On the ligand-field potential for f electrons in the angular overlap model

Chemical Physics 14 (1976) 393-401 0 North-Holland Publishing Company

ON THE LIGAND-FIELDPOTENTIALFOR f ELECTRONSIN THE ANGULAROVERLAPMODEL W.URLAND Institut fir anorganische und analytische Chemie, Heinrich-Buff-Ring S,

63 Giessen, FRG

Received 5 December 1975 The angular overlap model is applied to f electrons. General relationships are evaluated to express the iigand-field potential in angular overlap parameters for arbitmry ligand-fields. Comparison with the electrostatic model is presented for a specitk example.

1.

Introduction

The interpretation of the spectra and magneticprop erties of some compounds of the trivalent lanthanide ions by applying the crystal-fieldpoint-chargemodel (electrostatic model) has been difficult because of their low co-ordination symmetry. For such systemsit is generally impossibleto derive from magneticmeasurements unambiguousinformation about the crystal-field parameters [I 1. Easier to deal with because of the simple form of the crystal-fieldpotential are the compounds of high symmetry. So, if we considerf electrons,the mostextensiveexpansionof the crystal(hgand)-field potential V for the molecular symmetry C1 v = ROCOO Y; +qc22y2

2

+ R,(r,Y,4

+ c*1 y;

-I-c2&

+c2_1 y ;’

+ c2_2 q2>

+ c43Y,’ t CLOYSt c41 Y; t c4,,Y;

tc4_&

tc&$ _-2Y~2+cq_3Y~3tcq-4y1;4)

+R&~Y~+c~~Y; +QY;

contribution R c Y” for sphericalsymmeO try has been omitted [l] $00 . To derive (1) we made use of the usual expansion of V: where the

+cs4Y~+cs3Y~

+c61Yz

tc60Y~tc~_lY~ltc~_2Y~2

fc6_3~3+Cg_~Y~+Cg_5~5+Cs_6~6) (1) reduces for a cubic crystal-fieldin the point-charge model to:

(2)

radialfunction,ck4= expansion coefficient, Y{ = sphericalharmonic, k = quantum number I,4 = quantum number m; summation is over all possible functions which give non-vanishingmatrix elements G,m I Vl3,m’)). Apparently because of the above mentioned difticulties in the case of crystal-fieldswith low smetry magnetic properties only of lanthanide ions with high coordination symmetry have been studied so far [ 1,2] . For systems with low symmetry the application of the angular overlapmodel (AOM)of Schiffer and Jdrgensen [3] seems to be more adequate [4] The AOMand similarcovalent models have been applied to systemswith uncompleted d-shells14-71 as well as to systemswith uncompleted f-shells[S-14]. I3y employing the AOMwe derivein the following sections general relationshipswhich allowus to express the l&and-fieldpotential (1) in terms of angularoverlap parameters. As an example the ligand-fieldpotential of an octahedral complex is givenexplicitly. The expression for the potential in the AOMis compared with the one in the electrostatic model and the tRk =

* In the first equation of ref. [l j ChallgCd.

the signof Y? has to be

W.Urkmd/Ligand-field potentialfor f elechms

394

angularoverlapparameters are related to the crystalfield parameters.

2. ‘ihe application of the AOMto f orbit& As the AOMhas been extensively reviewed 13-7, 121 only a short description of the model will be given here. The AOMmay be seen as a fmt-order perturbation approach to covalent bonding Beingbased on covalen‘kythe AOMis thus in contrast to the crystal-field model resting on electrostatic interaction. With the assumptionof weak covalency the perturbation energiesof the central orbitals are proportional to the squares of metal-ligandoverlap integrals.Like in crystal-field theory the AOMassumesthe effects of severailigandsbonded to the central metal to be additive what is equivalent to the assumptionof no ligandligand overlap.We can write the antibonding energy E* of a given f orbital as: E* = k[&(A) F:"1*

=P_~(F~))~.

