j.n. murrell, j. tennyson and m.a. kamel- many-body contributions to the intermolecular potential in...

8/3/2019 J.N. Murrell, J. Tennyson and M.A. Kamel- Many-body contributions to the intermolecular potential in alkali halide cr

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MOLECULAR P~YsIcs, 1981, VOL. 42, No. 4, 747-755

M a n y - b o d y c o n t r i b u t i o n s t o t h e i n t e r m o l e c u l a r p o t e n t i a l i na l k a l i h a l i d e c r y s t a l s a n d c l u s t e r sby J. N. MURRELL and J. TENNYSON~-

School of Molecular Sciences, University of Sussex,Brighton BN1 9QJ, England

and M. A. KAMELDepartment of Physics, Faculty of Education,

Ain Shams University, Cairo, Egypt

(Received 9 October 1980 ; accepted 19 Nov e m be r 1980)SCFMO calculations have been made on lithium fluoride clusters bothwithin the crystal and as isolated species. Calculations have been madewith different bases in order to separate exchange, charge-transfer and in-duction energies. Calculations confirm the conclusion from perturbationtheory that charge-transfer is an important contribution to three-bodyenergies and gives a small four-body energy. The three-body energies inthe crystal environment are much smaller than for the isolated clusters.

I . INTRODUCTIONThe considerable interest in the static and dynamic properties of ioniccrystals, both perfect and defective lattices, and ionic melts, provides a con-

tinu ing incentive to obtain accurate interionic potential functions. There hasindeed been no lack of effort on this f ro nt ; the first alkali halide potentialswhich gave a good fit to the lattice energy were obtained by Huggins andMayer in 1933 [1].

The total potential for a finite cluster of ions can be written as a many-bodyexpansion. The first term will be the sum of pair potentials, the second werefer to as the three-body term, etc. The pair potentials contain the Coulombenergy, which is dominant at large distances and the exchange energy whichprovides the short range repulsion. There will be other smaller terms, somelong range and some short, such as dispersion, induction and charge transfer,whose relative importance depends on the internuclear separation. There arehowever problems relating to whether these energies make separate and additivecontributions to the two-body energy [2]. For polar molecules and particularlyfor ions it has generally been considered that the inductive energies are themost importa nt contribut ion to the higher n-body terms. These energies areassociated with the polarization of one atom or molecule in the total electricfield arising from its neighbours. As the componen t fields must be combinedvectorially the polarization depends on the relative positions of the neighbours.

t Current address: Institut voor Theoretische Chemie, Katholicke Universiteit,Toerniooveld, Nijmegen, The Netherlands.0026-8976/81/4204 0747 802.00 9 1981 Taylor&Francis LtdM.P. 2 m


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7 48 J . N . M u r r e l l e t a l .E a c h n e i g h b o u r d o e s n o t t h e r e f o r e m a k e a s e p a r a t e c o n t r i b u t i o n t o t h e i n -d u c t i v e e n e r g y , t h a t i s, t h e i n d u c t i v e e n e r g y is n o t p a i r a d d i ti v e .I n a p e r f e c t a l k a l i h a l i d e c r y s t a l l a t t i c e t h e e l e c t r o s t a t i c f i e l d a t t h e l a t t i c es i t e s i s z e r o s o t h a t t h e d i p o l a r i n d u c t i o n t e r m ( d i p o l e p o l a r i z a t i o n i n a u n i f o r mf i e l d ) is z e ro . T h u s , t h e s t a t ic p r o p e r t i e s o f t h e p e r f e c t l a t t ic e c a n b e i n t e r -p r e t e d q u i t e w e l l u s i n g p a i r p o t e n t i a l s o n l y , o r e f f e c ti v e p a i r p o t e n t i a l s w h i c hi n c o r p o r a t e s m a l l m a n y - b o d y t e r m s . T h e r e i s, h o w e v e r , e v i d e n c e f r o m l a tt ic ed y a n a m i c s a n d n e u t r o n s c a t te r in g t h at m a n y - b o d y t e r m s c a n n o t b e i g n o r e dc o m p l e t e l y [ 3 ].F o r i s o la t e d i o n ic c l u st e rs t h e r e i s n o d o u b t t h a t m a n y - b o d y e n e rg i e s ca n b eq u i t e l a r g e b u t a r e c e n t S C F M O s t u d y o f t h e c l u s t e r s [ L i~ C 1] ( ~- 1) + ( n = 1 to 4 )b y B o u n d s a n d H i n c h l i f f e [4 ] h as s h o w n t h a t t h e s e a r e c o n f i n e d al m o s t e n t i re l yt o t h e t h r e e - b o d y t e r m a n d t h e y a r e a s l a r g e a s t h e n o n - C o u l o m b i c p a r t o f t h et w o - t e r m b o d y . W e m i g h t e x p e c t t h a t fo r p r o b l e m s in v o l v i n g t h e i m p e r f e c ti o n i c l a t t i c e , s u c h a s d e f e c t s t r u c t u r e , i o n i c i m m o b i l i t y a n d p a r t i c u l a r l y m o l t e ns a lt d y n a m i c s , t h r e e - b o d y e n er g i e s w o u l d h a v e t o b e c o n s id e r e d . N e v e r t h e l e s st h e m a j o r i t y o f s u c h p r o b l e m s h a v e b e e n t a c k l e d w i t h a p a ir p o t e n t ia l .T h e o n l y w i d e l y u s e d a l t e r n a t i v e t o t h e p a i r p o t e n t i a l is t h e s h e l l m o d e l[ 5 ]. I n t h i s e a c h i o n is d i v i d e d i n t o a n i n n e r c o r e w h i c h h a s t h e n u c l e a r m a s sa n d a p o i n t c h a r g e X , a n d a m a s s l e ss o u t e r s h e l l o f c h a r g e Y ( X + Y is th e n e tc h a r g e o f t h e i o n ) . T h e c o r e a n d s h e ll a r e c o u p l e d b y a h a r m o n i c p o t en t ia l .C o r e - c o r e , s h e l l - s h e l l a n d c o r e - s h e l l i n t e r i o n i c p o t e n t i a l s a r e d e f i n e d a n du s u a l l y o n l y t h e l a t te r h a s n o n - C o u l o m b te r m s . T h e p a r a m e t e r s e n t e r i n g t h em o d e l a r e d e t e r m i n e d f r o m i o n p o l a r iz a t io n o r f r o m d i e le c t ri c co n s t a n ts .W e h a v e n o p e r s o n a l e x p e r i e n c e of u s i n g t h e s h el l m o d e l b u t n o t e t h a t i t h a si t s l i m i t a t i o n s . W i t h p a r a m e t e r s o p t i m i z e d f o r so l id s ta t e p r o p e r t i e s i t g i v e s ap o o r r e p r e s e n t a t i o n o f t h e d i a t o m i c a lk a l i h a l i d e e q u i l i b r i u m p r o p e r t i e s [ 3] .M o r e o v e r i n m o l e c u l a r d y n a m i c s c a lc u l a ti o n s sh e ll m o d e l s a r e m u c h m o r e t i m ec o n s u m i n g t h a n c a l c u l a t i o n s b a s e d o n n o n - p o l a r i z a b l e i o n p a i r p o t e n t i a l s [ 3 ] .W e s e e n o r e a s o n w h y , f o r s u c h c a l c u l a t i o n s , t h e i n t r o d u c t i o n o f p o l a r i z a t i o ne n e r g i e s b y t h r e e - b o d y t e r m s s h o u l d n o t b e m o r e a c c u r a t e a n d c o m p u t a t i o n a l lyc h e a p e r . ( I n c o m p a r i s o n w i t h th e s h el l m o d e l m o r e t e r m s h a v e t o b e c a lc u l a te dt o o b t a i n t h e p o t e n ti a l b u t t h e n u m b e r o f d y n a m i c a l v a ri a b le s is h a l v e d .)I n t h i s p a p e r w e u s e a s t a n d a r d S C F M O t e c h n i q u e t o in v e s ti g a te t h e p a i ra n d m a n y - b o d y p o t e n t i a l s o f a l k a l i h a l i d e s i n t h e c r y s t a l l a t t i c e a n d a s i s o l a t e dc l u s t e rs . O u r m a i n i n t e r e s t i s i n i n t e r p r e t i n g t h e m a n y - b o d y t e r m s , p a r ti c u l a r lyt h e i r d e p e n d e n c e o n t h e g e o m e t r y o f t h e i n t e ra c t i n g i on s . O n l y w h e n t h is h a sb e e n a c h i e v e d c a n w e h a v e a r a t i o n a l a p p r o a c h t o a c c u r a t e p o t e n t i a l s e i t h e re m p i r i c a l o r ab in i t io . W e b e g i n w i t h a f e w o b s e r v a t i o n s o n t h e p e r t u r b a t i o nt h e o r y o f i n t e r i o n i c p o t e n t i a l s .

