Computation of Field-Swept EPR Spectra for Systems with Large

5 Computation of Field-Swept E P R Spectra for Systems with Large Interelectronic Interactions

Downloaded by UNIV OF PITTSBURGH on February 29, 2016 | http://pubs.acs.org Publication Date: June 1, 1974 | doi: 10.1021/bk-1974-0005.ch005

R. L .

BELFORD,

P.

H.

DAVIS, G.

G.

BELFORD,

and T. M .

LENHARDT

Center for A d v a n c e d C o m p u t a t i o n , University of Illinois, U r b a n a - C h a m p a i g n ,

Ill.

61801

Abstract. The c o m p u t a t i o n of field-swept epr spectra f r o m a physical model often is a particular p r o b l e m for m e t a l s y s t e m s having large interelectronic interactions, because o f t h e l a r g e field range c o v e r e d . T h i s contribution p o i n t s out t h a t theoret ical f o r m u l a t i o n s c a n be d e v i s e d specifically f o r this p u r p o s e and s k e t c h e s t h e development and u s e o f two s u c h methods - one e x a c t and one a p p r o x i m a t e . 1.

Introduction

As s e v e r a l o t h e r contributions to this volume a t t e s t , e l e c t r o n p a r a m a g n e t i c r e s o n a n c e i s among t h e most i m p o r t a n t t o o l s f o r s t u d y i n g e l e c t r o n i c i n t e r a c t i o n s between m e t a l c e n t e r s . Large z e r o - f i e l d s p l i t t i n g s , leading t o epr t r a n s i t i o n s over a wide range o f magnetic f i e l d i n t h e t y p i c a l f i x e d - f r e q u e n c y s p e c t r u m , c h a r a c t e r i z e many h i g h - s p i n m u l t i c e n t e r s y s t e m s . In t h e s e c a s e s , computation o f p r e d i c t e d epr s p e c t r a from t r i a l H a m i l t o n i a n p a r a m e t e r s i s c o m p l i c a t e d by t h e f a c t t h a t t h e H a m i l t o n i a n i s a f u n c t i o n o f t h e s p e c t r o s c o p i c sweep v a r i a b l e , the m a g n e t i c f i e l d . Suppose a l l t h e H a m i l t o n i a n p a r a m e t e r s which c h a r a c t e r i z e a system are given. Then t h e p r o b l e m i s t o f i n d a l l v a l u e s o f t h e m a g n e t i c f i e l d , x, s u c h t h a t two e i g e n v a l u e s o f t h e H a m i l t o n i a n d i f f e r by t h e s p e c t r o m e t e r e n e r g y , w. In t h e u s u a l c a s e o f l i n e a r Zeeman t e r m s , a s y s t e m o f η b a s i s s t a t e s may g i v e r i s e t o as many as n ( n - l ) / 2 d i f f e r e n t t r a n s i t i o n fields. A l t h o u g h t h e two e n e r g y e i g e n v e c t o r s a s s o c i a t e d w i t h e a c h t r a n s i t i o n f i e l d a r e o r t h o g o n a l , n e i t h e r need be o r t h o g o n a l t o any o f t h e o t h e r n -n-2 e i g e n v e c t o r s i n v o l v e d i n t r a n s i t i o n s at t h e o t h e r f i e l d s . V a r i o u s methods t o t r e a t t h i s p r o b l e m a r e commonly employed ( ] ) . One example i s c o n s t r u c t i o n o f a map o f e n e r g y l e v e l s as f u n c t i o n s o f f i e l d by e i g e n s y s t e m s o l u t i o n on a f i n e magnetic f i e l d g r i d (2). A n o t h e r i s t h e c o n v e n t i o n a l p e r t u r b a t i o n treatment i n which the magnetic f i e l d ( o r f i e l d s h i f t ) i s t h e p e r t u r b a t i o n p a r a m e t e r and t h e e n e r g y l e v e l s and t h u s f r e quency a r e d e v e l o p e d as power s e r i e s i n t h e f i e l d ( s h i f t ) ; t h e 2

40

In Extended Interactions between Metal Ions; Interrante, Leonard V.; ACS Symposium Series; American Chemical Society: Washington, DC, 1974.


5.

BELFORD E T A L .

Field-Swept

EPR

41

Spectra

f r e q u e n c y s e r i e s must be t e r m i n a t e d and t h e c o r r e c t r o o t o f t h e p o l y n o m i a l w( x ) - w e x t r a c t e d (.3,4). W h i l e a l l t h e s e methods c a n be made t o work, and some do v e r y w e l l i n some c a s e s , t h e y a r e g e n e r a l l y adapted w i t h minimal m o d i f i c a t i o n from t e c h n i q u e s ap p r o p r i a t e t o c a l c u l a t i o n o f energy l e v e l s f o r systems c h a r a c t e r i z e d by f i x e d Hami1 t o n i a n s . As s u c h , one m i g h t e x p e c t them t o be g e n e r a l l y l e s s e f f i c i e n t o r l e s s p r e c i s e and d i r e c t t h a n methods e s p e c i a l l y d e v e l o p e d t o h a n d l e t h e f i x e d - f r e q u e n c y f i e l d swept p r o b l e m . We have begun t o l o o k i n t o t h e p o s s i b i l i t y o f d e v i s i n g s u c h methods, and h e r e we o u t l i n e two o f them. The f i r s t , the e i g e n f i e l d formulation ( J j , displays a l l the t r a n s i t i o n f i e l d s a s t h e r e a l e i g e n v a l u e s , x, o f a g e n e r a l i z e d e i g e n v a l u e e q u a t i o n AZ=xBZ. S e c t i o n 2 d i s c u s s e s t h e e i g e n f i e l d t e c h n i q u e b r i e f l y ; i t has been d e s c r i b e d i n some d e t a i l r e c e n t l y (V) a s has i t s a p p l i c a t i o n t o time-dependent slow-motion o r e x c i t o n hopping p r o b l e m s ( 5 ) . The e i g e n f i e l d a p p r o a c h i s e x a c t and s t r a i g h t f o r w a r d b u t s l o w f o r p r o b l e m s w i t h many b a s i s s t a t e s . The s e c o n d method, d e v e l o p e d i n S e c t i o n 3, i s a p e r t u r b a t i o n f o r m u l a t i o n d e vised to predict transition fields. 2.