(4)

Considerationof inequivalent A-bondingeffects in two directions perpendicular to the metal-ligandaxis leads to the more general angular factors Dit, where Asnow represents u-, ns- and nobonding, see for example [5]. The D(I) form a matrix expressingthe rotation of a set of met$wavefunctions into the l&and axis frame (see fig. 1). The D(o matrix can be either constructured using euIerianangles(0, p, $) or direction cosinesas in (5). x’

yi

2’

(9

The direction cosinesgive the mutual orientation of the metal (x,y,z) and the l&and frame (x’JJ’J’). In this paper the matrix for rotation of f orbitals Do is evaluated,using direction cosines.The real f-electron wavefunctions(7) are taken from ref. [15]. By replacingthe matrix (5) by the eulerian transformation (6) the FQ matrix for equivalent ‘h-bondingmay be constructed.

k

is a proportionality constant dependent on factors includingthe valence state ionization energiesof metal and ligand.h designatesthe bonding symmetry with respect to the metal-ligandaxis.S&Q.) is the maximal overlap integral for antibondinginteraction between metal and l&and functions of a given type u, x, etc., and given bond Iength.J+z),the angularoverlap integral, ls a property of the orientations of the overlapping orbitals and appliesto equivalent&bonding. The superscript labels the I quantum number of the metal functions. ek is the angularoverlapparameter for a particular bonding type.

;e

(6)

Thd matrix can be used generally for complexeswith axlally symmetric ligands.Here the mutual orientation of the metal and ligand frames is given completely by the eulerian rotations through 8, upand 0. The real f electron wavefunctions ]3,i\@are: 13,u) = la> =mr-31(2z2

-3X2 - 3y2)z,

13,ns)= Ins) = JijGr-3afiq4z2

-x2 -JJ)y,

]3,ac)= Inc) = @zrr’36
-x2 - y2).X,

)3,&S)= 18s)=@jGi3#4&(297), 13,Sc)= l&z)= JVG P33JEz(x2 13,@J= I& = *r-3&Ko(3xZy 13,+X)= I@= &-G-3:m(xS

Fig.1.

- y2), - y3), - 392).

(7)

The DQ matrix expressed in direction cosines is given in table 1. From table 1 wesee that u bonding involves

W.~Urhnd/Ligand-fieldpotential

- $11 S&,(o)

[afi72(37; [M$(Y:

-

To derive (9) the relationship

with Ins),

“kq = (-l)qc;_g,

with Id,

(c* is complex conjugated to c) has been used.

W!

Welike to express now the Rkck4 coefficients in angular overlap parameters. This can be achieved in three steps. (a) The Rkckq coefficients have to be expanded in one-electron matrix elements(3,m I Yl3,m’). This may be done by applying the followingequation:

with l&c),

Y~W&(O>withW, and

- 37;Ns;&7)

39s

with I o>,

with I&), [lG7&

for f electrorls

with I&*

The F(O matrix is givenin table 2. An equivalentmatrix is given in ref. [16]. The generalmatrix element of covalent interaction withligands between real f orbitab is [5] :

(3,ml VI3,m’) = kc QR kc&l)m~

(11)

l&ads

which has been derived from (3) and (12) [17] : where 13,u)and 13,~)are the realf orbitals 13,@and t is the runningindex for the bonding symmetry A$. I$ and $7 are the matrix elements of table 1. Eq. (8) contains (4) as a special case. From (8) we see that the covalent interaction between metal and l&andis separated into t = u, ns, nc, A$,6c, gs, and gc types. For each type the antibondmgmatrix elements are calculated by summingthe contributions from all ligands. Each contribation of a symmetry independent Iigand with the numberj is scaledwith the angularoverlap parameter eASfor the correspondingbonding type.

The general expansion of the ligand-fieldpotential (3) for f electrons is: B=ROcO,,Y;

t RZc2,,Y; + R2czl(Y; - y;‘) + Rzczz(Y; + 5”) +R&Y;

(12) (b) The second step is to express the matrix elements in the complex I3,m l= Im ) basis in those in the real basis 13,u)= lu>. The bases are related as in (13):

-
3. The ligand-fieIdpotential

+R4q&

(3,mlYk4(3,m’)=(-l)“‘~(~

- 5’) + R4c42(Y; + Yi2)

-L a (nsl - -Jfi (ncl,

-L 16s)+$ ISd, 12)= * ‘2I=-d -kSsl t &cl, d _-iI@)_ 13)@

(31=

-&I fi

<-II= -&,l+-&I, d y I-i=-;;:wt

t-21= -&&I+ -L&l, 4 4 -L Ilpc), l-3)= -2 Iid t+2 w, fi

-+I,

c-31= +

+R4ca3(Yz -Y,-3)+R4c44(Y; +c4)

