INTERPRETATION OF DATA OBTAINED FROM MÖSSBAUER, ESR, SUSCEPTIBILITY, OPTICAL AND XPS MEASUREMENTS

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INTERPRETATION OF DATA OBTAINED FROM MÖSSBAUER, ESR, SUSCEPTIBILITY, OPTICAL

AND XPS MEASUREMENTS

A. Trautwein

To cite this version:

A. Trautwein. INTERPRETATION OF DATA OBTAINED FROM MÖSSBAUER, ESR, SUSCEP-

TIBILITY, OPTICAL AND XPS MEASUREMENTS. Journal de Physique Colloques, 1980, 41 (C1),

pp.C1-95-C1-102. �10.1051/jphyscol:1980116�. �jpa-00219646�

JOURNAL DE PHYSIQUE Colloque Cl, suppl6ment au no 1, Tome 41, janvier 1980, page C1-95

INTERPRETATION OF DATA OBTAINED FROM M~SSBAUER, ESR, SUSCEPTIBI LITYr OPTICAL AND XPS MEASURWNTS

A. Trautwein

Fachbereich Angewcmdte Physik, UniversitZEt des SaarZandes, 6600 SaarbrUcken 11, W-Germany.

A b s t r a c t . - Molecular o r b i t a l c a l c u l a t i o n s a r e d e s i g n e d t o c a l c u l a t e t h e e l e c - t r o n i c and magnetic s t r u c t u r e f o r m o l e c u l e s o r c l u s t e r s . With t h e c a l c u l a t e d e l e c t r o n i c s t r u c t u r e ( e i g e n v a l u e s & and e i g e n v e c t o r s y o f t h e r e l e v a n t e l e c t r o - n i c Hamiltonian) i t i s i n p r i n c i p l e p o s s i b l e t o e v a l u a t e s p e c t r o s c o p i c d a t a from c o r r e s p o n d i n g e x p e c t a t i o n v a l u e s

<yil

8 1

?gf> ^,

^where

^Ti

r e p r e s e n t s t h e i n i t i a l and ?gf t h e f i n a l e l e c t r o n i c s t a t e , and 1s t h e a p p r o r i a t e o p e r a t o r

(i .e.

d(P)

f o r d e r i v i n g e l e c t r o n and s p i n d e n s i t i e s , ( 3 1 q

-

-T"$+/T' f o r e l e c t r i c f i e l d g r a d i e n t s , $ f o r d i p o l e moments and t r a n s i t i o n s , ~ + 2 g f o r magnetic mom- m e n t s ) . I n c l u d i n g e n e r g y s p a c i n g and Boltzmann p o p u l a t i o n o f e l e c t r o n i c s t a t e s i n t h e c a l c u l a t i o n l e a d t o t e m p e r a t u r e dependent d a t a . E x p e r i m e n t a l d a t a o b t a i - ned from Mijssbauer, ESR, s u s c e p t i b i l i t y , o p t i c a l and XPS measurements a r e r e l a - t e d t o t h e e x p e c t a t i o n v a l u e s mentioned above. The c a l c u l a t i o n a l p r o c e d u r e w i l l be d e s c r i b e d , and comparison o f c a l c u l a t e d and measured d a t a w i l l be p r e s e n t e d .

1. I n t r o d u c t i o n . - The mutual i n f l u e n c e o f motion i s j u s t a s it would b e i f t h e nu- e l e c t r o n i c s t r u c t u r e c a l c u l a t i o n s and c l e i were a t r e s t a t t h e p o s i t i o n t h e y oc- s p e c t r o s c o p i c measurements h e l p s t h e t h e o - cupy a t t h a t same i n s t a n t (Born-Oppenhei- r e t i c i a n t o t e s t t h e r e l i a b i l i t y o f t h e mer a p p r o x i m a t i o n ) . Then t h e e l e c t r o n i c v a r i o u s a p p r o x i m a t i o n s u s u a l l y i n v o l v e d i n system i s d e s c r i b e d by t h e Hamiltonian ( i n h i s c a l c u l a t i o n s and t h e e x p e r i m e n t a l i s t a . u . )

t o i n t e r p r e t h i s d a t a . Molecular o r b i t a l

H

A =

~ ( - f v ? - ; & , ) + ~ -

n ^N

"

^{4 1}

(MO) c a l c u a l t i o n s a r e d e s i g n e d t o c a l c u l a - r.4 ;>j 1;

