the wavefunctions which are not accurately known. One must therefore use approximate forms to describe them. The best results have been obtained in the ligand field calculations by using McLean linear combinations of Slater type orbitals P
(41) where p is the number of Slater type orbitals used for each value of n’ in the HartreeFock expansion and where
LIGAND FIELD CALCULATION
311
OF LnO
TABLE II Multiplicative Coefficients Used for the Evaluation of the &, Coefficients of Double Zeta Slater Type Orbitals of the LnIII Ions
s7s
IL’=7
0.51
0.55
%s
FL’=6
1.21
1.44
‘6~
El’=6
1.04
1.17
%d
n’=6
0.86
0.85
‘5d
II’=5
1.08
1.17
55f
IL’=5
0.97
0.84
S4f
n’=4
1.00
1.00
with
Unfortunately these expansions have not been calculated yet for the Ln 2+ions. The tables of McLean (10) yield the radial wavefunctions for elements with 2 = 55 to 2 = 92. For some of those elements the wavefunctions are calculated for both the neutral and the singly ionized atoms. By comparing the calculated wavefunctions for the same orbital of the neutral and of the singly ionized atoms of these particular elements we have deduced empirical rules which allow us to estimate the double zeta functions of the LnIII ions for the orbitals 6s, 73, 6p, 5d, 6d, 4f, and 5f (Table II). One can notice that the {nfcoefficients for the inner 4for 5d orbitals are not much modified through the neutral atom to the singly ionized atom, contrary to those of the outer 6p and 6s orbit& which are more changed because they are more affected by the screening effect of the removed orbital. The wavefunctions of the orbitals ns, (n - l)d, and (n - 2)f have been evaluated by extrapolating the observed change through the neutral atom to the singly ionized atom, neglecting contributions of weight less than 0.05. Orbitals (n + l)s, np, nd, and (n - 1)fwhich are involved in excited configurations have been calculated using the orthogonalization formula Rsl,,(r)
= AR:?(r)
+ RRz+i,l(r).
(43)
Expansion of formula (40) using a Slater type radial wavefunction leads to 1 1 R k (nl, n’I’) = 2e2 \l(2n)l(2n,)! (c + #‘+‘Rp’ xy[n+n’+k+
l,(1.+T’)R,]+(1.+~)~R~r[n+n’-k,(r+T’)R,l},
(44)
312
CARETTE AND HOCQUET TABLE III
Lower Energy Levels (cm-‘) for Each Configuration through the LnlII Series According to W. C. Martin et al. (6) for Observed Values and according to L. Brewer (I I) for Calculated Values (in the Last Case Estimated Uncertainties are Specified) 4fN
4?‘5d
4fN-‘6s
4fQ
4fN-‘7s
4fN-‘6d
4fN-*5d2
4fN-*Sdbs
0
3 277 I* 84, 15 262
26 283
33 856
Cl
8 972 1, 500 22 ml 18 039
16 976
22 89,
33 385
5 708
where y and I’ are the incomplete gamma functions x e-V’-’ dt r(a, x) = J0 and co qa,
x) =
sx
e-V-’
The spherical coefficient B$( n ‘I’,n I) is predominant can notice the following ordering, B:(4f,
4f) > B:(5d,
dt .
in all the configurations and one
5d) > B$(6s, 6s) > B:(6p, 6p),
(45)
which is directly related to the more or less inner character of the orbitals and controls the reordering of the energy levels. The nonspherical coefficients am very weak for the 5f orbital but much more important for the 5 d and 6p orbitals. This has the result that the splitting of the levels arising from the configurations 4 f Nor 4 f N-‘6s is much weaker than that of the levels arising from the configurations 4 f N-‘5 d or 4 f N-‘6p.
LIGAND FIELD CALCULATION
313
OF LnO
Large off-diagonal radial coefficients are mixing states pertaining to different configurations; these coefficients are responsible for strong configuration interactions. IV. LOWER ELECTRONIC ENERGY LEVELS OF THE LANTHANIDE
MONOXIDES
Before performing the ligand field calculation, weighted mean energies of the ion states have been evaluated according to the formula
E(&L)
=
c
Ja
(2;a +
1)
C (2Ja + l)E(Sd,Ja).
