Application of Molecular Orbital Calculations to MÃ¶ssbauer and NMR

1 Application of Molecular Orbital Calculations to Mössbauer and NMR Spectroscopy of Halogen-Containing Compounds

Downloaded by 80.82.77.83 on May 18, 2017 | http://pubs.acs.org Publication Date: July 1, 1981 | doi: 10.1021/ba-1981-0194.ch001

MICHAEL GRODZICKI, SIEGFRIED LAUER, and ALFRED X. TRAUTWEIN Angewandte Physik, Universität der Saarlandes, 66 Saarbrücken 11, West Germany ANNABELLA VERA Universita degli Studi di Parma, 43100 Parma, Italy

Self-consistent field and charge molecular orbital (MO) calculations are applied to a series of fluorine-, chlorine-, bromine-, and iodine-containing molecules. Calculated orbi tal energies and dipole moments are used for testing and comparing the MO theories. The calculation procedures for deriving (1) electron charge densities (O), (2) electric field gradient tensors, and (3) internal magnetic fields are de scribed in detail. In connection with calculated (O) values and measured isomer shifts δ, the relative change of nuclear charge radius, δR/R, is derived for iodine. Together with the various contributions to the electric field gradient, the quadrupole polarization of electronic cores γ(r) and the nuclear quadrupole moments Q for chlorine, bromine, and iodine are discussed. For one specific compound, N(C H ) FeI , the internal magnetic fields at the iron and iodine nuclei are evaluated simultaneously from the magnetic M O structure of this compound. All calculated data are com pared with experimental results. p

p

2

5

4

4

M

o l e c u l a r o r b i t a l ( M O ) c a l c u l a t i o n s p r o v i d e us w i t h e l e c t r o n i c a n d magnetic structure properties of a molecule, f r o m w h i c h suitable

spectroscopic data c a n be derived for comparison

with

experimental

Môssbauer a n d N M R w o r k . C o m p a r i n g e x p e r i m e n t a l h y p e r f i n e d a t a w i t h 0065-2393/81/0194-0003$08.75/0 © 1981 American Chemical Society

Stevens and Shenoy; Mössbauer Spectroscopy and Its Chemical Applications Advances in Chemistry; American Chemical Society: Washington, DC, 1981.

4

MOSSBAUER SPECTROSCOPY A N D ITS C H E M I C A L APPLICATIONS

c o m p u t e d e l e c t r o n i c d a t a y i e l d s n u c l e a r p r o p e r t i e s s u c h as 8R/R c h a n g e of n u c l e a r c h a r g e r a d i u s ) a n d Q ( n u c l e a r q u a d r u p o l e

(relative moments).

B e y o n d this, t h e m u t u a l f e e d b a c k of t h e o r y a n d e x p e r i m e n t h e l p s the q u a n t u m c h e m i s t test t h e r e l i a b i l i t y of t h e v a r i o u s a p p r o x i m a t i o n s i n v o l v e d i n his c a l c u l a t i o n s a n d h e l p s t h e spectroscopist

i n t e r p r e t his

measured data. I n the p r e s e n t w o r k , self-consistent

field

( S C F ) a n d self-consistent

c h a r g e ( S C C ) M O c a l c u l a t i o n s are a p p l i e d to a series of h a l o g e n - c o n t a i n i n g compounds.

I n the s e c o n d s e c t i o n , some of

the

approximations

i n h e r e n t to these c a l c u l a t i o n s are d e s c r i b e d , a n d e x p e r i m e n t a l o r b i t a l energies

and

dipole

moments

are

compared

with

calculated

values

derived from S C C - X a - M O , S C C - I E H - M O (iterative extended H u c k e l ) , Downloaded by 80.82.77.83 on May 18, 2017 | http://pubs.acs.org Publication Date: July 1, 1981 | doi: 10.1021/ba-1981-0194.ch001

S C F - M O ( c l o s e to the H a r t r e e - F o c k l i m i t ) , a n d G a u s s i a n 7 6 - M O v e r sions. T h e f o l l o w i n g sections d e a l w i t h i s o m e r shift a n d e l e c t r o n c h a r g e d e n s i t y at t h e M o s s b a u e r n u c l e u s , w i t h e l e c t r i c field g r a d i e n t s , a n d w i t h t h e i n t e r p r e t a t i o n of m e a s u r e d m a g n e t i c h y p e r f i n e fields at the i r o n a n d iodine nuclei. Molecular

Orbital

Calculations

T h e one-electron e q u a t i o n t h a t m a y d e s c r i b e t h e e l e c t r o n i c s t r u c t u r e of a m a n y - p a r t i c l e system is ( i n a.u.)