+ yz;“,

+ R6cas(Y; - ~s)fR6c66(Y;

+ 5%

Gpsl t~~cl. (13)

t R6c60Y; + R6csl(Y; - Y;‘) +R6c62(Y; + q2) + R6cG30’; - q31 +Q&

3 IhA

(9)

With these relationshipsit is straightforward to deduce transformationslike (14): ~2}VIO~=-~~GslVlo~t~~SclVlo~. & fl

(14)

W.UrlandfLignnd-fieldpotentioi

396

forf electrons

Table 1

D(f)matrixin direction cosines

Lmd 10’) Metal

&T24-&-1)

J1sQhY3

tazc57:-u+27*7B%

k~7$--lI~7*7m

J57zia383rz+o3738*+8373~2)

L/&I6Y&l)

iI&-O+h383

4’a*(5723-1147,73Q3

~(Q3837l+Q373Pr+83Y3al)

@717273

~~,7273+&r7,73+837172)

di%Qr7273+Q27173+Q37172)

01(8273+8372)+81@273+0372)

lo)

:7&--3)

im,

Mmc4-1)

I*8

16s)

-VT& M-1)

+Yl(Q283+Q382) 16C)

$\/Isr3(7:-7;)

m(7:Bs-r:B3+2!3,7*73

id&d37:-7;)

w

&%:03-7:Q3+2Q,7O3

&i@217:-7$)+4fi7,72&

3a38373t2Ql8l73thlrIP3 +28,7lQ3

-&27273)

-2827273)

idifQ2(7:-7:)++fi7,72Q~

~(Q,p,72+p,r,s+cr~7~~2 -Q28272)

I@

w&1(7:-37:)

h/i58

1 (7:-7;,-4&,72192

dfiQ,&-7i)-t&ldQ2

di%Q,Lh7,-9827!-Q18272 -Q272Pd

From eq. (11) and transformationslike (14) the coefficientsR c can be writtenin termsof one-electron k kq matrix elementsin the real basis.Weobtain the matrix of tabIe 3, (c) The final step is to relate the coefticientsRkckq to the angularoverlapparameterse, by meansof eq. (8). This cannot be done in general.Becauseof (8) the resultantrelationshipsare dependenton the particuiax

geomeuy, the nwnber of symmetry-independentligands and the involvedbondingtypes and may be established by a computerprogramme.The case of octahedralsymmetry willbe treated in the next section.

Table2 FCQmatrixin eulerian angles Lignd Mk?td

IO’)

Id)

Iid,

Iid,

397

W.Urklrtd/Ligand-fieId potentialfor f elecrrons

4. Application The relationships of sections 2 and 3 may be applied to a specific example. For convenience we choose for computation the l&and-field potential of an octahedron formed by six identical, axially symmetric ligands. The axis of quantization (z-axis) may lie along the threefold axis (see fig. 2).

16c’)

Ilpb)

The ligand-field potential VOhfor the octahedral ligand arrangement shown in fig. 2 is by symmetry arguments: VOh=ROcooY~ +R4c40Yt +R,cb3(Y,3 - q3) tRscsoY;

+R&(Y;

t R&&Y;

+ Ygd).

Ilpc’)

- Yi3) (15)

- _

*6 ‘66

of the Rpkq

expanded in one-electrok

I 1 I I I I I I

Tablei. ,’ The wcfficients matrix elements

..^

.

p Y

$ -

-

-

-li -

-t - .v - 2

-

-

” “_

7

-

R

-

L -

.J

-

A -

5Q 2L e

0

-

,

.A

$1

-

-

Vi

Pi

Ji

-

0

-

-

-

-

Ri

*Ei

-

-

Oi -

-Gl

V

ii

-

-

Ti __

-

.” ..” “...

1

I

-

-

z -

-

-

3/f -

0

A

A

e d - -d ---

G=~fl,H=~

-

4 -

-

-C -

-

-

-

-

-

-

-0 -

4 -8

* n

-i,

z

VI

RI

Ji

-

V

t

If.

..“.

2

8 a

-

-

-

-

i

-

-

ii1

0

.,I

-

8

R

.. -

-

-

> -

-

-

-

.Oi -

-

-

. , 1.p

3

3

-

R

.J -

V

h

Gi -C

$ -Pc.