(46)
J.
Crude calculations performed using the 4fN, 4fN-‘5d, 4fN-‘6s, and 4fN-‘6p configurations, neglecting spin-orbit and electrostatic interactions, show that the lower energy levels for most of the lanthanide monoxides arise from the 4fN-‘6s configuration, except for EuO and YbO the ground states of which arise from the configuration 4fN. This particular behavior is due to the large energy separation between the 4f N configuration and the others in EuIII and YbIII (Table III). Except for the two special cases La0 ’ and GdO ( 7), with respectively an empty 4f shell and a half-filled 4fshell, the lower complex of states arising from the configuration 4fN-‘6s must be studied in a J-J coupling scheme. A multiconfiguration calculation performed in the case of CeO has shown that the configurations other than 4fN-‘6s are of little weight on the composition of the eigenvectors as well as on the values of the energies of the lower states, so that the problem can be restricted to a one configuration calculation. On the other hand one can notice that the B!j coefficients have, for a 4forbital, a less important effect than that of the electrostatic coefficients and even than that of the spin-orbit coefficient (41, as a result of which the complex of states arising from the configuration 4fN-‘6s can be ascribed to the ground term of 4fN-‘. The Hamiltonian matrix of the free ion depends only on the two parameters sbrand G3(4f, 6s) to the exclusion of terms depending on the inner 4f electrostatic interaction. In this case, the matrix elements of the electrostatic interaction 4f6s and of the core spin-orbit interaction are given by the following formulas (5):
(4fN-‘ScLcJc, 63, JaIK,14fN-1ScLcJc, 65, Ja) = + G3(4L -
6s)
(2J, + 1)
tL,(L,
+ 1) - S(&
+ 1) - Jc(Jc +
111, (47)
for diagonal matrix elements, with the plus sign if J, = J, + $ and minus if J, = J, - I.
27
(4fN-‘S&J,
= Ja T :, 6s, JaIH,,14fN-‘S,L,J,
= J, f 4, 6s, Ja) = T1’J”f;; a
X i(/csc + L, + J, + 1.5)($
+ L, - J. + 0.5)(L,
+ J, - SC + OS)(S,
+ J, - L, + 0.5) (48)
’ The ligand field calculation of the energy level scheme of La0 will be the subject of a separate paper.
”
(Cm- ’ )
Energy
Eiganfunctions
2
2
1
1
1
n (cm’ ’ )
Energy
0.2115.5
5 > + 0.2214.5 4 > -
5 > -
0.,614.5
6 > + 0.3915.5
6 > + 0.1911.5
ll.1015.9
0.1116.5
0.9014.5
0.4415.5
5 > -
0.8815.5
7 > +
0.8715.5
0.i217.5
7 > -
0.9816.5
6 > -
0.0817.5
6 > + 0.1915.9
> -
0.9716.5
, 7 > + 0.06!6.5
8 > + 0.0717.5
7 > + 0.1116.5
0.9917.5
0.9917.5
EigcnfunctionP
Calculated Lower Electronic Levels and Eigenfunctions of the Lanthanide Monoxides
TABLE IV
5 z
5 >
5 >
6 >
7 >
7 >
8 >
8 >
8 > -
0.2017.5
7 > -
1.*cl,,.5
+
0.9816.5
>
7
8 > +
0.3517.5 0.33b7.5
+
7
>
* > -
0.9217.5 0.9417.5 0.0717.5
0.1316.5
0.1616.5 7
7
7
>
,
,
TABLE IV-Conlinued
3
3
3
3
ul
w
n
E=nlY (cm- ’ ,
Eiqenfunctions
TABLE IV-Continued
903 3 399 1 155 428 0
5.5 6.5
812
634
5.5
1
5.5
3
4.5
4.5
Energy (Cm-‘)
4.5
n
5 > 6 > +
0.9415.5 0.9615.5
Eigenfunctions
0.2616.5
0.3114.5
>
-
5 a + 6
0.0914.5
0.1013.5
3 4
,
>
3 > +
-0.6112.5
6 a 6 > +
0.7215.5
0.5016.5
-
5 > -
0.5915.5
>
5
0.5511.5
+
3 > +
0.7513.5
-
I
0.7213.5 >
8 > +
0.9617.5
>
7
0.9117.5
TABLE IV-Continued
4.5 6.5
0.9315 0.8216
> + Cl.1415 > - 0.5716
5.5
5.5
> - 0.1914 > + 0.0515
> >
3.5 4.5
1 273
4.5
1 269
835
685 757
6.5
6.5
7.5
131
0
7.5 7.5
8.5
I
I
6.5
975
5 208
6.5
1 598
5.5
1 090
5 320
5.5
8 715
5 597
5.5
6.5
8 847
5.5
5.5
9 111
5.5
189
1 821
4.5
12
5 606
4.5
5.5
5 902
4.5
TABLE
IV-Continued
7 > + 9 > 9 >
0.9917.5 1.0017.5
8 > 1.0016.5 0.9917.5
7 > + 7 >
0.99t7.5 0.9917.5