(-A +V(r)) w h e r e V(r) (MO)

(1)

frft

is a l o c a l p s e u d o p o t e n t i a l , a n d ^fc("r) are m o l e c u l a r o r b i t a l s

*fcW =

Z vj

Rt)

c

(2)

v

jk

Rv) is a n a t o m i c w a v e f u n c t i o n c h a r a c t e r i z i n g t h e (n

J9

l

j9

m )-th ;

a t o m i c o r b i t a l of t h e v-th a t o m ; f o r t h e p r e s e n t w o r k , S l a t e r - t y p e o r b i t a l s (STO)

are

used.

Multiplication

of

Equation 1

from

the

left

by

4>i *{r — Rv>) a n d i n t e g r a t i o n o v e r t h e e l e c t r o n i c coordinates y i e l d s t h e iv,)

secular equation

E ^ - * ^ ) ^ -

(3)

0

vj w i t h t h e H a m i l t o n i a n m a t r i x elements H^

v

ments

a n d t h e o v e r l a p m a t r i x ele

s%

v

Htf

= j

(

t

-

i t ) [ — A + V (?) ]

(?-

R ) d*r v


(4)

1.

Halogen-Containing

GRODZICKI E T A L .

S$> = J

(r-

5

Compounds

IV)

(t-t )d'r

(5)

y j ^ f - B , )

(6)

v

W i t h t h e short notations A +

H£(t)

V (P) NB

=

V(?)

-

\ [V™

(?-

it) +

f„ (?-

(7)


t h e H a m i l t o n i a n m a t r i x elements c a n b e w r i t t e n as

= |

(«f + « / )

+ \ (V$> + y y )

sp

y ^ = J * ^ ( T - E o ^ W

(8)

- y £ M ^ - « » ) W

F

)

( r (9)

w h e r e c",- is t h e v a l e n c e o r b i t a l i o n i z a t i o n p o t e n t i a l of t h e v - t h a t o m o r i o n . F r o m E q u a t i o n 8 the e x t e n d e d H i i c k e l t y p e e q u a t i o n s are o b t a i n e d b y the approximations

Vy

=

+

l(VX'

;

V™)Sy

tr +

V f - k H F

(10)

yielding

T = |

W

+ h-)s^.

I n the I E H calculations, the Cusachs approximation ( I )

(ii) is a d a p t e d f o r

t h e p r o p o r t i o n a l i t y constant ft ft-2-

(12)

\Stf\

a n d the d i a g o n a l H a m i l t o n i a n m a t r i x elements are d e s c r i b e d i n terms of v a l e n c e o r b i t a l i o n i z a t i o n p o t e n t i a l s w h i c h d e p e n d o n t h e effective c h a r g e Q

v

of t h e v - t h a t o m

(2): %

H

=

~

(*jo + v

*n Q») v


(13)

6

MOSSBAUER SPECTROSCOPY A N D ITS C H E M I C A L

APPLICATIONS

So f a r w e h a v e s u c c e s s f u l l y a p p l i e d t h i s m e t h o d t o t h e i n t e r p r e t a t i o n o f electric a n d magnetic hyperfine parameters of F e - c o n t a i n i n g compounds 5 7

(3). I n a d d i t i o n t o the I E H m e t h o d , a m e t h o d w a s u s e d t h a t e x p l i c i t l y i n c l u d e s n e i g h b o r c o n t r i b u t i o n s t o the H a m i l t o n i a n .

Recently w e have

d e s c r i b e d t h i s m e t h o d i n d e t a i l ( 4 ) , a n d h a v e a p p l i e d i t t o a series o f small molecules i n c l u d i n g second-row

elements ( 5 ) . T h e m a i n f e a t u r e

of t h i s m e t h o d i s t h a t V(r) i s d e s c r i b e d i n terms o f a n X«-like m o d e l potential w h i c h depends

o n effective a t o m i c charges, t h u s m a k i n g t h e

application of an S C C iteration procedure possible.


