Fermi surface of the noble metals
RAJEEV AHUJA, A K SOLANKI, T NAUTIYAL and S A U L U C K
Physics Department, University of Roorkee, Roorkee 247 667, India MS received 9 January 1989
Abstract. We report calculations of extremal areas of four Fermi surface (FS) orbits of the noble metals using the linear muffin tin orbital method in the atomic sphere approximation.
Our calculations indicate that the l = 3 potential parameters and increase in the number of k-points in the Brillouin zone (BZ) summation from 240 to 916 have no significant effect on the FS. All calculations were performed self-consistently including up to l = 2 potential parameters and with 240k points in the BZ summation. Calculations were performed with the exchange, correlation potentials (XCP) of Barth-Hedin, Barth-Hedin modified by Janak, Slater X,, and the Vosko-Wilk-Nussair. Results compared with other theoretical calculations indicate that none of the above XC potentials give an accurate representation of the FS for all the noble metals. We feel that the inclusion of the non-locality of XCP may give a better account of the FS geometry.
Keywords. Fermi surface; linear muffin tin orbital; exchange correlation potential.
PACS Nos 71.25; 71.45
1. Introduction
The band structure and fermi surface (FS) of noble metals is so well established that experimentalists use it to standardize their apparatus and theoreticians use it to debug their programmes. In fact the amount of work is so voluminous that it would be impossible to make a complete list of references (see e.g. Springford 1980; Ziman 1969;
Cochran and Haering 1969). Recent interest in this area was rekindled by the density functional (DF) theory which gave a prescription for calculating accurately the one- electron potentials used to obtain the ground-state properties such as FS geometry (Hohenberg and Kohn 1964; Kohn and Sham 1965). Recently Jepsen
e t a l(1981) calculated the band structure and FS of the noble metals using the linear augmented plane wave (LAPW) method with potentials constructed using the local approximation (LA) to the D F formalism and calculated self-consistently by the atomic sphere approximation (ASA) to the linear muffin tin orbital (LMTO) method.
Relativistic band shifts were included but spin-orbit coupling was neglected. Jepsen
et al(1981) have shown that their potentials give the FS comparatively satisfactorily although it is not possible to obtain a satisfactory account of the optical excitation energies if these are interpreted as single particle energy differences. In order to place the d-bands correctly, many-body corrections would be needed (Jepsen
et al1981).
Our own attempts in this direction have met with considerable success. We have
used a modified form of the combined interpolation scheme (Smith and Mattheiss
1974) with the parameters chosen to fit the FS as well as the optical gaps. Prior to this
the interpolation scheme was used to fit either one of these. Our success was with the
831
noble metals (Singh et ai (unpublished)), Pd (Bordoloi and Auluck 1988a), Pt (Bordoloi and Auluck 1988b), ferromagnetic Ni (Prasad et al 1977) and ferromagnetic Fe (Nautiyal and Auluck 1985, 1986). With our band structures, we have calculated mass enhancement factors and the dielectric function (Ahuja et al 1988; Bardoloi and Auluck 1988) and these are in good agreement with the data. The reason for this satisfactory agreement is because the interpolation scheme has a very large number of parameters (16 to be precise) and it would be improper to draw any reasonable conclusions from such calculations.
We would like to take the cue from the work ofJepsen et al and address ourselves to the question: can a first principle band calculation give an accurate representation of the FS geometry? (In this paper we consider only the noble metals). This is a valid question because the D F formalism should give the correct FS which is a ground-state property. In fact the de Haas-van Alphen (dHva) experiments measure extremal areas.
It would therefore be meaningful to calculate extremal areas. Jepsen et al have suggested that the FS of noble metals can be well represented by (i) the neck radius and (ii)the ratio of kr[lOO]/kr[llO]. We argue that FS geometry means extremai areas which includes many k vectors and not just two or three radii. One of our aims is to ascertain if Jepsen et al criterion is indeed correct i.e. that the two quantities above are sufficient to characterize the FS geometry of the noble metals.
