J. Tennyson and J.N. Murrell- A study of the variational convergence of the F centre energy levels of LiF

8/3/2019 J. Tennyson and J.N. Murrell- A study of the variational convergence of the F centre energy levels of LiF

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MOLECULAR PHYSICS, 1981, VOL. 42, No . 2,297-305

A s t u d y o f t h e v a r i a t i o n a l c o n v e r g e n c e o f t h e F c e n t r ee n e r g y l e v e l s o f L i F

by J . TEN NY SO N and J . N. MURRE LLSchool of Molecular Sciences, University of Sussex,

Brighton BN1 9QJ, England(Received 30 June 1980; accepted 2 September 1980)

SCFMO calculations have been made on the F centre states of LiF using acrystal fragment whose potential closely approximates that of the full crystal.The effect of including electrons of up to third neighbours has been studied.The most extensive calculation gives a binding energy for the 2S state in theunrelaxed lattice of 5"25 eV and a 2S~ 2p transition energy of 4"6 eV. Latticerelaxation has been calculated in a model appropriate to all lithium halides inwhich only nearest neighbour electrons are considered. For LiF an outwarddisplacement of nearest neighbour ions by 2"5 per cent of the lattice parameterhas been calculated. For the other halides the predicted displacements aresmaller.

1. INTRODUCTIONThe optical properties of F centres in alkali halides have been the subject of muc h

theoretical work. Early model based calculations were made by I nui and Ue mur a [1],Kojim a [2], Gou rar y and Adrian [3], and Woo d and Joy [4]. Variational calculationshave since been performed by Martino [5], Bartram et al. [6] and Wood and Opik[7, 8]. However, none of these are full ab initio as they all employ model or pseudopotentials for the ions surrounding the vacancy. The agreement with experimentaldata that is reached in these calculations may be due to the excellence of thewavefunc tions or to a fortunate choice of potential. I n fact by the presen t standardsof molecular calculations the variational wave functions were a l l quite modest.

A more recent calculation by Chaney and Lin [9] used a LCAO variationalscheme with a large basis. However, no basis functions were specifically optimizedfor the defect. Mor e seriously the Slater exchange approxi mation was used and onlya single iteration in a self-consistent-field cycle was performed. As this singleiteration chang es the bin din g ene rgy of the gro und state by - 0'089 E h (2"4 eV) it isimpossible to estimate accurately the converged SCF energies.

Most studies of the F centr e have assumed that the ions surro undi ng the vacancyare at the sites of the per fec t lattice. However, it is clear that the optical tr ansiti onenergies are sensitive to the geometrical parame ters of the lattice; the large differencebetween the F-centre absorption and emission energies is explained by latticerelaxation [10]. Kojima [2] in an early calculation deduced that in the F-centregro und state of LiF the nearest neighbo ur cations were displaced inwards by 7'4 percent of the lattice paramete r. Wood and Korr inga [11] calculated an outwar ddisplacement of 1 per cent for LiC1 and Bartram et al. [6] calculated an inwarddisplace ment for most alkali halides but Li F was an exception with a small outwarddisplacement of approximately 1 per cent.

0026-8976/81/4202 0297 $02"009 1981 Taylor & Francis Lt d


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2 98 J . T e n n y s o n a n d J . N . M u r r e l lI n t h i s p a p e r w e u s e a s t a n d a r d m o l e c u l a r ab ini t io S C F M O p r o g r a m t o c a lc u la t e

t h e e n e r g y l e v e ls o f t h e F c e n t r e o f L i F a n d t o d e t e r m i n e t h e g r o u n d s t a t e n e a r e s t -n e i g h b o u r r e l a x a t io n . S u c h p r o g r a m s a r e g e n e r a l ly e s t a b l i s h e d w i t h a m a x i m u m s iz et o t h e b as i s (N ) a n d t o t h e n u m b e r o f c h a r g e c en t r e s (C ) . T h e p r o g r a m w e u s ed ,A T M O L 3 [ 12 ], w a s o r i g i n a l l y e s t a b l i s h e d w i t h N = 1 2 7 , C = 5 0 , b u t d u r i n g t h ec o u r s e o f t h e p r e s e n t w o r k t h e s e w e r e i n c r ea s e d t o N = 2 5 4 a n d C = 1 0 0 . ( Ap r e l i m i n a r y r e p o r t o n t h e e a r l y c a lc u l a t i o n s w a s g i v e n i n [ 1 2 a] . ) A l t h o u g h t h e sen u m b e r s m i g h t a p p e a r l a r g e t h e r e s t r i c t i o n o n N w o u l d p r e v e n t th e a c h i e v e m e n t o fH a r t r e e - F o c k a c c u r a c y f o r c l u s te r s o f h e a v y i on s . C o m p u t a t i o n a l t i m e i n c r e a s e s v e r yr a p i d l y w i t h b a s i s s iz e a n d i t w o u l d b e d i f f ic u l t t o j u s t i f y c a l c u l a t i o n s w i t h m o r e t h a na b o u t 1 50 f u n c t i o n s . F o r t h i s r e a so n o u r c a l c u l a t i o n s h a v e b e e n m a d e o n L i F i n o r d e rt o e x p l o r e th e v a r i a t i o n a l l i m i t m o s t f u l ly , a l th o u g h s o m e d e d u c t i o n s a b o u t o t h e rl i t h i u m h a l i d e s c a n b e m a d e i n a f i r s t n e i g h b o u r a p p r o x i m a t i o n . C o m p u t a t i o n a l t i m ei s r e l a t i v e l y i n s e n s i t i v e to t h e n u m b e r o f c e n t r e s b u t a l t h o u g h t h e r e s t r i c t i o n o fC = 10 0 a l l o w s u s t o r e p r e s e n t u p t o e i g h t h n e i g h b o u r s i n th e L i F l a t ti c e a s p o i n tc h a r g e s t h i s g i v e s a c l u s t e r w h o s e r a d i u s i s o n l y t h r e e l a t t ic e s p a c i n g s .

