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The effect of different exchange approximations on a pseudopotential calculation of phonon dispersion relations in beryllium
Volume 31A, number 3
PHYSICS LETTERS
9 February 1970
THE
EFFECT OF DIFFERENT EXCHANGE APPROXIMATIONS ON A P S E U D O P O T E N T I A L CALCULATION OF PHONON DISPERSION RELATIONS IN B E R Y L L I U M *
W. F. KING Laboratory for Electrophysics, The Technical University, Lyngby, Denmark DK 2800
and P. H. C U T L E R Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Received 16 December 1969
A comparison is made of the Slater, Kohn-Sham and Hartree-Fock approximations for exchange used in a first principle pseudopotential calculation of the photon dispersion relations in beryllium.
We have computed the phonon d i s p e r s i o n r e l a t i o n s for b e r y l l i u m u s i n g H a r r i s o n ' s n o n - l o c a l f i r s t p r i n c i p l e pseudopotential theory [1] and with t h r e e different a p p r o x i m a t i o n s for the c o n duction b a n d - c o r e e l e c t r o n exchange. S l a t e r [2] o r i g i n a l l y proposed an a v e r a g e d l o c a l potential to a p p r o x i m a t e the m o r e complicated H a r t r e e - F o c k (HF) exchange contribution; in the S l a t e r a p p r o x i m a t i o n the exchange c h a r g e density" is r e p l a c e d by the c o r r e s p o n d i n g value p o s s e s s e d by a ( n o n - u n i f o r m ) f r e e - e l e c t r o n gas whose local d e n s i t y , p ( r ) , is equal to the a c t u a l d e n s i t y at the point r considered. In S l a t e r ' s notation, the a p p r o x i m a t i o n to the HF exchange p o t e n t i a l is
rex(k, r )
1
= - 2 e 2 (3p(r)/T;)~
1+7/ 1-77
which equals ½ for k = k F and 1 at the bottom of the F e r m i d i s t r i b u t i o n . When a v e r a g e d over o c cupied s t a t e s F(~/) equals ¼. Kohn and Sham [3] have questioned this a p p r o x i m a t i o n and suggested that the value o f F ( 7 / ) at the top of the F e r m i d i s t r i b u t i o n is m o r e a p p r o p r i a t e . T h i s r e s u l t s in the u s u a l Slater exchange potential b e i n g m u l t i * Research sponsored in part by the Air Force Office of Scientific Research, Grants No. 213-66 and 69-1704. 150
N
~_~ 24 o_ "a ~1.6 -~ ~" g ~ o.8
F07),
w h e r e T/ is the r a t i o of the e l e c t r o n momentum, k, to the value at the F e r m i surface, and F(~7) = ½ + l ~ l n 4~/2
32-
0
/s
f -ps
sJ
,aA ~
I.E
0.9
,
; 0.2
,
;
,
;
0.6 Q/Qmox
,
I
,
I.O
0
, ~ ,,1~~ 0.2
fOOOI ] T O.6 O/Q me x
1.0
Fig. 1. The longitudinal (L) and transverse (r) phonon dispersion relations in beryllium for waves propagating in the [1000] direction. Experimental points are from Schmunk [10]. The theoretical curves were calculated using the following exchange approximations for the conduction-band-core exchange: Slater (solid line), Kohn-Sham (dot-dash), and Hartree-Fock (dashed). plied by ot =3. C a l c u l a t i o n s [4,5] on f r e e a t o m s u s i n g the KS exchange a p p r o x i m a t i o n yield o n e - e l e c t r o n e n e r gies in b e t t e r a g r e e m e n t with HF r e s u l t s than the e n e r g i e s obtained in the Slater approximation. However, r e c e n t work by P a y n e [6] indicates that, for m e t a l s , b e t t e r r e s u l t s obtain with ~ > 1.