Our reasons for choosing the interpolation scheme were (i) it is very fast (ii) it gives the band structure and (iii)it can be used to calculate FS geometry. The Korringa- Kohn-Rostoker (KKR) method of parametrizing, which has now been made very fast, has been used either for FS geometry (Shaw et al 1972) or for band structure (Chen and Segall 1975) but not for both. Since calculations of extremal areas take a lot of computer time, it would be nice to have an ab initio band structure method which satisfies our three requirements. The L M T O method seems to fit this bill (Andersen
1975). In our work, we have used Skriver's codes (Skriver 1984).
In this paper we report calculations of the extremal areas for the noble metals using the L M T O method in the ASA. We have studied the effects of (i) varying the number of k points in the irreducible-Brillouin zone (BZ) summations (ii) including the f - b a n d parameters and (iii) varying the exchange-correlation (XC) potentials with the view of ascertaining which one is the most appropriate for FS work. The effects of relativistic shifts have also been studied.
In §2 we discuss our calculational method and a criterion for determining the FS fit.
In §3 we give our results for Cu, Ag and Au with our conclusions in §4.
2. Method of calculation
We have used the L M T O method in the ASA proposed originally by Andersen (1975) and given in detail by Skriver (1984) to calculate the energy eigenvalues and eigenvectors. Since this is a very widely used method, we will refer the reader to the well-known references. Our primary reason for choosing this method is that this is very fast; as fast as the empirical methods with the advantage of being an ab initio method. In our calculations we have included the correction terms. The calculations are done to self-consistency which we take to be that the change in the potential parameters is in fifth decimal place. We believe that this will converge the energy eigenvalues to within 10 -4 Ry. Starting from the parameters given in Skriver's book, this takes about half a dozen more iterations.
Using these self-consistent parameters, we have calculated FS orbital areas and masses using Stark's (private communication) area-mass routine. The area/masses of the computed surfaces in a plane normal to a direction (i.e. the magnetic field) were found by numerical integration of the radii calculated at a fixed interval of rotation in that plane. In calculations reported here the stepping angle 30 was taken to be 5 °.
Making this 2½ ° changed the calculated areas by less than ½%. We then calculated the shift in Fermi energy AE e required to bring the calculated extremal area in agreement with the experiment
AE r _ 1 Aexpt- Aca, ¢
It m b
where m~ is the calculated mass for the orbit. AE r is calculated for four orbits; Belly (Bill), Belly (BI00), Dog-bone (D110) and Neck (Nlll). The numbers denote the direction of the magnetic field (Lee 1969). The extreme AE r gives the error. This is meaningful since we are calculating energy eigenvalues. It would be futile to aim for an extreme AEr less than the accuracy of the eigenvalues. We feel that this is a better way of characterizing error than by percentages (Janak 1972, Janak et al 1972, 1975). We would also like to stress that the area-mass codes calculate eigenvalues at each k point on the FS and uses no fitting procedures. There is no need to do the fitting as LMTO is very fast. Our calculations are compared with the very accurate data of Coleridge and Templeton (1972). In fact the main reason for choosing the noble metals for this study was the existence of such accurate dHva data. Another being that the FS of noble metals is very simple.
3. Calculations and results
(a) General considerations
Jepsen et al have calculated the band structure and FS of the noble metals using the LA PW method with the potentials constructed using the local approximation to the DF formalism and calculated self-consistently by the ASA to the LMTO method.
Relativistic band shifts were included but spin orbit coupling was neglected. They included the s,p,d and f potential parameters with the BZ summations being performed over 715 k points. We would first like to address ourselves to (i) the effect of neglecting the f-potential parameters and (ii) the effect of varying the number of k points in the BZ summations. These are done for copper.
Consider first the effect of truncating the l-expansion in the potential parameters.