- 51 ⁽¹⁾

t e t h e e l e c t r o n i c and magnetic s t r u c t u r e o f m o l e c u l e s o r c l u s t e r s . I n t h e p r e s e n t communication t h e b a s i c p r i n c i p l e of t h e c a l c u l a t i o n a l p r o c e d u r e ( s e c t i o n 2 ) i s d e s c r i b e d , some common approximate methods a r e mentioned ( s e c t i o n 3 ) , m o l e c u l a r ex- p e c t a t i o n v a l u e s a r e d e f i n e d ( s e c t i o n 4 ) , and c a l c u l a t e d e l e c t r i c f i e l d g r a d i e n t s

( s e c t i o n 5 )

,

e l e c t r o n c h a r g e d e n s i t i e s a t t h e n u c l e u s ( s e c t i o n 6 )

,

i n t e r n a l magnetic f i e l d c o n t r i b u t i o n s ( s e c t i o n 71, magnetic s u s c e p t i b i l i t y and g - f a c t o r s ( s e c t i o n 81, e n e r g i e s ( s e c t i o n 9 ) , d i p o l e moments and XPS i n t e n s i t i e s ( s e c t i o n l o ) a r e p r e s e n t e d f o r some compounds and compared w i t h expe- r i m e n t a l d a t a .

2 . B a s i c P r i n c i p l e . - A m o l e c u l a r system c o n s i s t i n g o f N n u c l e i and n e l e c t r o n s i s d e s c r i b e d by t h e t o t a l H a m i l t o n i a n , which i n c l u d e s motion and i n t e r a c t i o n o f n u c l e i and e l e c t r o n s . However, we c a n r e s t r i c t o u r s e l v e s t o t h e e l e c t r o n i c p a r t of

Htota3

t h i s assumes t h a t t h e e l e c t r o n s a d j u s t themselves t o new n u c l e a r p o s i t i o n s s o r a - p i d l y , t h a t a t any i n s t a n t t h e e l e c t r o n

where Z k and a Rk a r e t h e c h a r g e and t h e po-

s i t i o n v e c t o r of n u c l e u s k , and

gi

i s t h e p o s i t i o n v e c t o r o f e l e c t r o n i. Each o f t h e n e l e c t r o n s may be a s s o c i a t e d w i t h a one- e l e c t r o n f u n c t i o n

yi,

N c a l l e d s p i n o r b i t a l , which i s a p r o d u c t o f a spa_tial and a s p i n p a r t :

G i ( ? , $ )

⁼

%(;I X ; ( S > .

The t o t a l n - e l e c t r o n wavefunction i s now b u i l t up a s an a n t i s y m m e t r i z e d p r o d u c t o f m o l e c u l a r s p i n o r b i t a l s

Such a n a r r a y o f s p i n o r b i t a l s , known a s S l a t e r d e t e r m i n a n t , i s t h e s i m p l e s t wave- f u n c t i o n which s a t i s f i e s t h e antisymmetry p r i n c i p l e . The e l e c t r o n i c p a r t o f t h e t o - t a l e n e r g y of t h e system i s d e r i v e d from e q s . ( 1 ) and ( 2 ) by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980116

C1-96 JOURNAL DE PHYSIQUE

According to the variational principle,the best molecular orbitals are obtained by va- rying the one-electron functions

yi

until E achieves its minimum value.The condition for the minimum of energy leads to the dif- ferential equations, also termed Hartree- Fock (HF) equations

where

- Ei

are orbital energies, and where ^A J . is the Coulomb operator and K is the

I j

exchange operator. The orbitals

yi

obtai- ned from the solution of these equations

(4) are referred to as HF-MO's.

For molecular systems the direct solution of eq. (4) is impracticable. It is more convenient to approximate the HF-MO's by a linear combination of atomic basis orbi- tals

qP

^(LCAO)

where the (&may be Slater type orbitals (STO' s ) or Gaussian type orbitals (GTO' s) or linear combinations of STO's or GTO's.

The coefficients cp; are now optimized so as to give minimum energy. This procedure leads to a set of equations, which are al- gebraic (Roothan equations) l and for any molecular size much easier to handle than the HF equations which are differential equations :

with the Fock matrix

the overlap matrix between AO's

(P,,

^and

$,

the bond order or density matrix

the one-electron energy matrix

and the four-center Coulomb and exchange integrals J( P V

, ^{A 6}

^{) and K( P A , v b ) .}

The Roothan equations (6) can be transfor- med to a standard eigenvalue problem,which yields MO energies E i as well as LCAO-MO

coefficients c Ai. Since the Fock matrix depends on the LCAO-MO coefficients c i

,

c5i through PAC

,

the solutions of (6) can achieve selfconsistency by stepwiseim- provement of c ~ ; : with an initial guess of cai the Fock matrix Fp, is evaluated and the Roothan equations are solved; this leads to a new set of coefficients ca;

,

and so on.