W e c a n j u s t i f y th e v a l i d i t y o f a c l u s t e r m o d e l f o r th e g r o u n d s t a te a n d f i rs t e x c i t e ds t a te o f t h e F c e n t r e v a c a n c y e l e c t r o n b y t h e f a c t t h a t t h e s e e n e r g i e s a r e w e l l a b o v e t h eo c c u p i e d b a n d e n e r g ie s o f t h e p e r f e c t c r y s t a l a n d b e l o w t h e c o n d u c t i o n b a n d .H o w e v e r , i t i s c l e a r t h a t t h e s e e n e r g i e s w i l l b e s e n s i t i v e t o th e C o u l o m b p o t e n t i a l o ft h e c r y s t a l a n d h e n c e t h e c l u s t e r m u s t r e p r o d u c e t h i s a c c u r a t e l y . T h e f i rs t s e c t i o n o ft h e p a p e r i s c o n c e r n e d w i t h t h i s p r o b l e m .

2 . T H E C R Y ST A L P O T E NT I ALT a b l e 1 s p e c if i es t h e s t r u c t u r e o f th e c r y s t a l lo c a l t o t h e v a c a n c y . T h e i t h s h e ll h a s

n i e q u i v a l e n t i o n s a t d i s t a n c e a D i f r o m t h e c e n t r e w h e r e a is t h e l a t t ic e p a r a m e t e r .O u r i n i t ia l s t u d i es w e r e m a d e w i t h th e p r o g r a m h a v i n g C = 5 0 a n d w e r e t h e r e f o r e

r e s t r i c t e d t o o n l y f o u r s h e l l s o f i o n s . D u e t o t h e v e r y s l o w c o n v e r g e n c e o f t h eM a d e l u n g e x p a n s i o n o f t h e p o t e n t i a l t h e C o u l o m b p o t e n t i a l in th e v a c a n c y i s v e r yp o o r l y r e p r e s e n t e d b y n e t c h a r g e s o f _ 1 o n t h e l a t t i c e s i t e s . W e c a n a c h i e v e a c o r r e c tp o t e n t i a l a t th e v a c a n c y c e n t r e b y t a k i n g a n e f f ec t i v e c h a r g e o n t h e f o u r t h n e i g h b o u r so f - 0 " 1 2 8 6 ( w h i c h i s i n d e p e n d e n t o f l a tt i c e p a r a m e t e r ) . H o w e v e r , t h i s p o t e n t i a l ,w h i c h w e c a l l m o d e l 1 , i s t o o a t t r a c t i v e f o r e l e c t r o n s a w a y f r o m t h e c e n t r e a s c a n b es e e n f r o m t a b l e 2. T h e p o t e n t i a l a t a /2 ( t o w a r d s n e a r e s t n e i g h b o u r s ) i s i n e r r o r b yo n l y 0 "0 7 p e r c e n t , b u t a t 3 a / 2 t h e e r r o r i s 2 4 p e r c e n t .

W i t h t h e l i m i t C = 1 0 0 w e w e r e a b l e t o t a k e u p t h e e i g h t h n e i g h b o u r s a n d i t w a sn o w p o s s i b l e to o b t a i n a C o u l o m b p o t e n t ia l , m o d e l 2 , w h i c h w a s c o r r e c t a t t h e c e n t r e

T a b l e 1 . S t r u c t u r e o f t h e c r y s t a l l a t ti c e l o c a l t o th e d e f e c t . T h e r e a r e N i e q u i v a l e n t i t hn e i g h b o u r s a t d i s t an c e s D fa f r o m t h e c e n t r e . T h e n e t c h a r g e s r e f e r t o t h e t w o m o d e l sd e s c r i b e d i n t h e t e x t .i 1 2 3 4 5 6 7 8N i 6 12 8 6 24 24 12 6D~ 1 2 3 4 5 6 8 9