2. I)ERTURBATION THEORY OF INTERIONIC POTENTIALST h e m o s t d e t a i l e d a n a l y s i s o f t h e a l k a l i h a l i d e p a i r p o t e n t i a l u s i n g p e r t u r b a -t i o n t h e o r y w a s g i v e n b y B r u m e r a n d K a r p l u s [6 ] u s i n g t h e e x c h a n g e p e r t u r b a -t i o n t h e o r y o f M u r r e l l a n d c o - w o r k e r s [7 , 8 ]. T h e y s h o w e d t h a t to s e c o n do r d e r i n t h e i n t e r a c t i o n p o t e n t i a l s a n d s e c o n d o r d e r i n o v e r l a p a p o t e n t i a l i s

o b t a i n e d t h a t c a n b e r e p r e s e n t e d q u a l i t a t i v e l y b y a m o d i f i c a t i o n o f t h e R i t t n e rf u n c t i o n [ 9] . T h i s t h e y c a ll e d t h e d i s t o r t i o n - d i p o l e f u n c t i o n ; i t h a s t h e


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I n t e r m o l e c u l a r p o t e n t i a l s i n a l k a l i h a l i d e c r y s t a l s 7 4 9f o l l o w i n g f o r m ( i n a t o m i c u n i t s ) :

VmxI)D(R) = (A + B (% - c% )/2R ~) exp ( - R / p )- - R - 1 - ( o ~ x - ~ m ) / 2 R 4 - C / R 6, (1 )

w h e re ~ x a n d ~ m a re t h e f r e e i o n p o la r i z a b i l i t ie s a n d A , B , C a n d p a r e c o n s t a n tp a r a m e t e r s o f t h e s y s t e m .

A n R - 7 t e r m i n t h e R i t t n e r f u n c t i o n w a s d r o p p e d o n t h e g r o u n d s t h a t i tw o u l d a r is e f r o m a h i g h e r o r d e r p e r t u r b a t i o n . T h e t e r m i n B is a n a d d i t i o n a n dh a s it s o r i g in s in s e c o n d - o r d e r e n e r g i e s w h i c h ar e o v e r l ap d e p e n d e n t . S e v e r a la s s u m p t i o n s w e r e m a d e b y B r u m e r a n d K a r p l u s [6 ] i n a r r iv i n g a t t h i s f u n c t i o na n d p r o b a b l y t h e m o s t s e ri o u s w a s th a t t h e B - t e r m h a s th e s a m e e x p o n e n t i a ld e p e n d e n c e o n R a s t h e A - t e r m ( w h i c h r e p r e s e n t s f i r s t - o r d e r e x c h a n g e ) .

A n a l t e r n a t i v e a p p r o a c h t o s e c o n d - o r d e r t e r m s w h i c h a r e o v e r l a p d e p e n d e n ti s t h ro u g h c h a rg e - t r a n s f e r s t a t e s [7 , 1 0 ] . I f t h e v a r ia t i o n a l m e th o d i s u s e d too b t a i n t h e w a v e f u n c t i o n o f t h e i n t e r a c t i o n s y s t e m s w i t h a s m a l l b a s is o f f u n c t i o n st h e n c h a r g e - t r a n s f e r s t a te s a r e t h e m o s t e f f i c ie n t f u n c t i o n s t o p r o v i d e t h e s ee n e rg i e s .