Eigenfield

Equations f o r Exact

Calculation

A r e c e n t p a p e r ( ] ) d e s c r i b e s a new ( g e n e r a l i z e d e i g e n s y s t e m ) f o r m u l a t i o n f o r e x a c t c a l c u l a t i o n o f r e s o n a n c e f i e l d s and i n t e n s i t i e s f o r a m o l e c u l e o r l a t t i c e s i t e d e s c r i b e d by a t i m e independent H a m i l t o n i a n which i s a polynomial f u n c t i o n o f f i e l d ; i t a l s o d e s c r i b e s some p r o p e r t i e s o f t h i s f o r m u l a t i o n , w h i c h we c a l l t h e e i g e n f i e l d e q u a t i o n . The e i g e n f i e l d c o n c e p t c a n a l s o be used t o f a c i l i t a t e s o l u t i o n o f r e l a x a t i o n master e q u a t i o n s , which p r o v i d e i n t e n s i t y as a f u n c t i o n o f f i e l d f o r m o l e c u l e s i n slow m o t i o n o r f o r e x c i t o n h o p p i n g among l a t t i c e s i t e s i n a c r y s t a l (5). H e r e , we r e s t r i c t o u r d i s c u s s i o n t o a s i m p l i f i e d d e v e l o p ment o f t h e s t a t i c e i g e n f i e l d e q u a t i o n f o r s y s t e m s w i t h l i n e a r Zeeman t e r m s a n d t o a p r a c t i c a l a p p l i c a t i o n t o a s y s t e m o f i n t e r a c t i n g t r a n s i t i o n metal ions. A s we s h a l l s e e , t h e e i g e n f i e l d approach i s very a t t r a c t i v e because i t i s a d i r e c t , s t r a i g h t f o r w a r d , e x a c t method, b u t a t p r e s e n t i t has t h e d r a w b a c k t h a t com p u t a t i o n t i m e i n c r e a s e s much t o o r a p i d l y w i t h t h e number o f b a s i s states. A c c o r d i n g l y , u n t i l b e t t e r numerical t e c h n i q u e s a r e de v e l o p e d f o r computer s o l u t i o n o f t h e e i g e n f i e l d e q u a t i o n s , t h e method i s o f p r a c t i c a l u s e o n l y f o r s y s t e m s h a v i n g j u s t a few (~10) b a s i s s t a t e s . I t does p r o v i d e r e s o n a n c e f i e l d s and t r a n s i t i o n v e c t o r s o f h i g h a c c u r a c y w h i c h we have employed a s t e s t s t a n d a r d s o f e x c e l l e n c e f o r a p p r o x i m a t e methods. 2.1 Development o f E i g e n f i e l d E q u a t i o n . F o r c o n v e n i e n c e , we u s e t h e m a t r i x r e p r e s e n t a t i o n t h r o u g h o u t . M a t r i x o p e r a t o r s , v e c t o r s , and s c a l a r s a r e d i s t i n g u i s h e d by c o n t e x t , n o t by s p e c i a l symbols. The H a m i l t o n i a n , H, f o r t h e p r o b l e m i s t h e sum o f two p a r t s — a z e r o - f i e l d t e r m , F, ( e x c h a n g e c o u p l i n g , d i p o l a r and


42

E X T E N D E D INTERACTIONS B E T W E E N

METAL

IONS

p s e u d o d i p o l a r c o u p l i n g , h y p e r f i n e c o u p l i n g , e t c . ) and a Zeeman t e r m , xG ( e l e c t r o n i c and n u c l e a r Zeeman e n e r g i e s ) . Here χ d e notes t h e magnitude o f t h e magnetic f i e l d . F o r example, t h e c o u p l i n g o f two Cr i o n s ( S=3/2) may be d e s c r i b e d bv a s j j i n H a m i l t o n i ^ n ^ H = F + x G i n w h i c h F=D( S , + S - V 2 ) + 3 , · ^ S 4 J S ,·Τ and G={3e.g.S, where e d e n o t e s a u n i t v e c t o r i n t h e d i r e c t i o n o f the magnetic f i e l d . Now f o r a f i x e d s p e c t r o m e t e r e n e r g y , w, r e s o n a n c e l i n e s w i l l o n l y o c c u r between two s t a t e s , u-»v, when t h e f o l l o w i n g two c o n d i tions are s a t i s f i e d together: 2