0.1517.5
0.1617.5
0.1617.5
0.1617.5
7 > +
8 b -
7 > +
8 > +
0.0516.5
0.0516.5
0.0416.5
0.0516.5
6 >
f
>
7 >
7 >
TABLE IV-Continued
320
CARETTE AND HOCQUET TABLE V Spin-Orbit Coefficients sb/( cm-‘) Used for the Ligand Field Calculation &III
bf
hII
NdIII
757
640
780
SmIII
TbIII
1671
DyIII
I600
ErIII
HOI11
1900
2170
2300
TrnIII
2500
for off-diagonal matrix elements; and 6s, JaIHso14fN-‘UcJc, 6s, Ja)
(4fN-‘ScLcJc,
= +[z,+
[Jc(J, + 1) - L,(Lc + 1) - S,(S, + l)]
(49)
with the plus sign if N - 1 d 7 and minus if N - 1 3 7. These matrix elements are necessary to build up the Hamiltonian matrix of the free ion LnIII when the configuration 4fN-‘6s has not been experimentally studied. For EuIII the configurations 4f65 d, 4f66s, and 4f’ have been analyzed in a RussellSaunders coupling scheme, contrary to the configuration 4 f 66p which has been studied in a J-J coupling scheme. The observed states arise from the configuration 4 f ‘( *S)
TABLE VI Calculated and Observed Lower Energy Levels of CeO
Energy
state
Xl
Calculated r&=0.5
R
z,=2
(Cm-')
Observed
X2
2 3
0 197
0 191
Wl
1
1 026
1 003
%
2
1 160
1 133
912
"1 "2
01
1 756 1 919
1 657 1 823
1 679 1 870
"1 X3 XL % Wb
0+ 4 3 3 2
1 2 2 3 2
1 2 2 2 3
1 2 2 2 2
"3 "4
2 1
3 944 3 724
3 726 3 733
3 463 3 642
Tl "2
01
4 109 4 391
3 980 4 202
3 821 4 133
"3
0+
4 476
4 352
4 458
978 495 146 268 994
886 111 410 991 146
0 80 812
932 040 141 617 772
TABLE VII
179
77
(X,3.5
(2,2.5 (3,4.5
(3)5.5
(1,l.S (2,3.5
(2)4.5
(1)2.5
(2,S.S
(1,3.5
(1,4.5
(1,5.5
X(6.5,
Calculated and Observed Lower Energy Levels of CeO, PrO, SmO, GdO, TbO, DyO, HoO, and ybo
322
CARETTE AND HOCQUET
and from the core states 4f6( ‘F) for the configurations 4 f ‘nl. The states of the configuration 4f 66p lie in a range extending from 78 982 cm-’ to 90 155 cm-’ above the ground state 4f’( ‘S) . As this configuration acts as a secondary component among the states in competition for the ground state, the mean energy 80 000 cm-’ has been assigned to all the states arising from the 4f N-‘6p configuration. The calculated ground state of EuO is then a *I; state arising from the configuration 4f’ ( ’ S) with a complex of states arising from the configuration 4f6( ‘F) 6s lying 8 000 cm-’ above it. Finally, for YbIII the four configurations 4f14, 4f136s, 4f135d, and 4f136p have been analyzed in a J-J coupling scheme. A ligand field calculation performed on the 37 states arising from the configurations 4f14( ’ S) and 4f13( ‘F)nl yields a ground state fl = 0 with a complex of states arising from the configuration 4 f I3(*F)nIlocated 900 cm -’ above it. Results of the ligand field calculation through the lanthanide series are collected in Table IV. The eigenfunctions displayed are the contributions of the three eigenvectors of more important weights, I J,, Ja) in the J-J coupling scheme and I L&, n,l,; La&) in the LS coupling scheme (i.e., for EuO and GdO). The values of the spinorbit coefficient used in this calculation are given in Table V. A common value was used through the lanthanide series for the electrostatic interaction parameter G3( 4f, 6s) fixed to 150 cm-’ in conformity with the value obtained by Dulick (I) for this parameter for CeO. V. DISCUSSION