F i r s t - o r d e r e x c h a n g e e n e r g ie s i n v o l v e t h e o v e r l ap o f o c c u p i e d o r b i t al s o f th eg r o u n d s t a t e s o f t h e i n t e ra c t i n g s y s t e m s . T h i s o v e r l ap h a s r o u g h l y a n e x--p o n e n t i a l d e p e n d e n c e o n R a n d i s a c c o u n t e d f o r i n e x p r e s s io n ( 1 ) b y t h e At e r m . C h a r g e - t r a n s f e r e n e r g ie s , i n c o n t r a st , d e p e n d o n t h e o v e r la p o f a no c c u p i e d o r b i t a l o f t h e d o n o r ( t h e h a l o g e n a n i o n ) a n d a v a c a n t o r b i t a l o f t h ea c c e p to r ( t h e m e ta l c a t i o n ) . I n o th e r w o rd s , f o r t h e alk al i h a l id e m o le c u le t h ec h a rg e t r a n s f e r s t a t e is a c o v a le n t s t a t e o f t h e s y s t e m a n d th e a c c e p to r o rb i t a li s t y p i c a l ly a v a l e n c e o rb i ta l o f t h e n e u t r a l m e ta l . T h e o v e r l a p o f t h e s e d o n o ra n d a c c e p t o r o r b i t a l s i s m u c h l a r g e r ( f o r t h e s a m e v a l u e o f R ) t h a n t h e o v e r l a p

9 e s p o n s i b l e f o r e x c h a n g e e n e r g y a n d h e n c e w e e x p e c t t h e c h a r g e t r a n s f e r e n e r g yt o d e c a y e x p o n e n t ia l l y w i t h R a t a m u c h s l o w e r ra t e th a n d o e s t h e e x c h a n g ee n e r g y .

L e t u s n o w c o n s i d e r t h e p o l a r i z a t i o n o f a s y s t e m D ( a t o m , m o l e c u l e o r i o n )in t h e e l e c t r i c f i e ld o f a ll n e ig h b o u r s . P o la r i z a t io n c a n b e r e l a t e d to t h e e x c i t a -t i o n o f e l e c t r o n s f r o m o c c u p i e d t o v a c a n t o r b i t a l s a n d t h e e x c i t a t i o n ~a-->d~a, isa s s o c i a t e d w i t h a c o n t r i b u t i o n t o t h e i n d u c t i o n e n e r g y

- [ I 4 a 4 a " Z V K d v J ~ /A E a a ' ; (2)K C D

V K i s t h e p o t e n t i a l e n e r g y o f a n e l e c t r o n i n t h e f ie l d of s y s t e m K . T h e p a i r -a d d i t i v e p a r t o f ( 2 ) i s

- E ' [ I ~a q ~ a V ' c d v ] ~ / A E a a ' (3 )K

a n d t h e n o n - p a i r a d d i t i v e p a r t i s- E ' Z ' I d v I ( 4 )

K L r

I t h a s b e e n p o i n t e d o u t b y C l a v e r i e [ 1 0] t h a t t h e n o n - p a i r a d d i t i v e p a r t o f t h ei n d u c t i o n e n e r g y a ri s es i n t h r e e - b o d y t e r m s b u t n o t i n h i g h e r t e rm s . F o rn e u t r a l a t o m s V K d e c a y s e x p o n e n t i a l l y a w a y f r o m t h e c e n t r e o f K a n d h e n c eth e in t e g ra l s a p p e a r in g in (3 ) a n d (4 ) a r e g e n e ra l ly s m a l l . F o r i o n s V zc d e c a y sa s R - a a n d t h e r e f o r e t h e i n d u c t i o n e n e r g i e s w i ll b e m u c h l a rg e r .

2D2


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7 5 0 J . N . M u r r e l l et al .A n e x p r e s s i o n f o r t h e c h a r g e - t r a n s f e r e n e r g y c a n b e o b t a i n e d w h i c h i s

a n a l o g o u s t o ( 2 ) a n d h a s b e e n i n v e s t i g a t e d f o r i t s n o n - p a i r a d d i t i v e f e a t u r e s b yW i l li a m s a n d c o - w o r k e r s [ 1 1 ]. A s s o c i a t e d w i t h t h e t r a n s f e r o f a n e l e c t r o nf r o m ~ a o f D t o S a o f A t h e r e i s a c o n t r i b u t i o n t o th e e n e r g y o f

- - [I (~cl~a -- Sad~ d 2) X V K dT")]/AEda" (5 )K ~D( E x p r e s s io n ( 5 ) b e c o m e s i d e n t i c a l w i th ( 3 ) i f ~ a i s a n o r b i t a l o f D ( s a y (~ a ') b e -c a u s e i n t h i s c a se Saa wi l l b e z e r o . )

E a c h i n t e g r a l i n ( 5 ) c o u p l e s t h e d o n o r a n d a c c e p t o r w i t h t h e p o t e n t i a l o f th ea c c e p t o r o r t h e p o t e n t i a l o f s o m e t h i r d a t o m ( K r A ) . T h e p a i r - a d d i t i v et e r ms in ( 5 ) a r e

- - [ I (~ d ~ a - S a d dd 2 ) V 'A d v ]2 /m e d a (6 )a n d t h e r e w i l l b e t h r e e - b o d y t e r m s w h i c h c o u p l e ~ a a n d ~ a t o t h e p o t e n t i a l o f at h i r d a t o m K . I n c o n t r a s t t o t h e i n d u c t i o n e n e r g y t h e r e w i ll a l so b e f o u r - b o d yt e r m s o f t h e t y p e

- Z " ~ ." I (C ~ a d? .- -S .a d ?a z) v ~ : d v " I ( 4 a ( ~ . - - S . a d P a ~ ) V L d v / A e a . (7 )K L C Kb u t n o h i g h e r t e r m s .