Volume 31A, number 3
PHYSICS LETTERS
It is evident that t h e r e is, as yet, no unique or c l e a r l y s u p e r i o r method for attacking this p r o b lem. Slater has r e m a r k e d that a g r e e m e n t with HF one e l e c t r o n e i g e n v a l u e s m a y not n e c e s s a r i l y be the c r i t e r i o n one should u s e , p a r t i c u l a r l y for e n e r g y - b a n d theory. E n e r g i e s c a l c u l a t e d by the HFS method a r e only slightly l e s s than those f r o m the HF method; however, H e r m a n and S k i l l m a n [8] have shown v e r y good a g r e e m e n t can be obtained b e t w e e n HFS o n e - e l e c t r o n e n e r gies and e x p e r i m e n t a l values. F o r this r e a s o n , the wavefunctions u s e d i n the p r e s e n t c a l c u l a t i o n s w e r e g e n e r a t e d f r o m the H e r m a n and S k i l l m a n HFS p r o g r a m . In the p r e s e n t work effective p o t e n t i a l s w e r e obtained for the c o n d u c t i o n - b a n d - c o r e e l e c t r o n exchange with both the S l a t e r and KS a p p r o x i m a tions. Since the exchange i n t e r a c t i o n is m o s t i m p o r t a n t in the c o r e r e g i o n , w h e r e the v a l e n c e electrons assume predominant atomic-like chara c t e r , the a t o m i c 2s wave function can be u s e d a s a f i r s t a p p r o x i m a t i o n to the conduction e l e c t r o n wavefunction. We have u s e d the 2s atomic function with the Be ++ c o r e wavefunction to c a l culate the HF exchange t e r m by a method d e s c r i b e d by H a r t r e e [9]; this then p r o v i d e s a t h i r d effective exchange potential. Using each of the t h r e e different exchange a p p r o x i m a t i o n s we cons t r u c t e d the e n e r g y - w a v e n u m b e r c h a r a c t e r i s t i c F(q) with H a r r i s o n ' s n o n - l o c a l p s e u d o p o t e n t i a l t h e o r y [1]. (The d e t a i l s of t h e s e c a l c u l a t i o n s , which a r e i d e n t i c a l with the exception of the exchange a p p r o x i m a t i o n s , will b e published e l s e where. ) The r e s u l t s for the [0001] d i r e c t i o n a r e s u m m a r i z e d in fig. 1. In both the longitudinal and t r a n s v e r s e c a s e s , the highest d i s p e r s i o n f r e q u e n c i e s r e s u l t f r o m the u s e of the HF exchange a p p r o x i m a t i o n , the i n t e r m e d i a t e f r o m the KS a p p r o x i m a t i o n , while the b e s t a g r e e m e n t is obtained with the S l a t e r exchange. It is evident f r o m t h e s e c a l c u l a t i o n s that although the S l a t e r a p p r o x i m a t i o n with its i n h e r e n t l i m i t a t i o n s
9 February 1970
such as (i) o v e r e s t i m a t i n g the e x c h a n g e potential [3-11], and (ii) w r o n g b e h a v i o r in the l i m i t of l a r g e i n t e r n u c l e a r s e p a r a t i o n s [12], it n e v e r t h e l e s s gives b e s t a g r e e m e n t with m e a s u r e d d i s p e r s i o n r e l a t i o n s for Be. A p p a r e n t l y the i n t e r a t o m i c f o r c e s in b e r y l l i u m a r e such that it is j u s t t h e s e p r o p e r t i e s that a r e c o m p e n s a t e d for in the Slater approximation. Since the s a m e f i r s t p r i n c i p l e pseudopotential approach, but with the Kohn-Sham exchange a p p r o x i m a t i o n has b e e n u s e d to obtain excellent a g r e e m e n t with m e a s u r e d d i s p e r s i o n c u r v e s of divalent hcp Mg [13], it f u r t h e r e m p h a s i z e s i n this application the i n a p p r o p r i a t e n e s s of a unique choice for the exchange a p p r o x i m a t i o n [6]. We would like to thank the staff of the computation c e n t e r at the T e c h n i c a l U n i v e r s i t y of D e n m a r k for t h e i r a s s i s t a n c e and cooperation with t h e s e calculations.
References 1. W.A. Harrison, Pseudopotentials in the theory of metals, (W. A. Benjamin, Inc., New York 1966). 2. J.C. Slater, Phys. Rev. 81 (1951)385. 3. W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A 1133. 4. B.Y. Tong and L. J. Sham, Phys. Rev. 144 (1966) 1. 5. R.D. Cowan, A.C. Larson, D. Liberman, J.B. Mann and J. Waber, Phys. Rev. 144 (1966) 5; F. Herman, J. Van Dyke, and I. Ortenburger, Phys Rev. Letters 2L (1969) 807. 6. H. Payne, Phys. Rev. 157 (1967) 515. 7. J. C. Slater, Phys. Rev. 165 (1968) 658. 8. F. Herman and S. SkiUman, Atomic structure calculations, (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963). 9. D.R. Hartree, The Calculation of atomic structures, (John Wiley and Sons, Inc., New York, 1957). 10. Experimental points are from R. E. Schmunk, Phys. Rev. 149 (1966) 450. 11. J. Callaway, Solid state physics, Vol. 7, (Academic Press, New York, 1958), p. 107. 12. J. Robinson, F. Bassini, K. Knox and J. Schrieffer, Phys. Rev. Letters 9 (1962) 215. 13. To be published.