3. Approximations.- Eq. (6) describes clo- sed-shell cases. A generalization of Root- han's equations for open-shell cases Is gi- ven by separate representation of exchange

-

_L

interaction for or and R spins in eq. (6)

.

Calculations in which all integrals are worked out explicitly using Roothan's pro- cedure are termed ab initio calculations.

The amount of computation time requiredfor such accurate calculations, especially for large systems, can became extremely high.

Thus, for many chemical and physical pro- blems, where qualitative or semiquantitati- ve knowledge of the MO's is sufficient to extract the necessary information, approxi- mations to the Roothan's equations may be considered. A number of approximate MO the- ories have been developed, as for example Hfickel methods3-' (nonextended

,

^extended,

extended and iterative), zero differential overlap methods6-' (CNDO,INDO,NDDO,NINDO)

,

pseudo-potential methods lo-' (one-elec- .tron approximation of two-electron contri-

butions by model potentials such as in the X, method).

In the present contribution the electronic structure and expectation values for mole- cules and clusters have been derived on the basis of the iterative extended Huckel theory (IEHT), in some cases including a li- mited configuration interaction calculati- on' ; and for some smaller molecules a mo- del potential method12 was used.

4. Expectation Values.- In order to compa- re experimental data with calculated va- lues we define the expectation value of a spatial one-electron operator 8

[= d '

($1 for electron charge and spin density, ( 3pq-r2&) / r 5 for electric field gradient,

2

^{for di-}

pole manent)

4 0 ) =

5,

^n;

Ilyie(;)

^{yi(;) d ;}

,

⁽¹¹⁾

where ni i s t h e number of e l e c t r o n s i n MO

Yi.

S u b s t i t u t i n g t h e LCAO s o l u t i o n ( 5 ) i n - t o (11) y i e l d s

From eq. ( 1 2 ) it i s o b v i o u s t h a t a l l mole- c u l a r e x p e c t a t i o n v a l u e s c o n s i s t of a mole- c u l a r p a r t Ppu

,

which i s d i r e c t l y d e r i v e d from t h e M O ' s

Yi

( s e e e q . ( 5 ) and ( 9 ) ) , and of a n atomic p a r t t o > a t , which can b e e v a l u a t e d from t h e b a s i s A O ' s

<p,

^and

(P, .

I n Mdssbauer s p e c t r o s c o p y where we measure a t a s p e c i f i c s i t e k w i t h i n a m o l e c u l a r sy- stem we may d i s t i n g u i s h t h r e e d i f f e r e n t c o n t r i b u t i o n s t o

<

o

>

at

,

depending on

( i ) whether b o t h

+,,

^and

^qy

b e l o n g t o s i - t e k ( d i r e c t o r v a l e n c e c o n t r i b u t i o n ) ( i i ) whether o n l y

9,

b l o n g s t o s i t e k , b u t

+,

t o a l i g a n d o n ) , and

-

(iii) whether b o t h ,

qp

^{and &t} b e l o n g t o l i g a n d s ( l a t t i c e o r l i g a n d c o n t r i b u t i - o n ) .

-

5 . E l e c t r i c F i e l d G r a d i e n t and Quadrupole S p l i t t i n g . - A s a f i r s t example f o r c a l c u l a - t i n g e x p e c t a t i o n v a l u e s a c c o r d i n g t o eq.

( 1 2 ) we d e r i v e e l e c t r i c f i e l d g r a d i e n t s ( E FG). The c o r r e s p o n d i n g t e n s o r o p e r a t o r f o r a l l n e l e c t r o n s and N n u c l e i i n t h e mole- c u l e i s

I n c a s e t h a t c o r e e l e c t r o n s of t h e Msssbau- e r i s o t o p e a r e e x c l u d e d from t h e LCAO ba- s i s s e t i n o r d e r t o save computer t i m e ( i . e .

2 6

f o r 5 7 ~ e : I s

...

^{3p ) ,} e q . ( 1 3 ) h a s t o b e m o d i f i e d by t h e a p p r o p r i a t e S t e r n h e i m e r

s h i e l d i n g f u n c t i o n y ( r )

Case ( i ) and (iii) of s e c t i o n 4 r e d u c e s t o t h e well-known e x p r e s s i o n s

and

I t i s worth n o t i n g a t t h i s p o i n t t h a t R f o r i r o n i s n o t 0 . 3 2 , which i s v e r y o f t e n used i n t h e l i t e r a t u r e , b u t 0.12 i f 3d 5 b e l o n g s

t o t h e c o r e I 4 , I 5 , and 0.08 i f a l l 3d e l e c - t r o n s a r e i n c l u d e d i n t h e LCAO b a s i s s e t 15

.