T h e c o n t r i b u t io n s t o t h e t w o - b o d y i n d u c t i o n a n d c h a rg e t ra n s f e r t er m s , ( 3)a n d ( 6 ), a r e a ll o f t h e s a m e s ig n a n d w i l l a c c u mu la t e t o s t a b i l i z e t h e s y s t e m .T h e t h r e e - a n d f o u r - b o d y t e r m s , h o w e v e r , c a n b e o f e i t h e r si g n a n d t h e i r n e tc o n t r i b u t i o n w i l l b e l a r g e r o r s m a l l e r d e p e n d i n g o n t h e s i t u a t i o n .

T h e r e a r e ot h e r c o n t r ib u t i o n s t o t h e m a n y - b o d y e n e r gi es f r o m e x c h a n g e a n dd i s p e r s i o n b u t t h e s e a r e e x p e c t e d t o b e s m a l l a n d c a n b e i g n o r e d f o r i o n i cs y s t e ms [ 11 ] .

3. CALCULATIONSWe shall compare calculations on lithium fluoride clusters both isolated and

within a crystal environment. We wish to analyse the induction and charge-transfer energies in both situations and do this by a suitable choice of basis.

To calculate the energy in the crystal environment we follow a procedureused before [12] to study the SCF energy levels of the F-centre defect and definea finite lattice whose Coulomb potential in the central region closely approxi-mates the Coulomb potential in the unit cell of the infinite lattice. SCFMOcalculations are then made on a small number of ions contained within an innerzone at the centre of the finite lattice.

T h e c o o r d in a t e s o f i o n s i t e s i n t h e f i n i t e l a t t i c e a r e g iv e n i n t a b l e 1 a n d t h ef ig u r e s h o w s th e i o n p o s i t i o n i n t h e ( 0 , 0 , 0 ) p la n e . On ly t h e e l e c t r o n s o f t h ef o u r c e n t r a l i o n s i n t h i s p l a n e h a v e b e e n i n c l u d e d i n th e S C F c a l c u la t io n s , o t h e rs i t e s e n t e r in g t h e c a l c u l a t i o n o n ly a s p o in t c h a r g e s .

T h e C o u l o m b p o t e n t i a ls a l o n g t h e x a n d y a x e s o f t h e f ig u r e a r e z e r o b y s y m -m e t r y a s i n t h e i n f in i t e l a t ti c e . T h e c h a r g e s o n t h e o u t e r two s h e l ls l i s t e d i nt a b l e 1 w e r e c h o s e n t o m a k e t h e C o u l o m b p o t e n t i a l a t t h e f o u r c e n t r a l s i te se q u a l t o t h e M a d e l u n g p o t e n t ia l o f t h e i n f i n i te l a t t ic e a n d t o m a k e t h e p o t e n t i a la t t h e e ig h t p o in t s w i th c o o r d in a t e s ( O , + R + R ) a n d ( + 25, O , + R ) e q u a l t oz e r o a s i t s h o u ld b e in t h e i n f in i t e c r y s t a l . W i th t h e s e c h a r g e s , 0 .7 21 1 a n d


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Intermolecular potentials in alkali halide crystals 7 5 1

~ u

1- V"1 L i+

)Y

Rep resen ta t ion o f the z = 0 p lane o f the l a t t i ce used in the ca lcu la t ions .Ta ble 1 . Specif icat ion of the f in i te la t t ice used for the crys tal calculat ions . R is half thelat t ice d is tance which for L iF is 3.7965 a0. r is the d is tance of the app ropr iate shel lf rom the cen t r e o f the l a t t i ce .

Coordinates/R C h a r g e N u m b e rr2/R2 [x], [Yl, [z[ ]q[ of cen tres2 1, 1, 0 1 46 1, 1, 2 1 8

10 3, 1, 0 1 814 3, 1, 2 1 1618 1, 1, 4 1 818 3 , 3 , 0 1 422 3, 3, 2 1 826 5, 1, 0 1 826 3, 1, 4 0.7211 1630 5, 1, 2 0.2476 16

0 " 2 4 7 6 a s s h o w n i n t a b l e 1 , t h e C o u l o m b p o t e n t i a l i n t h e r e g i o n o c c u p i e d b y t h ec e n t r a l i o n s i s v e r y c l o s e t o t h a t i n t h e u n i t c e l l o f t h e c o m p l e t e c r y s ta l .

C a l c u l a t i o n s h a v e b e e n m a d e w i t h th r e e d i f f e r e n t b a s e s. F i r s t l y w e u s e d a ne x t e n d e d b a s i s o f g a u s s i a n f u n c t i o n s ; ( 6s , 4 p ) f o r l i t h i u m [ 1 3, 14 ] a n d (S s, 4 p )f o r f l u o r i n e [ 1 3 ] . T h i s b a s i s a l lo w s f o r c h a r g e t r a n s f e r , w h i c h i s p r i m a r i l yf r o m t h e 2 p o r b it a l s o f f l u o r i n e to t h e 2 p o r b it a ls o f l i t h i u m b u t t h e a m o u n t o fp o l a r i z a t i o n a l l o w e d b y t h e b a s i s i s v e r y s m a l l. L i + h a s a v e r y l o w p o l a r i z a b i l i t ya n d m o s t o f t h e p o l a r i z a b i l i t y o f F - i s a s s o c i a t e d w i t h p - d e x c i t a t i o n s . T oi n t r o d u c e p o l a r i z a ti o n a n d h e n c e i n d u c t i o n e n e r gi e s, w e a d d e d t o th e f l u o r i n eb a s is a s i n g le g a u s s ia n d o r b i t a l w i t h e x p o n e n t ~ - - 0 . 2 4 w h i c h m a x i m i z e s t h e F -p o l a r i z a b i l i t y ( a e a l c = 2 " 6 e ~ a 0 2 E h - - 1 ) . C a l c u l a t i o n s w e r e a ls o m a d e a t t h em i n i m a l b a s is l e v e l, ( i s ) f o r l i t h i u m a n d 2 s, l p ) f o r f l u o r i n e , t h e c o n t r a c t i o n


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7 52 J . N . M u r r e ll e t a l .c o e f f ic i e n t s b e i n g o b t a i n e d b y c a l c u l a t io n s o n L i + a n d F - u s i n g t h e e x t e n d e db a s is . I n m i n i m a l b a s is t h e r e is n o c h a rg e t r a n s f e r o r i n d u c t i o n e n e r g y .