N e t c h a r g e sM o d e l 1 + 1 - 1 + 1 - 0 ' 1 2 8 6 0 0 0 0M o d e l 2 + 1 - -1 + 1 - 1 + 1 - 1 + 0 . 5 1 70 - 0 " 2 5 7 4


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F c e n t r e e n e r g y l e v e l s o f L i F 2 9 9T a b l e 2 . C o m p a r i s o n o f t h e e x a c t a n d m o d e l e le c t r o s ta t i c p o t e n t i a l s V ( r ) a n d V ( r ) + [ a - r [ - 1a t d i s t a n c e r f r o m t h e c e n t r e a l o n g t h e l i n e t o t h e n e a r e s t n e i g h b o u r s ( t h e [ 10 0]d i r e c t i o n ) . A l l p o t e n t i a l s a r e t o b e d i v i d e d b y a t h e l a t t i c e p a r a m e t e r .

E x a c t M o d e l 1 M o d e l 2r r ( r ) g ( r ) + I - r 1 -1 V ( r ) g ( r ) + Ia - r I - ' g ( r ) g ( r ) + Ia - ~ [ = l0 1-7476 0'74 76 1-7477 0.74 77 1'74.76 0-7476a /2 2 0 2 '001 5 0"0015 2'0001 0"0001a o ~ - 0 " 7 4 7 6 co - 0 ' 7 0 9 9 co - 0 - 7 4 7 63a/2 0 "6 66 7 - 1 " 3 3 3 3 0 ' 82 5 2 - 1 " 1 7 4 8 0 "6 49 8 - 1 " 3 5 0 2

a n d a t n e a r e s t n e i g h b o u r s b y c h o o s i n g e f f e ct iv e c h a r g e s o n t h e s e v e n t h a n d e i g h t hn e i g h b o u r s e q u a l t o 0 '5 1 7 0 a n d - 0 " 2 5 7 4 r e s p e c t i v e l y . T a b l e 2 s h o w s t h a t t h ep o t e n t i a l i s i n e r r o r b y o n l y 2 ' 5 p e r c e n t a t 3 a / 2 .

3 . B A s i s S E T SC a l c u l a t i o n s w e r e m a d e w i t h c o n t r a c t e d g a u s s i a n f u n c t i o n s . F o r t h e f l u o r in e

a t o m v a n D u i j n e v e l d t [1 3] h as d e r i v e d an (8s , 4 p ) b a s i s c o n t r a c t e d t o (3s , 2:0) ; th i ss y m b o l i s m s t a n d s f o r th r e e s a n d t w o p g a u s s i a n f u n c t io n s . T h i s d o u b l e - z e t a b a s is w a su s e d f o r t h e c e n t r a l a n i o n v a c a n c y . A f u r t h e r c o n t r a c t i o n t o (2s , l p ) w a s m a d e w i t hc o e f f i c ie n t s d e r i v e d f r o m a c a l c u l a t io n o n F - . F o r L i w e t o o k th e v a n D u i j n e v e l d t[ 1 3] ( 8 s) c o n t r a c t e d t o (3 s) p l u s a ( 4 p ) c o n t r a c t e d t o ( 2 p ) g i v e n b y W i l l i a m s a n dS t r e i t w i e s e r [ 1 4] . T h i s (3s , 2 p ) b a s i s w a s u s e d i n s o m e c a l c u l a t i o n s f o r t h e n e a r e s tn e i g h b o u r s b u t f o r m o s t c a l c u la t io n s a f u r t h e r c o n t r a c t i o n t o (2s , l p ) w aS e m p l o y e d .T h i s w a s b a s e d u p o n a L i + c a lc u l a t io n , t h e o c c u p i e d a n d f i r s t t w o v i r t u a l o r b i t a l sb e i n g s e l ec t e d .

I n a d d i t i o n t o t he F ( 3 s , 2 p ) f u n c t i o n s a t th e v a c a n c y c e n t r e w e a d d e d d i f fu s e s a n dp g au ss ia ri~ f u n c t i o n s w h o s e e x p o n e n t s w e r e o p t i m i z e d i n m o d e l 1 c a l c u l a t i o n s inw h i c h n e a r e s t n e i g h b o u r e l e c tr o n s w e re e x p l i c i t ly c o n s i d e r e d b u t o n l y i n a m i n i m a lb a s i s r e p r e s e n t a t i o n . T h e d i f fu s e s f u n c t i o n s a r e a se t o f e v e n t e m p e r e d [ 15 ]g a u s s i a n s , w i t h e x p o n e n t s c~, c~/~ a n d c~/? . W e f o u n d n o s i g n i f i c a n t i m p r o v e m e n t i nu s i n g s u c h a se t fo r th e p f u n c t i o n s o v e r a s in g l e f u n c t i o n o p t i m i z e d a t e a c h v a l u e o ft h e l a t t ic e p a r a m e t e r a . T a b l e 3 d e f i n e s t h e s e b a s e s .