2


z

2 Z

ζ

2 Z

2

( F+xG-e)u=0 , Γ2-11 L

1

( F+xG-e-w)v=0

T h a t i s , i f e i s t h e e n e r g y o f t h e l o w e r s t a t e , e+w must be t h e energy o f t h e upper s t a t e . The two c o u p l e d e q u a t i o n s c a n be c o n v e r t e d i n t o a s i n g l e one w h i c h c o n t a i n s o n l y one e i g e n v a l u e , x, by means o f o u t e r - p r o d u c t a l g e b r a . The o u t e r p r o d u c t , A 0 B, o f a n mXn m a t r i x A and an m'Xn m a t r i x Β i s a n mm'Xnn m a t r i x : ( ®B) , . , =(A©B) . . . ^ B . The o u t e r 1

1

A

i

i

; k

k l

( n l i

( n l +

l i n l k

n l + k l

i k

i l k l

p r o d u c t o p e r a t i o n has t h e f o l l o w i n g u s e f u l p r o p e r t i e s : ( I ) i t i s a s s o c i a t i v e ; (2) Α 0 Β i s H e r m i t i a n i f and o n l y i f A a n d Β a r e b o t h H e r m i t i a n ; and (3) ( A 0 B)( C 0 D)=( AC) 0 ( BD) ( 6 ) . The two e q u a t i o n s [2-1] c a n be o u t e r - m u l t i p l i e d on t h e r i g h t and l e f t , r e s p e c t i v e l y , by I v and l u ( I d e n o t e s a u n i t m a t r i x ) , c o n v e r t e d by u s e o f p r o p e r t y (3) o f t h e o u t e r p r o d u c t , and s u b t r a c t e d t o y i e l d the eigenfield equation: [2-2]

(F0I-I0F+wI0I)(u0v)

= x( I 0 G-G 0

I ) ( u 0 v)

2.2 P r o p e r t i e s o f E i g e n f i e l d Equation. Here we summarize some o f t h e more i m p o r t a n t p r o p e r t i e s o f [2-2] and i t s s o l u t i o n s ; a more d e t a i l e d d e s c r i p t i o n i s found i n r e f . J_. E q u a t i o n [2-2] i s a g e n e r a l i z e d e i g e n v a l u e e q u a t i o n o f t h e f o r m AZ=xBZ, w h e r e t h e o p e r a t o r s A and Β a r e H e r m i t i a n m o t r i c e s o f o r d e r n ( f o r H o f o r d e r n) and t h e e i g e n v e c t o r s , Z, a r e t r a n s i t i o n v e c t o r s u 0 v of dimension n . A c c o r d i n g l y , t h e r e a r e n e i g e n f i e l d values. The f o r m o f Β i n s u r e s t h a t i t i s a t l e a s t η-fold s i n g u l a r ; t h e r e fore, η o f the eigenvalues are i n f i n i t e . They c o r r e s p o n d t o u n r e a l i z a b l e t r a n s i t i o n s f r o m an e n e r g y l e v e l t o i t s e l f . In gen e r a l , i f t h e e i g e n v a l u e s o f t h e Zeeman o p e r a t o r , G, a r e g r o u p e d i n t o d e g e n e r a t e s e t s , Β wi11 have s i n g u l a r i t y e q u a l t o t h e sum o f s q u a r e s o f t h o s e d e g e n e r a c i e s ( E m ) ; t h e r e f o r e t h e r e w i l l be (n -Em ) n o n i n f i n i t e e i g e n f i e l d s ( J J . Normally, these w i l l occur i n p a i r s o f equal magnitude but o p p o s i t e s i g n , r e f l e c t i n g the p h y s i c a l f a c t that r e v e r s a l o f t h e magnetic f i e l d d i r e c t i o n does n o t change t h e r e s o n a n c e s p e c t r u m ( c f . r e f . J_ c o n c e r n i n g t h e t i m e - r e v e r s a l symmetry o f F and G ) . Thus, t h e r e a r e 2

2

2

2

2

2


BELFORD E T

5.

2

2

2

2

Field-Swept

AL.