151
PHYSICS LETTERS
9 February 1970
THE
EFFECT OF DIFFERENT EXCHANGE APPROXIMATIONS ON A P S E U D O P O T E N T I A L CALCULATION OF PHONON DISPERSION RELATIONS IN B E R Y L L I U M *
W. F. KING Laboratory for Electrophysics, The Technical University, Lyngby, Denmark DK 2800
and P. H. C U T L E R Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Received 16 December 1969
A comparison is made of the Slater, Kohn-Sham and Hartree-Fock approximations for exchange used in a first principle pseudopotential calculation of the photon dispersion relations in beryllium.
We have computed the phonon d i s p e r s i o n r e l a t i o n s for b e r y l l i u m u s i n g H a r r i s o n ' s n o n - l o c a l f i r s t p r i n c i p l e pseudopotential theory [1] and with t h r e e different a p p r o x i m a t i o n s for the c o n duction b a n d - c o r e e l e c t r o n exchange. S l a t e r [2] o r i g i n a l l y proposed an a v e r a g e d l o c a l potential to a p p r o x i m a t e the m o r e complicated H a r t r e e - F o c k (HF) exchange contribution; in the S l a t e r a p p r o x i m a t i o n the exchange c h a r g e density" is r e p l a c e d by the c o r r e s p o n d i n g value p o s s e s s e d by a ( n o n - u n i f o r m ) f r e e - e l e c t r o n gas whose local d e n s i t y , p ( r ) , is equal to the a c t u a l d e n s i t y at the point r considered. In S l a t e r ' s notation, the a p p r o x i m a t i o n to the HF exchange p o t e n t i a l is
rex(k, r )
1
= - 2 e 2 (3p(r)/T;)~
1+7/ 1-77
which equals ½ for k = k F and 1 at the bottom of the F e r m i d i s t r i b u t i o n . When a v e r a g e d over o c cupied s t a t e s F(~/) equals ¼. Kohn and Sham [3] have questioned this a p p r o x i m a t i o n and suggested that the value o f F ( 7 / ) at the top of the F e r m i d i s t r i b u t i o n is m o r e a p p r o p r i a t e . T h i s r e s u l t s in the u s u a l Slater exchange potential b e i n g m u l t i * Research sponsored in part by the Air Force Office of Scientific Research, Grants No. 213-66 and 69-1704. 150
N
~_~ 24 o_ "a ~1.6 -~ ~" g ~ o.8
F07),
w h e r e T/ is the r a t i o of the e l e c t r o n momentum, k, to the value at the F e r m i surface, and F(~7) = ½ + l ~ l n 4~/2
32-
0
/s
f -ps
sJ
,aA ~
I.E
0.9
,
; 0.2
,
;
,
;
0.6 Q/Qmox
,
I
,
I.O
0
, ~ ,,1~~ 0.2
fOOOI ] T O.6 O/Q me x
1.0
Fig. 1. The longitudinal (L) and transverse (r) phonon dispersion relations in beryllium for waves propagating in the [1000] direction. Experimental points are from Schmunk [10]. The theoretical curves were calculated using the following exchange approximations for the conduction-band-core exchange: Slater (solid line), Kohn-Sham (dot-dash), and Hartree-Fock (dashed). plied by ot =3. C a l c u l a t i o n s [4,5] on f r e e a t o m s u s i n g the KS exchange a p p r o x i m a t i o n yield o n e - e l e c t r o n e n e r gies in b e t t e r a g r e e m e n t with HF r e s u l t s than the e n e r g i e s obtained in the Slater approximation. However, r e c e n t work by P a y n e [6] indicates that, for m e t a l s , b e t t e r r e s u l t s obtain with ~ > 1.