We have performed calculations with 240 k points in the BZ summation in two ways by including (i) l = 0, 1,2 terms and (ii)l = 0, 1, 2, 3 potential parameters for copper.
Our results for the four FS orbits are given in table 1. For these calculations, we have used the Barth-Hedin (BH) exchange-correlation (XC) potential (Barth and Hedin 1972). A look at table 1 shows that the results do not change significantly by including the l = 3 terms. The extreme AE r, which is determined by the NI 11 and BI 11 orbits, reduces from 12.5 mRy to I 1.5 mRy.
Calculations were next performed using 240 k points, 505 k points and 960 k points in the B Z summations. These results are also given in table 1. Once again we have used the BH XC and included the s, p and d potential parameters. We obtain an extreme AE r of 12.5 mRy (240 k), 11-8 mRy (505 k) and 12.3 mRy (916 k). There is no
Table !. Calculated extremal areas for copper ~,, = 2"669 au, K, = 0"7213 au-i.
Name of the orbit B i l l N l l l B100 DI10 K~-[100] K N
A Kr[110] K s
Experimental Area AEe pts. in Bz Up to XC
1-5523 0-0581 1 ' 6 0 2 6 0"6707 1'11 0'189
1.5414 f r o 7 0 1 1-6633 0.6496
240 I = 3 BH 1.134 0-208
-l- 0-0018 - 0-0097 - 0"0014 -- 0.0056 1 "5403 04)695 1-6057 0"6469
240 I = 2 BH 1-139 0-207
+ 0-0028 - 0.0097 -- 0'0007 -- 0'0060 1"5339 0"0681 1"6011 0-6530
505 I = 2 BH 1'140 0-205
+0.0038 -0"0080 +0-0003 -0.0046 1"5346 0-0684 1 '6019 0"6523
916 I = 2 BH 1.140 0-205
+0'0038 -0-0085 +0'0002 -0"0048 1-6051 0 . 0 9 3 7 1-6791 0"5841
715 l = 3 HL 1-149 0-240
-0"0125 -0"0270 -0"0182 -0-0238 1"5394 0 . 0 6 9 9 1 ' 6 0 6 6 0.64809
240 l = 2 BHJ 1"140 0.208
+ 0.0040 - 0-0097 - 0.0009 - 0-0059 1.5547 0 - 0 5 4 6 1.6029 0.6610
240 I = 2 VWN 1-104 0-183
- 0-0006 + 0.0030 - 04)001 - 0-0027
240 I = 2 X~ 1.5528 0 . 0 5 7 0 1.6031 0-6590 1.108 0-187
,, = 0-77 - 0.0001 + 0-0009 - 0-0001 - 0-0032
significant change. H e n c e all f u r t h e r c a l c u l a t i o n s are p e r f o r m e d with 240 k p o i n t s a n d i n c l u d i n g l = 0, 1, 2 t e r m s only.
(b) Copper
A l o o k at table 1 indicates t h a t the B H X C p o t e n t i a l does n o t give a g o o d r e p r e s e n t a t i o n for the F S of copper. A n e x t r e m e AE F is a r o u n d 10 m R y , l a r g e r t h a n a n e x t r e m e AE r of a b o u t 0.1 m R y o b t a i n e d b y us with the i n t e r p o l a t i o n scheme (Singh et al ( u n p u b l i s h e d ) ) or by the K K R p a r a m e t r i z a t i o n (Shaw et al 1972). W e have also c a l c u l a t e d A E r for the p a r a m e t e r s of J e p s e n et al a n d o b t a i n 14-5 m R y . M o r e o v e r we have to lower E F by a b o u t 15 m R y from the value given b y J e p s e n et al. T h i s m a y be a t t r i b u t e d to the fact t h a t they used the L A P W m e t h o d for c a l c u l a t i n g eigenvalues.