For heavy Mtjssbauer i s o t o p e s ( i o d i n e ) it

,.

_V

might even happen t h a t w i t h i n a b o u t 2 A

~ ( r ) h a s n o t y e t r e a c h e d i t s a s y m p t o t i c v a l u e 'jr,. R- and *-values f o r s e v e r a l e l e m e n t s and i o n s have been e v a l u a t e d un- d e r v a r i o u s a p p r o x i m a t i o n s i n o u r group 15

.

The e v a l u a t i o n of t h e o v e r l a p c o n t r i b u t i o n t o t h e EFG, c a s e ( i i ) i n s e c t i o n 4 , r e q u i - r e s t h e knowledge o f

r ( r ) .

F i g u r e 1 shows t h e S t e r n h e i m e r s h i e l d i n g f u n c t i o n f o r Fe ²⁺

w i t h ( w i t h o u t ) 3d 6 e l e c t r o n s

.

F o r t h e p r e s e n t c a l c u l a t i o n s , o f c o u r s e , t h e ( r ) c u r v e r e p r e s e n t e d by t h e broken l i n e h 8 s t o be c o n s i d e r e d , b e c a u s e t h e 3d e l e c t r o n s a r e a l r e a d y i n c l u d e d i n o u r LCAO b a s i s s e t . The c o n t r i b u t i o n ( i i ) i s f i n a l l y d e r i v e d

1 6 from a n u m e r i c a l i n t e g r a t i o n p r o c e d u r e

.

I n T a b l e 1 we l i s t t h e v a r i o u s c o n t r i b u t i - ons t o V _{z z} f o r f e r r o c e n e , Fe (C5H5) 2,and the iron-pentacyanide-derivatc , \ f e ( C N ) 5 ~ ~ 3 ] 3-

,

and compare V _{z z} (

q

=0) w i t h

A

E~ (exp. )

.

-12

0 1 2 3 L 5 r in ou.

F i g . 1 : S t e r n h e i m e r s h i e l d i n g f u n c t i o n

r ( r )

f o r ~ e * + w i t h 3 d 6 e l e c t r o n s ( s o l i d l i n e ) a n d w i t h o u t 3 d e l e c t r o n s ( b r o k e n l i n e ) . Ta- k e n f r o m r e f . 1 5

T a b l e 1: V a r i o u s c o n t r i b u t i o n s t o t h e EFG c o m p o n e n t V Z z ( i n mms-l) f o r F e ( C g H g ) 2 and [ ~( C N ) C ~ N H ~ ] e 3 - ; f o r Fe (CgHg) 2 t h e asymme- t r y p a r a m e t e r q i s z e r o , f o r [ ~( C N ) e ~ , N H ~ ] 3-

t a k e s t h e v a l u e s 0 . 1 4

zrc con~lbyrlen.

I

r . ( ~ ~ ~ ~ l ~

1

~ . ( C N I ~ S ! J ~ J ~ - v ~ ~ l a n c * , t r a Pa36 I+ ^{4.29 w.35}

--.~--- Crm 1lg.d

COX..

th. tot.1 vz= I. to.24.

lbl COod, M.L., Butt4n.. J . , Poyt, 0 . . m, W.Y. Aced. Soi. 1)9 119741 193.

(01 U f l * ~ , U., W.vwlrthr w., Pluck, E . , xuhn. PI, ~I-I.uN(, 8 . . Physik 2 (1963) 311

from .-.41 119- fson P e 4 p 1igand cram li9.na- ll',.nd W t.1 V=,

r14.67 r7.70

+ 0.34 + 1.43

- ^5.01

+ 1.66"~

-0.05 -0.22 Q.15 Q . 6 9

C1-98 JOURNAL DE PHYSIQUE

Further examples for EFG calculations as described above, which are included inthis volume, are on garnets (see G. Amthauer et al)

,

^{and on Fe} (alkyl-dithiocarbamate) 2C1, I

(see V.R. Marathe et al).

Also in this volume Lauer et a1 present a contribution, which describes the evalua- tion of Sternheimer factors and functions of various elements and ions.

