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Volume 103A, number 4 PHYSICS LETTERS 2 July 1984

E N E R G Y R E L A T I O N S I N T H E D E N S I T Y F U N C T I O N A L T H E O R YO F d D I M E N S I O N A L H E A V Y P O S IT I V E IO N SN . H . M A R C HTheoretical Chem istry Department, University o f Oxford, 1 South Parks Road, O xford OX1 3T G, U KReceived 15 May 1 984

By means o f the virial theorem, the total energy E, eigenvalue sum E s and chemical potential ta are related in a heavypositive ion in d dimensions. The scaling properties o f these quantities in two dimensions are thereby established.

I n r e c e n t w o r k [ 1 ] , t he e ige nva lue sum E s a nd th ec he m ic a l po t e n t i a l # ha ve be e n s tud i e d f o r N e l e c t r onsm o v i n g i n d e p e n d e n t l y i n b a r e p o i n t c h a rg e p o t e n t i a ls ,a s a f unc t ion o f d im e ns iona l i t y d . I n pa r t i c u l a r , f o r t her a t i o E s / N i n t h e a s y m p t o t i c l i m i t o f la rg e N , t h ef o l low ing r e su l t w a s e s t a b l i she d [ 1 :E s / N P = d ( 4 - d ) / ( 4 + 2 d - d 2 ) . ( 1 )The pu r p ose o f t he p r e se n t l e t t e r i s t o i nc lude t he s e l f -c ons i s t e n t f i e ld , w h ic h s c r e e ns t he ou t e r m os t e l e c t r onsf r o m t h e f u ll a t t ra c t i o n o f t h e b a r e p o i n t c h a r g e o fm a g n i t u d e Z e sa y .

B e low , u se w i ll be m a de o f t he de ns i t y f u nc t iona lt h e o r y a t t h e l e v e l i n w h i c h e x c h a n g e a n d c o r r e l a ti o ne f f e c t s a r e ne g l e c t e d . Th i s i s know n to be a n a de qua t ea p p r o x i m a t i o n f o r d i m e n s i o n a l i t y d = 3 i n t h e l i m i to f su f f i c i e n t ly he a v y pos i t ive ions [ 2 ] Th e Eu le r e qua -t i o n e x p re s s in g t h e c o n s t a n c y o f t h e c h e m i c a l p o t e n -t i a l ta i n spa c e m a y be w r i t t e n [ 3 ]p = 5 T / S o ( r ) + V n( r + V e ( r . ( 2 )H e r e T I p ] i s t he k ine t i c e n e r gy a s a f unc t iona l o f t hee l e c t r o n d e n s i t y o ( r ) , V e i s t he s c r e e n ing c on t r i bu t iono f t h e e l e c t r o n c l o u d t o t h e t o t a l H a r t re e p o t e n t i a l e n -e r gy , w h i l e Vn r e p r e se n t s t he ba r e nuc l e us . W e nowm u l t i p ly e q . ( 2 ) b y p a n d in t e g r a t e o ve r a ll spa c e. U s -ing t he no r m a l i z a t i on c o nd i t i on t ha t t he r e a r e N e l ec -t r ons i n t he c ha r ge c loud , p lu s t he c ons t a nc y o f / a i ns p a ce , w e f i n d i m m e d i a t e l yN p = / ' p 5 T~- -~ ) d , r + Ven + 2Uee , (3),J186

w he r e Uen is t h e e l e c t r o n - n u c l e a r p o t e n t i a l e n e r g yf p V n d r w h i l e t h e e l e c t r o n - e l e c t r o n p o t e n t i a l e n e r g y

1 1 U ee I s ~ f P V e d r , t he f a c to r ~ a s u sua l p r e ve n t ing do u -b l e c o u n t i n g o f e l e c t r o n - e l e c t r o n i n t e ra c t io n s .