W i t h the view to a s c e r t a i n i n g h o w the v a r i o u s t r e a t m e n t s of X C i n f l u e n c e the e i g e n v a l u e s a n d the FS, we have r e p e a t e d the c a l c u l a t i o n s with the B a r t h - H e d i n XC p o t e n t i a l u s i n g J a n a k ' s (BHJ) p a r a m e t e r s ( J a n a k 1975) (which m a k e it the s a m e as the H e d i n L u n d q v i s t (HL) X C p o t e n t i a l ( H e d i n a n d L u n d q v i s t 1971), Siater X~ p o t e n t i a l (Slater 1971) a n d the recent m o s t a c c u r a t e V o s k o - W i l k - N u s s a i r ( V W N ) X C p o t e n t i a l (Vosko et al 1980). J a n a k et al (1972, 1975) have s h o w n t h a t the X~ m e t h o d with
= 0-77 gives a g o o d fit to the F S of copper. W e have therefore t a k e n ~ = 0.77 i n the X, m e t h o d .
T h e calculated F S areas for v a r i o u s X C p o t e n t i a l s for c o p p e r are given i n table 1.
The values of extreme AEr for the various XC are 4.l mRy (X= method with a = 0.77), 13.7mRy (BHJ), 5.7mRy (VWN) and 12"5 mRy (BH). The X, method with a = 0.77 gives the best fit to the FS data. The price we pay is that ~ is an adjustable parameter so obviously the fit is better. Note the recent XC of VWN is also as good as the X~
results. The BH and BHJ do not give good representation for the FS of copper.
At this juncture we would like to compare our results with those of Janak et ai who have performed self-consistent calculations for Cu and Ag using the KKR method.
They have taken the N111 and B111 orbits to decide the fit to the FS. Our calculations support this. A look at table 1 indicates that the extreme AE r is indeed governed by the N l l l and B i l l orbits for all the XC used except for the X~ and for the VWN exchange-correlation where it is governed by the NI 11 and D110 orbits. Janak et al found that a = 0.77 in the X~ method provides the best fit to the FS. Using their results we obtain the extreme AEr to be 6.7 mRy (for the four orbits) which is slightly larger than the AE F obtained by us. We have not explored the possibility of varying n any more because we feel that variation with ~ of the FS orbits will not be much different from that obtained by Janak et al. All the extreme AE r are much larger than the AE r of about 0-1 mRy obtained by the interpolation scheme (Singh et al) or by the KKR parametrization (Shaw et al 1972).
Jepsen et al have determined the goodness of the FS geometry by calculating only two parameters (i) neck radius kN and (ii) an anisotropy parameter A = kr[lOO]/kF[lO0]. We have also calculated these and these are given in table 1 for the various XC potentials. Although the change in these parameters is small for the various XC potentials nevertheless the X~ (~ = 0.77) gives values that are closest to the experimental values. We are surprised that Jepsen et al contention that two parameters characterize the Cu FS has been well borne out by our calculations. It is surprising to note the difference in the values of k~ and A obtained by Jepsen et al and by us using their potential parameters. This could be due to the fact that they used the LAPW method.
(c) Silver
We have performed similar calculations for silver. Results are given in table 2. The extreme AE r u~ing the various XC are 16.4mRy (BH), 16.3 toRy (BH with f - b a n d parameters), 25"5mRy (Jepsen's parameters), 15.6toRy (BHJ), 15.3mRy (BH with 505 k points), 0.9 mRy (X~ with • = 0.77) and 3"8 mRy (VWN). As in the case of copper, the extreme AEr is again governed by the NI 11 and BI I l orbits except for the X~ case where it is B I l l and BI00 orbits. All results are for 240k points and including l = 0, 1, 2 terms only, unless stated otherwise. Once again, we find that the X~ method (with at = 0.77) gives the best fit to the data. In fact this is the kind of agreement one gets with the KKR parametrization (Shaw et al 1972) or interpolation scheme (Singh et al). Amongst the other XC potentials, the VWN is the best. We have also calculated the neck radius k N and anisotropy parameter A. Again we find that the X~ method (0t = 0.77) gives values for these in good agreement with experiment. It is pleasing to note that k N and A for Jepsen et al potentials agree with their values. For copper this was not the case. Calculations by Janak et al give an extreme AE r of 8.5 mRy. Thus we are led to the conclusion that the LMTO method in the ASA gives a better representation to the FS of Cu and Ag using the same ~t in the X~ method. The VWN fares best considering that it has no adjustable parameter.
