GENERALIZED HELLMANN-FEYNMAN AND VIRIAL THEOREMS IN DENSITY FUNCTIONAL THEORY AT FINITE TEMPERATURES

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GENERALIZED HELLMANN-FEYNMAN AND VIRIAL THEOREMS IN DENSITY FUNCTIONAL

THEORY AT FINITE TEMPERATURES

Xu Xi-Shen, Zhang Wan-Xiang

To cite this version:

Xu Xi-Shen, Zhang Wan-Xiang. GENERALIZED HELLMANN-FEYNMAN AND VIRIAL THEO-

REMS IN DENSITY FUNCTIONAL THEORY AT FINITE TEMPERATURES. Journal de Physique

Colloques, 1984, 45 (C8), pp.C8-23-C8-26. �10.1051/jphyscol:1984805�. �jpa-00224303�

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JOURNAL DE PHYSIQUE

Colloque C8, supplément au n ° l l , Tome k5, novembre 1984 page C8-23

GENERALIZED HELLMANN-FEYNMAN AND VIRIAL THEOREMS IN DENSITY FUNCTIONAL THEORY A T FINITE TEMPERATURES

Xu X i - s h e n and Zhang Wan-xiang

Institute of Applied Physios and Computational Mathematics, P.O. Box 8009, Beijing, China

Ré s umë - Partant de l'expression du grand potentiel et des équations self consistantes à un électron de la théorie de la fonction de densité, nous avons étendu les travaux de Slater et Janak afin de montrer que les théorèmes de Hellmann-Feynman et du viriel étaient valides pour la matière condensée à température finie dans l'approximation des noyaux fixes.

A b s t r a c t - For a condensed n a t t e r w i t h t h e n u c l e i f r o z e n , s t a r t i n g from t h e grand p o t e n t i a l and t h e s e l f - c o n s i s t e n t o n e - e l e c t r o n e q u a t i o n s o f 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 y a t f i n i t e t e m p e r a t u r e s , we have e x t e n d e d S l a t e r ' s and J a n a k ' s works t o show t h a t t h e Hellmann-Feynman and t h e v i r i a l theorems s t i l l hold g o o d .

S i n c e Hohenberg and Kohn / 1 / , Mermin / 2 / , and Kohn and Sham / V proposed s u c c e s s i v e l y 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 y (DFT) a t z e r o and a t f i n i t e

t e m p e r a t u r e s , t h e t h e o r y has been e x t e n s i v e l y a p p l i e d t o s t u d y v a r i o u s problems c o n c e r n i n g t h e e l e c t r o n i c s t r u c t u r e s and o t h e r s t a t e p r o p e r t i e s o f condensed m a t t e r and d e v e l o p e d c o n s i d e r a b l y ( s e e r e v i e w a r t i c l e s / 4 , 5 / ) . Janak / 6 / , f o l l o w i n g S l a t e r ' 3 method 111, d e r i v e d t h e v i r i a l theorem f o r DFT a t z e r o t e m p e r a t u r e . We e x t e n d t h e s e work3 and show t h a t t h e Hellmann-Feynman and t h e v i r i a l theorems hold good i n DFT a t f i n i t e T a s w e l l .

1 - FUNDAMENTAL EQUATIONS OF DFT AT FINITE TEMPERATURES

Let u s c o n s i d e r a condensed m a t t e r . In t h e Born-Oppenheimer a p p r o x i m a t i o n , f o r t h e e l e c t r o n system a t temperature T and of c h e m i c a l p o t e n t i a l fi , t h e grand p o t e n t i a l n. o f DFT a t f i n i t e T i s g i v e n a s a f u n c t i o n a l o f t h e e l e c t r o n d e n s i t y n ( r , T ) ( i n Hartree atomic u n i t s , ii = e = m = 1 ) / 5 / :

< i m\-\ *- , f *•* n ( r . T ) 1 (( - „», n C r . T ) n ( r ' . T ) . 1 _.' Z*2P T f

+ F³( n ( r , T ) ) + F^{x c}( n ( r * , T ) } - j d? n ( r , T ) / t . ( 1 ) The first line is the Coulomb potential energy & , which is the sum of Coulomb

interaction contributions for electrons and the nuclei, electrons with one another (Hartree term), and nuclei with one another. F fn(r,T)1 is defined as the free energy of a non-interacting electron system with the same electron density n(r,T). Hence, F ^(r.T)} represents the contribution of exchange=

correlation interaction to the free energy.

From Oibbs variational principle, the following self-consistent one-electron equations are obtained / 5 / :

n(r.T) = ^ i / i ^ i ^ Y i ^ ) , (3)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984805

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C8-24 JOURNAL DE PHYSIQUE

where

rxc

^ist h e exchange-correlation c o n t r i b u t i o n t o t h e e f f e c t i v e one=

e l e c t r o n p o t e n t i a l V e f f

,

w h i l e f i is t h e Fermi d i s t r i b u t i o n f u n c t i o n :

In d e r i v i n g t h e above e q u a t i o n s , we have used t h e f o l l o w i n g e x p r e s s i o n f o r Fs

,

which i s t h e sum of t h e k i n e t i c energy K and t h e e n t r o p y term -T3:

1 2

P~ =

Ziii 5

^d?y;(P)(- ²

^v

⁾y i ( l ) + IT &(fi I n ii + ( l - i i ) ~n (1 -ii

11.