T a b l e s 2 , 3 a n d 4 s h o w t h e c a l c u l a t e d e n e r g i e s i n t h e t h r e e b a s e s f o r f r e ec l u s t er s a n d f o r c l u s t e rs w i t h i n t h e c ry s t al . I n a ll c a s es t h e C o u l o m b e n e r g yh a s b e e n r e m o v e d , t h i s b e i n g c a l c u l a te d o n t h e a s s u m p t i o n o f p o i n t c h a r g e s a tt h e n u c l e i ( i. e. a s s u m i n g 1 0 0 p e r c e n t i o n i c c h a r a c te r a n d n o o v e r l a p p i n g e l e c t r o nd e n s i ti e s ) . F o r t h e c r y s t al c al c u l a ti o n th i s C o u l o m b e n e r g y is t h e s a m e fo r a llc l u s t e r s b e c a u s e a l a t t i c e s i t e i s e i t h e r o c c u p i e d b y a n i o n o r b y t h e a p p r o p r i a t ep o i n t c h a r g e . A l l c l u s t er s a r e b a s e d o n t h e s q u a r e a r r a y s h o w n i n t h e f i g u r ew i t h l a t t ic e s p a c i n g a = 3 . 7 9 6 5 a 0. T h u s t h e L i + - F - s e p a r a t i o n i s e q u a l to aa n d t h e L i + - L i + a n d F - - F - d i s t a n c e s a re 2 2/2 a . T h e e n e r g i e s h a v e b e e na n a l y s e d in t o m a n y b o d y t e r m s : E ( 2 , n ) , f o r ex a m p l e , i s t h e t w o - b o d y e n e r g yi n t h e n - a t o m c l u s t e r , t h e C o u l o m b e n e r g y n o t b e i n g i n c l u d e d .

I n t h e m i n i m a l b a s is a ll t h e t w o - b o d y e n e rg i e s a re p o s i t iv e a n d t h i s c a n b ea t t r i b u t e d t o t h e e x c h a n g e e n e rg y . T h e r e i s l it tl e d i ff e r e n c e b e t w e e n th ec r y s t a l a n d f r e e c l u s t e r e n e r g i e s a n d t h r e e - b o d y t e r m s a r e o n l y a f e w p e r c e n t o ft w o - b o d y . I n o t h e r w o r d s , i n t h e a b s e n c e o f c h a r g e t r a n s f e r a n d i n d u c t i o ne n e r g i es t h e t w o - b o d y e n e r g y d o m i n a t e s a n d i s v e r y s im i l a r i n a n d o u t s i d e t h ecrys ta l .F r o m t a b l e 3 i t c a n b e s e e n t h a t c h a r g e - t r a n s f e r a n d t h e s m a l l a m o u n t o f i n -d u c t i o n f r o m t h i s b a s is l e a d t o n e g a t i v e t w o - b o d y e n e r g i es e x c e p t i n t h e c a s e o f( F - ) 2 . H o w e v e r , t h e e x c h a n g e e n e r g y i n c r ea s e s r a p i d l y a t s h o r t e r d i s t a n c e s a n dt h e r e m u s t o f c o u r s e b e a b a l a n c e b e t w e e n a t t r a c ti v e a n d r e p u l s i v e f o r c e s i n th ee q u i l i b r iu m c r y s t a l l a t t i c e .T h e t w o - b o d y e n er g i e s a re s u b s t a n t i a ll y m o r e n e g a t i v e fo r t h e f r e e c lu s t e rt h a n f o r t h e c r y s ta l . W e s h o u l d e x p e c t th i s fo r t h e i n d u c t i o n e n e r g y a s t h e f ie l di s z e r o a t t h e l a t t i c e s i t e s i n t h e c r y s t a l. I t i s a p p a r e n t ly a l s o t r u e f o r t h ec h a r g e - t r a n s f e r e n e r g y a n d t h i s c a n b e a t t r i b u t e d t o t h e f a c t t h a t i n t h e c r y s t a lt h e i o n s a re s it t in g i n t h e d e e p C o u l o m b h o l e p r o d u c e d b y t h e l a tt i c e w h i c hf a v o u r s a h i g h i o n i c c h a r a c t e r f o r t h e s y s t e m . I n t h e c r y s t a l w e f in da b o u t 9 9 p e r c e n t i o n ic c h a r a c t e r f o r L i - b u t a s a f r e e c lu s t e r i t i s o n l y a b o u t8 5 p e r c e n t i o n ic . ( T h e s e n u m b e r s a r e a p p r o x i m a t e b e c a u s e n e i t h e r t h e d i p o l em o m e n t n o r t h e o r b i ta l p o p u l a t io n s g i v e an u n a m b i g u o u s m e a s u r e o f i o ni cc h a r a c t e r . )P r e v i o u s c a l c u l a t i o n s o n L i F [ 15 , 1 6] h a v e s h o w n t h a t t h e r e i s a p p r e c i a b l ei o n i c - c o v a l e n t m i x i n g e v e n a t a s e p a r a t i o n o f 1 1 a 0 w h i c h i s th e i o n i c - c o v a l e n td i a b a t ic cr o s s i n g p o i n t . T h i s i s e v i d e n c e t h a t t h e c h a r g e - t r a n s f e r c o n t r i b u t i o nt o t h e e n e r g y f a ll s o f f s l o w l y w i t h d i s t a n c e e v e n t h o u g h i t is f o r m a l l y a n e x p o n e n -t ia l ly d e c a y i n g te r m . I n c o m p a r i s o n t o t h e r e s u l ts o f t a b l e 2 t h e t h r e e - b o d yt e r m s i n t a b l e 3 ar e q u i t e l a r ge a n d c a n c e r t a i n ly n o t b e n e g l e c t e d i n c o m p a r i s o nw i t h t h e n o n - c o u l o m b i c p a r t o f t h e t w o - b o d y e n e r g y . T h e r e s u l ts a r e e n t i r e lyc o n s is t e n t w i t h t h o s e o f B o u n d s a n d H i n c h l if f e : w e f in d t h e f o u r - b o d y t e r m t ob e u n i m p o r t a n t .I n c r e a s in g t h e p o l a r i z a b i l i t y o f t h e f l u o r id e i o n g iv e s t h e r e s u l t s i n t a b l e 4 .T h e t w o - b o d y e n e r g y f o r ( F - ) 2 i s s u b s t a n t i a l ly l o w e r e d ( t h e r e i s n e g l ig i b l ec h a r g e - t r a n s f e r f o r t h i s s p e c i e s ) b u t t h a t f o r L i - i s n o t . T h i s i s a n i m p o r t a n tp o in t t h a t h a s b e e n n o t e d i n c a l c u l a t i o n s o n H~ (s Z~ +). W h e n a b a s i s i s u s e dw h i c h g i v es a n a p p r e c i a b l e a m o u n t o f c h a r g e - t r a n s f e r t h e n o n l y a s m a l l e x tr as t a b il i z a ti o n c a n b e o b t a i n e d f r o m i n d u c t i o n t e r m s [ 1 7] .