Table 2. Calculated extremal areas for silver ~s = 3.005 au, K, = 0-6386au- 1.
Name of the orbit B i l l N l l l BI00 DII0 A K s / K s
Experimental Area AE r pts. in Bz Up to XC
1.2311 0 . 0 2 3 8 1"2684 0-5381 1"09 0-136
AREA 1 . 2 2 4 4 0 . 0 3 7 5 1 ' 2 6 6 4 0-5258
240 I = 3 1.104 0-171
BH + 0.0023 - 0-0140 - 0-0007 - 0.0043 1.2249 0 . 0 3 6 0 1"2687 0-5252
240 I = 2 BH 1-106 0-168
+ 0-0022 - 0-0142 - 0-0001 - 0"0045 1"2220 0 - 0 3 5 2 1-2660 0"5286
505 / = 2 BH 1.114 0-176
+ 0.0031 - 0.0122 + 0.0008 -- 0.0034 1'2189 0 - 0 4 1 5 1-2645 0"5248
715 I = 3 HL 1.106 ff166
+0'0045 0'0210 +0.0014 -0"0049
!'2235 0"0362 1 "2678 0-5266
240 I = 2 BHJ 1'107 0'168
+0-0026 -0"0130 +0-0002 -0"0041
240 l = 2 VWN 1"2358 0"0218 1 ' 2 6 9 2 0'5373 1-080 0-130 -0-0017 +0-0021 -00003 -0-0003
240 I = 2 X, 1-2340 0"0243 1 ' 2 6 8 7 0"5354 1-083 0-138 a = 0-77 - 0.0010 - 0"0005 - 0.0001 - 0"0009
(d)
Gold
T a b l e 3 s u m m a r i z e s o u r results for gold. T h e e x t r e m e A E r for the v a r i o u s X C p o t e n t i a l s a r e 3-2 m R y (BH), 3-6 m R y (BH w i t h f - b a n d p a r a m e t e r s ) , 2-8 m R y ( J e p s e n ' s p o t e n t i a l ) , 3 - 5 t o R y ( B H with 5 0 5 k points), 3 - 4 t o R y (BHJ), 1 9 . 5 m R y ( V W N ) a n d
! 5"8 m R y (X~ m e t h o d with at = 0.77). U n l i k e the cases o f C u a n d A g t h e e x t r e m e
AE v
is n o w n o t g o v e r n e d b y the s a m e t w o orbits. I n fact for v a r i o u s X C p o t e n t i a l s used, t h e o r b i t s a l w a y s vary. T h e fact t h a t the X~ m e t h o d d o e s n o t give a g o o d fit to the F S o f g o l d suggests the n e e d to v a r y ~. W e h a v e p e r f o r m e d c a l c u l a t i o n s for v a r i o u s ~t's a n d p l o t t e d the results in figure I. T h e g r a p h i l l u s t r a t e s t h a t even in the X~ m e t h o d , different values o f 0t give different o r b i t s w h i c h c o n t r o l the e x t r e m e A E r . F r o m the g r a p h we find t h a t 0t = 0.693 w o u l d yield a n e x t r e m eAEe
o f 3.5 toRy, w h i c h is the s a m e as with BH a n d B H J X C p o t e n t i a l s . O n c a l c u l a t i n gks
a n d A we find t h a t t h o s e X C p o t e n t i a l s which give the best fit t o the F S g e o m e t r y a l s o give v a l u e s o f these p a r a m e t e r s in a g r e e m e n t with the e x p e r i m e n t s . A g a i n theks
a n d A c a l c u l a t e d b y us using J e p s e net al
p o t e n t i a l p a r a m e t e r s d o e s n o t a g r e e with t h e i r o w n values.(e) Relativistic corrections