EPR

Spectra

43


(n -Em )/2 physically d i s t i n c t transitions. Moreover, A i s p o s i t i v e - d e f i n i t e ( i . e . , a l l o f i t s e i g e n v a l u e s a r e g r e a t e r t h a n 0) i f and o n l y i f t h e s p e c t r o m e t e r e n e r g y i s g r e a t e r t h a n t h e t o t a l span o f t h e z e r o - f i e l d energy l e v e l s . Only i n t h i s c a s e a r e a l l e i g e n f i e l d v a l u e s g u a r a n t e e d t o be r e a l . O b v i o u s l y , complex e i g e n f i e l d s o l u t i o n s , l i k e the i n f i n i t e ones, are p h y s i c a l l y s p u r i o u s ; t h u s t h e r e a c t u a l l y may be f e w e r t r a n s i t i o n s t h a n (n -Em )/2. I n t e n s i t i e s a r e r e a d i l y o b t a i n e d from e i g e n f i e l d t r a n s i t i o n vectors ( J j . 2.3 Example o f E i g e n f i e l d E q u a t i o n . The e i g e n f i e l d method y i e l d s a l l t r a n s i t i o n f i e l d s and i n t e n s i t i e s w i t h no s u b s t a n t i a l d i f f i c u l t i e s ; i t i s e f f e c t i v e l y a u t o m a t i c , and we have been u s i n g i t r o u t i n e l y f o r small problems ( e . g . , t r i p l e t s t a t e s , S=l; coupled doublets — i . e . , binuclear C u ( I I ) s i t e s , S|=S =l/2; q u a r t e t s , S=3/2) b o t h f o r g e n e r a t i o n o f f i x e d - o r i e n t a t i o n s p e c t r a and f o r s i m u l a t i o n o f powder s p e c t r a . S i n c e t h e w o r k and s t o r a g e requirements i n c r e a s e very r a p i d l y w i t h s i z e of b a s i s s e t , the l a r g e s t p r o b l e m w h i c h we have y e t h a n d l e d by d i r e c t s o l u t i o n o f the e i g e n f i e l d equations i s that of a coupled Cr " -Cr" " " p a i r , f o r w h i c h F and G a r e 16x16 m a t r i c e s and A and Β c o n s e q u e n t l y a r e 256x256 m a t r i c e s . That w i l l be t h e e x a m p l e d e s c r i b e d h e r e . We t a k e two i d e n t i c a l S=3/2 i o n s on t h e ζ a x i s w i t h i s o t r o p i c Zeeman t e r m s and a x i a l i n t e r - and i n t r a - i o n z e r o - f i e l d t e r m s . The p a i r b a s i s s e t c a n be c o n s t r u c t e d as t h e o u t e r p r o d u c t o f b a s i s s e t s f o r t h e two S=3/2 p a r t i c l e s , and t h e 16x16 p a i r H a m i l t o n i a n as t h e o u t e r p r o d u c t o f 4x4 s i n g l e - p a r t i c l e f a c t o r s . T h e r e f o r e , we can s e t up t h e e n t i r e p r o b l e m i n t e r m s o f 4x4 m a t r i c e s . Our e x ample i s as f o l l o w s : 2

+

f+

f

H

[2-3] F=D( S [2-4]

2

©

z

2

I+I ©

S - 2 . 51 0 z

G=g0( c o s 9 [ S

0

z

I+I ©

I)+d( 2 S

z

0

S »S © z

S -S ©

x

S j +s i n G [ S ©

x

y

I+I ©

x

S) y

)

The m a g n e t i c f i e l d i s i n t h e xz p l a n e a t an a n g l e θ t o t h e ζ axis. Here I i s t h e 4x4 u n i t m a t r i x , and S i s a diagonal z

m a t r i x w i t h v a l u e s (-3/2,-1/2,+1/2,+3/2), and follows:

[2-5]