7/9

I n t e r m o l e c u la r p o t e n t i a l s i n a l k a l i h a l i d e c r y s ta l s 7 5 3T a b l e 2 . C a l c u l a t e d e n e r g i e s o f i o n c l u s t e r s w i t h m i n i m a l b a s i s . T h e t o p f i g u r e i n e a c h

c a s e r e f e r s t o c l u s t e r s w i t h i n t h e c r y s t a l a n d t h e b o t t o m ( i n p a r e n t h e s i s ) t o t h e f r e ec l u s t e r . E(k, n) i s t h e k - b o d y e n e r g y ( e x c l u d i n g C o u l o m b e n e r g y ) f o r t h e n - i o nc l u s t e r .

S t r u c t u r e E/Eh E ( 2 , n)/E~ E ( 3 , n)/Eh E ( 4 , n)/EhLi + - 7 . 2351 4 - - - - - -

( - 7 . 2 3 5 1 4 ) - - - - - -F - - 9 9 . 0 7 8 7 2 - - - - - -

( - 9 9 . 0 7 8 9 6 ) - - - - - -( L i + ) 2 - 1 4 . 4 7 0 2 8 0 - - - -

( - 1 4 . 4 7 0 2 8 ) ( 0 ) - - - -( F - ) 2 - 1 9 8 . 1 5 6 1 3 0 . 0 0 1 3 1 - - - -

( - 1 9 8 . 1 5 6 8 6 ) ( 0 . 0 0 1 0 6 ) - - - -L i + F - - 1 0 6 - 3 0 9 8 2 0 . 0 0 4 0 4 - - - -

( - 1 0 6 . 3 0 9 5 2 ) ( 0 . 0 0 4 5 8 ) - - - -(L i+ ) 2F - " 113"5 4092 0"0080 8 0 - -

( - 1 1 3 . 5 4 0 1 9 ) ( 0 . 0 0 9 1 6 ) ( - 0 . 0 0 0 1 1 ) - -L i + ( F - ) 2 - 2 0 5 " 3 8 3 3 5 ) 0 " 0 0 9 3 9 - 0 . 0 0 0 1 6 - -

( - 2 0 5 " 3 8 2 0 1 ) ( 0 " 0 1 0 2 2 ) ( - 0 " 0 0 0 1 7 ) - -( L i + ) z F - ) 2 - 2 1 2 . 6 1 0 5 7 0 ' 0 1 7 4 7 - 0 . 0 0 0 3 2 0

( - 2 1 2 " 6 0 9 3 8 ) ( 0 " 0 1 9 3 8 ) - 0 . 0 0 0 5 6 ( 0 )

T a b l e 3 . C a l c u l a t e d e n e r g i e s of i o n c l u s t e r s w i t h t h e b a s i s Li ( 6 s , 4p), F ( 8 s , 4 p ) . A n a l y s i sa s in t ab le 2 .

S t r u c t u r e E/Eh E ( 2 , n)/Ell E ( 3 , n)/Eh E ( 4 , n)/EhL i + - 7 . 2 3 5 2 0 - - - - - -

( - 7 . 2 3 5 2 0 ) - - - - - -F - - 9 9 . 3 6 1 4 3 - - - - - -

( - 9 9 . 3 6 1 6 0 ) - - - - - -( L i + ) 2 - 1 4 . 4 7 0 5 1 - 0 - 0 0 0 1 1 - - - -( - 1 4 . 4 7 0 7 3 ) ( - 0 . 0 0 0 3 4 ) - - - -( F - ) 2 - 1 9 8 " 7 2 2 7 8 0 " 0 0 0 0 8 - - - -

( - 1 9 8 . 7 2 3 5 8 ) ( - 0 . 0 0 0 3 8 ) - - - -L i + F - - 1 0 6 . 6 0 0 7 0 - 0 . 0 0 4 0 7 - - - -

( - 1 0 6 . 6 1 9 8 5 ) ( - 0 . 0 2 3 0 5 ) - - - -( L i + ) 2 F - - 1 1 3 . 8 3 7 1 2 - 0 . 0 0 8 2 5 0 . 0 0 2 9 6 - -

( - 1 1 3 . 8 6 2 1 0 ) ( - 0 . 0 4 6 4 4 ) ( 0 . 0 1 6 3 4 ) - -L i + ( F - -) ~ - 2 0 5 . 9 6 5 7 0 - 0 " 0 0 8 0 6 0 " 0 0 0 4 2 - -

( - 2 0 5 . 9 9 3 4 0 ) ( - 0 . 0 4 6 4 8 ) ( 0 . 0 1 1 4 8 ) - -( L i + ) 2 ( F - ) 2 - 2 1 3 . 2 0 2 9 6 - 0 . 0 1 6 3 1 0 . 0 0 6 7 6 - 0 . 0 0 0 1 5

( - 2 1 3 . 2 3 2 4 5 ) ( - 0 . 0 9 2 9 2 ) ( 0 . 0 5 5 6 4 ) ( - 0 . 0 0 1 5 7 )


8/9

754T a b l e 4 .