Ea r l ie r , t he w r i t e r [ 4 ] s t ud i e d t he k ine t i c t e r m ine q . (3 ) , a n d d e m o n s t r a t e d b y m e a n s o f s o m e s p e c if ice x a m p l e s t h a t f o r la rg e N t h e l o ca l d e n s i ty , T h o m a s -F e r m i , c o n t r i b u t i o n g a v e t h e d o m i n a n t t e r m , t h e n o n -loc a l c o r r e c t i ons a f f e c t i ng on ly l ow e r o r de r t e r m s inN . Thus , de no t ing t he t o t a l k ine t i c e ne r gy in t he l oc a lde ns i t y l im i t b y T , e q . ( 3 ) f o r N p s im p l i f i e s i n d d i -m e n s i o n s t oN p = ( 1 + 2 / c O T + U en + 2 U ee . ( 4 )I t i s obv iou s t ha t t he e ige nva lue sum E s i s s im p ly T +Uen + 2Uee and us ing th is r e sul t in eq . (4) y ie ldsN l a = ( 2 / d ) T + E s . ( 5 )

A t t h i s s ta ge , w e invoke the v i r ia l t he o r e m , w h ic hf o r t he p r e se n t p r ob l e m r e a ds [ 5 ]2T + ( d - 2 ) ( U en + U e e ) = 0 , ( 6 )w h i l e t he t o t a l e ne r gy E i s e v ide n t lyE = T + Uen + Ue e . (7)E l i m i n a ti n g t h e p o t e n t i a l e n e r g y t e r m s b e t w e e n e q s.( 6 ) a nd ( 7 ) y i e ld s t he nT = [ ( 2 - d ) / ( 4 - d ) ] E , ( 8 )w h ic h i nc lude s t he f a m i l i a r r e su l t T = - E f o r t h r e ed im e ns ions . B y subs t i t u t i ng f o r T f r om e q . ( 8 ) i n to

0 . 3 7 5 0 6 0 1 / 8 4 / $ 0 3 . 0 0 E l s e v ie r S c ie n c e P u b l is h e rs B . V .( N o r th - H o l l a nd P hys i c s P ub l i sh ing D iv i s ion )

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Volume 103A, number 4 PHYSICS LETTERS 2 July 1984eq. (5) , t he resu l tE = E s - NO )a 4 - d) /2 a - 2) 9)

a ( E s _ N O )o l l o w s , w h i c h r e d u c e s t o t h e r e s u l t E =g i v en p r e v i o u s ly b y t h e w r i t e r f o r d = 3 [ 6 ] .

A s w e l l a s b e i n g c o r r e c t a s y m p t o t i c a l l y f o r l a r g e Ni n t h e f a m i l i a r t h r e e - d i m e n s i o n a l c a s e , t h e r e s u l t ( 9 )c a n b e a p p l i e d f o r d / > 5 . F o r d = 4 , t h e n u m e r a t o re v i d e n t l y v a n is h e s ; a n i m m e d i a t e c o n s e q u e n c e o f t h ev i r ia l t h e o r e m i n t h e f o r m ( 7 ) l e a d i n g t o E --- 0 . F o r d= 2 , c o n s i d e r e d i n m o r e d e t a i l b e l o w , b o t h t h e n u m e r -a t o r a n d d e n o m i n a t o r i n e q . ( 9 ) b e c o m e z e r o . I t i sw o r t h a d d i n g h e r e r e l at i o n s f o r t h e p o t e n t i a l e n e r g yc o n t r i b u t i o n s :U e e - N O = [ ( d 2 _ 2 d - 4 ) / d ( 4 - d ) ] E ,U e n + N o = [ ( 4 + 4 d - d 2 ) / d ( 4 - d ) ] E . ( 1 0 )

O f c o u r s e , t h e e x p l i c i t s o l u t i o n o f t h e c a s e d = 3h a s a lr e a d y b e e n o b t a i n e d b y n u m e r i c a l in t e g r a t i o n o ft h e a p p r o p r i a t e n o n - l i n e a r T h o m a s - F e r m i e q u a t i o na n d t h e t o t a l e n e r g y is k n o w n t o t a k e t h e f o r m [ 3 ]13 = Z 7 /3 f 3 ( N / Z ) , ( 1 1 )w h e r e / ' 3 i s a n u m e r i c a l l y k n o w n f u n c t i o n , w i t ha n a l y t i c r e p r e s e n t a t io n s n e a r N / Z = 0 a n d a l s o a r o u n dt h e n e u t r a l a t o m l i m i t w h e r e N / Z t e n d s to u n i t y [ 7 ] .F o r t h is t h r e e - d i m e n s i o n a l c a s e , M a r c h a n d P u c c i [ 8 ]h a v e f u l ly d e s c r i b e d h o w o n e c a n c h a r a c t e r i z e t h et o t a l g r o u n d s t at e e n e r g y b y m e a n s o f th e c h e m i c a lp o t e n t i a l .