W i t h a view to d e t e r m i n e h o w relativistic c o r r e c t i o n s w o u l d influence the s h a p e o f the F S we h a v e d e c i d e d to s t u d y g o l d b e c a u s e h e r e the relativistic c o r r e c t i o n s a r e the largest. W e h a v e c a l c u l a t e d the r e l a t i v i s t i c b a n d s (neglecting s p i n - o r b i t c o u p l i n g ) a l o n g s y m m e t r y d i r e c t i o n s . O u r c a l c u l a t i o n s d e m o n s t r a t e t h a t the n e c k r a d i u s is a l m o s t u n c h a n g e d b y the relativistic b a n d shifts. H o w e v e r these i n c r e a s e k v [ 1 0 0 ] b y 0-3% a n d d e c r e a s e
kv[1
10] b y 1.4%. T h i s c h a n g e s A t o 1"226. T h i s will t e n d t o r e d u c eTable 3. Calculated extremal areas for gold ~,~ = 3-002 au, K s = 0-6392 a u - 1.
N a m e o f the orbit B I l l N l l l B I 0 0 D I I 0 A Ks~Ks
Experimental Area AEF pts. in Bz U p to X C
1-,1909 0-0406 1'2837 0 ' 5 1 3 0 1"20 0"179
240
240
505
715
240
240
240
240
240
240
1"2033 0.0443 1.3010 0.5087
I = 3 B H 1"198 0-186
- 0.0036 - 0.0051 - 0-0049 - 0.0015 1-2023 0'0439 1-3014 0.5071
I = 2 B H 1"203 0.185
- 0.0035 - 0.0048 - 0.0053 - 0-0021 1.1987 0-0432 1'2988 0"5108
= 2 B H 1"205 0 ' 1 8 4
- 0-0023 - 0-0037 - 0-0043 - 0.0008
1.2601 0.0568 1'3570 0.4627
= 3 H L 1-184 0.'210
- 0"0214 - 0-0207 - 0.0215 - 0.0187 1.2015 0.0442 1"3022 0"5078
= 2 B H J 1.204 0.186
- 0-0031 - 0-0051 - 0.0053 - 0.0019
1.2160 0.0327 1.2994 0.5196
= 2 V W N 1-58 0.160
- 0 . 0 0 8 0 + 0 - 0 1 1 5 - - 0 " 0 0 4 9 + 0 . 0 0 2 4
= 2 X~ 1"2137 0"0347 1'2993 0"5180 1"164 0.164
a = 0 " 7 7 - 0 " 0 0 7 2 + 0 " 0 0 8 6 - 0 ~ 0 4 8 + 0 " 0 0 1 8
= 2 X . 1'2185 0"0304 1"2990 0.5217 1"151 0 ' 1 5 4 a = 0 . 8 0 - 0 " 0 0 8 9 + 0 " 0 1 4 9 - 0 - 0 0 4 9 + 0 - 0 0 3 2
= 2 X~ 1"2047 0-0421 1-3012 0.5107 1-193 0.181
= 0.71 - 0"0041 - 0"0022 - 0"0051 - 0"0080
I = 2 X,, 1"2024 0.0441 1'3023 0.5078 1"161 0"185
= 0-693 - 0"0033 - 0-0049 - 0.0053 - 0.0018
the BI00 area slightly but will leave the extreme AEF almost unchanged. Hence relativistic corrections are not sufficient to reduce AEF significantly.
4. Conclusions and discussions
We have reported results of accurate calculations of four FS orbits for the noble metals within the ASA to the LMTO method. These calculations have been performed by (i) including or neglecting f-band parameters, (ii) varying the number of k-points in the BZ summations and (iii) using different XC potentials. Our results indicate that the f-band potential parameters have only a marginal influence on the FS. This is not surprising because f-bands are at least -,, 5 eV above the EF. We also find by changing the number ofk points in the BZ summation from 240 to 916 does not change the FS significantly.