F o r one a t o m this p o t e n t i a l is g i v e n b y

( r ) — r

+

&

r

[ 7 J

x 2 p a t

to

d

+ J

x

p

x

&t

to

dx

J

*[!^ ]

1/3

65« | ^ - P a t a ( tr(r )) I

A s s u m i n g t h a t t h e a t o m i c c h a r g e d e n s i t y p (r) &t

depends

(14)

exponentially

on r P . t ( r ) - ^ e - " , 07T

, - , ( Q )

(15)

w h e r e N is the n u m b e r o f electrons, o u r a t o m i c m o d e l p o t e n t i a l has t h e form

y

a t

(r)

^

- N

+

v

e ^

- a'rjN^e^*;

J = 1.5 «

3 y ^

/

3

(16)

V

a t

( r ) is a f u n c t i o n o f t h e effective a t o m i c c h a r g e ^ i a Q, N, a n d rj. W e r e p r e s e n t the m o l e c u l a r p o t e n t i a l V

m o

i ( r ) b y the superposition

of a t o m i c p o t e n t i a l s ( E q u a t i o n 16) FmoiW - E v . t ( | ^ - B « | )

(17)

GB

E q u a t i o n 17 i m p l i e s

GB

w h i c h overestimates t h e e x c h a n g e c o n t r i b u t i o n b e c a u s e V , m o i ( 0 s h o u l d e x

be replaced b y


1.

Halogen-Containing

GRODZICKI E T A L .

Fex.

m o l

(r) ~

(?) = ( £

Compounds

P

7

RJ ) * "

x

(19)

se

H o w e v e r , E q u a t i o n 17 has c o n s i d e r a b l e c o m p u t a t i o n a l advantages

over

E q u a t i o n 19. I t is w e l l u n d e r s t o o d that t h e X

a

e x c h a n g e p o t e n t i a l takes too l a r g e

values f o r l a r g e distances r f r o m the o r i g i n ( 6 , 7 ) ; therefore, w e m a y a c c o u n t for this effect a n d for the a p p r o x i m a t i o n i n h e r e n t i n E q u a t i o n 18 b y r e p l a c i n g ??/3 i n the exchange p a r t of E q u a t i o n 16 b y 77, w h i c h also has o b v i o u s c o m p u t a t i o n a l advantages. E v a l u a t i n g the p o t e n t i a l m a t r i x elements V* , w e e x p a n d the a t o m i c v

potentials V t(|r*—

I ) , c e n t e r e d at Jt^, i n t o s p h e r i c a l h a r m o n i c s , c e n

a


t e r e d at the o r i g i n o f the m o l e c u l a r f r a m e , V^(\t^Rj)-^r^V (r,B ) haLt

Y* (RJ

m

lm

(20)

Y {r) lm

lm

I n this p r o c e d u r e , t h e first t e r m i n E q u a t i o n 16, w h i c h is p r o p o r t i o n a l to 1 / r , takes t h e f o r m (—

l + 1

)

9

w h i l e the r e m a i n i n g

r a d i a l terms are e v a l u a t e d u s i n g t h e B a r n e t t - C o u l s o n f - e x p a n s i o n nique (8).

A n analogous e x p a n s i o n is a p p l i e d f o r t h e one-center

s i o n o f the a t o m i c w a v e f u n c t i o n s (4).

T h e n , the t w o - c e n t e r

can b e evaluated analytically i n closed form.

tech expan

integrals

T h e three-center i n t e g r a l s

are d e r i v e d f r o m

-

?Z

E se^W

t

U%"(r,RJ

(21)

W%»&J

1= 1

w h e r e the Z-summation m a y b e t e r m i n a t e d f o r n o n l i n e a r m o l e c u l e s a t I = 1 or 2, as has b e e n s h o w n p r e v i o u s l y ( 5 ) . T h e p a r a m e t e r s o f the S C C - X x]> = x

v

j0

a

m e t h o d are d e t e r m i n e d b y

+ x

n

Q% + x

v

j2

Q

(22)

2

v

w h e r e x m a y b e r e p l a c e d b y t h e i o n i z a t i o n p o t e n t i a l c, t h e Slater e x p o n e n t f, a n d t h e p o t e n t i a l p a r a m e t e r 77, r e s p e c t i v e l y . free p a r a m e t e r s , e

jtl

Vi,z> f;\o> fi,i>

a

n

and c

; > 2

are taken from

T o reduce the number of atomic data

( 9 ) , while

d £,,2 are d e t e r m i n e d f r o m r e x p e c t a t i o n v a l u e s ( 5 ) o f

atomic H a r t r e e - F o c k w a v e functions (10).

T h u s , the o n l y free p a r a m e t e r s

are 770 a n d c w h i c h h a v e b e e n d e t e r m i n e d b y fitting e x p e r i m e n t a l i o n i z a 0

t i o n p o t e n t i a l s a n d d i p o l e m o m e n t s of t h e h a l o g e n h y d r i d e s ( H F , H C 1 , H B r , and H I ) . T h e parameters for the I E H calculations have been determined i n a s i m i l a r f a s h i o n . T h e r e s u l t i n g p a r a m e t e r sets are l i s t e d i n T a b l e I .