Volume 31A, number 3
PHYSICS LETTERS
It is evident that t h e r e is, as yet, no unique or c l e a r l y s u p e r i o r method for attacking this p r o b lem. Slater has r e m a r k e d that a g r e e m e n t with HF one e l e c t r o n e i g e n v a l u e s m a y not n e c e s s a r i l y be the c r i t e r i o n one should u s e , p a r t i c u l a r l y for e n e r g y - b a n d theory. E n e r g i e s c a l c u l a t e d by the HFS method a r e only slightly l e s s than those f r o m the HF method; however, H e r m a n and S k i l l m a n [8] have shown v e r y good a g r e e m e n t can be obtained b e t w e e n HFS o n e - e l e c t r o n e n e r gies and e x p e r i m e n t a l values. F o r this r e a s o n , the wavefunctions u s e d i n the p r e s e n t c a l c u l a t i o n s w e r e g e n e r a t e d f r o m the H e r m a n and S k i l l m a n HFS p r o g r a m . In the p r e s e n t work effective p o t e n t i a l s w e r e obtained for the c o n d u c t i o n - b a n d - c o r e e l e c t r o n exchange with both the S l a t e r and KS a p p r o x i m a tions. Since the exchange i n t e r a c t i o n is m o s t i m p o r t a n t in the c o r e r e g i o n , w h e r e the v a l e n c e electrons assume predominant atomic-like chara c t e r , the a t o m i c 2s wave function can be u s e d a s a f i r s t a p p r o x i m a t i o n to the conduction e l e c t r o n wavefunction. We have u s e d the 2s atomic function with the Be ++ c o r e wavefunction to c a l culate the HF exchange t e r m by a method d e s c r i b e d by H a r t r e e [9]; this then p r o v i d e s a t h i r d effective exchange potential. Using each of the t h r e e different exchange a p p r o x i m a t i o n s we cons t r u c t e d the e n e r g y - w a v e n u m b e r c h a r a c t e r i s t i c F(q) with H a r r i s o n ' s n o n - l o c a l p s e u d o p o t e n t i a l t h e o r y [1]. (The d e t a i l s of t h e s e c a l c u l a t i o n s , which a r e i d e n t i c a l with the exception of the exchange a p p r o x i m a t i o n s , will b e published e l s e where. ) The r e s u l t s for the [0001] d i r e c t i o n a r e s u m m a r i z e d in fig. 1. In both the longitudinal and t r a n s v e r s e c a s e s , the highest d i s p e r s i o n f r e q u e n c i e s r e s u l t f r o m the u s e of the HF exchange a p p r o x i m a t i o n , the i n t e r m e d i a t e f r o m the KS a p p r o x i m a t i o n , while the b e s t a g r e e m e n t is obtained with the S l a t e r exchange. It is evident f r o m t h e s e c a l c u l a t i o n s that although the S l a t e r a p p r o x i m a t i o n with its i n h e r e n t l i m i t a t i o n s
9 February 1970
such as (i) o v e r e s t i m a t i n g the e x c h a n g e potential [3-11], and (ii) w r o n g b e h a v i o r in the l i m i t of l a r g e i n t e r n u c l e a r s e p a r a t i o n s [12], it n e v e r t h e l e s s gives b e s t a g r e e m e n t with m e a s u r e d d i s p e r s i o n r e l a t i o n s for Be. A p p a r e n t l y the i n t e r a t o m i c f o r c e s in b e r y l l i u m a r e such that it is j u s t t h e s e p r o p e r t i e s that a r e c o m p e n s a t e d for in the Slater approximation. Since the s a m e f i r s t p r i n c i p l e pseudopotential approach, but with the Kohn-Sham exchange a p p r o x i m a t i o n has b e e n u s e d to obtain excellent a g r e e m e n t with m e a s u r e d d i s p e r s i o n c u r v e s of divalent hcp Mg [13], it f u r t h e r e m p h a s i z e s i n this application the i n a p p r o p r i a t e n e s s of a unique choice for the exchange a p p r o x i m a t i o n [6]. We would like to thank the staff of the computation c e n t e r at the T e c h n i c a l U n i v e r s i t y of D e n m a r k for t h e i r a s s i s t a n c e and cooperation with t h e s e calculations.
References 1. W.A. Harrison, Pseudopotentials in the theory of metals, (W. A. Benjamin, Inc., New York 1966). 2. J.C. Slater, Phys. Rev. 81 (1951)385. 3. W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A 1133. 4. B.Y. Tong and L. J. Sham, Phys. Rev. 144 (1966) 1. 5. R.D. Cowan, A.C. Larson, D. Liberman, J.B. Mann and J. Waber, Phys. Rev. 144 (1966) 5; F. Herman, J. Van Dyke, and I. Ortenburger, Phys Rev. Letters 2L (1969) 807. 6. H. Payne, Phys. Rev. 157 (1967) 515. 7. J. C. Slater, Phys. Rev. 165 (1968) 658. 8. F. Herman and S. SkiUman, Atomic structure calculations, (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963). 9. D.R. Hartree, The Calculation of atomic structures, (John Wiley and Sons, Inc., New York, 1957). 10. Experimental points are from R. E. Schmunk, Phys. Rev. 149 (1966) 450. 11. J. Callaway, Solid state physics, Vol. 7, (Academic Press, New York, 1958), p. 107. 12. J. Robinson, F. Bassini, K. Knox and J. Schrieffer, Phys. Rev. Letters 9 (1962) 215. 13. To be published.
151