6. Electron Density at the Mossbauer Nu- cleus and Isomer Shift.- From eq. (1 2) we derive the electron charge density at the nuclear site of a Mossbauer isotope as

It turns out that the overlap and ligand contributions to

9

⁽⁰⁾

,

case (ii) and (iii) in section 4 , are in most cases negligible

T a b l e 2 : R e l a t i v i s t i c e l e c t r o n c h a r g e d e n s i t i e s

1

-

^{i n a;3} f o r Fe 3d5 a n d Fe 3d6 c o n f i g u r a - t i o n . Taken from r e f . 1 8

o r b i t a l ,U Fe 3d ⁵ Fe 3d 6

1 s 6798.7 6798.9

2 s 644.1 644.1

3 s 90.1 88.4

Post et all have derived

3

(of values for ocbahedral FeF6- and Fe(CN)6-clusters from ab initio MO calculations. Fig. 2 shows the linear proportionality between their calculated

9

(0) values and experimental isomer shift

( 6 )

data according to the relation

with a mean shape of 0( = -0.24ko.02 a.3

-

¹

rnms

.

because of the relatively steep decay of the ligand wavefunctions. However, consi- derable influence upon ?(o) comes from Ppp and

I$),

⁽⁰⁾

1 ,

^{case (i)}

.

Taking 5 7 ~ e again as an example, we consider p = Is,

2s, 3s, 4s. (Relativistic contributions from p-electrons are minor, additionally they are absolutely negligible with re- spect to differences in ?(o) between two compounds) 7. Rewriting eq. ( 16) approxi- mately by

we distinguish three contributions to

(3

(0) :

(i) I5qp(o)

1

values, given in Table 2 for Fe 3d and 3d6 configuration, indicate the amount of potential shielding of nuclear potential, and its influence upon

9

⁽⁰⁾

by adding one 3d electron to Fe 3d (poten- 5 tial contribution to

9

^{(0) )}

.

(ii) The de- viation of the elements Pw ( p = I s, 2s, 3s) from 2 describes the amount of inter- action of core orbitals with ligand orbi- tals (overlap distortion contribution to

9

(0) )

.

Due to the extremely large number for

1 el

⁽⁰⁾

I

even very small deviations

Of Pls,ls from 2 may have influence upon

y

(0) to a considerable extend. (iii) The bond order matrix element P4s,4s is a measure of the valence contribution to

cj (0)

.

F i g . 2 : C a l c u l a t e d r e l a t i v i s t i c e l e c t r o n c h a r g e d e n s i t i e s v e r s u s measured isomer s h i f t s . Taken from r e f . 19

In MO calculations with limited basis set (i.e. without Fe Is, 2s, 3s) the overlap distortion contribution to

7

(0) is evalu- ated from a separate reorthonormalization procedure l 7 18. For FeF2 which was repre- sented in the IEHT-MO calculations by a [ F ~ F ~ ] 4- cluster, we calculated pressure dependent charge densities

9

(01, the va- rious contributions of which are shown in Fig, 3 together with the calculated change of Fe 3d and 4s population with pressure?O In Fig. 4 we finally compare pressure de- pendent isomer shifts with total charge

d e n s i t i e s (3 ^{( 0 ) .} The l i n e a r dependence ac- c o r d i n g t o eq. ( 1 8 ) i s w e l l r e p r e s e n t e d i n Fig. 4 ; u t a k e s a v a l u e of -0.22+0.02 a. 3

-

1 mms

.

15066.5J ^{Y , , ,}

0 LO 80 120 3.6

p In kbar

Fig. 3: C a l c u l a t e d (IEHT-MO) p r e s s u r e dependences f o r FeF2 of a ) Fe 3d and Fe 4 s p o p u l a t i o n , b) t h e o v e r l a p c o n t r i b u t i o n s t o

9

(01 from Fe I s , 2s and 3s o r b i t a l s , c ) t h e t o t a l o v e r l a p c o n t r i b u t i o n

9

O V ( ~ )

,

t h e p o t e n t i a l c o n t r i b u t i o n

y2'

⁽⁰⁾

,

t h e valence c o n t r i b u t i o n

gvai

^to)

^,

and t h e t o t a l e l e c - t r o n charge d e n s i t y

?

^{O ) A l l}

p

( 0 ) v a l u e s i n a,?

Taken from r e f . 20

p in kbar

7 5

F i g . 4: Linear dependence between experimental pressure-dependent isomer s h i f t s and c a l c u l a t e d r e l a t i v i s t i c e l e c t r o n charge d e n s i t i e s o f FeF2.

Taken from r e f . 20. Experimental isomer s h i f t s a t 300 K r e l a t i v e t o m e t a l l i c i r o n a r e taken from r e f . 21

7. I n t e r n a l Magnetic F i e l d . - The v a r i o u s

A

c o n t r i b u t i o n s t o t h e i n t e r n a l f i e l d , H i n t ,

measured a t a Mossbauer n u c l e u s , c o n s i s t

A

of t h e c o n t a c t f i e l d H', t h e o r b i t a l f i e l d

GL,

t h e s p i n - d i p o l a r f i e l d

;;SD,

and t h e -'ST

s u p e r t r a n s f e r r e d f i e l d H

-.