8

MOSSBAUER SPECTROSCOPY A N D ITS C H E M I C A L APPLICATIONS

Table I.

Final Parameter Set Used for S C C - X and I E H Calculations

a

0


Orbital

to

it

2.5 (2.6) 2.3 (2.2)

0.1

2.15 (2.5) 1.89 (2.1)

0.1

2.45 (3.3) 1.95 (2.5)

0.1

2.50 (3.0) 1.98 (2.4)

0.1

1.57 (1.61) 1.46 (1.61)

0.1

1.2 (1.2)

0.4

to

F

2s

F

2p

CI

3s

40.80 (34.0) 16.84 (18.0)

25.57 (18.0) 20.54 (15.0)

3.54

27.20 (29.0) 13.19 (13.0)

15.78 (15.5) 13.19 (10.0)

1.63

24.75 (26.0) 11.83 (11.5)

14.14 (14.0) 11.69 (9.0)

1.36

19.72 (25.0) 10.74 (10.5)

12.24 (14.0) 10.06 (8.0)

1.09

14.96 (25.0) 9.52 (10.0)

17.54 (11.0) 14.69 (11.0)

3.54

10.34

3.54

1.63

Br

4s

Br

4p

I

5s

I

5p

C

2s

C

2p

H

Is

10.34 (13.6)

20.67 (12.8)

N N

2s 2p

(30.0) (11.5)

(12.0) (12.0)

(1.95) (1.95)

0 0

2s 2p

(33.0) (15.2)

(15.0) (15.0)

(2.27) (2.0)

s s

3s Zp

(22.0) (11.5)

(9.6) (9.6)

(1.82) (1.82)

Fe Fe Fe

3d 4s 4p

(7.0) (7.5) (6.5)

(8.0) (8.0) (8.0)

(2.87) (1.4) (1.4)

3.54

0.5

2.5

0.4

2.4

0.4

2.3

0.4

2.4

0.5

2.4

0.8

0.3

3p

1.09

2.7 0.3

CI

1.36

•no

0.3

0.3

0.3

"Energies in e V ; values in brackets correspond to I E H calculations, values with out brackets to S C C - X calculations. a

B e s i d e s t h e M u l l i k e n p o p u l a t i o n analysis, w e u s e t h e p o p u l a t i o n analysis w h i c h y i e l d s " d i p o l e - c o r r e c t e d p o p u l a t i o n n u m b e r s " n

v

= Y , H V'

p

^ ^ %

v

-

Z

%

v

)

ij

w i t h the b o n d order matrix


(

2

3

)

1.

Halogen-Containing

GRODZICKI E T A L .

£ " = Z >

P

k

9

Compounds

( (r) Q (r) ^

GVo(W)

( * > , « , ) dr

(44)


0

Z*

W

«/ o ^

(47)

{l';r,R )dr v

48

T h e three-

center i n t e g r a l s , E q u a t i o n 4 1 , c o u l d b e h a n d l e d i n a n analogous

manner.

H o w e v e r , i n v i e w o f the e x p e c t e d smallness of these c o n t r i b u t i o n s ( s e e T a b l e V I I I ) , w e use t h e f o l l o w i n g a p p r o x i m a t i o n :

Q iLM

=

Q 8i8 M V

Ov'v

Jt

w h e r e SjJ" i s t h e o v e r l a p m a t r i x a n d Q '? v

8

SJtM

t w e e n t h e t w o s f u n c t i o n s o f t h e ( / , v)-block

QZ.M

- -Z 4

( ^ 49

is the matrix element b e o f Q {? w h i c h i s V

M

E 7 ("') % (»'> u

w

v

d 2s -> d 3s - » d 4s - » d 5s - > d 2p^f 3p^f 4 p ^ / 5p->/ 3d-*g 3d-»s 4d->gr 4d - » s 2p-»p 3p - » p 4p -> p 5p - » p 3d-» d 4d-*d Total

HPRD

10*R

1.133 0.211 0.116 0.082 0.099 0.485 0.256 0.218 0.049 0.297 -0.049 0.270 0.151 -0.246 -1.761 -5.211 2.289 -0.460 -1.751

-0.616 -0.017 -0.005 -0.009 -0.111 -0.147 -0.049 -0.028 -0.024 -0.034 0.022 -0.026 -0.055 -2.441 -2.947 -6.539 1.716 0.012 0.893