⁽⁷⁾

I1

-

THE GENERALIZED HELIMANN-FEmAN THEOREM

I f t h e i m p l i c i t dependence o f n(^r,T) on

3,

could be ignored, t h e n d i f f e r e n t i a - t i n g Eq.(l ) w i t h r e s p e c t t o

2,

w i l l l e a d t o

"

-

The terms on t h e r i g h t hand s i d e r e p r e s e n t t h e f o r c e s , computed by c l a s s i c a l e l e c t r o s t a t i c s , e x e r t e d by t h e t o t a l e l e c t r o n i c c h a r g e d i s t r i b u t i o n and by a l l n u c l e i except t h e d t h on t h e 4 t h nucleus. Hence t h e l e f t hand member may be t h e t o t a l f o r c e e x e r t e d by t h e whole system on t h e Nth nucleus. The Hellmann-Feynman theorem a f f i r m s t h a t t h e above s t a t e m e n t 4 i s t r u e i r r e s p e c t i v e o f t h e i g n o r a n c e o f t h e i m p l i c i t dependence o f n ( ? , ~ ) on R , .

A c t u a l l y , however, f i ( t h r o u g h Ei),

Ti

^and ^lp; ^alldependent on

@,

i m p l i c i t l y . In o r d e r t h a t t h e theorem may hold in t h i s c a s e , we must show t h a t t h e addi- t i o n a l terms a r i s i n g from such dependences e x a c t l y cancel. Indeed, it i s easy t o show from t h e fundamental e q u a t i o n s t h a t t h e terms a r i s i n g from -Ifti and

y ;

may be reduced t o

- - Z i f i J " b " Y ' )

H P i

^{L i Y i} ⁺Y f ~ , ( V j f ; Y i ) l

= - z i f i L i V $ R { d ~ y l i % = 0 i ( 9 )

which v a n i s h on account of t h e n o r m a l i z a t i o n o f t h e

Ti's.

On t h e o t h e r hand, we can show t h a t t h e a d d i t i o n a l terms a r i s i n g from v & f i a r e

-z,(v- f

- p )

+ I ~ T ~n(-)f fi = 0 : ( 1 0 )

R, i i

which v a n i s h on account o f Eq.(6). T h i s is n o t a s u r p r i s e t o u s because such r e s u l t s a r e j u s t what t h e v a r i a t i o n a l p r i n c i p l e r e q u i r e d .

Thus, we have proved t h a t t h e g e n e r a l i z e d Hellmann-Feynman theorem h o l d s i n t h e DFT at f i n i t e T.

I11

-

THE VIXIAL THEOREM

The v i r i a l theorem r a y be o b t a i n e d , f o l l o w i n g S l a t e r / 7 / , by o p e r a t i n g on E ~ . ( 2 ) w i t h ($. V ), m u l t i p l y i n g from l e f t by f i y ;

,

i n t e g r a t i n g o v e r a l l s p a c e , and then summing o v e r a l l occupied s t a i e s . Using t h e c o n j u g a t e of Eq. ( 2 ) t o r e e x p r e s s t h e term y;(Veff

-

E i ) ( r . V Y i ) , we f i n d

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Now s u b s t i t u t i n g t h e following i d e n t i t y /7/

i n t o Eq.( 11 ) and n o t i n g t h a t t h e i n t e g r a l of t h e divergence term v k i s h e s

because t h e wave f u n c t i o n s vanish s u f f i c i e n t l y f a r o u t s i d e t h e system, we o b t a i n :

2K = ' j d i ? n ( ? , T ) ( $ - v v e f f )

.

^{( 1}^{7 )}

On s u b s t i t u t i n g t h e expression ( 4 ) f o r Vepf

,

a f t e r some manipulations and u s i n g t h e g e n e r a l i z e d Hellmann-Feynman theorem, Eq.(13) becomes

-z=(?&*

~ $ ~ ) f i [ n ( + , ~ ) ] ³2K + f

-

JdG 0(:,~)(3. v p X c )

.