A number of experimental results on lanthanide monoxide spectra are now available, especially from laser spectroscopy, which allow us to check the predictions of the ligand field theory. The first molecule of this series which has been extensively studied was CeO (12) for which a complete low-energy level diagram has been derived from laser induced fluorescence experiments. Table VI gives the experimental values of the energies of the 16 low energy levels of CeO as well as the corresponding theoretical values with an effective ligand charge Z/e equal to 2e and OSe. Despite the factor four variation of the radial coefficients B$ n ‘I’, nl), the change of the energies of the levels with respect to 2, is minor. This result justifies the fact that we have taken 2, = 2 in Eq. (2). This choice is not connected with any hypothesis on the actual ionicity of the lanthanide monoxide molecules. In the same way a comparison of experimental results with our ligand field calculationscanbeachievedforPrO(13),SmO(14),Gd0(7, 15),Tb0(16),Dy0(17), Ho0 (18)) and YbO (19) (see Table VII). This table also gives the corresponding values as resulting from ligand field calculations performed by M. Dulick et al. (20) on all the lanthanide monoxides. A nonambiguous assignment of the observed low energy levels of these molecules is possible. The labelling of the states used here for CeO, PrO, and Ho0 is one which has been developed for CeO (12). States with tl - 4 are labelled Xi, R’i, Vi, Ui, and Ti, respectively, = J,,J,l,J.-2,J.-3,andJ, where the subscript i identifies each state from the lowest in each stack of levels with constant J, - Q. The J, value taken into account for this labelling is that of the eigenvector 1J,, J.) with the more important weight entering in the composition of the molecular eigenvector. This method of labelling the states in the J-J coupling
LIGAND FIELD CALCULATIONOF LnO
323
scheme breaks down when there is sufficient mixing between eigenstates with different J, such that it is no longer possible to assign a well-defined J, to the state. This is the case in SmO, TbO, and DyO where the labelling is changed to reflect the order of states of a given Q, (i)fl being the ith state with Q from the lowest. The ground state is labelled X ( Q) . Dulick has used direct radial functions generated by the Hartree-Fock method in the numerical evaluation of the radial integrals, except in the case of CeO for which he has used empirical estimates obtained from a least-squares fit of the 16 low-lying electronic states, assigned in the CeO laser induced fluorescence spectra, to the 4f6s levels of the ligand field model. Apart from this last case for which a comparison is meaningless, Table VII shows that Dulick’s values are closer to those observed for PrO, DyO, and Ho0 whereas ours are closer for SmO, TbO, and YbO. These discrepancies, in the absence of any fundamental differences, must be ascribed to the choice of the radial eigenfunctions on the one hand and of the G3 parameter on the other hand. The largest discrepancy between Dulick’s values and ours occurs for EuO where our value for the separation of the configurations 4f66s and 4 f' is 7 937 cm-’ as compared to Dulick’s value of 3 299 cm-‘. The separation between the 4f N-‘6s and 4 f N configurations is very sensitive to the value of the 6s eigenfunction. An increase of 3% of the { coefficients of the 6s double zeta Slater type orbital of Yb2+ results in a change of the separation between the first O- state and the ground state 0’ from 921 cm-’ to 2 024 cm-‘. Only the ground state has been observed in the EuO molecule ( 19) and it is therefore impossible to know what is the more realistic value. The fact that, in most cases, the separations X2 - X1, W2 - W,, etc. . . are much smaller in our calculations than in Dulick’s is a consequence of fixing the G3 parameter at 150 cm-’ instead of 300 cm-‘. VI.