B

0.517 0.194 0.111 0.073 -0.012 0.338 0.207 0.190 0.025 0.264 -0.027 0.244 0.096 -2.687 -4.708 -11.750 4.005 -0.448 -0.858 -14.2

-10.4

-3.8

• T h e various contributions to R due to different angular and radial excitations are included. Physical Review

w i t h p a r a m e t e r s a, b, c

l9

d

ly

r

u

c , d , r ( T a b l e VII) y i e l d i n g a n excellent 2

2

2

agreement w i t h the numerically computed y ( r ) . Comparison of Calculated V

zz

and Measured e qQ. 2

A l o n g t h e lines

d e s c r i b e d i n t h e t w o p r e v i o u s sections, t h e e l e c t r i c field g r a d i e n t V

zz

the

1 2 7

I,

7 9

at

B r , a n d C 1 n u c l e i for several molecules w a s calculated. T h e 3 5

v a r i o u s c o n t r i b u t i o n s to this g r a d i e n t at t h e

1 2 7

I n u c l e u s are g i v e n

t o g e t h e r w i t h t h e o c c u p a t i o n n u m b e r s of t h e 5s, 5p v a l e n c e o r b i t a l s Table V I . Sternheimer Shielding Factors for N e u t r a l and Negative Halides W i t h and Without the ns, np Valence Orbitals (41) Neutral Element F CI Br I

With Valence Orbitals -7.1 -25.4 -66.0 -136.0

Atom Without Valence Orbitals +0.08 —1.2 -6.2 -16.9

Negative With Valence Orbitals -22.3 -55.4 -133.0 -254.0

Ion Without Valence Orbitals +0.08 — 1.2 -6.1 -16.2

S. Lauer (Diplomarbeit)



1.

Halogeti-Containing

GRODZICKI E T A L .

n«, np , n , z

and n

Px

in Table VIII.

Py

21

Compounds

T h e g e n e r a l t r e n d reflected i n this

t a b l e is t h a t t h e v a l e n c e c o n t r i b u t i o n V ° ° ( E q u a t i o n 3 8 ) is t h e c o n t r i b u t i o n to the t o t a l e l e c t r i c t r i b u t i o n Vf

(Equation

z

contribution

39)

v

41)

is

completely

i m p o r t a n t feature is t h a t the l i g a n d c o n t r i b u t i o n V™ the core contribution V ™

r e

con

three-center

negligible.

Another

( E q u a t i o n 40)

and

( E q u a t i o n 33) nearly cancel each other i n a l l

cases. T h i s finally results i n a t o t a l V

zz

contribution

dominant

g r a d i e n t , w h i l e the o v e r l a p

is r e l a t i v e l y s m a l l a n d t h e

(Equation

V£

field

t h a t is n e a r l y e q u a l to the v a l e n c e

V™. zz

T h e calculated V

zz

values f o r

r u p o l e c o u p l i n g constants

1 2 7

I vs. the e x p e r i m e n t a l n u c l e a r q u a d

are p l o t t e d i n F i g u r e 5.

(e qQ) 2

the s o l i d l i n e c o r r e s p o n d s to t h e n u c l e a r q u a d r u p o l e

T h e s l o p e of

moment

Q(

1 2 7

I)

w h i c h takes t h e v a l u e —0.62 b a r n . T h i s v a l u e does not c o n t a i n r e l a t i v i s t i c corrections

of

< r " > w i t h i n the v a l e n c e c o n t r i b u t i o n V™. 3

priate correction factor i n < r " > 3

c o m p a r i n g H a r t r e e - F o c k (10)

r e

i =

The

R r e i < r ~ > has b e e n d e r i v e d 3

appro from

a n d r e l a t i v i s t i c H a r t r e e - F o c k - S l a t e r results


22

MOSSBAUER SPECTROSCOPY A N D ITS C H E M I C A L

Table V I I .

-8.7 0.08 -1.2 -6.1 -16.2

0.075 0.023 -0.12 -0.16 -0.18

(41)

a

b

4.25 17.0 7.1 6.0 5.0

1.30 0.2 0.7 0.9 1.2

R

Fe * F" cr Br" r

and R

Self-Consistent Sternheimer Factors

Element 2

APPLICATIONS

A l l values correspond to the situation that valence orbitals (Table I) are omitted from the self-consistent Sternheimer procedure. a

for

. )

The


3

d a s h e d l i n e i n F i g u r e 5 takes care of t h e r e l a t i v i s t i c c o r r e c t i o n , r e s u l t i n g in

Application of Molecular Orbital Calculations to MÃ¶ssbauer and NMR

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