C a l c u l a t i o n s o n d i a t o m i c L i F g a v e t h e r e s u l ts s h o w n i n t a b le 4 . T h e s e a r ec o m p a r e d w i t h H a r t r e e - F o c k a n d e x p e r i m e n t a l v a lu e s i n t h e sa m e t a b le . I t c a n b es e en t h a t b o t h t h e d o u b l e - z e t a a n d m i n i m a l b a s e s g i v e s a ti s fa c t o ry r e su l t s b u t i ts h o u l d b e m e n t i o n e d t h a t a m i n i m a l b a s i s o p t i m i z e d f o r n e u t r a l a t o m s r a t h e r t h a ni o n s g i v e s v e r y p o o r r e s u l t s .

4. RESULTS FOR THE UNRELAXED LATTICEI n i t ia l c a l c u la t i o n s w e r e m a d e t o in v e s t ig a t e th e d e p e n d e n c e o n b a s i s f u n c t i o n s

m o s t c l o s e l y l o c a l i z e d t o th e d e f e c t , i . e. t h e d e f e c t c e n t r e a n d n e a r e s t n e i g h b o u r i o n s .T a b l e 5 s h o w s t h e b i n d i n g e n e r g i e s o f t h e f i r s t t w o s t a t e s o f t h e F c e n t r e o b t a i n e d b ys e p a r a t e S C F c a l c u la t io n s . T h e s e s t a t es h av e d i f f e r e n t s y m m e t r i e s a n d c a n b ed e s i g n a t e d 2 S a n d 2 p ; a s y m m e t r y e q u i v a l e n c i n g p r o c e d u r e w a s f o l l o w e d fo r t h e 2 ps t a te [ 1 8] . T h e b i n d i n g e n e r g i e s a r e r e l a t iv e t o t h e v a c a n c y w i t h o u t a n e l e c t r o n( u s u a l l y c a l l e d th e F ~ c e n t r e ) a n d t h e t o t a l S C F e n e r g y o f t h i s is g i v e n . C a l c u l a t i o n sw e r e m a d e o n l y i n m o d e l 1 w i t h a l l s i t es e x c e p t n e a r e s t n e i g h b o u r s r e p r e s e n t e d b y


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3 0 0T a b l e 3 .

J . T e n n y s o n a n d J . N . M u r r e l lM i n i m a l b a s i s f u n c t i o n s a n d d e f e ct c e n t r e f u n c t i o n s u s e d i n th i s p a p e r . T h e d o u b l ez e t a f u n c t i o n s f o r F a n d L i a r e g i v e n i n [ 1 3 , 1 4 ].

S y m m e t r yC o n t r a c t i o n C o n t r a c t i o n

E x p o n e n t c o e f f ic i e n t E x p o n e n t c o e f f ic i e n tL i F -"S 0 ' 6 3 0 6 0 ' 3 3 8 8 7 " 6 0 8 9 0 " 3 6 4 91 " 9 2 12 0 ' 4 7 1 8 2 2 - 2 2 1 8 0 - 4 6 2 4

6 " 1 7 6 7 0 " 2 4 6 9 6 9 ' 4 0 2 3 0 " 2 2 5 22 2 " 0 8 2 7 0 - 0 7 7 5 2 4 6 ' 2 3 6 3 0 " 0 6 5 99 7 " 1 55 1 0 ' 0 1 6 2 1 0 8 2 ' 6 7 6 4 0 "0 1 3 46 4 7 - 0 6 3 6 0 " 0 21 3 7 2 1 3 ' 1 3 8 0 0 - 0 0 1 8

0 ' 0 2 8 1 0 ' 6 2 0 6 0 " 4 0 1 2 0 " 6 88 10 ' 0 7 2 5 0 ' 4 4 5 0 1 " 3 5 7 0 0 ' 4 1 6 70 - 0 2 4 0 0 - 5 0 8 9 0 ' 5 3 3 0 0 " 3 48 10 ' 2 7 5 0 0 ' 1 3 8 9 1 " 3 5 1 2 0 ' 3 1 7 20 " 5 0 0 2 0 ' 8 3 4 8 4 ' 9 9 3 5 0 ' 1 4 6 71 "53 4 3 0 '0 2 2 8 2 2 "7 4 7 6 0 '0 2 7 9

D e f e c ts 0 ' 1 3 7 0 1 ' 0s 0"184 9 1 .0s 0 .2 4 9 6 1 '0p(a= 3"80 ao) 0-01 66 1"0p ( a = 4 ' 8 6 a o ) 0 ' 0 1 04 1"0p(a = 5 ' 2 0 a o ) 0 ' 0 1 0 2 1 ' 0p(a = 5 .6 7 a o ) 0 .0 1 0 1 1 '0