Table 4 gives all our results for the noble metals with various XC potentials. It is
obvious that no single XC potential gives a good representation for the FS of the
noble metals. The X~ method with a variable ~t gives the best agreement with the
experimental data. The value of ~t for Au comes to be 0.693 while for Cu and Ag it is
0"77. The BH and BHJ XC potentials work well for Au but not for Cu and Ag. The
.015
v
.01C
I l l
.OOS
Dl10
>-
OC 0
<3
- .OOS
I
(~r~ o17o
C ~ -
BIO0 FI []
I L
0 . 7 5 0 . 8 0
Figure I. Shift in the Fermi energy AE r required to bring the calculated extremal area in agreement with the experiment for the four Fermi surface orbits as a function of the parameter a in the X, exchange-correlation potential for gold.
V W N XC potential gives the Ag FS satisfactorily but not for Cu and Au. Thus no single XC gives the FS of all the noble metals satisfactorily. We are a bit surprised that the V W N XC potential which is the most reliable with an estimated maximum error of I m R y gives such a large extreme AE r. The best agreement works out to be an extreme AEr of 3 - 4 mRy (0"9 mRy for Ag is surprising and stands out from the others).
It is worth noting that these AE r are smaller by 50% than those from the accurate and
T a b l e 4 . Extreme A E r (in mRy) for the noble metals--A summary.
Name of the metal pts. inBz Up to XC
Copper Silver Gold
240 I = 3 BH 11-5 16.3 3.6
240 I = 2 BH 12.5 16.4 3.2
505 l = 2 BH 11.8 15-3 3.5
916 l = 2 BH 12.3 - - - -
715 I = 3 HL 14-5 25'5 2"8 240 1 = 2 X~ 4"I 0"9 15.8
(a - 0.77)
240 I = 2 VWN 5-7 3"8 19"5 240 I = 2 BHJ 13.7 15-6 3"4 240 I = 2 X. - - - - 3"5
(~ - 0'693)
detailed calculations of J a n a k et al and hence are the smallest ever obtained by an ab initio band calculation. They are still an order of magnitude larger than the AEr = 0.1 m R y obtained by the empirical methods. Clearly the empirical m e t h o d s with the m a n y adjustable parameters can give a better representation for the FS geometry in noble metals and no ab initio method can hope to compete with them.
We have also studied suggestion of Jepsen et al that the FS of noble metals could be characterized by ks~ks and A = kF[.lOO]/kr[110]. Here again we find that there is no consistent picture. F o r Cu and Ag, the XC potentials which give good agreement with the experimental areas also give values of k s / k s and A in agreement with the experiment. F o r Au, this was not found to be the case. We also find that the values of A and k s / k s using Jepsen et al potential parameters do not always agree with the values quoted by Jepsen et al. We feel that this m a y be due to the fact that they used a more accurate L A P W method.
H o w can one improve the results. Three possibilities exist (Shaw et al 1972):
relativistic correction, non-muffin-tin corrections and the non-local XC potentials.
We have calculated energy eigenvalues and FS dimensions for Au, which has the largest relativistic corrections and find that these corrections are not enough to reduce AE r sufficiently. According to the work of Rudge (1969) and Koelling et al (1970) the non-muffin-tin and relativistic corrections are not sufficient to explain the large values of AE p. W a n g and Rasolt (1977) have investigated the effect of non- locality in XC potential in Cu and concluded that non-locality has the potential of resolving the discrepancy between first principles calculations local theory and experiment. We plan to look into this aspect in the future.
5 . A c k n o w l e d g e m e n t s
We are grateful to the Roorkee University Regional C o m p u t e r Centre for allowing us the use'of their D E C 2050 system. This work was supported by CSIR, India.
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