The c o n t a c t f i e l d H' a r i s e s from t h e con- t r i b u t i o n of t h e s p i n d e n s i t y a t t h e nu- c l e u s 2 2

I n c a s e t h a t t h e s p i n d e n s i t y o r i g i n a t e s from s - e l e c t r o n s ( w h i c h have f i n i t e c h a r g e d e n s i t y

9

^{( 0 )} a t t h e n u c l e u s ), eq. ( 2 0 ) c a n be r e p r e s e n t e d by ² 3

w h e r e < S s > i s t h e e f f e c t i v e s p i n produced by t h e s e s - e l e c t r o n s . On t h e o t h e r hand, i f t h e s p i n d e n s i t y o r i g i n a t e s from p- o r d - e l e c t r o n s , t h e s p i n - p a i r e d c o r e - s - e l e c - t r o n s may become s p i n - p o l a r i z e d due t o ex- change i n t e r a c t i o n w i t h t h e p- o r d - e l e c - t r o n s . F o l l o w i n g a g a i n Watson and F r e e - man23, H~ t h e n i s r e p r e s e n t e d by

HC

=

2

~ ' ' ' ~ 4 sP,d>

+ 2 H'*'

4 sp,d> ) (22) where H c o r e

and Elva1 r e p r e s e n t t h e amount of s p i n - p o l a r i z a t i o n o f c o r e - and valence-

s - e l e c t r o n s of t h e Mossbauer i s o t o p e . Both, H~~~~ and Hval, have t o b e a d j u s t e d t o t h e a c t u a l number o f d- and p-elec- t r o n s , b e c a u s e p o t e n t i a l s h i e l d i n g e f f e c t s have c o n s i d e r a b l y i n f l u e n c e upon t h e s p i n - p o l a r i z a t i o n of t h e c o r e - and v a l e n c e - s - e l e c t r o n s 2 3 . From MO c a l c u l a t i o n s t h e d- and p - p o p u l a t i o n and t h e e f f e c t i v e s p i n - d e n s i t i e s (Ss) and<S

>

a r e e ~ t r a c t e 2 : ' ~ ~

p , d

t h u s making t h e c a l c u l a t i o n o f H~ straight-

forward. ₄

The o r b i t a l f i e l d H~ a r i s e s from t h e orbi- t a l motion o f t h e e l e c t r o n s

The t o t a l o r b i t a l a n g u l a r momentum Lp o f t h e e l e c t r o n s around t h e MBssbauer nu- c l e u s i s d e r i v e d a c c o r d i n g t o eq. ( 1 2), w i t h u s u a l l y n e g l e c t i n g c a s e s (ii) and (iii) i n s e c t i o n 4 .

The s p i n - d i p o l a r f i e l d

csD

^{i s} t h e c o n t r i -

c1-100 JOURNAL DE PHYSIQUE

b u t i o n from t h e magnetic moment, which i s a s s o c i a t e d w i t h t h e s p i n o f t h e e l e c t r o n s o u t s i d e t h e n u c l e u s

Comparing e q s . (24) and ( 1 3 ) it i s o b v i o u s t h a t t h e s p i n - d i p o l a r f i e l d i s d e r i v e d i n a s i m i l a r f a s h i o n a s t h e EFG.

The s u p e r t r a n s f e r r e d f i e l d

ZsT

may b e eva- l u a t e d a l o n g t h e l i n e s d e s c r i b e d by Sa-

A -

watzky e t a l Z b , however, w a s found t o be n e g l i g i b l e f o r t h e a p p l i c a t i o n s p r e s e n t e d h e r e .

I n T a b l e 3 we summarize t h e v a r i o u s con- t r i b u t i o n s t o

~i~~

f o r i r o n - d i t h i o c a r b a m a - t e compounds27. The comparison w i t h expe- r i m e n t a l v a l u e s a t t h e i r o n n u c l e u s 28,29

I

H i n t ( ~ e ( d t c ) 2 ~ 1 ) =32.9f 1 .O T, Hint ( ~ e ( d t c )

I ) = 1 7.2-1 2.4 T

,

and a t t h e i o d i n e nucleus3?

(Fe ( d t c ) 21) =5.3-6.7 T , i n d i c a t e s t h a t -

- t h e agreement between c a l c u l a t e d and mea- s u r e d magnetic d a t a i s q u i t e s a t i s f a c o t r y .