T a b l e 4 . C a l c u l a t e d e c i u i l i b ri u m b o n d l e n g t h a n d d i s s o c i a ti o n e n e r g y ( to i o ns ) o f d i a t o m i cL i F . a 0 = 5 - 2 9 2 x 1 0 11 m , E h = 2 7 "2 1 e V i s e q u i v a l e n t t o 2 6 2 6 k J m o l - ~

B a s i s r Ja o De/E hM i n i m a l 3 -0 5 0 - 2 90D o u b l e z e t a 2 "9 9 0 ' 3 2 1H a r t r e e - F o c k [ 16 ] 2 ' 96 0 -2 9 6E x p t . [1 7 ] 2 "96 0 "2 9 0

T a b l e 5 . C a l c u l a t e d t o t a l e n e r g y o f t h e F ~ c e n t r e a n d b i n d i n g e n e r g i e s o f t h e 2 8 a n d 2 pF c e n t r e s t a t es . A l l c a l c u l a t i o n s a r e f o r t h e u n r e l a x e d l a t t i c e w i t h l a t t i c e p a r a m e t e ra = 3 ' 7 9 6 a 0. M i n . = m i n i m a l b a si s, D Z = v a l e n c e d o u b l e z et a b a si s, d i f . = d i f f u s ef u n c t i o n s a t d e f e c t c e n t r e .

B a s is E n e r g i e s / E h A E ~ J e VN e a r e s t D e f e c tn e i g h b o u r s c e n t r e F ~ 2 S 2 pM i n . F - D Z - 4 8 ' 5 8 6 2 0 '1 5 3 6 0' 03 5 3 3-2M i n . F - D Z + d i f . - 4 8 ' 5 8 6 2 0 '2 0 01 0 '0 9 4 5 2"9D Z F - D Z - 4 8 ' 5 8 6 4 0 "1 88 5 0 '1 0 2 2 2-3D Z F - D Z + d i f . - 4 8 " 5 8 6 4 0 '2 0 2 2 0 '1 0 0 7 2"8


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F centre energy levels of LiF 301the point charges specified in table 1. Both minimal and double zeta bases were usedfor nearest neighbour electrons as described in the table.

Th e ene rgy of the F, state (six Li + in the C ou lo mb field of poin t charges) is largelyunaffec ted by the basis. T he energies of bot h the zS and 2p states are loweredsignificantly by using DZ rather than m inimal basis for the Li when no diffusefunction s are included at the defect centre. How ever, with diffuse functions there islittle i mpro veme nt on using the DZ basis. I n other words a minimal basis at the firstneighbours and DZ plus diffuse functions at the centre appears to be nearlysaturated. For this reason, and with reasonable c omputa tiona l economy, we shalladopt this basis for the more extensive calculations which we shall describe.

Tabl e 6 shows calculations of the 23 and 2p bindin g energies in models 1 and 2with successive inclusion of the neig hbour ing ion electrons. The fir st calculations inthe table correspond to the point charge model of Gourary and Adrian [3] as nonei ghbo ur ion electrons are explicitly considered in the S CF calculations. However,the first neighbour lithium 2s and 2p basis functions were included to give addedflexibility to the wavefunctions. T he relatively high binding energy, similar to thatobtai ned by Gour ary and Adr ian, reflects the lack of orthogonal ity to the Li lsorbitals.Table 6. Calculated binding energy (Eh)'Ofthe 2S and 2p states and the resulting excitationenergy. L is the number of shells neighbouring the defect whose electrons have beenexplicitly included in the SCF calculations. The experimental value of AE~p is 5'08 eV[22].

Model 1 Model 2L "2S 2p AE~p/eV 2S 2p AE~p/eV0 0.299 0.130 4.6 0-299 0-129 4-61 0'202 0"101 2"8 0'199 0;068 3-62 0.186 0"088 2.7 0"195 0"052 3.93 0.180 0.062 3.2 0"193 0"023 4"6

The most extensive calculations included first, second and third neighbourelectrons and, to economize, the lithium 2s and 2/0 functions on third neighbourswere omit ted from the basis. The bindi ng ener gy of the 2S state appears to convergequite quickly with L and there is little difference between the results of the twomodels. The re is a wide variation in previous calculation of this binding energy andin many instances it is considerably overestimated. For example, a recent SCF X~calculatio n gives a bin din g ener gy of - 1"16 E h [19]. Phot oele ctron ejection studie shave not been made on LiF but for RbI [20] they indicate that the 2S state is boun dby approxi mately 0"11 E h and the 2p state is only just bound. Our results areconsistent with this view.