W e s h a ll c o n c l u d e b y d i sc u k si n g t h e t w o - d i m e n s i o n a lc a s e in a li t t le m o r e d e t a i l. A s s e e m s t o h a v e b e e nn o t e d f ir st b y L e n n a r d - J o n e s a n d W o o d s [ 9 ] , a n d e x -p l o i t e d b y s u b s e q u e n t w o r k e r s [ 1 0 ] , t h e d i ff e r e n t ia le q u a t i o n f o r t h e s e l f - c o n s i s t e n t p o t e n t i a l i s t h e n l i n e a r .K v e n t s e l a n d K a t r i e l [ 1 0 ] h a v e s o l v e d t h is e q u a t i o ni n t e r m s o f m o d i f i e d B e ss el f u n c t i o n s a n d u s i n g t h e irr e s ul ts w e h a v e e s t a b l is h e d t h a t , c o r r e s p o n d i n g t o e q .( 1 1 ) , t h e t w o - d i m e n s i o n a l s c a li n g p r o p e r t y o f t h et o t a l e n e r g y o f h e a v y p o s i ti v e i o n s isE 2 ( Z , N ) = Z 2 f 2 ( N / Z ) . ( I 2 )U s in g t h e t h e r m o d y n a m i c r e l a t io n [ 3 ]

O = ( a E / a N ) z , ( 1 3 )i t f o l l o w s t h a t t h e c h e m i c a l p o t e n t i a l h a s t h e s c a l i n gb e h a v i o u ro ( Z , N ) = Z f ~ ( N / Z ) . ( 1 4 )R e t u r n i n g t o t h e v i r i a l t h e o r e m ( 6 ) w e n o t e t h a t w h e nd = 2 t h is im p l i e s T = 0 w h i c h i s t o b e i n t e r p r e t e d t om e a n t h a t , t o l e a d in g o r d e r i n N f o r l ar g e N , t h e p o t e n -t i a l e n e r g y U = U e n + U e e d o m i n a t e s T i n t h e t w o - d i -m e n s i o n a l c a s e ; i. e . E = U in t h e a s y m p t o t i c l i m i t .F r o m e q . ( 5 ) i t t h e n f o l lo w s t h a t , f o r d = 2 ,E s = N O = Z N f ' 2 ( N / Z ) = Z 2 [ ( N / Z ) f ' 2 ( N / Z ) ] , ( 1 5 )w i t h t h e s a m e t y p e o f s c a li n g t h e r e f o r e a s i n e q . ( I 2 )f o r t h e t o t a l e n e r g y E 2 . F o r t h e n e u t r a l a t o m c a se N= Z a n d d = 2 , K v e n t s e l a n d K a t r i e l [ 1 0 ] a r g u e t h a tt h e c h e m i c a l p o t e n t i a l O is z e r o , w h i c h w o u l d t h e ny i e l dE = U= -U ee = 1~ U e n . ( 1 6 )T h e p o t e n t i a l e n e r g y r e l a t io n s e x h i b i t e d i n e q . ( 1 6 )a r e , f o r d = 2 , t h e r e s u l t s o f p u t t i n g O = 0 i n e q s . ( 1 0 )f o r g e n e r a l d i m e n s i o n a l i t y .

P a r t o f t h i s w o r k w a s c a r r ie d o u t d u r i n g a v is it t oU C S B , C a l i fo r n i a. T h e w r i t e r w i s h e s t o t h a n kP r o f e s s o r J . R . S c h r i e f fe r f o r h i s v e r y k i n d h o s p i t a l i ty .R e f e r e n c e s

[1] N.H . M arch (1984), to be published.[2] J.M.CI Scott, Philos. Mag. 43 (195 2) 859 .[3] S. Lundqvist and N.H. March, eds., Theory o f the in-homogeneous electron gas (Plenum, New York, 1983).[4] N.H. March, J. Chem. Phys. 74 (1981) 2376.[5] M. Parrinello and N.H. M arch, J. Phys. C9 (1976) L147.[6] N.H. March, J. Chem. Phys. 72 (1980) 1994.[7] Y. Tal and M. Levy, Phys. Rev . A23 (1981 ) 40 8;N.H. March, J. Chem. Phys. 76 (1982) 1430;J .P. G rou t , N.M. March and Y. Tal . J. Ch em. Phys. 79(1983) 331.[8] N.H. March and R. Pucei, J . Chem. Phys. 78 (1983) 2480.[9] J.E. Lennard-Jones and H .J. Woo ds, Proc. R. Soc. 120(1928) 727.[10] G.F. K ventsel and J. K atriel, Phys. Rev. A24 (1981)2299.

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