( F u r t h e r d e t a i l s o f t h e c a l c u l a t i o n con- c e r n i n g t h e e l e c t r o n i c and magnetic s t r u c - t u r e o f F e ( d t ~ ) ~ C l , I i s p r e s e n t e d i n V.R.

M a r a t h e ' s e t a 1 c o n t r i b u t i o n i n t h i s vo- lume. )

Table 3: C a l c u l a t e d magnetic d a t a ( s e e t e x t ) f o r Fe (dtcf 2X ( w i t h X=C1, I ) a t t h e i r o n and i o d i n e nu- c l e u s , r e s p e c t i v e l y . Values of H a r e g i v e n i n T e s l a . Taken from r e f . 27

I r o n i n . Fe ( d t c ) 2X i o d i n e i n F e ( d t c I 2 I

H i n t

36.7 19.6

( a ) T h e a c t u a l v a l u e f o r <SZ> w h i c h i s u s e d i n t h e c a l c u l a t i o n o f H i n t c o n t r i b u t i o n s i s 1 . 1 4

-

$ , s i n c e a t 1 . 4 K , w h e r e m e a s u r e m e n t s

h a v e b e e n p e r f o r m e d , o n l y t h e

?L

2 K r a m e r s d o u b l e t i s p o p u l a t e d .

(b) C a l c u l a t e d w i t h b o n d o r d e r m a t r i x e l e - m e n t s i n s t e a d o f o r b i t a l c h a r g e s q 3 d a n d 9 4 s .

( c ) C a l c u l a t e d f r o m t h e LCAO c o e f f i c i e n t s o f t h e s i n g l y o c c u p i e d M O s . ( I n c l u d e s f a c - t o r

:-;

^{s e e ( a ) )}

^.

( d ) O v e r l a p c o n t r i b u t i o n w h i c h c o m e s f r o m a M u l l i k e n p o p u l a t i o n a n a l y s i s o f s i n g l y o c - c u p i e d M O s . ( I n c l u d e s f a c t o r

4;

^{s e e ( a ) )}

.

( e ) C o r e p o l a r i z a t i o n c o n t r i b u t i o n . ( f ) D i r e c t 5 s c o n t r i b u t i o n .

( g ) ^{A l l} H$ c o n t r i b u t i o n s a r e n e g l i g i b l e . 8 . Magnetic S u s c e p t i b i l i t y and Gyromagne- t i c Tensors.- Magnetic s u s c e p t i b i l i t y and gyromagnetic t e n s o r s ( X and G ) d e s -

Pq Pq

c r i b e t h e i n t e r a c t i o n o f t h e m a g n e t i c ho- ment o f an e l e c t r o n i c system w i t h an e x t e r -

a

n a l l y a p p l i e d magnetic f i e l d H.

The t e m p e r a t u r e dependence o f X i s d e r i - ved from t h e e x p e c t a t i o n v a l u e

;R~S> ^,

which i s a Boltzmann a v e r a g e o v e r a l l e l e c t r o n i c s t a t e s .

a A

The G-tensor i s d e r i v e d from tL+2S> by r e - w r i t i n g t h e i n t e r a c t i o n

i n t e r m s o f a n e f f e c t i v e s p i n . 5

-

i n gene- r a l can be asymmetric; i n t h i s c a s e a sym- m e t r i c t e n s o r i s c o n s t r u c t e d by 3 1

D i a g o n a l i z a t i o n o f G c o r r e s p o n d s t o an o p e r a t i o n i n

R

3 , whlch t r a n s f o r m s t h e used Pq b a s i s s e t i n t o t h e main a x e s system o f G From t h e main components G,,G2,G3 t h e n t h e pq'

well-known g - f a c t o r s a r e d e r i v e d

(From t h i s p r o c e d u r e t h e s i g n of gi c a n n o t b e d e t e r m i n e d . )

F o r K3Fe(CN)6, f o r which a l s o K6ssbauer p a r a m e t e r s have been d e r i v e d a l o n g t h e p r o - c e d u r e d e s c r i b e d s ~ f a r ~ ~ , t h e powder s u s -

-

¹

c e p t i b i l i t y

?!

=3(

X I

+ X2+

X3)

h a s been eva- l u a t e d a s f u n c t i o n of t e m p e r a t u r e , and was compared w i t h e x p e r i m e n t a l r e s u l t s ( F i g . 5 ) . The c a l c u l a t e d and measured g - f a c t o r s f o r K3Fe(CN)6 a r e l i s t e d i n T a b l e 4 . From i n - s p e c t i o n of F i g . 5 and T a b l e 4 one r e a l i - z e s t h a t t h e o r e t i c a l and e x p e r i m e n t a l da- t a a r e i n r e a s o n a b l e agreement.