Th e convergenc e of the 2p state with L is slower than that of the 2S state and thedifference between the two models is significant when measur ed against the 2 S ~ 2 ptransition energy. The 2p state is much more diffuse than the 2S and the model 1potential becomes progressively too attractive as the distance from the defect centreincreases. F or this reason we believe that the results of model 2 are more reliable. AMulliken population analysis of our best wavefun ctions (model 2 with L = 3) gives2 x 10 - 6 in each thir d n eig hbo ur Li + (1 s) orbita l for the 2S state but 74 x 10-6 in eachorbital (symmetry equivalent) for the 2p state.


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302 J. Ten nys on and J. N. Murr ellFrom this wavefunction of the 2S state we have calculated the spin densities

at the first and third neigh bour Li nuclei and find the values 6'0 x 10 -2 ao 3 and2"6 x 10 -4 ao 3 respectively. These numbe rs are bo th domin ated by the Li + (ls)contributions to the wavefunction. We have not calculated the second neighbourfluorine spin density because we expect its absolute value to be inaccurate andwe can make no comparison of relative values for different shells. From ENDORstudies on Li F, Holt on and Blum [21] deduce spin densiti es of 2"3 x 10-2 ao 3 and2"9 x 10- 4 ao 3 at first and t hir d ne igh bou r L i + , value s which are in reasona ble agree-ment with our calculations.

The experiment al absorption band associated with the 2S-- ,2P transition is broadand s truct urele ss and for LiF has its maxi mum at 5"08 eV at 5 K [22]. By the F ra nc k-Condon principle this maxi mum should correspond to the vertical excitation energyat the eq uil ibr ium geo metry of the 2S state. None of our calculations gives a value ashigh as e xper ime nt but our bes t calculation only differs by 0"5 eV, an amo unt whichmig ht be e xpect ed in a molecul ar calculation at this level of treat ment. However, wehave not yet examined the relaxation of the lattice.

Exa min ati on of the virtual levels of the F~ state suggests that other excit ed statesof the F centre might be bound. Whe n second neighbour electrons are included inthe c alcula tion a second s virtual level is found to be close in energy to the first p level.Th ir d neighb our electrons raise the energy of this orbital slightly.

Fo r m ode l 2 with L = 3 the first thre e vir tual levels of the F~ state are at - 0" 185 (s),-0" 023 (p) and +0"012(s). As is seen from ta ble 6 the first two are very close to thebin din g energies of the 2S and 2p states. Hence we de duce t hat an excited 2S state liesclose to the ionization continuum.

Because 2S-42S transitions are symmetry forbidden, any absorption bandassociat ed with the second excited state is difficult to see. A second 2S state has,however, been observed in potas sium halides by use of an electric field to lower thesymmetry (the Stark effect) [23] and is found to be just above the 2p state.

5. LATTICE DISTORTIONA compl ete tr eat ment of the lattice relaxation around an F centre is a lengt hy

problem. Firstly, ionic displacements around a vacancy are not confined to thenearest neighbours, and secondly for the 2p state they are not totally symmetric, asthe 2p state is subject to a Jahn-Teller distortion. In this study we consider onlytotally symmetr ic displacements of the nearest neighbour ions. The potentials aretherefore calculated as a function of a parameter k, which is the displacementoutwards of a nearest neigh bour Li + ion. Calcul ations were made in the firstnei ghbo ur appr oxim ati on (L = 1) using model 1 for the F~, 2S and 2 p states. For theF~ state we can make an inde pend ent esti mate of the potenti al curve from empi ricalfunction s and this allows us to test the rel iabili ty of the molec ular SC F calculations.

In this approximation, in which only the defect basis functions and nearestnei ghb our Li + functions are used, it is possible to calculate the relaxation in all thelithi um halides by using appropriate values for the lattice parameters, Table 3 givesthe diffuse p functions optim ized at the four lattice para mete rs relevant to LiF , C1, Brand I respect ively. The figure shows the results of these calculations. We note thatfor lattices except L iF the 2S state potential is very flat and in all cases the pot entia lcurves for the F~ and 2p states are nearly parallel.


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F c e n t r e e n e r g y l e v e l s o f L i F 303The potential cur ves arise fr om a balance between an outw ard force on the Li +

ions from the Coul omb potential of the lattice and an inward force arising from theelectron. For the F~ state only the Coul omb force is present and as the odd electronin the 2p state is only weakly bound the inward force in this case is also small.We can see from table 2 that model 1 underestimat es the Coulo mb potential at thenearest ne ighbo ur sites and this deficiency becomes worse as the distance f rom thecentre increases. However, this error is partly compensated by the absence ofexchange repulsion between the Li ions and their nei ghbou ring anions. Aconseque nce of this omission is that the outwar d displac ement of the Li in the F~state would continue until there is coincidence with an anion centre. Thuscalculations with only nearest neighbours included in the SCF procedure will notgive a correct equilibrium position for the F~ state. The important question,however, is whether the potential is correct for small displacements from theundistorted lattice.