4 ”

-

_i,

-

-

-

-

-

-

-

-

-

-

8 ”

A

,,,.

W.Urland/Lt

CSclVW=

Co-ordinates

399

pozenrkifor f electrons

GislVl6s)=$(5ea + 5e, + e6 + 7e&

af the figands

Gp~lVllpc1= ik(20e0 + 15en + *a GpslVlgs) = 1(5en + $

+ 3eJ,

GcI@id = -@sl Vim)= h&e0

? Fig. 2.

From table 3 we obtain for the expansion coefficients: ROcOO= $&Wlo>

+ h.Wlnc~

= lfi(Jea

CScl~7rs)

= Gslv!7rc~

+ fen + es - ib$,

+ en - 4e6 + JeJ, = 4pwlu~

= C&%x~

= 0.

(13

The expressionsfor the diagonalone-electron matrix elements may be checked by a diagonalsum rule [3,12], saying that for each bonding type t and for each of the 2Z+lmetal orbitals the sum of the coefficients (Djo)2 of the angular overlap parameters er equals the number N of the Uganiis:

+hlVlns) t GclV16c) + (6slVl6s)

21+1

+ (cpclPl$X)+ (rpSlVl~S)), R,c,, = 3446W’la)

GpclVlu)

+ 25eJ,

c

t=1

+ (nclVlnc)

2lil @)’

= N,

c

1

(D$)2 = N.

(18)

For our example (18) becomes:

+ (mlVjns>-7(6clv16c) - 7WVl6s) + 3GpClvl~C~ + 3Gpslyl@), R,c,, =&&-t/%tklVlnc)

4 (F;9)2 = 6,

+ i&clVlns)

+i&GslVlnc) + ~Wlns) + (6/@)(~lvlu) - iW%Wlu)), R6cG0= 34%422(ul ?‘lu)- ;(~clVlnc)

W?

-$(nslVl?rss>+ ~Ciclv16c~ + &sslVl6s~ -i&ciVlW) - roGpslyllps~), R6ch3 = ~~(-&S’l~c~ + #iGslVh> + @slVlns>

Eqs. (15~(17) lead by substitution to the final expression for the ligand-fieldpotential:

t &Whd

VOh= j$N12e0 + 24en + 24es + 24eV)Yi

7

- 44~1 Vllpd+ i@slVlu)),

Rscs6 = ~~(&S’l~c~

+Gt--2eo - se, + 9 e&- 2eV)Yi - i&l iI&

- :(&I~)).

+m(2eD (16)

Application of (8) using the co-ordinatesof fe. 2 and the matrix elements of table 2 leads to the following expressionsfor the one-electron matrix elements WflU): (ffl?W

= &Sea + 6en + 3Oc, + ‘tip),

bclVinc) = (nslVlns)=$(e, + 7en + 5e, + 5eP),

+ $e, - yes t BV)(Yz - Yi3)

+@(#e0-4en+Ses + 2$4W%9eo i & &ZS$%($ea

-&eV)Yi - kn + Be6 - G&Y,’

- G3)

-4e, +le, -3eV)(Y,” + Y$ (20)

The above relationship for Vohin the AOMmay be compared with the one in the electrostatic model Cl81:

W.Urhd/&and-field

+ (z&+41,5)r_y&Y~ + &/%$Y~ + (zi?r6&z7)~~~Y~ + yVZzqs(y66

-

C3)]

+ Jd2m(Y,3

- Yi3,

+ -$,I,

cm

are the usual constant expressionsand are defined for examplein ref. [18]*. Wesee that the appropriate ratios of the coefficients in (20) and (21) coincide: where ~t&?~

poten tip1for

f electrons

Wegavea generalrecipe to compute the Iigand-fieldpo; tential for arbitrary symmetry, any number of different ligandsand any bonding type. In the case of octahedral symmetry we gavedef&te expressionsForthe figandfield potential in the AOM.Wethen comparedwith the electrostaticmodei and related the angularoverlapparametersto the crystal-fieldparameters. Confmingourselvesto u bondingonly like in ref. [S], we reduce the number of ligand-fieldparametersfrom two b4 and p6 in ref. [ 181) to one, whichis eo. This can be done for all kindsof co-ordinationsymmetry, assumingidenticalligandsand equal bond lengths. The reduction of the number of ligand-field parameters is