^{( 1 4 )}

For a condensed m a t t e r under a h y d r o s t a t i c p r e s s u r e p and occupying a volume V, one must c o n s i d e r a l s o t h e e x t e r n a l f o r c e s which balancing t h e i n t e r n a l f o r c e s on t h e n u c l e i t o keep them i n equilibrium. p h e s e e x t e r n a l f o r c e s r e s u l t in a c o n t r i b u t i o n -7pV. which would make

-

Z , ( i i ; V ~ ~ ) n vanish a l t o g e t h e r . Hence, in such circumstances, Eq.(14) t u r n s o u t t o be t h e g e n e r a l expression of t h e v i r i a l theorem in DFT a t f i n i t e T which we required:

The v i r i a l theorem may a l s o be simply derived by t h e method based on an argument from t h e s c a l i n g of t h e wave function /7/. Let R be a d i s t a n c e determining t h e s c a l e of t h e system. When we adopt t h e r e p r e s e n t a t i o n by dimensionless q u a n t i t i e s , t h e wave f u n c t i o n s can be w r i t t e n in t h e form

and they s a t i s f y t h e normalization condition:

Now by dimensional a n a l y s i s , t h e grand p o t e n t i a l A

,

E q . ( l ) , may be w r i t t e n a s :

2 2

n(n(:.r)] = R-'F,(

{;-I)

⁺^i3-'F2(

[ 3 ) )

⁺P ~ ~ ( ~ ( + , T ) ]

+ k ~ & ( f ~ l n f i + (1-fi) In (1-fi)]

- pzifi

⁹ ^{( 1 8 )}

where t h e f i r s t two terms correspond t o t h e k i n e t i c energy K and t h e Coulomb p o t e n t i a l energy 3 r e s p e c t i v e l y . Of course, t h e f i r s t t h r e e t e r n s dependent i m p l i c i t l y on f i as well. However, we have a l r e a d y pointed o u t t h a t afl/afi= 0 by t h e v a r i a t i o n a l p r i n c i p l e . Therefore, in what follows we need n o t consider t h e dependence of f i on R

.

Now l e t u s c a l c u l a t e -R(d/dR)h

.

F i r s t l y , i t i s easy t o show t h a t

Secondly, a s r e g a r d s t h e d i f f e r e n t i a t i o n of Fxc

,

i t may be c a l c u l a t e d by t h e expansion fonnulas o f a n a l y t i c Punctional, and i t i s easy t o f i n d t h a t

( - ~ x c ) ~ ~ R l =

-p

O ( + , T ) ( S . V ~ ~ ~ )

.

Thus we g e t

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JOURNAL DE PHYSIQUE

Noting t h a t V o e ~ ~ , we have

- a d / a H ) n = - 3 ~ ( & n ) , , ~ = ~ P V ;

and t h a t in t h e Born-Oppenheimer a p p r o x i m a t i o n , we must p u t

v~

⁼⁰

.

R a F i n a l l y we o b t a i n t h e v i r i a l theorem j u s t i n t h e form o f ~ ~ . ( 1 5 ) .

In t h e l o c a l DFT, t h e e x c h a n g e - c o r r e l a t i o n c o n t r i b u t i o n t o t h e f r e e energy may be w r i t t e n as:

~ ~ ~ ( n ( f , ~ ) ] a j d f '

~(G,T)

f X c ( n ( t , T ) )

,

^{( 2 3 )}

t h u s , t h e v i r i a l theorem can be transformed t o

3 p V = 2K +

3 -

7 5 d * n ( ? , T ) ( f x c b ) - p l c ( n ) )

.

We remark t h a t f o r t h e exchange p o t e n t i a l e i t h e r i n t h e n o n - l o c a l Hartree-Pock form o r i n t h e l o c a l form o f S l a t e r ' s X d method, t h e c o n t r i b u t i o n t o t h e v l r i a l i s d u s t t h e t o t a l exchange energy

P x .

We a l s o remark t h a t , when n e g l e c t i n g t h e c o r r e l a t i o n e f f e c t s o r n e g l e c t i n g t h e e x c h a n g e - c o r r e l a t i o n e f f e c t s c o m p l e t e l y , one o b t a i n s t h e g e n e r a l e x p r e s s i o n o f t h e v i r i a l theorem o f a system w i t h Coulomb i n t e r a c t i o n in t h e Hartree-Pock approximation o r i n t h e H a r t r e e approximation r e s p e c t i v e l y /8/, 1.e.

L a s t l y , we may mention t h a t all t h e s e r e s u l t s a r e much t h e same in form as t h a t in t h e T 0 c a s e , t h e o n l y d i f f e r e n c e l i e s in t h e replacement o f t h e e l e c t r o n d e n s i t y n($) by t h e t e m p e r a t u r e dependent form n($,T), i.e.,

n ( f ) = z i y ; ( i ' r ) ~ ~ ( $ 1 =a+ n(+,T) =

L

ifiy;(f.) y i ( * )

.

^{( 2 6 )}

where t h e o n e - e l e c t r o n wave f u n c t i o n s

yi

and

Y / ;

and t h e Fermi d i s t r i b u t i o n f u n c t i o n f i a r e o b t a i n e d by s o l v i n g t h e s e l f - c o n s i s t e n t o n e - e l e c t r o n e q u a t i o n s w i t h t e m p e r a t u r e , i . e . , Eqs. ( 2 ) - ( 6 ) .

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