0 . 0

L i F L i C I L i B r L i l

- ( 1

]0 : , o :o - o : , - o : i o : , o :o - o : , k / x o : , o : o - o : , o h o :o - o : ,SCF energies of F~([[]), 2S(C)) and 2P(A) states calculated for model 1, L = 1, as a function ofthe outward displacement k of nearest neighbour Li + ions. Calculations have beenmade for lattice spacings appropriate to each lithium halide crystal.

Empirical pair potentials give a good representation of the properties of perfectcrystals. I t is possible to use these to calculate the relaxation in the F~ state, ass umin gthat the pair potential is unch ange d by the for mation of the defect. We used the pairpotential first suggested by Huggins and Ma yer [24] which separates Coul ombic andexponential repulsive terms. The repulsive term for a separation r is written

b . C i j exp [ ( r i - [- r j - r ) / p ] , ( 1 )where r i and r i are the effective radii of the t wo ions, and C i j is the so-called Paul ingcoefficient [25] which depend s on the charge and num ber of outer electrons of the


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304Table 7.

J. Tennyson and J. N. MurrellParameters for the Huggins-Mayer repulsive potential (1), as deduced by Fumiand Tosi [26].

Li + F- C1- Br- IrJa o 1"727 2"566 3-039 3"281 3"636b =0"00583 En P=0"6414ao

two ions. T he paramete rs b and p have been optimized for the alkali halides and intable 7 we give values deduced b y Fu mi and Tosi [26]. We have investigated the useof other published parameters but they made little difference to the final results.

To determine the Coulomb energy for the F~ state we start with the potentialexperienced by one ion within its Wig ner -Se itz cell which arises from the rest o f theperfe ct crystal. A t the cell cen tre the poten tial is _+ 1"7476/a; the sign dependi ng onthe charge of the ion. The probl em of calculating the summed c ontrib ution from allions in a NaC1 lattice at a general point withi n the cell has been solved by Hajj [27].This potential, written M ( q ) / a is a slowly varying function of q near the site centre(q=0) and for small values it can be approximated by

M ( q ) = ___[1"7476 + 3-6q4]. (2)In the relaxed lattice we are conside ring six Li + ions mov ing simultane ously;

hence we must subtract from (2) the con tribut ion from the other five Li + ions andreplace it by the full Co ulom b potential of the cluster o f six. We m ust also remov e thecontribution from the anion which is missing at the centre. The full expression forthe C oulo mb potential o f the defect for a relaxation k is

4 4 ) 1a ~ - ~ + k - q ( a 2 + ( a + k ) 2 ) 1/2- + ~ +~2(a+k) (3 )plus terms independent of k.

The total energies of the F= state which are obtained from the empirical potentialand the SCF calculation cannot be compared because they refer to different sizedfragments. However, the relevant quantity for determining the equilibrium positionis the slope of the potential. In this re spect we find almost exact agreement bet weenthe empirical and SCF calculations, the two being indistinguishable on the scale ofthe figure. This perhaps implies that we have a fortuitous cancellation of errorsbetween our model Coul omb potential and the L i+ -X - exchange repulsion terms.We therefore assumed that the corrections which should be made to the S CF curvesfor the 2,9 and 2 p states are negligible comp are d with o ther e rrors in our cal culations.

Our calculations predic t an outw ard displacem ent of the Li ions (k positive) o f0'09 a 0 in Li F. Thi s is 2"5 per cent o f the lattice pa ramet er. Because the 2S statepotentials are so flat for the other halides our predicted movements are subject tolarge uncertainties in view of the a pproxima tions of our model. T her e is a smallmovement outward (0'02a0) for LiC1 and inward for LiBr (0'02a0) and LiI(- 0" 04 a0). Kojima calculated a much larger displacement for LiF of the oppositesign. Ou r results are, however, in reasonable agreeme nt with those of Korring a [11]and Bartram et al. [6].


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F c e n t re e ne rgy l e v e l s o f L i F 3056. CONCLUSIONS

T o o b t a i n v a r i a ti o n a l l y a c c u r a t e r e s ul t s o n th e L i F F c e n t r e h a s p r o v e d t o r e q ui r ea r a t h e r h e a v y c a l c u l a t i o n b y m o l e c u l a r s t a n d a r d s . O n l y m o d e l 2 w i t h L = 3 g i v es ag o o d e x c i t a t i o n e n e r g y b u t i t c a n b e s e e n f r o m t a b l e 6 th a t i t i s t h e 2 p s t a t e w h i c h i ss l ow t o c o n v e r g e . A s w e h a v e u n d e r e s t i m a t e d t h e e x c i t a ti o n e n e r g y a n y v a r ia t i o n ali m p r o v e m e n t w o u l d h a v e t o b e m o r e i n t h e 2 8 t h a n t h e 2 p s t a te ; p e r h a p sr e o p t i m i z a t i o n o f t h e d i f f u s e o r b i ta l s i n t h e m o d e l 2 , L = 3 c a l c u l a t i o n .