From (20) and (21) we may expressthe crystaLfIeld parametersof ref. [18] in angularoverlapparameters. We obtain: po = ze2+$r = )(eo t 2eR+ 2es + 2eq), p4 = ze2(r4)/n5= -&(-2eo - $e, + Yes - 2eJ, pa =re2G6)1,7=j#e0

- 4eT + le, - &e,).

due to the powerful feature of AOMto classifybonding interaction between ligandand metal. The other powerful feature whichmakes AOMsuperior to the electrostatic model is the fact that the angularoverlapparameter can be estimatedby approximateformulae [4,8]. We thus conclude that in future the applicationof AOMwill simplifythe interpretation of the magnetic properties and spectra of lanthanideions with low symmetry.

(23)

Also, we may reiate the angularoverlapparametersto the crystal-fieldparametersdF(r) of Elliott and Stevens Acknowledgement [19] usingthe appropriaterelationshipsgivenin ref. I thank ProfessorR. Hoppe for his support of this [181: work. For discussionsI am grateful to Dr. M. Gerloch _4$~~,=6+., =$(e a t 2e If t2e 6 t2e ) from the Universityof Cambridge(England). Ip’ Ai(r4) = -&p4 = &(-2eo - $en + ye, - 2e,), A;@) = h$

= &(teo - 4en + teg - fi$),

Ai(r4, = -&?!p4

= w(-2en

A;(+, = &h,o,

= -+j&($e,

A6(+3 = 3~ 6

6

References

- !e,, + lpeb - 2eg),

= $j-#e ~ - 4”,+%e6

- 4en + fe, - $eq), - i%J.

415.

[2] L.M. Engelhardt andB.N.F&is, J. Chem. Sot. A (1968)

(24)

Analogousexpressionslike (23) and (24) could be ob-tained for any kind df co-ordinationsymmetry using the generaIrelationshipsof sections2 and 3;

5. Conclusion h this paper we have applied the AOMto f orbitals. ‘-Eq. (21) hai be& obtained from (5) in ref. [18] by gutting

. SO& = 4 and addingthe contribution @&)lZ&‘e for sphes@alsymmetry.

(11 L.M. EngeIhardt and B.N. Figgis, J. Chem. Sot. A (1970)

1258. 13) C.E. Sch&Terand C.K. Jbrgensen, MoL Phys 9 (1965) 401. I41 M. Ge+xh and R.C. Slade. Ligand-field parameters (Cam-

bridge Univ. Press, 1973). CE. Sch%fer, Structure and Bonding 5 (1968) 68. C.E. Sctiffer, Proc. Roy. Sot. A297 (1967) 96. C.E. Sch&Ter,Pure and Applied Chem. 24 (1970) 361. CK. J&gensen, R. Pappalardo, and H.-H. Schmidtke, J. Chem. Phys. 39 (1963) 1422. [9] W.C. Perkins and C.A. Crosby, J. Chem. Phys 42 (1965) [5] [6] [7] [8]

407: [lo] J.D. Axe and C. Buns, Phys. Rev. 152 (1966) 331. [ 111 G. Bums, Phys Lett. 2SA (1967) 15. [ 121 C.K.J&men, godem Aspectsof LigandField Theory (Nor&-Ho&ml,Amsterdam,1971).

W.Urland/Ligand-field potentialfor f electrons [13] D. KuseandC.K. Jqkgensen,Chem.Phys Letters l(1967)

114. [ 141 H.-D.Ambqer, R.D. Fischer and B. Kanellakopulos, Theor. Cbim. Acta 37 (1975) 105. [ISI S.E. Hamung and C.E. Schafer, Stiucture and Bonding 12 (1972) 201. ’ [16] S.F.A. Kettle and A.J.P. Pioli, Inorg. Chim. Acta 1 (196?) 275.

401

[I?] M.Rotenberg, R. Bivins,N. Metropolis,J.K. Wooten Jr., The 3-j and sj Symbols (MIT Press, C’ambridgs,Mass., 1959). [lS] M. Gcrloch and D.J. Mackey, J. Chem. Sot. A (1970) 3030. [I91 RJ- Elliott and K.W.H.Stevens, Proc. Roy. Sec. 215 A (lY52)437,218 A(1953) 553,219 A (1953) 387.