CONCLUSION
The ligand field theory was able to predict the lower energy level scheme of the lanthanide monoxides. Further developments of this method can be envisaged now in three main directions, say, (i) testing the ligand field calculated molecular wavefunctions by evaluating molecular quantities which are sensitive to the wavefunction compositions, such as electric dipolar moments, LandC factors, etc. . . . ; (ii) generalizing the multiconfigurational calculations in order to assign the excited states of the lanthanide monoxides, and CeO seems to be a good candidate for checking the method on account of the large amount of available experimental data; (iii) widening this type of calculation to the actinide series for which the inner 5f shell plays a part analogous to the 4 f shell in the lanthanide series. RECEIVED: February 18, 1988 REFERENCES 1. M. DULICK, Ph.D. thesis, M.I.T., 1982. 2. R. W. FIELD, Ber. Bunsenges. Phys. Chem. 86,771-779
(1982).
324
CARETTE
AND
HQCQUET
3. U. FANO AND G. RACAH, “Irreducible Tensorial Sets,” Academic Press, San Diego, CA, 1959. 4. Z. R. GOLDSCHMIDT, “Spectroscopic and Group Theorical Methods,” Racab Memorial Volume, NorthHolland, Amsterdam, 1968. 5. B. G. WYBOURNE, ‘Spectroscopic Properties of Rare Earths,” Interscience, New York, 1965. 6. W. C. MARTIN, R. ZALUBAS, AND L. HAGAN, “Atomic Energy Levels-The Rare Earth Elements,” NSRDSNBS60, 1978. 7. P. CARETTE, A. HOCQUET,M. DOUAY, AND B. PINCHEMEL,J. Mol. Spectrosc. 124,243-271 (1987). 8. B. R. JUDD, “Operator Techniques in Atomic Spectroscopy,” McGraw-Hill, New York, 1963. 9. C. W. NIELSON AND G. F. KOSTER, “Spectroscopic Coefficients for the pN, dN, fN Configurations,” MIT Press, Cambridge, MA, 1963. 10. A. D. MCLEAN AND R. S. MCLEAN, At. Data Nucl. Data Tables 26, 197-381 (1981). II. L. BREWER,J. Opt. Sot. Amer. 61, 1666-1682 (1971). 12. C. LINTON, M. DULICK, R. W. FIELD, P. CAREITE, P. C. LEYLAND, AND R. F. BARROW, J. Mol. Spectrosc. 102,441-497 (1983). 13. M. DLJLICKAND R. W. FIELD, J. Mol. Spectrosc. 113, 105-141 (1985). 14. C. LINTON, Guo BUJIN, R. S. RANA, AND J. A. GRAY, J. Mol. Spectrosc. 126, 370-392 (1987). IS. Yu. N. DMITRIEV, L. A. KALEDIN, E. A. SHENYAVSKAYA,AND L. V. GURVICH, Acta Phys. Hung. 55,
467-479 (1984). 16. A. N. KULIKOV, L. A. KALEDIN, A. I. KOBYLIANSKY, AND L. V. GURVICH, Canad. J. Phys. 62, 18551870 (1984). 17. C. LINTON, D. M. GAUDET, AND H. SCHALL, J. Mol. Spectrosc. 115, 58-73 (1986). 18. Y. C. LILJAND C. LINTON, J. Mol. Spectrosc. 104,72-88 (1984). 19. S. MCDONALD, Ph.D. thesis, M.I.T., 1985. 20. M. DULICK, E. MURAD, AND R. F. BARROW, J. Chem. Phys. 85, 385-390 (1986).