Fig.5: C a l c u l a t e d (IEHT-MO-CI) powder s u s c e p t i b i l i - t y ( s o l i d l i n e ) of K ~ F ~ ( C N ) ~ a s a f u n c t i o n of tem- p e r a t u r e . Taken from r e f . 32. The e x t r a p o l a t i o n

(broken l i n e ) l e a d s t o a Weiss c o n s t a n t h = 40 K , which i s comparable with t h e v a l u e eexp = 56 K from t h e d a t a o f F i g g i s e t a 1 ( r e f . 33)

Table 4 : g - f a c t o r s ( s e e t e x t ) f o r K3Fe (CN) 6. Taken from r e f . 32

c a l c . r e s u l t s exp. ESR r e s u l t s ( a )

Fel Fe

2 Fe

1 Fe2

Y1: 2.39 2.30 2.36 2.31

( a ) Baker, J . M . , Bleaney, B . , Bowers, K.D., Proc.

Roy. Soc. (1956) 1205

9 . Energies.- The orbital energies which

were derived using the IEHT approximation are in qualitative agreement with those re- ported from experimental work (see Table 51

Table 5: O r b i t a l e n e r g i e s ( i n cm-l) a s d e r i v e d from IEHT c a l c u l a t i o n s and o p t i c a l experiments. Taken from r e f s . 20, 32

compound IEHT exp.

K7Fe (CN) 32000 3 5000

1 loo 9300

Table 6 : C a l c u l a t e d and experimental i o n i z a t i o n po- t e n t i a l s ( i n ~y = 109737.3 cm-l) .Taken from r e f .36 molecule o r b i t a l exp. ( a ) c a l c .

H2° a 1 -2.37 -2.31

-0.93 -0.92

C H ~ O H ' 3 a ' -1.38 -1.27 l a " -1.29 -1.11

4 a ' -1.12 -1.11

5 a ' -0.95 -0.96

2a" -0.82 -0.84

( a ) Siegbahn, K . , e t a 1 i n "ESCA Applied t o Free Molecules", North-Holland, Amsterdam (1969)

More accuracy has been achieved with our

-

compared to the IEHT method

-

more elabo- rate model-potential MO calculations 12,34

A s an example we present energies for H20

and CH30H in Table 6.

lo. Dipole moments and XPS intensities.- Dipole moments are derived according to eq. (12) with the operator

a=?.

XPS inten- sities result from 34

where fjrrare XPS cross-sections taken from Gelius and siegbahn3'. For the two com- pounds from above, again using the model- potential MO method1 2, the calculated and measured dipole moments and XPS intensi- ties are summarized in Table 7.

Table 7: C a l c u l a t e d and experimental XPS i n t e n s i - t i e s and d i p o l e moments (Debye) .Taken from r e f . 36.

molecule o r b i t a l X P S ( ~ )

exp XPScalc <;

>

H,O a , 44 .o 44 .o

l a " 3.2 3 .o

2.00 (c)

4 a ' 2 .o 1.7

5 a ' 3.3 1.7

2a" 1 . 6 5.5

( a ) s e e r e f . (a) of Table 6. ( b ) exp. ( c ) c a l c . 1 1 . Summary.- On the basis of the IEHT me- thod, with a fixed set of parameters ( ! )

,

electric field gradients, electron densi- ties at the Mossbauer nucleus, internal magnetic fields, suscepitibility, g-fac- tors and energies have been derived, which are in reasonable agreement with experi- ment. This shows that even a relatively simple MO method is able to reproduce sat-

isfactorily experimental results, and thus leads to a deeper understanding of the electronic and magnetic structure of the compounds under study.

The more sophisticated model-potential cak culations require much more computer space and time; it is still questionable whether the results justify the effort.

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft. The author grate- fully acknowledges the fruitful coopera-

c1-102 JOURNAL DE PHYSIQUE

t i o n over several y e a r s , w h i c h l e d t o m a n y of t h e r e s u l t s presented here, w i t h Pro- f e s s o r s I . D 6 z s i , A.J. F r e e m a n , U . G o n s e r , F . E . H a r r i s , D r s . S . K . D a t e , J . P . D e s c l a u x , J . M . F r i e d t , M. G r o d z i c k i , A . K o s t i k a s , Y.

M a e d a , V . R . M a r a t h e , R. R e s c h k e , R . Z i m - m e r m a n n , and t o M r . R. B l a s and M r . ^S

..

^Iau-

e r .

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