J . N . Murrell e t a l .C a l c u l a t e d e n e r g i e s o f i o n c l u s t e r s w i t h t h e b a s i s L i ( 6 s , 4 p ) , F ( 8 s , 4 p , l d ) .A n a l y s i s a s i n t a b l e 2 .

S t r u c t u r e E/Eh E ( 2 , n)/Eh E ( 3 , n)/Eh E ( 4 , n)/EnL i + - 7 . 2 3 5 2 0 - - - - - -

( - 7 . 2 3 5 2 0 ) - - - - - -F - - 9 9 . 3 6 1 4 3 - - - - - -

( - 9 9 . 3 6 1 6 0 ) - - - - - -( L i + )~ - 1 4 .4 7 0 5 1 - 0 . 0 0 0 1 1 - - - -

( - 1 4 . 4 7 0 7 3 ) ( - 0 . 0 0 0 3 4 ) - - - -( F - ) 2 - 1 9 8 .7 2 5 98 - 0 - 0 0 3 1 2 - - - -

( - 1 9 8. 73 0 23 ) ( - 0 . 0 0 7 0 3 ) - - - -L i + F - - 1 0 6 . 6 0 1 8 6 - 0 " 0 0 52 3 - - - -

( - 1 0 6 . 6 2 1 5 3 ) ( - 0 . 0 2 4 7 3 ) - - - -( L i + ) 2 F - - 1 1 3 .8 3 9 2 5 - 0 " 0 1 0 5 7 0 . 0 0 3 1 5 - -

( - 1 1 3" 8 66 4 0) ( - 0 ' 0 4 9 8 0 ) ( 0 .0 1 5 4 0) - -L i + ( F - ) 2 - 2 0 5" 9 70 3 9 - 0 ' 0 1 3 5 8 0 . 0 01 2 5 - -

( - 2 0 5 - 9 9 7 4 3 ) ( - 0 . 0 5 6 4 9 ) ( 0 . 01 7 46 ) - -( L i + )2 ( F -) 2 - 2 1 3 . 2 0 8 4 4 - 0 . 0 2 4 1 5 + 0 . 0 0 8 8 0 - 0 . 0 0 0 1 7

( - 2 1 3 . 2 3 5 7 3 ) ( - 0 . 1 0 6 2 9 ) ( 0. 06 5 72 ) ( - 0 . 0 0 1 5 5 )

T h e t h r e e - b o d y i n d u c t i o n e n e r g y s h o u l d b e s m a l l fo r t h e c l u s t e r ( L i + ) 2 F -b e c a u s e t h e p o l a r i z i n g f ie l d s f r o m t h e l i t h i u m i o n s a r e o r t h o g o n a l . W e n o t et h a t E(3, 3 ) f o r t h i s s p e c i e s is v e r y s i m i l a r i n t a b l e s 3 a n d 4 . F i n a l l y w e n o t et h a t t h e f o u r - b o d y t e r m s a r e v e r y s i m i l a r i n t h e s e t w o t a b l e s c o n f i r m i n g o u ra n a l y s i s i n w 2 t h a t f o u r - b o d y t e r m s a r i s e ' f r o m c h a r g e - t r a n s f e r .

T a b l e 5 g i v e s t h e e n e r g i e s o f d e f e c t v a c a n c i e s o f t h e c r y s t a l w h i c h h a v e b e e no b t a i n e d b y r e m o v i n g a n i o n f r o m o n e o f t h e f o u r c e n t r a l s i t e s . A l t h o u g ht h e s e v a c a n c i e s l o w e r t h e s y m m e t r y o f t h e e l e c t r o s t a t ic f i e ld i n w h i c h t h e c l u s t e rs i ts t h e t w o - b o d y a n d t h r e e - b o d y e n e r g i e s a r e s t i l l m u c h c l o s e r t o t h e v a l u e so b t a i n e d f o r t h e p e r f e c t c ry s t a l t h a n f o r t h e f r e e c l u s t e r . I t a p p e a r s t h e r e f o r e

T a b l e 5 . C a l c u l a t e d e n e r g i e s o f c l u s te r s a d j a c e n t t o v a c a n c y d e f e c ts . T h e v a c a n c y isl o c a t e d i n o n e o f t h e f o u r c e n t r a l s i t e s a n d t h e c l u s t e r o c c u p i e s t h e o t h e r t h r e es i t e s. R e s u l t s a r e g i v e n fo r m i n i m a l (i ), e x t e n d e d ( i i) a n d p o l a r i z a t i o n ( ii i ) b a s e s a sd e s c r i b e d i n t h e t e x t .V a c a n c y S t r u c t u r e E/En E ( 2 , 3 ) / E h E ( 3 , 3 ) / E h

F - ( L i + ) ~ F - ( i) - 1 1 3 . 5 4 0 6 5 0 - 0 0 8 3 6 0( ii ) - 1 1 3 . 8 4 1 3 1 - 0 . 0 1 1 5 0 0 .0 0 3 21

( ii i) - 1 1 3 . 8 4 2 3 0 - 0 - 0 1 0 7 5 0 .0 0 34 1L i + L i + ( F - ) ~ (i ) - 2 0 5 . 3 8 2 6 1 0 . 0 0 9 2 2 - 0 . 0 0 0 2 5

( ii ) - 2 0 5 . 9 7 4 5 8 - 0 . 0 1 3 5 0 - 0 . 0 0 0 1 8( ii i) - 2 0 5 . 9 9 4 1 4 - 0 . 0 1 8 9 5 0 .0 0 16 0


9/9

l n t e r m o l e c u l a r p o t e n t i a l s i n a l k a l i h a l i d e c r y s t a l s 7 5 5t h a t t h e s y m m e t r y o f t h e c r y s t al c a n b e b r o k e n q u i t e s e v er e ly w i t h o u t c h a n g i n gg r e a t l y t h e t w o - b o d y a n d t h r e e - b o d y t e r m s a n d p a r t i c u l a r l y b y n o t i n c r e a s i n gt h e r e la t iv e i m p o r t a n c e o f t h r e e - b o d y t o t w o - b o d y t e r m s .