^0 S

χ

=1/2

v

S

and

S are ^

X

0

/3

0

\ 0

0

"/3

0

2

0

/3

0

-2

0

/3

0

0

"/3

0

/3

0

0^

/3

0

2

0

2

0

0

0 /3

0y 0

P a r a m e t e r s f o r t h e t h r e e t e s t s a t 0=1.5°, 51

and

as

88.5° were


44

EXTENDED INTERACTIONS B E T W E E N

IONS

w/3=6802.6523 g a u s s , g=1.9759, η/β=1006.25 g a u s s , and d/$= -107.74 g a u s s . Now F and G a r e r e a d i l y ( a n d i n a computer program, q u i t e e a s i l y ) c o n s t r u c t e d , as a r e A and B. For example, t h e f i r s t t e r m o f F l e a d s t o a d i a g o n a l m a t r i x c o n t r i b u t i o n t o A, D ( S (χ) I ® I © I - I ® I ® S ® I ) , i t s diagonal elements being (D/4) (64 9's, 128 I ' s , 64 9 ' s j - (D/4) ( 1 6 r e p e a t s o f t h e p a t t e r n 4 9's, 8 l ' s , 4 9 ' s ) . N o t e h e r e t h a t G has 2 n o n d e g e n e r a t e e i g e n v a l u e s (:ί^β), 2 t w o f o l d (+2gβ), 2 t h r e e f o l d ( + 0 ) , and 1 f o u r f o l d ( 0 ) d e g e n e r a t e sets. Thus t h e r e must be e x a c t l y 2+8+18+16=44 i n f i n i t e f i e l d s . The computer program y i e l d e d 44 f i e l d s between 3 x l 0 and 3 x l 0 i n absolute value. I n no c a s e was t h e computed i n t e n s i t y f o r one o f these ' i n f i n i t e - f i e l d t r a n s i t i o n s as g r e a t as 1 0 " (strongest line=l). A l s o , s i n c e f o r o u r t e s t c a s e s D and d a r e s u f f i c i e n t l y s m a l l compared w i t h w (w>|9D/21+|9dl) so t h a t F i s p o s i t i v e d e f i n i t e , t h e r e s h o u l d be e x a c t l y ( 2 5 6 - 4 4 ) / 2 = l 0 6 r e a l p o s i t i v e e i g e n f i e l d s and 106 n e g a t i v e images. The computer r e s u l t s a l s o showed t h i s f e a t u r e , w i t h t h e p l u s - m i n u s p a i r s g e n e r a l l y a g r e e i n g t o 14 d i g i t s . At 0=51° t h e 106 e i g e n f i e l d s ranged f r o m 404 t o 3716 g a u s s . A u s e f u l f e a t u r e o f t h e e i g e n f i e l d program i s t h a t i t p r o d u c e s f i e l d v a l u e s and i n t e n s i t i e s f o r a l l l i n e s a t once. Here, i t i s i n t e r e s t i n g t o see t h a t s e v e r a l l i n e s a r e p r e d i c t e d i n t h e 400-900 g a u s s r e g i o n w i t h i n t e n s i t i e s ~. 01 t o .03$ o f t h o s e o f t h e most s t r o n g l y a l l o w e d l i n e s and ~1-3$ o f t h o s e o f t h e w e a k e r l i n e s i n t h e 1000 g a u s s r e g i o n . These a r e e s s e n t i a l l y d o u b l e s p i n - f l i p t r a n s i t i o n s i n v o l v i n g e x c i t a t i o n o f one i o n and dee x c i t a t i o n o f i t s p a r t n e r ; u n d e r some c i r c u m s t a n c e s t h e y c o u l d be o b s e r v e d e x p e r i m e n t a l l y and w o u l d be s e n s i t i v e t o t h e i n t e r m o l e c ular coupling. Most a p p r o x i m a t i o n methods w o u l d p r o v i d e o n l y t h e ' n o r m a l l i n e s , w i t h t h e r e s u l t t h a t e x p e r i m e n t a l i s t s might i g n o r e t h e abnormal ones a l t o g e t h e r . The computer program u s e d t o s o l v e t h i s g e n e r a l i z e d e i g e n v a l u e p r o b l e m was w r i t t e n by A. Sameh and C. Chang o f t h e U n i v e r s i t y o f I l l i n o i s ' Center f o r Advanced Computation. Their method ( SQ,Z") i s a v e r s i o n o f t h e Q.Z a l g o r i t h m o f M o l e r and S t e w a r t ( 7 ) , w h i c h we have u s e d ( J j f o r s m a l l e r p r o b l e m s . The SQZ a l g o r i t h m ( 8 ) r e q u i r e s t h a t A and Β b o t h be r e a l , s y m m e t r i c m a t r i c e s ( w i t h one o f them p o s i t i v e d e f i n i t e ) and t h e n t a k e s a d v a n t a g e o f symmetry t o g e n e r a t e s o l u t i o n s more r a p i d l y ( a n d o f t e n more a c c u r a t e l y ) t h a n Q.I. Even t h e n , t h e example d i s c u s s e d h e r e r e q u i r e d a b o u t t e n m i n u t e s o f p r o c e s s i n g t i m e on an IBM 360/91 c o m p u t e r , w h e r e a s a s y s t e m o f two c o u p l e d i o n s o f S=l/2 e a c h would r e q u i r e o n l y a small f r a c t i o n o f a second. The s p a c e r e q u i r e m e n t s w e r e a l s o f o r m i d a b l e , t h e example g i v e n r e q u i r i n g more t h a n one m i l l i o n b y t e s . The r e a s o n f o r t h e t i m e and s p a c e d i f f i c u l t i e s i s t h a t the c u r r e n t s t a t e o f numerical a n a l y s i s o f gen e r a l i z e d e i g e n v a l u e p r o b l e m s i s as y e t q u i t e p r i m i t i v e ; n e i t h e r Q.Z nor SQZ t a k e s any a d v a n t a g e o f t h e s p a r s e n e s s o r s p e c i a l f o r m 2

2

Z

2

1 7


METAL

1

2 8

1

M


, 9

5.

Field-Swept

BELFORD E T A L .

EPR

45

Spectra

o f A a n d B. A d v e n t o f a n u m e r i c a l method w h i c h does s o w i l l make t h e e i g e n f i e l d f o r m u l a t i o n w i d e l y p r a c t i c a l f o r r e a s o n a b l y l a r g e systems. A measure o f t h e g o o d n e s s o f t h e e i g e n s y s t e m i s t h e r e s i d u al a s s o c i a t e d w i t h e a c h e i g e n f i e l d x^. The p r o g r a m c a l c u max.[(A-x B)Z ] I a t e d f o r e a c h f i e l d a r e s i d u a l R. — T T T T T i 1 — u n i t , where k


k

IMI

+

k

B

K l ll ll

I I A| J ^max^Spl A^p|, and t h e n o r m a l i z a t i o n o f i s chosen so that 02, we have 0

η

(η

,,

(Η -μ)ϋ< >-ΚΗ'-μ')υ - + Σ ( H j

[3-8] (Η°-μ-*#°)ν

(η,

ίη

=

4

(j )

(n

J > )u "

j

U

2

1,

-ΚΗ'-μ'-β«)ν " + Σ (H J =2

( j )

-e(j))v

( n

"

j )

=0

Each o r d e r i s r e p r e s e n t e d by a c o u p l e d p a i r o f e q u a t i o n s . Taking t h e inner product o f t h e f i r s t equation i n each p a i r w i t h u° and t h e s e c o n d w i t h v°, s u b t r a c t i n g , and r e a r r a n g i n g w i t h t h e aid o f [3-1], provide equations f o r the successive c o r r e c t i o n s t o the f i e l d . [3-9]

J

n

For convenience,

n

we l e t (u^ ^,u°)=( v^ ^,v°) = 6 no 1

x'=[&y-KF' -F' )+x°(G' -G )]/(G° -G° ) ^ uu vv' uu vv' vv uu' v

v

[3-10] ,

l

0

0

G

„ - G v„ v. ^ ' uu „ r ' vvvv ) ,,_ F' u'u -F'v ' v ^ u ' u v- 'G v„ ' ) + x ( G u'u < (G -G° ) vv uu' \/\/ llll' l v

l l |

l

u

x

etc.