I t is g e n e r a l l y e x p e c t e d t h a t e l e c t r o n c o r r e l a t i o n ( b e y o n d t h e S C F l e ve l ) h a s as m a l l e ff e c t o n t h e d i f f e r en c e i n e n e r g y b e t w e e n s t a te s w h i c h h a v e t h e s a m e n u m b e ro f e l e c t r o n p a i r s . R e c e n t c a l c u l a t io n s [ 28 ] o n t h e 21-I a n d 2 E s t a te s o f H C N + s h o wt h a t c o n f i g u r a t i o n i n t e r a c t i o n s t a b il i z es t h e 2 E m o r e t h a n t h e 2 FI b y 0 "3 e V . F o rH N C + , h o w e v e r t h e 2 E is s ta b i l i z ed b y 1"3 e V . W e c a n t h e r e f o r e e x p e c t e r r o r s o f u pt o ~ 1 e V i n o u r c a l c u l a t i o n s d u e t o n e g l e c t o f c o r r e l a t i o n a n d i n t h a t r e s p e c t o u r2 S - - * 2 P e x c i t a ti o n e n e r g y m u s t b e c o n s i d e r e d q u i te s a t is f a c to r y .

T h e a u t h o r s t h a n k D r . P . D . T o w n s e n d f o r s t i m u l a t i n g d i s c u s si o n s o n t h is to p i c .REFERENCES

[1] INUI, T ., and UEMURA, Y., 195 0, Prog. the or. Ph ys. , 5, 252.[2] KOJIMA, T., 1957 , J. l~hys. Soc. Japan, 12, 908, 918.[3] GOURARY, B. S ., a nd ADRIAN, F. J., 195 7, Phy s . Re v . , 105, 1180.[4] WOOD, R. F . , and JoY, H. W ., 1964 , Phy s . Re v . A, 136, 451.[5 ] MA RTINO, F . , 1968, Int . J . quant. Chem., 2, 217, 233.[6] BARTRAM,R. H ., STONEHAM, A. M ., and GRASH, P., 1968, Phy s . Re v . , 176, 1014.[7] OP IK, V. , and W OOD, R. F . , 1969 , Phy s . Re v . , 179, 772.[8] WOOD, R. F . , and OPIK , V. , 1969 , Phy s . Re v . , 179, 783.[9 ] CHAN EY, R. C . , and LIN, C . C . , 1976 , Phy s . Re v . B, 13, 843.[10] BEALLF O W L E R , W . , 1964, Phy s . Re v . A, 135, 1725.

[11] WooD, R. F., and KORRINGA, J . , 1961, Phys . Rev . , 123, 1138.[12] SAUNDERS,V . R . , an d G U E S T , M . F . , A T M O L 3 , d o c u m e n t e d b y t h e A t l a s C o m p u t i n gDiv i s io n , R u th e r f o r d Lab o r a to r y .[12a] MU RRELL, J . N . , and TENNYSON , J . , 1980 , Che m . Phy s . L e t t ., 69, 212.[13] VAN DU IJNEVELDT, F. B., 1975, I B M J l R es . D e v. , 945, 1.[14] WILLIAMS,J. E. , an d STREITWlESER, A. , JR., 1974, Chem. Phys . Le t t . , 25, 507.[15] RAFrENETTI, R. C., 1973, J . chem. Phys . , 59, 5936.[16] Mc LEA N, A. D ., and YOSHIMINE, M ., 1967, I B M J l R es . D e v . S u p p l.[17] C om put ed f ro m data in MCLEAN, A. D . , 1963 , J. chem. Phys. , 39, 2653.[18] GUEST, M . F. , and SAUNDERS, V. R., 1974, Molec . Phys . , 28, 819.[19] TANG KAI, A. H . , 1980 , Disser ta t ion , Un ivers i ty o f Uppsala .[20] APKER, L., and TAFT, E., 1959, Phys . Rev . , 79, 964.[21] HOLTON, W . G. , and BLUM, H. , 1962 , Phy s . Re v . , 125, 89.[22] DAWSON, R. K., and POOLEY, D., 1969, Phys . S tat . Sol . , 35, 95. -[23] GRASSANO,U . M ., MARGITONDO, G., and ROSEI, R., 1970, P h y s . R e v . B, 2, 3319.[24] HDGGINS, M . L., and MAYER, J. E ., 1933, J. chem. Phys. , 1, 643.[25] PAULINe, L., 1928, Z. Kristallogr., 67, 377.[26] FUM I, F. G. , and T osI , M . P. , 1964 , J. Chem. Phys. Solids, 25, 31.[27] H AJJ, F. Y ., 1972, J. chem. Phys. , 56, 891.[28] MURRELL, J. N., and AL-DERzI, A., 1980, J . chem. Soc . , Faraday II , 76, 319.

J. Tennyson and J.N. Murrell- A study of the variational convergence of the F centre energy levels of LiF

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