F u r t h e r c a l c u la t io n s w o u l d b e n e e d e d t o e s ta b l i s h t h e n a t u r e o f t w o - b o d ya n d m a n y - b o d y t e r m s i n o t h e r d i s t o r t e d c r y s t a l s t r u c t u r e s a n d i n p a r t ic u l a rw h e t h e r e f fe c ti v e t w o - b o d y t e r m s c a n b e o b t a i n e d w h i c h w i ll be a d e q u a t e f o ra p p r o x i m a t e d y n a m i c a l c a lc u la t io n s . W e h a v e o b t a in e d t w o - b o d y p o t e n t ia l sf o r s o m e s y m m e t r i c a l d i s t o r ti o n s of t h e c e n t r a l i o n s i n o u r la t ti c e . F o r e x a m p l e ,b y d i s p l a c i n g t h e t w o L i + i o n s a l o n g t h e ( 1, 1 , 0 ) a x i s w e h a v e o b t a i n e d a t w o -b o d y p o t e n t i a l f o r L i + - L i + . A l l w e c a n c o n c l u d e a t t h i s p o in t , h o w e v e r , i st h a t t h e s e p o t e n t i a l s h a v e a f u n c t i o n a l d e p e n d e n c e o n t h e i n t e r - i o n i c d i s t a n c ew h i c h i s s i m i l a r t o t h a t f o r t h e f r e e i o n p a i r b u t w e d o n o t k n o w w h e t h e r t h i sp o t e n t i a l c a n b e a p p l i e d i n le s s s y m m e t r i c a l s i tu a t i o n s .

4 . C O N C L U S I O N ST h e r e a r e t w o i m p o r t a n t c o n c l u s i o n s f r o m t h i s w o r k . F i r s t l y , t h e m a n y -

b o d y c o n t r i b u t i o n s t o t h e i n t e r - i o n i c p o t e n t i a l w i t h i n t h e c r y s t a l a r e s m a l l e r ,r e l a ti v e t o th e t w o - b o d y te r m , t h a n f o r f r e e i o n c lu s t er s . S e c o n d l y , c h a r g e -t r a n s f e r t e r m s m a k e a n i m p o r t a n t c o n t r i b u t i o n t o t h e t h r e e - b o d y e n e r g i e s a n df o r l it h i u m f l u o r i d e c l u s te r s a p p e a r to b e m o r e i m p o r t a n t t h a n i n d u c t i o n t e r m s .

I f t h i s s e c o n d p o i n t is f o u n d t o b e g e n e r a l l y t r u e f o r io n i c s y s t e m s t h e nt h e r e s e e m s t o b e l i tt le p o i n t i n m a k i n g f u r t h e r d e v e l o p m e n t s o f t h e s h e l l m o d e l ,w h i c h c a n o n l y s i m u l a t e i n d u c t i o n e n e r g ie s . A m o r e p r o f it a b l e a p p r o a c hw o u l d b e t o o b t a i n a n e x p l ic i t f u n c t i o n a l f o r m o f t h e w h o l e t h r e e z b o d y e n e r g y ,i n d u c t i o n a n d c h a r g e -t r an s f e r, b y a b - i n i t i o c a l c u l a t i o n s .

M . K a m e l t h a n k s th e B r i t is h C o u n c i l f o r f i n a n c ia l s u p p o r t w h i c h e n a b l e dt h e s e c a l c u l a t i o n s t o ~b e p e r f o r m e d a t t h e U n i v e r s i t y o f S us s e x .

REFERENCES[1] HUCGINS, M . L. , a nd MAYER, J. E., 1933, J. chem . Phys., 1, 643.[2] MURRELL, J. N ., 19 74, Orbital Theories of M olecules and Sol ids, e d it e d b y N . H .March (Clarendon Press) , p. 311.[3] SANGSTER,M. J . L . , and DixoN, M. , 1976, Adv. Phys . , 25, 247.[4] BOUNDS, D . G ., a nd H INCHLIFFE,A., 1980, Molec. Phys., 40, 989.[5] COCHRAN,W., 1971, Sol id St . Sc i . , 2 , 1 .[6] BRUMER, P. , an d KARPLUS, M ., 1973, J . chem. Phys . , 58, 3903.[7] MURRELL, . N ., RANBIC, M ., a nd WILLIAMS,D. R., 1965, Proc. R. Soc. A, 284, 566.[8] MURRELL, . N ., an d SHAW, G ., 1 967, J . chem. Phys . , 46, 1768.[9] RITTNER, E. S ., 1 951 , J . chem. Phys . , 19, 1030.[10] CLAVEmE,P., 1978, Intermolecular Interactions from Diatomics to Biopolymers, edi tedby B. Pullman (Wiley), p. 69.[11] WILLIAMS,D . R., SCHAAD, L. J., an d MURRELL,J. N., 1967, J. chem. Phys. , 47, 4916.[12] TENNYSON, J., an d MURRELL, . N ., 1981, Molec. Phys. , 42, 297.[13] VAN DUIJNEVELDT,F . B . , 1975, I B M J I R es., 945, 1.[14] WILLIAM S, . E ., an d STEEITWEISER, A., 197 4, Chem. Phys . Le t t . , 25, 507.[15] KA HN, L. R., a nd HAY, P. J. , 1974, J . chem. Phys . , 61, 3530.[16] BOTTER,B. J., KOOTER, J. A., an d MULBER, J. J. C., 1975, Chem. Phys. Let t . , 33, 532.

[17] MURRELL, . N ., a nd TEXEm A-Dms, J. J. C., 1971, Molec. Phys., 22, 532.

j.n. murrell, j. tennyson and m.a. kamel- many-body contributions to the intermolecular potential in...

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