,

H e r e we u s e t h e c o n d e n s e d n o t a t i o n (u ,Gu°)=G

, , etc. u u The c o m p l e t e p r e s c r i p t i o n f o r a p e r t u r b a t i o n s o l u t i o n i s now clear. One f i r s t c a l c u l a t e s ( u °, v°, M-,w°, bu) by e x a c t d i a g o n a l i z a t i o n o f H°. Then one f i n d s x by [ 3 - 9 ] , μ' f r o m one o f t h e e q u a t i o n s w h i c h i s a p r e c u r s o r t o [ 3 - 9 ] , and u and ν' f r o m [3-6], The p r o c e s s c a n c o n t i n u e t o any o r d e r . I t i s a l s o pos s i b l e , as i n t h e s t a n d a r d p e r t u r b a t i o n developments, t o d e r i v e 1

1



5.

BELFORD E T A L .

Field-Swept

EPR

47

Spectra

t h e f i r s t 2n+l c o r r e c t i o n s t o t h e e i g e n f i e l d f r o m t h e f i r s t η corrections t o the eigenvectors. A good s t r a t e g y o f c a l c u l a t i o n i s t o w o r k i n t h e b a s i s i n w h i c h H° i s d i a g o n a l ; t h e n many o f t h e r e l a t i o n s h i p s t a k e a s i m p l e form. I n o u r f i r s t programs u s i n g f r e q u e n c y - s h i f t p e r t u r b a t i o n s , we s t a r t by a p p l y i n g t o a l l m a t r i c e s i n t h e p r o b l e m t h e s i m i l a r i t y t r a n s f o r m a t i o n w h i c h d i a g o n a l i z e s H°. From t h i s p o i n t u n t i l t h e end (where e i g e n v e c t o r s a r e transformed back i n t o t h e o r i g i n a l b a s i s f o r o u t p u t ) we w o r k i n t h i s b a s i s a n d f i n d t h a t t h e programs t h e r e b y c a n be made v e r y e c o n o m i c a l . We have programmed t h e method up t o f o u r t h o r d e r i n v a l u e s a n d t h i r d order i n vectors. 3.3 Example o f F r e q u e n c y - S h i f t P e r t u r b a t i o n a n d C o m p a r i son w i t h E i g e n f i e l d R e s u l t s . We have u s e d a ^ r e q u e n c y - s h i f t p e r t u r b a t i o n program t o a n a l y z e t h e C r -Cr pair spectra de s c r i b e d i n t h e n e x t p a p e r i n t h i s volume ( 9 ) . The e i g e n f i e l d e x a m p l e s d e s c r i b e d i n S e c t i o n 2.3 p r o v i d e d s t a n d a r d s o f c o m p a r i son f o r t h e f r e q u e n c y - s h i f t p e r t u r b a t i o n program. For t h e s e s t u d i e s , o n l y t h e f r e q u e n c y s h i f t was t a k e n a s t h e p e r t u r b a tion — i.e., F'^^O. No a c t u a l d e g e n e r a c i e s o f H° c a u s e d trouble. The d i a g o n a l i z a t i o n s o f H° u s e d t h e w i d e l y a v a i l a b l e EISPAC m a t r i x e i g e n v a l u e package. G e n e r a l l y , t h e p e r t u r b a t i o n program, up t h r o u g h f o u r t h - o r d e r f i e l d c o r r e c t i o n s , r e q u i r e s o n l y a b o u t 1$ o f t h e p r o c e s s i n g t i m e expended by t h e e i g e n f i e l d program, and i t c a n be made more e f f i c i e n t . It also requires o n l y a f r a c t i o n (~10#) o f t h e s t o r a g e s p a c e u s e d by t h e e i g e n f i e l d program. For t h e most p a r t , agreement between t h e e x a c t ( e i g e n f i e l d ) and t h e p e r t u r b a t i o n f i e l d s was good t o 6, 7, o r 8 s i g n i f i c a n t d i g i t s ( i . e . , t h e d i f f e r e n c e s were ~ 1 0 ~ gauss o r l e s s ) . A t y p i c a l r e s u l t i s g i v e n b e l o w ; θ=51°: |

+ + +

3

M,

,m

x°=1022.94, χ'=59.5029, x"=-0. 1604, x = 0 . 0 0 6 6 , x = - 0 . 0 0 0 3 x=l082.288879. (Compare e x a c t = l 0 8 2 . 2 8 8 8 9 1 ) Intensity: 0.1013 (Compare exact=0.1013) F i n a l v e c t o r s ( a l l i n agreement w i t h t h e e x a c t the four places p r i n t e d ) :

result t o

u=(.0720, .0672, .1350, .2371, . 0672,-.0938,-.1747,-.4974, . 1350, -.1747, . 1826, . 3233, . 2371,-. 4974, . 3233,-. 1728) v=(.0026, . 0 0 8 6 , . 0 2 5 6 , . 0 6 1 2 , . 0 0 8 6 , .0259, .0707, .1571, .0256, .0707, .1812, .3827, .0612, .1571, .3827, .7780) I n a l m o s t a l l c a s e s , t h e r e s u l t s c o u l d be i m p r o v e d by a very simple, r a p i d e x t r a p o l a t i o n o f t h e t h i r d and f o u r t h order f i e l d c o r r e c t i o n s toward i n f i n i t e order. The i d e a b e h i n d t h e e x t r a p o l a t i o n i s t h a t t h e s u c c e s s i v e p e r t u r b a t i o n s may a p p r o x i mate a g e o m e t r i c s e q u e n c e , i . e . , x ^ ^ = r x ^ where r i s n e a r l y J

+

American Chemical Society Library 1155 13th St.Metal N. Ions; W. Interrante, Leonard V.; In Extended Interactions between ACS Symposium Series; American Society: Washington, DC, 1974. Washington, D.Chemical C. 20036

48

EXTENDED INTERACTIONS B E T W E E N

constant f o r j>3.

By t a k i n g x 5

6

t u r b a t i o n s e r i e s x^ ^+x^ ) + . . . examples

,,,,

/x

111

t o be t h i s

sums t o ( x

l l l l

2

ratio,

METAL

IONS

the per

,

) /(x" -x"").

Some

follow:

Σ ο χ ^ ) =1022.94+173.0747-2.0890+0.7314-0.2216=1194.4355 x( e x t r a p o l a ted) = 1194.4870 x( e x a c t ) = 1 1 9 4 . 4 8 5 7 . Σ

4 ( J) Χ

.94+97.7430-0.6986+0.0915-0.0157=1120.0602 x(extrapolated)=l120.0625 x(exact)=1120.0626.

=1022

EQX^ ^=1022.940+407.033+3.635-15.536+10.061=1428. 133 x(extrapolated)=l424.178 x(exact)=1425.149.


J

The l a s t c a s e i n t h e p r e v i o u s p a r a g r a p h i l l u s t r a t e s a s l o w c o n v e r g e n c e p r o b l e m w h i c h o c c u r r e d o c c a s i o n a l l y and which was i n a few c a s e s m a n i f e s t e d by d i v e r g e n c e o f t h e p e r t u r b a t i o n s e r i e s . These c a s e s o c c u r r e d when t h e c h a r a c t e r o f one o r b o t h o f t h e s t a t e v e c t o r s i n v o l v e d i n a t r a n s i t i o n was c h a n g i n g v e r y r a p i d l y w i t h f i e l d , s o t h a t t h e range o f c o n v e r g e n c e o f the p e r t u r b a t i o n method was p a t h o l o g i c a l l y s m a l l — i . e . , o n l y a s m a l l 6w c o u l d be t o l e r a t e d . T h i s t e n d s t o happen c l o s e t o t h e c r o s s i n g p o i n t o f two i n t e r a c t i n g s t a t e s . We s u f f e r no s e v e r e d i f f i c u l t i e s f r o m t h i s p r o b l e m , b e c a u s e i t i s e a s y t o r e c o g n i z e such c a s e s and t o c o r r e c t them by t r y i n g a new x°. However, f o r a p r a c t i c a l s e l f - c o n t a i n e d program, one would need t o d e v i s e an a l g o r i t h m f o r a u t o m a t i c r e c o g n i t i o n and a d j u s t m e n t . Incidentally, in the p a t h o l o g i c a l c a s e s t h e v e c t o r s and i n t e n s i t i e s a r e f a r w o r s e than the f i e l d s . As e x p e c t e d , h e r e t h e a p p r o x i m a t e method, though f a s t e r , i s much l e s s f o o l p r o o f t h a n t h e e x a c t ( e i g e n f i e l d ) method. In some c a s e s , t h e same t r a n s i t i o n was computed s t a r t i n g from two o r t h r e e r a t h e r d i s t a n t v a l u e s o f x°. I n most c a s e s , the agreement was r e a s o n a b l e . Of c o u r s e , l a r g e f i e l d s h i f t s from x° produced l e s s a c c u r a t e r e s u l t s t h a n s m a l l s h i f t s , the p a t h o l o g i c a l c a s e s m e n t i o n e d above e x c e p t e d . Some c o m p a r i s o n s f o l l o w : F o r θ=88.5°, x ( e x a c t ) = 2 8 5 8 . 5 4 3 0 , întensity=l.0007. T h r e e p e r t u r b a t i o n t r i a l s f r o m d i f f e r e n t s t a r t i n g f i e l d s produced J

Σοχ^ * ^ =2200+655. 8418+3. 7346- 1.3842+0. 4590=2858. 6512 x ( e x t r a p o l a ted)=2858.5269 ; I n t e n s i t y = 0 . 9 9 8 5 J

Σοχ^ * ) =2900-41.4622+0.0051+0.0001+0.0000=2858. 5430 x( e x t r a p o l a ted)=2858.5430 ; I n t e n s i t y = l . 0 0 0 0 J

Σοχ^ * ^ =3600-742.3981+0.6875+0. 1904+0.0483=2858. 5281 x(extrapolated)=2858.5445; Intensity=0.9990 A l l t h e t h i r d - o r d e r v e c t o r s a g r e e d t o w i t h i n a few p a r t s i n t h e f o u r t h d e c i m a l p l a c e f o r e a c h v e c t o r component; i n e a c h c a s e t h e t r a n s i t i o n a r o s e f r o m the 5th-»10th l e v e l s o f t h e s t a r t i n g Hami1tonian.


5.

BELFORD

Field-Swept

E T AL.

EPR

49

Spectra

3.4 D e g e n e r a t e S t a t e s . Suppose t h a t a f i e l d p o s i t i o n must be c o m p u t e d f r o m a n i n i t i a l l y R e g e n e r a t e s e t o f t r a n s i t i o n s . C o n s i d e r { U|,..., u }-*[ V|,... ,v }, where t h e s e t { u°) i s a p - f o l d Q

0

degenerate

s e t o f e i g e n s t a t e s o f H° and { v } a q - f o l d s e t . 0

[3-11]

H u°=yj,u!; Η^°=(μτΗ*°)ν]

T h e r e a r e s e v e r a l ways t o p r o c e e d . Normally, a degenerate per t u r b a t i o n t r e a t m e n t w o u l d c a l l f o r d i a g o n a l i z a t i o n o f H' w i t h i n d e g e n e r a t e b l o c k s o f H°. However, t h a t i s n o t a l w a y s p o s s i b l e > a p r i o r i , because H i s F'+x°G'+χ'ΰ , i n w h i c h x i s not y e t known. The d e s i r e d s e p a r a t i o n c a n be o b t a i n e d by d i a g o n a l i z i n g b o t h t h e f u } and t h e { v } s u b - b l o c k s o f F + x G ' and t h e n diagonalîzing G w i t h i n a n y r e m a i n i n g d e g e n e r a t e b l o c k s . I f any d e g e n e r a c i e s p e r s i s t , t h e y must be a t t e n d e d t o i n h i g h e r o r d e r . An a l t e r n a t i v e f o r d e g e n e r a t e z e r o * t h - o r d e r s t a t e s l e a d s t o a s m a l l e i g e n f i e l d problem. B e g i n n i n g w i t h [ 3 - 1 1 ] , one g e t s t h e f o l l o w i n g p a i r o f f i r s t - o r d e r equations analogous t o those f o r the nondegenerate case.


n e r e

1

0

0

1

0

,

,

,

0

l

E.b.(H -u, )u?+(H°-M )u =0 i

[3-12] (H -M, -6W)V .+(H -^-W°)V =0

Y_.c.

,

i

Taking the inner product left,

i

0

j

and t h e second

o f the f i r s t

e q u a t i o n w i t h u° on t h e

by v£, and d e n o t i n g t h e s u b m a t r i c e s

0

0

w i t h i n t h e { u } and { v } b l o c k s a s h

U

and h

V

of H

1

and t h e v e c t o r s o f

c o e f f i c i e n t s b. and c. a s b and c r e s p e c t i v e l y , we g e t t h e f o l lowing matrix

equations. u

[3-13] (Here

v

(h%"I )b=0; I

U

V

( h -(μ,'+ôw) I ) c=0

d e n o t e s t h e u n i t m a t r i x h a v i n g t h e same o r d e r a s h V

U

—

U

i . e . , p.) A p p l y i n g 0 I c t o t h e f i r s t and I b © t o t h e second o f t h e s e e q u a t i o n s and s u b t r a c t i n g y i e l d s [3-14] . (h 0 U

[3-14]

V

I -I

U

V

©

U

h +6wI ©

U

V

I ) (b © U

c) =0 U

The f o r m o f h i s now f ^ x ' g " , where f and g a r e s u b m a t r i c e s w i t h i n t h e { u } b l o c k o f F + x ° G and G ° , r e s p e c t i v e l y . Thus [3-14] c a n be r e a r r a n g e d t o [ 3 - 1 5 ] . 0

,

l

[3-15] U

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T h i s i s a s m a l l e i g e n f i e l d e q u a t i o n based on t h e t r a n s i t i o n s from the { u } t o f v } b l o c k s ; t h e e i g e n f i e l d s a r e t h e f i r s t - o r d e r 0

0


50

EXTENDED

INTERACTIONS

BETWEEN

METAL

IONS

t r a n s i t i o n f i e l d s f o r t h e s e t r a n s i t i o n s , and t h e e i g e n v e c t o r s are the c o r r e c t z e r o ' t h order t r a n s i t i o n v e c t o r s . Acknowledqements. T h i s r e s e a r c h was s u p p o r t e d by t h e P e t r o l e u m R e s e a r c h t h e N a t i o n a l S c i e n c e F o u n d a t i o n , and t h e Advanced R e s e a r c h j e c t s Agency.


Literature 1. 2. 3. 4. 5. 6.

7. 8. 9.

Fund, Pro

Cited.

B e l f o r d , G. G., Belford, R. L., and Burkhalter, J. F., J . Magn. Resonance (1973) 11, 251. Hempel, J . C., Morgan, L. O., and L e w i s , W. B., I n o r g . Chem. ( 1 9 7 0 ) 9, 2064. B y f l e e t , C. R., Chong, D. P., Hebden, J . Α., and M c D o w e l l , C. Α., J . Magn. Resonance ( 1 9 7 0 ) 2, 69. Abragam, A. and B l e a n e y , Β., "Electron P a r a m a g n e t i c Reso nance o f Transition I o n s , " C l a r e n d o n P r e s s , O x f o r d , 1970. B e l f o r d , R. L. and Belford, G. G., J . Chem. Phys. ( 1 9 7 3 ) 59, 853. K o r n , G. A. and K o r n , Τ. Μ., "Mathematical Handbook f o r Scientists and Engineers," § 13.2-10, M c G r a w - H i l l , New Y o r k , 1961. M o l e r , C. B. and S t e w a r t , G. W., SIAM J. Numer. Anal. (1973) 1O, 241. Chang, C.-C., M.S. T h e s i s i n Computer S c i e n c e , U n i v . o f Illinois a t Urbana-Champaign, 1974. D a v i s , P. H. and Belford, R. L., this volume.


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